Microscopic Interactions and Macroscopic Properties Final Report (1st

Deutsche
Forschungsgemeinschaft
Macromolecular
Systems:
Microscopic
Interactions and
Macroscopic Properties
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

Deutsche
Forschungsgemeinschaft
Macromolecular Systems:
Microscopic Interactions
and Macroscopic Properties
Final report of the
collaborative research centre 213,
``Topospezifische Chemie und Toposelektive
Spektroskopie von MakromolekÏlsystemen:
Mikroskopische Wechselwirkung und
Makroskopische Funktion'', 1984^1995
Edited by
HeinzHoffmann,MarkusSchwoererand
Thomas Vogtmann
Collaborative Research Centres

Deutsche Forschungsgemeinschaft
Kennedyallee 40, D-53175 Bonn, Federal Republic of Germany
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Printed in the Federal Republic of Germany

Contents
Preface ...................................................
List of Contributors ..........................................
XV
XVII
I
Mainly Solids
1 Model Systems for Photoconductive Material ..................... 3
Harald Meyer and Dietrich Haarer
1.1 Introduction . . ............................................. 3
1.2 Experimental techniques . . . ................................... 4
1.3 Transport models ............................................ 5
1.4 Results ................................................... 7
1.4.1 Moleculary doped polymers . ................................... 7
1.4.2 Side-chain polymers ......................................... 9
1.4.3 Conjugated systems .......................................... 10
1.5 Outlook .................................................. 12
References . . . ............................................. 13
2 Novel Photoconductive Polymers ............................... 15
Jörg Bettenhausen and Peter Strohriegl
2.1 Introduction . . ............................................. 15
2.2 Liquid crystalline oxadiazoles and thiadiazoles ...................... 16
2.2.1 The basic idea . ............................................. 16
2.2.2 Monomer synthesis .......................................... 17
2.2.3 Oligo- and polysiloxanes with pendant oxadiazole groups .............. 19
2.2.4 Photoconductivity measurements ................................ 21
2.3 Starburst oxadiazole compounds ................................. 22
2.3.1 Motivation . . . ............................................. 22
2.3.2 Synthesis of starburst oxadiazole compounds ....................... 24
2.3.3 Thermal properties .......................................... 27
References . . . ............................................. 30
V

Contents
3 Theoretical Aspects of Anomalous Diffusion in Complex Systems ...... 31
Alexander Blumen
3.1 General aspects ............................................. 31
3.2 Photoconductivity ........................................... 32
3.3 The Matheron-de-Marsily model ................................ 36
3.4 Polymer chains in MdM flow fields .............................. 38
3.5 Conclusions . . ............................................. 42
Acknowledgements .......................................... 42
References . . . ............................................. 43
4 Low-Temperature Heat Release, Sound Velocity and Attenuation,
Specific Heat and Thermal Conductivity in Polymers ............... 44
Andreas Nittke, Michael Scherl, Pablo Esquinazi, Wolfgang Lorenz,
Junyun Li, and Frank Pobell
4.1 Introduction . . ............................................. 44
4.2 Phenomenological theory for heat release . . ........................ 46
4.2.1 Generalities . . . ............................................. 46
4.2.2 The standard tunneling model with infinite cooling rate . .............. 47
4.2.3 Influence of higher-order tunneling processes and a finite cooling rate . . . . 49
4.2.4 The influence of a constant and thermally activated relaxation rate ....... 52
4.3 Experimental details ......................................... 54
4.4 Experimental results and discussion .............................. 55
4.4.1 Specific heat and thermal conductivity ............................ 55
4.4.2 Internal friction and sound velocity .............................. 58
4.4.3 Heat release . . ............................................. 62
4.5 Conclusions . . ............................................. 65
Acknowledgements .......................................... 66
References . . . ............................................. 66
5 Spectral Diffusion due to Tunneling Processes at very low Temperatures 68
Hans Maier, Karl-Peter Müller, Siegbert Jahn, and Dietrich Haarer
5.1 Introduction . . ............................................. 68
5.2 The optical cryostat .......................................... 69
5.3 Theoretical considerations . . ................................... 71
5.4 Temperature dependence . . . ................................... 73
5.5 Time dependence ............................................ 74
References . . . ............................................. 76
6 Optically Induced Spectral Diffusion in Polymers Containing Water
Molecules: A TLS Model System ............................... 78
Klaus Barth, Dietrich Haarer, and Wolfgang Richter
6.1 Introduction . . ............................................. 78
6.2 Experimental setup for burning and detecting spectral holes ............ 79
6.3 Reversible line broadening phenomena ............................ 80
6.4 Induced spectral diffusion . . ................................... 83
References . . . ............................................. 87
VI

Contents
7 Slave-Boson Approach to Strongly Correlated Electron Systems ....... 88
Holger Fehske, Martin Deeg, and Helmut Büttner
7.1 Introduction . . ............................................. 88
7.2 Slave-boson theory for the t-t'-J model ............................ 90
7.2.1 SU(2)-invariant slave-particle representation ........................ 90
7.2.2 Functional integral formulation ................................. 93
7.2.3 Saddle-point approximation . ................................... 95
7.2.4 Magnetic phase diagram of the t-t'-J model ........................ 97
7.3 Comparison with experiments .................................. 102
7.3.1 Normal-state transport properties ................................ 102
7.3.2 Magnetic correlations and spin dynamics . . ........................ 105
7.3.3 Inelastic neutron scattering measurements . . ........................ 106
7.4 Summary ................................................. 109
References . . . ............................................. 110
8 Non-Linear Excitations and the Electronic Structure of Conjugated
Polymers ................................................. 113
Klaus Fesser
8.1 Introduction . . ............................................. 113
8.2 Models ................................................... 114
8.3 Disorder .................................................. 116
8.4 Non-linear excitations ........................................ 117
8.5 Perspective . . . ............................................. 119
Acknowledgements .......................................... 120
References . . . ............................................. 120
9 Diacetylene Single Crystals ................................... 122
Markus Schwoerer, Elmar Dormann, Thomas Vogtmann,
and Andreas Feldner
9.1 Introduction . . ............................................. 122
9.2 Photopolymerization ......................................... 129
9.2.1 Carbenes .................................................. 129
9.2.2 Intermediate photoproducts . ................................... 131
9.2.3 Electronic structure of dicarbenes ................................ 131
9.2.3.1 Electron spin resonance of quintet states ( 5 DC n ) .................... 131
9.2.3.2 ENDOR of quintet states . . . ................................... 136
9.2.3.3 ESR and ENDOR of triplet dicarbenes 3 DC n ....................... 139
9.2.4 Flash photolysis and reaction dynamics of diradicals .................. 141
9.3 Holography . . . ............................................. 144
9.3.1 Theory ................................................... 145
9.3.2 Experimental setup .......................................... 147
9.3.3 General characterization . . . ................................... 149
9.3.4 Angular selectivity .......................................... 150
9.3.5 Prepolymerized samples ....................................... 152
9.3.6 Chain length, polymer profile, and grating profiles ................... 152
9.3.7 Multrecording . ............................................. 154
VII

Contents
9.3.8 Holography . . . ............................................. 154
9.4 Di-, pyro-, and ferroelectricity .................................. 155
9.4.1 Dielectric properties of diacetylenes .............................. 156
9.4.1.1 Correlation of polymer content and electric permittivity . .............. 156
9.4.1.2 Application to topospecifically modified diacetylenes . . . .............. 158
9.4.1.3 Additional applications ....................................... 159
9.4.2 Pyroelectric diacetylenes . . . ................................... 159
9.4.2.1 IPUDO ................................................... 159
9.4.2.2 NP/4-MPU . . . ............................................. 160
9.4.2.3 DNP/MNP . . . ............................................. 161
9.4.2.4 Spurious piezo and pyroelectricity of diacetylenes .................... 161
9.4.3 The ferroelectric diacetylene DNP ............................... 162
9.4.4 Summary ................................................. 166
9.5 Non-linear optical properties ................................... 167
9.5.1 Aims of investigation ........................................ 167
9.5.2 Experimental setup .......................................... 167
9.5.3 Theoretical approaches ....................................... 170
9.5.4 Sample preparation .......................................... 170
9.5.5 Value and phase of the third order susceptibility w (3) .................. 170
9.5.6 Relaxation of the singlet exciton ................................ 171
9.5.7 The w (3) tensor components . ................................... 172
9.5.8 Signal saturation ............................................ 173
9.5.9 Spectral dispersion, phase, and relaxation of w (5) ..................... 174
9.5.10 Conclusion . . . ............................................. 176
References . . . ............................................. 177
10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids .......... 181
Thomas Giering, Peter Geißinger, Wolfgang Richter, and Dietrich Haarer
10.1 Introduction . . ............................................. 181
10.2 Stochastic theory ............................................ 183
10.3 Rare gases ................................................. 186
10.4 Experimental . . ............................................. 187
10.5 Inhomogeneous absorption lines ................................. 188
10.6 Pressure effects ............................................. 190
10.7 Rare gas mixtures ........................................... 191
10.8 Summary ................................................. 194
Acknowledgements .......................................... 195
References . . . ............................................. 195
II
Mainly Micelles, Polymers, and Liquid Crystals
11 The Micellar Structures and the Macroscopic Properties of Surfactant
Solutions ................................................. 199
Heinz Hoffmann
11.1 General behaviour of surfactants ................................ 199
VIII

Contents
11.2 From globular micelles towards bilayers . . . ........................ 200
11.3 Viscoelastic solutions with entangled rods . ........................ 202
11.3.1 General behaviour ........................................... 202
11.3.2 Viscoelastic systems ......................................... 205
11.3.3 Mechanisms for the different scaling behaviour ..................... 209
11.4 Viscoelastic solutions with multilamellar vesicles .................... 211
11.4.1 The conditions for the existence of vesicles ........................ 211
11.4.2 Freeze fracture electron microscopy .............................. 212
11.4.3 Rheological properties ........................................ 213
11.4.4 Model for the shear modulus ................................... 217
11.5 Ringing gels . . ............................................. 220
11.5.1 Introduction . . ............................................. 220
11.5.2 The aminoxide system ........................................ 221
11.5.3 The bis-(2-ethylhexyl)sulfosuccinate system ........................ 224
11.5.4 PEO-PPO-PEO block copolymers ................................ 226
11.6 Lyotropic mesophases ........................................ 227
11.6.1 Introduction . . ............................................. 227
11.6.2 Nematic phases and their properties .............................. 228
11.6.3 Cholesteric phases and their properties ............................ 232
11.6.4 Vesicle phases and L 3 phases ................................... 233
11.7 Shear induced phenomena . . ................................... 236
11.7.1 General ................................................... 236
11.7.2 Under what conditions do we find drag-reducing surfactants? ........... 236
11.8 SANS measurements on micellar systems . . ........................ 239
11.9 A new rheometer ............................................ 243
References . . . ............................................. 247
12 Photophysics of J Aggregates .................................. 251
Hermann Pschierer, Hauke Wendt, and Josef Friedrich
12.1 Introduction . . ............................................. 251
12.2 Basic aspects of pressure and electric field phenomena in hole burning
spectroscopy of J aggregates ................................... 252
12.3 Experimental . . ............................................. 253
12.4 Results ................................................... 254
12.5 Discussion . . . ............................................. 256
12.5.1 Pressure phenomena ......................................... 256
12.5.2 Electric field-induced phenomena ............................... 258
Acknowledgements .......................................... 258
References . . . ............................................. 259
13 Convection Instabilities in Nematic Liquid Crystals ................ 260
Lorenz Kramer and Werner Pesch
13.1 Introduction . . ............................................. 260
13.2 Basic equations and instability mechanisms ........................ 264
13.2.1 The director equation ......................................... 264
13.2.2 The velocity field ........................................... 266
IX

Contents
13.2.3 Electroconvection ........................................... 267
13.2.3.1 The standard model .......................................... 267
13.2.3.2 The weak electrolyte model . ................................... 268
13.2.4 Rayleigh-Bénard convection . ................................... 269
13.3 Theoretical analysis .......................................... 269
13.4 Rayleigh-Bénard convection . ................................... 275
13.5 Electrohydrodynamic convection ................................ 278
13.5.1 Linear theory and type of bifurcation ............................. 278
13.5.2 Results of Ginzburg-Landau equation ............................. 279
13.5.3 Beyond the Ginzburg-Landau equation ............................ 281
13.5.3.1 Experimental results ......................................... 281
13.5.3.2 Theoretical results and discussion ................................ 282
13.6 Concluding remarks .......................................... 286
Acknowledgements .......................................... 288
Note added . . . ............................................. 289
References . . . ............................................. 290
14 Preparation and Properties of Ionic and Surface Modified
Micronetworks ............................................. 295
Michael Mirke, Ralf Grottenmüller, and Manfred Schmidt
14.1 Introduction . . ............................................. 295
14.2 Polymerization in normal microemulsion . . ........................ 295
14.2.1 Mechanism and size control . ................................... 295
14.2.2 Surface functionalization of microgels ............................ 298
14.3 Polymerization in inverse microemulsion . . ........................ 299
14.3.1 Preparation of ionic microgels .................................. 299
14.3.2 Properties of ionic microgels and interparticle interaction .............. 299
14.4 Conclusion and relevance to future work . . ........................ 303
References . . . ............................................. 304
15 Ferrocene-Containing Polymers ................................ 305
Oskar Nuyken,Volker Burkhardt, Thomas Pöhlmann, Max Herberhold,
Fred Jochen Litterst, and Christian Hübsch
15.1 Introduction . . ............................................. 305
15.2 Addition polymers ........................................... 306
15.2.1 Radical polymerization ....................................... 306
15.2.2 Radical copolymerization . . . ................................... 308
15.2.3 Anionic polymerization of VFc ................................. 308
15.2.3.1 Living polymerization ........................................ 309
15.2.3.2 Block copolymers ........................................... 312
15.2.4 Polymeranalogeous reactions ................................... 314
15.3 Polymers with ferrocene units in the main chain ..................... 315
15.3.1 Polycondensation ............................................ 315
15.3.2 Polymers by addition of dithiols to diolefins ........................ 316
15.3.2.1 Radical reaction ............................................ 316
15.3.2.2 Base catalyzed reactions . . . ................................... 316
X

Contents
15.3.2.3 Acid catalyzed reactions . . . ................................... 318
15.3.3 1,1'-dimercapto-ferrocene as initiator ............................. 319
15.3.4 Reductive coupling .......................................... 320
15.4 Mößbauer studies of polymers containing ferrocene .................. 320
References . . . ............................................. 323
16 Transfer of Vibrational Energy in Dye-Doped Polymers ............. 325
Johannes Baier, Thomas Dahinten, and Alois Seilmeier
16.1 Introduction . . ............................................. 325
16.2 Experimental . . ............................................. 326
16.3 Results and discussion ........................................ 327
16.4 Summary ................................................. 332
References . . . ............................................. 332
17 Picosecond Laser Induced Photophysical Processes of Thiophene
Oligomers ................................................ 333
Dieter Grebner, Matthias Helbig, and Sabine Rentsch
17.1 Introduction . . ............................................. 333
17.2 Experimental . . ............................................. 334
17.3 Spectroscopic properties of oligothiophenes ........................ 337
17.4 Results ................................................... 337
17.4.1 Picosecond-transient spectra of oligothiophenes in solution ............. 337
17.4.2 Time behaviour of transient spectra .............................. 339
17.4.3 Size dependence of spectroscopic properties of oligothiophenes ......... 341
17.4.4 Size dependence of the kinetic behaviour of oligothiophenes ............ 342
17.5 Discussion . . . ............................................. 343
References . . . ............................................. 343
18 Topospecific Chemistry at Surfaces ............................. 344
Hans Ludwig Krauss
18.1 Introduction . . ............................................. 344
18.1.1 The problem . . ............................................. 344
18.1.2 Preparative and analytical methods ............................... 344
18.1.3 Industrial applications ........................................ 345
18.1.4 The standard procedures of the Phillips process ..................... 345
18.1.5 Earlier work . . ............................................. 346
18.2 The support . . . ............................................. 346
18.2.1 Unmodified silica ........................................... 346
18.2.2 Modified silica ............................................. 347
18.2.3 Others .................................................... 348
18.3 Transition metal surface compounds .............................. 348
18.3.1 The metals . . . ............................................. 348
18.3.2 Impregnation and activation . ................................... 349
18.4 Coordinatively unsaturated sites ................................. 350
18.4.1 Reduction of saturated surface compounds . ........................ 350
18.4.2 Elimination of ligands ........................................ 352
XI

Contents
18.5 Physical properties of the coordinatively unsaturated sites .............. 353
18.5.1 Topologically different sites . ................................... 353
18.5.2 Optical and magnetic properties ................................. 353
18.6 Chemical properties of the coordinatively unsaturated sites ............. 355
18.6.1 Survey of catalytic reactions ................................... 355
18.6.2 Olefin polymerization ........................................ 356
18.6.3 Other catalytic reactions . . . ................................... 360
18.7 Deactivation . . ............................................. 361
18.8 Summary and outlook ........................................ 362
References . . . ............................................. 363
III
Biopolymers
19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers 369
Stefan Limmer, Günther Ott, and Mathias Sprinzl
19.1 Introduction . . ............................................. 369
19.2 Site-specific NMR spectroscopy of chemically synthesized RNA duplexes . . 370
19.2.1 Stability of tRNA-derived acceptor stem duplexes .................... 371
19.2.2 Manganese ion binding sites at RNA duplexes ...................... 374
19.2.3 Structural determination of short RNA duplexes by 2D NMR spectroscopy . 377
19.2.4 NMR derived model of the tRNA Ala acceptor arm ................... 379
19.2.5 Chemical shifts and scalar coupling as an indicator of RNA structure in the
vicinity of a G-U pair ........................................ 380
19.2.6 Structure of aminoacyl-tRNA and transacylation of the aminoacyl residue . . 382
19.3 Structure of elongation factor Tu ................................ 383
19.3.1 Sequence of Thermus thermophilus EF-Tu.......................... 384
19.3.2 Crystallization, X-ray analysis, and the tertiary structure . .............. 386
19.3.3 Nucleotide binding and GTPase reaction . . ........................ 387
19.3.4 Mechanism of GTP induced conformational change of EF-Tu ........... 388
19.3.5 Aminoacyl-tRNA in complex with EF-Tu 7 GTP ..................... 389
19.3.6
1 H NMR of yeast Phe-tRNA Phe EF-Tu7GTP complex . . .............. 391
19.3.7
13 C NMR studies of the Val-tRNA Val EF-Tu7GTP ternary complex ....... 393
19.3.8 Role of EF-Tu in complex with aminoacyl-tRNA ..................... 395
19.3.9 EF-Tu interaction with EF-Ts ................................... 395
19. 3.10 Site-directed mutagenesis of EF-Tu ............................... 396
19.4 Summary and conclusions . . ................................... 397
Acknowledgement ........................................... 397
References . . . ............................................. 398
20 Spectroscopic Probes of Surfactant Systems and Biopolymers ......... 401
Alexander Wokaun
20.1 Introduction . . ............................................. 401
20.2 Diffusion in surfactant systems .................................. 402
20.2.1 Structural characteristics of micellar solutions, cubic phases, and multilamellar
vesicles from NMR self-diffusion measurements .............. 402
XII

Contents
20.2.2 Probing of mobilities in multilamellar vesicles by forced Rayleigh scattering 405
20.2.3 Dimensionality of diffusion in lyotropic mesophases from fluorescence
quenching ................................................. 409
20.2.4 Summary of results .......................................... 412
20.3 Vibrational spectroscopy and conformational analysis of oligonucleotides . . 413
20.3.1 Spectroscopic characterization of right and left-helical forms of a hexadecanucleotide
duplex ........................................ 413
20.3.2 SERS spectra of deoxyribonucleotides ............................ 415
20.3.3 Studies of chromophore-DNA interaction by vibrational spectroscopy . . . . . 417
20.3.4 Summary of results .......................................... 418
20.4 Related projects carried out within the framework of the Collaborative
Research Centre ............................................ 419
20.5 Remarks and acknowledgements ................................ 421
References . . . ............................................. 422
21 Energy Transport by Lattice Solitons in a-Helical Proteins ........... 424
Franz-Georg Mertens, Dieter Hochstrasser, and Helmut Büttner
21.1 Introduction . . ............................................. 424
21.2 The model ................................................. 426
21.3 Quasicontinuum approximation ................................. 428
21.4 Velocity range for the quasicontinuum approach ..................... 431
21.5 Solitary waves for realistic parameter values ........................ 432
21.6 Iterative method and stability ................................... 434
21.7 Conclusion . . . ............................................. 437
References . . . ............................................. 438
IV
Appendix
22 Documentation of the Collaborative Research Centre 213 ............ 443
Markus Schwoerer and Heinz Hoffmann
22.1 List of Members ............................................ 443
22.2 Heads of Projects (Teilprojektleiter) .............................. 444
22.2.1 Projektbereich A: Gemeinsame Einrichtungen ...................... 444
22.2.2 Projektbereich B: Festkörper ................................... 444
22.2.3 Projektbereich C: Funktionale Systeme – Mizellen, Oberflächen
und Polymere . ............................................. 445
22.2.4 Projektbereich D: Biopolymere ................................. 446
22.3 Guests .................................................... 447
22.4 Co-workers . . . ............................................. 450
22.5 International Cooperation . . . ................................... 457
22.6 Funding .................................................. 459
XIII

Preface
At the end of 1983, about eight years after the inauguration of the University of Bayreuth
the Deutsche Forschungsgemeinschaft (DFG) agreed to establish the Collaborative Research
Centre 213, TOPOMAC, for the promotion of basic research in chemistry and physics of
macromolecular systems. The title of TOPOMAC in its full length, “Topospezifische Chemie
und Toposelektive Spektroskopie von Makromolekülsystemen: Mikroskopische Wechselwirkung
und Makroskopische Funktion” expessed the intention of the original applicants:
a productive cooperation between physicists, chemists and biochemists across the mutual
borders of their original research fields. Until the end of 1995 TOPOMAC was supported by
the Deutsche Forschungsgemeinschaft with 28 MDM.
The present book documents the achievements of TOPOMAC. It is not a minute addition
of all the results which have been published in periodical journals but rather a survey of
important research fields of members of TOPOMAC. The articles have been written towards
or after the end of the support period as both, review and original publications.
The book covers the fields of:
. Electronic, photoelectric, thermal, dielectric, optical and magnetic properties of macromolecular
solids (polymers and polymer crystals),
. Micellar structures, J-aggregates, liquid crystals, µ-gels, ferrocene-containing polymers
and topospecific chemistry at surfaces and in single crystals, and
. Biopolymers and surfactant systems as studied by site directed spectroscopy and site directed
chemistry and also by the theory of energy transport.
During the period of its support TOPOMAC had 30 members (Teilprojektleiter). Only
ten of them have been members for the entire period, mainly because twenty times a call
from other universities or research institutions reached one of the members of the Collaborative
Research Centre 213. Fourteen of them followed this call and left the Collaborative Research
Centre 213. Less than two years after the end of the period of support some of the remaining
former members of Collaborative Research Centre 213 together with young and
new faculty members began to continue the formal cooperation between chemists and physicists
in the field of macromolecular research. Their actual cooperation in the meantime
never had been terminated.
The editors would like to express the sincere thanks of the members of TOPOMAC to
the foreign guests of TOPOMAC, to the Deutsche Forschungsgemeinschaft, to the University
XV

Preface
of Bayreuth and also to the Freistaat Bayern. Our foreign guests stayed for long or short periods
between one year and one day. They have contributed in an essential and special manner
to the success on our research fields. They also strongly intensified the national and international
scientific relations of both, the members and the research students of TOPO-
MAC. The cooperation of the speakers of TOPOMAC with Dr. Funk from the Deutsche Forschungsgemeinschaft
office throughout the entire period of support was excellent. The help
of the former president of the university, Dr. K. D. Wolff and his chancellor, W. P. Hentschel
as well as the continuous support by the late Ministerialrat G. Grote and by J. Großkreutz
from the Bayerisches Staatsministerium für Unterricht, Kultus, Wissenschaft und Kunst has
been an essential stimulus for the scientific members of TOPOMAC. Last but not least we
thank Doris Buntkowski for her faithful and reliable work as our secretary.
The Editors
XVI

List of Contributors
Alexander Blumen
Theoretische Polymerphysik
Universität Freiburg
Herrmann-Herder-Straße 3
79104 Freiburg
Elmar Dormann
Physikalisches Institut
Universität Karlsruhe
Engesserstr. 7
67131 Karlsruhe
Pablo Esquinazi
Institut für Experimentelle Physik II
Universität Leipzig
Linnestraße 5
04103 Leipzig
Klaus Fesser
Fachbereich Physik
Universität Greifswald
Domstraße 10a
17489 Greifswald
Josef Friedrich
Lehrstuhl für Physik
Technische Universität München
85350 Freising-Weihenstephan
Holger Fehske
Theoretische Physik I
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Dietrich Haarer
Experimentalphysik IV
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Heinz Hoffmann
Physikalische Chemie I
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Lorenz Kramer
Theoretische Physik II
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Hans Ludwig Krauss
Heunischstraße 5 b
96049 Bamberg
Franz G. Mertens
Theoretische Physik
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Oskar Nuyken
Lehrstuhl für Makromolekulare Stoffe
Technische Universität München
Lichtenbergstraße 4
85747 München
XVII

List of Contributors
Sabine Rentsch
Institut für Optik und Quantenelektronik
Friedrich Schiller Universität Jena
Max-Wien-Platz 1
07743 Jena
Wolfgang Richter
Experimentalphysik IV
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Paul Rösch
Biopolymere
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Manfred Schmidt
Institut für Physikalische Chemie
Universität Mainz
Welder-Weg 11
55099 Mainz
Markus Schwoerer
Experimentalphysik II
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Alois Seilmeier
Physikalisches Institut
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Mathias Sprinzl
Biochemie
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Peter Strohriegel
Makromolekulare Chemie I
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Thomas Vogtmann
Experimentalphysik II
Universität Bayreuth
Universitätsstraße 30
95447 Bayreuth
Alexander Wokaun
Bereich F5
Paul Scherrer Institut
CH-5232 Villingen
XVIII

I
Mainly Solids
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

1 Model Systems for Photoconductive Materials
Harald Meyer and Dietrich Haarer
1.1 Introduction
The effect of photoconductivity, i. e. the increase of the electrical conductivity of a material
upon illumination with light of suitable photon energy, was first discovered in selenium by
W. Smith in 1873.
A typical application for these materials is the xerographic process [2, 3], which was
developed by C. F. Carlson in 1942 [1]. Here, a photoconducting film on top of a grounded
electrode is homogeneously charged by a corona discharge. Typical field strengths are up to
10 6 V/m. In a second step the image of the original document is projected onto the film. At
the areas where the film is illuminated charge carriers are generated in the photoconducting
film. One species traverses the film and recombines at the grounded electrode whereas the
oppositely charged species neutralizes the surface charges. With this step the original image
is transferred into an electrostatic image on the photoconductor film. Subsequently, small toner
particles are deposited on the photoconductor. They stick to the charged regions and
thus generate a real image. In the next step, this image is transferred onto paper and fixed
by thermally fusing the toner particles on the paper.
The process for a laser printer is similar except for the fact that the image is written
onto the photoconductor directly by a laser or a diode array. If the toner particles are fused
directly onto the photoconductor instead of transferring them onto paper the photoconductor
can be used as an offset printing master [4].
Potential systems for commercial use have to meet several requirements:
a) sensitivity in the visible region of the spectrum;
b) good charge transport properties, i. e. charge carrier mobility in excess of 10 –7 cm 2 /Vs
[3] and minor trapping effects;
c) good mechanical and dielectrical properties;
d) excellent film forming properties and possibility of manufacturing defect free large area
films;
e) mechanical flexibility for the use in small sized desktop devices.
The last two requirements cannot be met neither by organic nor inorganic crystalline
materials. Therefore both, organic and inorganic amorphous photoconductors, have been de-
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
3

1 Model Systems for Photoconductive Materials
veloped. Inorganic materials like e. g. a-Se, As 2 Se 3 or alloys of Se and Te show good transport
properties with mobilities in the range of 0.1 cm 2 /Vs at room temperature [7], and high
sensitivity for visible light together with moderate mechanical and dielectric properties. One
of their most important disadvantages compared to organic systems is that most of these materials
are highly toxic.
In the past 30 years, since H. Hoegl discovered photoconductivity in the organic polymer
poly(N-vinylcarbazole) (PVK) (Fig. 1.2) [5, 6], organic materials have almost completely
replaced their inorganic counterparts, although the effective mobility in these systems is
typically 5 orders of magnitude lower as compared to a-Se.
Organic photoconductors can be regarded as large or medium bandgap semiconductors.
A typical value for commercially applied systems is 3.5 eV for carbazole derivatives
(Section 1.4). This inherent lack of sensitivity in the visible region has been overcome by
using charge transfer systems [5] or multilayer systems with additional charge generation
layers. For a review see e. g. Ref. [3]. On the other hand, the large bandgap of organic materials
virtually eliminates the influence of thermally generated charge carriers and thus improves
the dielectric properties as compared to low bandgap materials.
Therefore the main target of the works, which will be described in the following, was
to identify the key factors which limit the charge carrier mobilities in organic systems and
to develop new high mobility materials.
1.2 Experimental techniques
Besides the xerographic discharge method [3, 10], the time-of-flight (TOF) technique is generally
used to study the charge carrier transport of thin organic films. Here, the photoconducting
film with a typical thickness of 10 mm is sandwiched between two electrodes (Fig. 1.1). Electron-hole
pairs are generated by the energy hn of a strongly absorbed laser pulse, which is irradiated
through one of the semitransparent electrodes. For most organic materials the charge
carrier generation process can be described by an Onsager model, either in its one or three dimensional
form [11–13].
The wavelength of the laser is chosen to ensure that the penetration depth is considerably
less than the sample thickness. Thus the charge carriers are generated close to the illuminated
surface. Under the influence of an externally applied electrical field the electronhole
pairs are separated and one species, depending on the polarity of the external field, immediately
recombines at the illuminated electrode. The opposite charged species drifts
through the sample, thus giving rise to a time dependent photocurrent I p (t).
For a quantitative analysis the knowledge about the electrical field inside the sample
is necessary. This can be achieved by performing the measurements in the small signal limit,
i. e. disturbations due to space charge effects are negligible and the electric field in the bulk
is determined by the externally applied field. It turns out, that this condition is fulfilled as
long as the generated photocharge Q tot is less than 10 % of the charge Q = CU, which is
4

1.3 Transport models
Figure 1.1: Principle setup for TOF experiments.
stored on the electrodes. With typical experimental parameters (sample thickness d =10mm,
sample area A =4cm 2 ,10 21 monomer units per cm 3 , voltage U = 300 V, sample capacitance
C = 1 nF) the above criterion leads to an upper limit for the photocharges of Q tot

1 Model Systems for Photoconductive Materials
neglected, or in other words, the detrapping always occurs to states at e d . The traps can be
caused by static disorder as well as by self trapping due to polaronic effects or both.
Mathematically equivalent with the MT model is the Continuous Time Random Walk
(CTRW) model [47–49], where the exponential trap distribution density g (e) in the MT
model,
"
g…"† /exp ; …2†
k B T 0
corresponds to the well-known algebraic distribution of hopping times in the CTRW model
[45],
…t† /t 1 ; …3†
where the disorder parameter a in Eq. 3 corresponds to the dimensionless temperature
(a = T/T 0 ).
A variety of experimental data [16, 25, 26, 33, 34] can be described by the empirical
formula,
ef f ˆ 0 exp
p
" 0 E
k B T ef f
; …4†
first proposed by Gill [33], where e 0 is the zero field activation energy, E the electrical field
and b the Poole-Frenkel factor. This equation describes the thermal activated release from
localized traps where the activation energy is lowered by an external field according to the
Poole-Frenkel effect [37].
The effective temperature T eff is related to the physical temperature T by
1
ˆ 1
T ef f T
1
T 0
:
…5†
Initially, the characteristic temperature T 0 simply was an empirical parameter. In
Section 1.4, however, we shall see that in certain cases this parameter can be interpreted microscopically.
An alternative approach [28, 50–54] is based on the assumption that the density of
states can be modelled by a Gaussian distribution. Charge carrier transport occurs via direct
hopping between the localized sites. In general, the differences between the two models are
too small to be detected experimentally. Since our data can be quantitatively explained
within the framework of multiple trapping we will restrict the discussion to this model.
6

1.4 Results
1.4 Results
As stated in Section 1.1, organic photoconductors can be regarded as large or medium bandgap
semiconductors. In contrast to inorganic crystalline semiconductors the electronic coupling between
adjacent sites is weak. Therefore the electronic states in these materials are primarily of
molecular character and the charge carrier transport has to be regarded as an incoherent hopping
process between two neighbouring sites. For the following discussion organic photoconductors
will be divided into three main groups namely molecularly doped polymers (Section
1.4.1), side-chain polymers (Section 1.4.2.), and conjugated systems (Section 1.4.3).
1.4.1 Molecularly doped polymers
In molecularly doped polymers (MDPs) the transport molecules are molecularly dispersed in
an inert matrix. Due to crystallization the maximum concentration of chromophores, which
can be achieved, is typically in the range of 50 mol%.
Typical chemical compounds include oxadiazole derivatives [14], pyrazolines [15–
18], hydrazones [19–22], carbazole derivatives [23–26], triphenylmethane (TPM) derivatives
[27, 28], triphenylamine (TPA) derivatives [29, 30], and TAPC [31], which can be regarded
as a dimer of TPA. The charge carrier mobilities at room temperature are typically
in the range from 10 –6 cm 2 /Vs for N-isopropylcarbazole [25] to 10 –4 cm 2 /Vs for p-diethylaminobenzaldehyde
diphenyl hydrazone (DEH) [22].
In order to study the effect of chemical substitutions of the respective transport molecule
N-isopropylcarbazole (NIPC) and derivatives thereof, with electron donor as well as acceptor
substituents (Fig. 1.2), have been investigated in a polycarbonate host [26].
As compared by 3,6-dibromo-N-isopropylcarbazole (DBr-NIPC) the effective hole
mobility of 3,6-dimethoxy-N-isopropylcarbazole (DMO-NIPC) is slightly decreased. The
reason for this behaviour is the fact that the cation, which is relevant for the transport process,
is stabilized by electron donating substituents like the methoxy group, as can be seen
from the shift of the ionisation potentials [26]. This leads to stronger localization of the
charge within the aromatic rings as compared to derivatives with moderate electron acceptors
like bromine. Therefore the spatial overlap of the electronic states of adjacent molecules,
which are involved in the transport process, is reduced.
With strong acceptors like the nitro groups in 3,6-dinitro-N-isopropylcarbazole (DN-
NIPC) the ionisation potential is already larger than for the host matrix polycarbonate. In
this case even the matrix acts as a trap for holes, thus preventing efficient charge carrier
transport [26].
Since the effective mobility is determined by the spatial overlap of the transport states,
the concentration dependence of the effective mobility can be described by
ef f / r 2 exp
2r
; …6†
r 0
7

1 Model Systems for Photoconductive Materials
N
H 3 CO
N
N
N
N
C 2 H 5
R
R
BTA
R: H: NIPC
Br: DBr-NIPC
OCH 3 : DMO-NIPC
NO 2 : DN-NIPC
Si
Si
O
CH 3
PMPS
n
Ca n
CH CH 2
PVK
(CH 2 ) n
Ca
n
R
Polysiloxan
R
n
Ca:
N
R:
O
DPOP-PPV
Figure 1.2: Chemical structure of photoconducting materials. Abbreviations see text.
H
PPV
where r is the mean distance between adjacent transport molecules and r 0 the wave function
decay parameter. For carbazole derivatives r 0 is typically in the range 1.3–1.6 Å, depending
on the substituted groups [26].
In recent years, also percolation models have been successfully applied to experimental
data [55] as an alternative approach to model the transport properties of doped disordered
systems. For polycarbonate doped with a derivative of benztriazole (BTA) (Fig. 1.2) it has
been shown that the concentration dependence of the effective mobility can be described at
low concentrations by [55]
ef f ˆ 0 …p p c † t …7†
for p>p c , where p is the concentration of transport molecules, p c the percolation threshold,
and t the critical exponent. The experimental values (p c = 0.095 and t = 2.46) are consistent
with 3D-continuum percolation calculations [56].
Since this system shows good transport properties over a wide range of concentrations,
the influence of extrinsic traps could be studied [57]. Here, a sample containing 25 wt% of
BTA in a polycarbonate host was doped with small amounts of the organic dye astrazone
orange (AO). It turned out that the effective mobility is reduced by a factor of 2 when the
8

1.4 Results
concentration of AO is increased to 0.5 wt%. This finding becomes also important when
thinking of new applications like organic electroluminescent devices. One way to increase
the efficiency in these devices is to molecularly disperse the luminescent moiety into a
charge transport material. But there is a trade-off between efficient trapping of the charge
carriers – which is improved by higher concentrations of the luminescent groups – and concentration
quenching of the fluorescence, which can be avoided by keeping the concentration
as low as possible. The above described measurements [57] show that organic dyes can be
very efficient traps for charge carriers. Therefore the concentration of the luminescent moiety
can be as low as 2% or less [61], thus avoiding concentration quenching without loosing
high trapping efficiency.
1.4.2 Side-chain polymers
In side-chain polymers, where the chromophore is covalently bonded to a polymer main
chain, the concentration of transport molecules can be increased as compared to MDPs without
causing crystallization. Because the underlying physical mechanism of charge carrier
transport (nearest neighbour hopping between weakly coupled, localized states) is the same
for both MDPs and side-chain polymers, the transport properties are qualitatively similar.
This finding offers the opportunity to tailor the thermal and mechanical properties of
the photoconductor without seriously affecting the transport properties. This can be achieved
either by variation of the spacer length between the chromophore and the backbone or by variation
of the backbone itself. A typical example for this group is PVK [12, 32, 34, 35]. Here,
the transition from dispersive to non-dispersive transport could be observed [34]. Due to the
fact, that the TOF curves have been measured over 10 decades in time, it was possible by
means of a numerical inverse Laplace transform [34] to calculate from the measured photocurrent
the trapping rate distribution, which is a measure for the density of localized states.
In polysiloxane derivatives with pendant carbazolyl groups [58, 59] (Fig. 1.2) the effect
of a variation of the spacer length was investigated. Starting from a spacer length of 3 or 5
methylene units up to 6 or 11 groups, the glass transition temperature drops from its initial value
of 51 8Cor78Cdownto–58Cor–458C for the C 11 spacer. It turns out, that the zero field
activation energy e act for the hole transport remains unchanged and is the same for NIPC or
PVK, e act = 0.51 eV [58]. This indicates, that the activation energy in side-chain polymers
with pendant carbazolyl groups is dominated by intrinsic properties of the carbazolyl moiety
and is not much affected by other factors like the morphology of the polymer [58].
However, the absolute values for m eff differ by roughly one order of magnitude, where the
compound with the shortest spacer shows an effective mobility in the range of 10 –6 cm 2 /Vs
(T = 300 K, E =3710 5 V/cm). For the materials with a spacer length of 5 and 6 methylene
units the mobility is lower by one order of magnitude and therefore comparable to the mobility
in NIPC or PVK. A striking feature, when comparing NIPC with PVK, is the fact that
polycarbonate doped with 20 wt% of NIPC shows the same hole mobility at room temperature
as compared to PVK where the carbazole concentration amounts to 86 wt%. Obviously,
the covalent attachment of the chromophore to a stiff backbone, like PVK with its glass
transition temperature of 227 8C, induces a mutual orientation of adjacent pendant groups,
9

1 Model Systems for Photoconductive Materials
which reduces the electronic coupling between the two sites. Therefore it is desirable to
induce a well-defined amount of flexibility in both, the spacer and the backbone, to allow
for mutual reorientation of the chromophores. For a polysiloxane backbone with carbazolyl
transport groups the optimum spacer length n is equal to 3.
1.4.3 Conjugated systems
In contrast to the two main groups described above the charge carrier transport in conjugated
systems occurs via the polymer backbone. Various substituents are used to tailor the
mechanical properties and the processibility. For instance, the chain can be a skeleton with
quasi-s-conjugation like in polysilanes [36, 38–42] or polygermylenes [39].
The second group of main chain polymers contains a p-conjugated backbone like
polyacetylene, polythiophene, polypyrrole, polyphenylene, poly(phenylenevinylene) (PPV),
and their derivatives. For a review see [43, 44].
Both types of conjugated systems have been investigated. In the case of poly(methylphenyl
silane) (PMPS), a quasi-s-conjugated polymer (Fig. 1.2), non-dispersive transport
has been found down to a temperature of 250 K [60]. For lower temperatures, the transport
is dispersive, which indicates that the transport in these materials is controlled by traps.
Compared to the materials described above, the depths of the relevant traps are much lower.
The zero field activation energy turns out to be as low as 0.37 eV as compared to 0.5 eV for
materials containing carbazole. This gives a hole mobility at room temperature of roughly
10 –3 cm 2 /Vs, which is 3 orders of magnitude higher than for the materials described above.
The high mobility and the transparency in the visible region makes PMPS also useful for
applications in multilayer electroluminescent devices [61–63].
Besides the s-bonded PMPS, we investigated oligomeric conjugated compounds based
on carbazole [64]. First experiments with a trimer of a carbazole derivative showed that this
material is soluble in most of the common organic solvents. It forms a low molecular weight
glass and shows no tendency to crystallize even after several months. In this material the
p-system has been extended over three monomer units, which results in remarkably high
hole mobilities as compared to the monomer model compound NIPC. First TOF experiments
show an increase in mobility by roughly two orders of magnitude as compared to unconjugated
carbazole systems, which reaches 2 7 10 –4 cm 2 /Vs [64].
The second group of p-conjugated polymers contains one p z -orbital per carbon atom
which stands perpendicular to the plane of the s-bonded skeleton of the main chain. The interaction
between these orbitals leads to electronic states, which are delocalized – at least
partly – along the conjugated chain. As compared to the intrachain coupling, the intrachain
coupling is lower by typically one or two orders of magnitude [66] thus leading to a large
anisotropy of the electronic and optical properties. These materials can be regarded as basically
one-dimensional organic semiconductors with bandgaps in the visible region of the
spectrum, e. g. 1.5 eV (trans-(CH) x ) [43], 2.0 eV (polythiophene), 2.5 eV (polyphenylene vinylene),
and 3.0 eV [poly(para-phenylene)] [67].
We investigated the charge transport properties of the p-conjugated polymer poly
(para-phenylenevinylene) and its soluble substituted derivative DPOP-PPV (Fig. 1.2). In
10

1.4 Results
contrast to e. g. (CH) x , PPV can be prepared in undoped form. If oxygen is carefully heated
out, PPV exhibits excellent film forming properties as well as thermal stability under ambient
conditions. In recent years, PPV has drawn additional attention due to its use in organic
electroluminescent devices [45].
In the following we will show that the charge carrier transport in these materials can
be described by the conventional models mentioned before, which have been developed for
the characterization of disordered molecular systems.
Based on the MT model a method has been developed to calculate the distribution of
capture rates from the measured photocurrent [68, 69]. In contrast to Ref. [34], the method
is based on a Fourier transform technique with better numerical stability as compared to the
inverse Laplace transform used in Ref. [34].
With this approach the capture rate density o c is given by
! c …" ! †k B T ˆ 2
p
Q tot sin
t mic
I…!†
!
! ; …8†
where Q tot is the total photogenerated charge, t mic the microscopic transit time, and I (o) the
Fourier transform of the measured photocurrent, and f the phase shift (Eq. 9). For DPOP-
PPV, t mic = 7.5610 –10 s for the given experimental conditions.
ˆ tan 1 Im I…!†
Re I…!†
…9†
The trap depth e o is related to the frequency o by the expression
! ˆ r 0 exp
" !
; …10†
k B T
where r 0 is the attempt-to-escape frequency (r 0 =10 10 s –1 for DPOP-PPV).
In DPOP-PPV the effective mobility is field dependent and thermally activated according
to Gill’s formula (Eq. 4) [33] with a characteristic temperature T 0 = 465 K (Fig. 1.3).
The calculated trap density g (e) is exponential down to a pronounced cutoff energy e d ,
g…"† /exp
"
: …11†
k B T 0
This cutoff energy corresponds exactly to the measured activation energy of the effective
mobility. Furthermore, the initially simply empirical parameter T 0 in Gill’s formula
(Eq. 4) could be correlated with the decay constant k B T 0 of the trap distribution.
By calculating the total number of trapping events it can be seen that the typical distance
which a charge carrier travels between two trapping events is of the order of 4 Å. This
value is comparable to the intrachain distance and indicates that the transport in these materials
can be best described by conventional hopping between closely neighbouring sites. No
evidence for bandlike transport has been found.
11

µ eff
[cm 2 /Vs]
1 Model Systems for Photoconductive Materials
10 -2
10 -3
10 -4
350 K 300 K 270 K
230 K
500 kV/cm
350 kV/cm
200 kV/cm
75 kV/cm
0kV/cm
10 -5
10 -6
10 -7
10 2
10 0
T 0
= 465K
10 -2
10 -4
10 -6
10 -8
10 -10
0 2 4 6 8
10 -8
2.5 3.0 3.5 4.0 4.5
1/T [1000/K]
Figure 1.3: large frame: effective mobility m eff of DPOP-PPV as a function of temperature for some
electrical fields, data points for zero field were extrapolated from field dependent measurements; inset:
Arrhenius fits for different electrical fields. For details see text. Data from [68, 69].
For the case of PPV the dispersion of the photocurrent is too large for transit time determination.
It can be concluded that even at room temperature the release times from the
deepest traps, which control the transport properties, are in the range of 1 s or less. This corresponds
to an effective mobility of less than 10 –8 cm 2 /Vs [69]. Based on TSC measurements
from different groups [70], we conclude that this behaviour is primarily caused by the
existence of grain boundaries.
It is reasonable to assume that in conjugated polymers with a rigid backbone the preferred
orientation of the main chain will be parallel to the film surface. Therefore, the
charge carrier transport through the film has to occur perpendicular to the backbone, i. e.
perpendicular to the direction with high intrinsic mobility, which would mean a principle
limit for this class of materials for this kind of applications.
1.5 Outlook
As described above, disordered organic materials have been developed with effective hole
mobilities of up to 10 –3 cm 2 /Vs together with good mechanical and dielectric properties.
However it seems that for disordered systems a considerable improvement of this value will
12

References
be difficult to achieve, because for all investigated material groups the charge carrier transport
is limited by the localized nature of the electronic states and by the hopping mechanism
of the transport process.
Based on recent other works [8, 9], we think that for developing even higher mobility
materials it is essential to improve the structural order of the material. This may open new
application fields, e. g. the use of active organic materials in semiconductor devices. One
promising way is the use of highly ordered liquid crystals. With these materials, however,
charge carrier mobilities up to 0.1 cm 2 /Vs or even more are realistic, which would make
them comparable to single crystalline systems [65].
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50. P.M. Borsenberger, L. Pautmeier, H. Bässler: J. Chem. Phys., 94(8), 5447 (1991)
51. G. Schönherr, H. Bässler, M. Silver: Philos. Mag. B, 44(3), 369 (1981)
52. H. Bässler, G. Schönherr, M. Abkowitz, D.M. Pai: Phys. Rev. B, 26(6), 3105 (1982)
53. L. Pautmeier, R. Richert, H. Bässler: Philos. Mag. Lett., 59(6), 325 (1989)
54. L. Pautmeier, R. Richert, H. Bässler: Synth. Met., 37, 271 (1990)
55. H. Domes, R. Leyrer, D. Haarer, A. Blumen: Phys. Rev. B, 36(8), 4522 (1987)
56. D. Stauffer: Introduction to Percolation Theory, Taylor and Francis, London (1985)
57. H. Domes: PhD thesis, Universität Bayreuth (1988)
58. H. Domes, R. Fischer, D. Haarer, R. Strohriegl: Makromol. Chem., 190, 165 (1989)
59. H. Schnörer, H. Domes, A. Blumen, D. Haarer: Philos. Mag. Lett., 58(2), 101 (1988)
60. H. Kaul: PhD thesis, Universität Bayreuth (1991)
61. H. Suzuki, H. Meyer, J. Simmerer, J. Yang, D. Haarer: Adv. Mater., 5, 743 (1993)
62. H. Suzuki, H. Meyer, S. Hoshino, D. Haarer: J. Appl. Phys. in press (1995)
63. J. Kido, K. Nagai, Y. Okamoto, T. Skotheim: Appl. Phys. Lett., 59, 2760 (1991)
64. C. Beginn, J.V. Grazulevicius, P. Strohriegl,J. Simmerer, D. Haarer: Macromol. Chem. Phys., 195,
2353 (1994)
65. N. Karl: in: K. Sumino (ed.): Defect Control in Semiconductors, Elsevier Science Publishers,
North Holland (1990)
66. P. Gomes da Costa, R.G. Dandrea, E.M. Conwell: Phys. Rev. B, 47(4), 1800 (1993)
67. G. Leising, K. Pichler, F. Stelzer: in: H. Kuzmany, M. Mehring, S. Roth, (eds.): Springer Series in
Solid-State Sciences, Vol. 91: Electronic Properties of Conjugated Polymers III, Springer, Berlin
(1989)
68. H. Meyer: PhD thesis, Universität Bayreuth (1994)
69. H. Meyer, D. Haarer, H. Naarmann, H.H. Hörhold: Phys. Rev. B, in press (1995)
70. M. Onoda, D.H. Park, K. Yoshino: J. Phys. (London), Condens. Matter, 1(1), 113 (1989)
14

2 Novel Photoconductive Polymers
Jörg Bettenhausen and Peter Strohriegl
2.1 Introduction
Electrophotography is the only area in which the conductivity of sophisticated organic materials
and polymers is exploited in a large scale industrial process today. Photoconductors are
characterized by an increase of electrical conductivity upon irradiation. According to this definition
photoconductive materials are insulators in the dark and become semiconductors if
illuminated. In contrast to electrically conductive compounds photoconductors do not contain
free carriers of charge. In photoconductors these carriers are generated by the action of
light.
The discovery of photoconductivity dates back to 1873 when W. Smith found the effect
in selenium. Based on this discovery C. F. Carlson developed the principles of the xerographic
process already in 1938.
Photoconductivity in organic polymers was first discovered in 1957 by H. Hoegl, who
found that poly(N-vinylcarbazole) (PVK) and charge transfer complexes of PVK with electron
acceptors like 2,4,7-trinitrofluorenone act as photoconductors [1].
Besides the application of photoconductive polymers in photocopiers these materials
are also widely used in laser printers in the last years. The third area in which photoconductors
are applied is the manufacturing of electrophotographic printing plates.
The organic photoconductors used in practice are based on two types of systems. The
first one are polymers in which the photoconductive moiety is part of the polymer, for example
a pendant or in-chain group. The second group involves low molecular weight compounds
imbedded in a polymer matrix. These so-called moleculary doped polymers are
widely used today.
One interesting class of photoconductive materials are oxadiazoles. It is known for a
long time that derivatives of 1,3,4-oxadiazole are good photoconductors. Compound 1 for
example is described in patents and was frequently applied in photocopiers [2].
1 2
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
15

2 Novel Photoconductive Polymers
The oxadiazole 1 is a hole transport material since the electron withdrawing effect of
the oxadiazole group with three electronegative heteroatoms is overcompensated by two
electron donating amino groups.
Within the last years, oxadiazoles like 2-(biphenyl)-5-(4-tert.butylphenyl)-1,3,4-
oxadiazole (PBD) 2 have been frequently applied in organic light emitting diodes [3]. Here
the electron withdrawing oxadiazole unit dominates the electronic properties and the oxadiazole
compounds act as electron injection and transport layers. Furthermore, 2,5-diphenyloxadiazoles
have been used as building blocks in thermostable polymers and they are highly
fluorescent as well.
In this paper the synthesis and characterization of a number of novel low molecular
weight oxadiazole derivatives and polymers is described. The compounds show different molecular
shapes, e. g. rod-like and star-shaped structures have been realized.
In addition to the compounds described here, a series of different main chain and side
group polymers with oxadiazole moieties are presently synthesized in our research group [4].
2.2 Liquid crystalline oxadiazoles and thiadiazoles
2.2.1 The basic idea
The aim of this work is the preparation of photoconductive compounds with high carrier mobilities.
There are a number of indications that the carrier mobility, i. e. the speed of the
charge particles within the sample, depends on the geometric arrangement of the photoconductive
moieties.
At room temperature for instance, the mobility in single crystals of aromatic compounds
like anthracene or perylene is very high, i. e. in the range of 10 –1 cm 2 /Vs [5]. In
amorphous polymers the carrier mobilities are orders of magnitude smaller and typical values
are in the range of 10 –6 and 10 –8 cm 2 /Vs [6].
So we were interested to investigate if the higher order in the liquid crystalline state
leads to higher mobilities in comparison to less ordered amorphous solids. In liquid crystalline
polymers a macroscopic orientation of the photoconductive groups in the mesophase
can be achieved by means of electric or magnetic fields. The orientation can be frozen in by
lowering the temperature below the glass transition. By this, materials with a degree of order
between a perfect single crystal and totally disordered amorphous polymers are obtained.
With this in mind we started the synthesis of a series of liquid crystalline model compounds
and polymers in which the side groups possess both mesogenic and photoconductive
properties.
16

2.2 Liquid crystalline oxadiazoles and thiadiazoles
2.2.2 Monomer synthesis
In the past years we have synthesized a variety of compounds with oxadiazole and thiadiazole
moieties [7]:
3 X=S 4 X=O
The rod-like thiadiazole derivative 3 is liquid crystalline and shows a smectic as well
as a nematic LC phase with the following phase behaviour: c 103 s a 180 n 203 i. An acrylate
monomer was prepared by hydroboration of the terminal double bond and subsequent
esterification with acryloyl chloride and then polymerized. The resulting polymer 5 exhibits
a mesophase which has not yet been identified.
5
Unfortunately the clearing point of the polymer is at 246 8C, where the material starts
to decompose. Therefore orientation in an electric field was not possible.
In contrast to the thiadiazole the oxadiazole 4 is neither liquid crystalline as monomer
nor as polymer. The reason for the lack of mesogenic properties is that the substitution of
the sulphur by oxygen introduces a bend into the molecule which prevents the formation of
a LC phase.
Recently liquid crystalline oxadiazoles (Scheme 2.1) have been described [8]. Here
the oxadiazole is coupled to at least two benzene or cyclohexane rings. Thereby an increase
of the mesophase stability is achieved.
Scheme 2.1: Liquid crystalline oxadiazole compounds [8].
17

2 Novel Photoconductive Polymers
6 7
8
9
10-13
10 R=Ph-OC 6
H 13
12 R = Ph-N(CH 3
) 2
11 R=C 7
H 15
13 R = Ph-N(CH 3
)C 6
H 13
Scheme 2.2: Monomer synthesis.
Our aim was to synthesize a series of different oxadiazole monomers which are functionalized
for the preparation of polymers. Scheme 2.2 shows the synthesis of the monomeric
1,3,4-oxadiazoles. The oxadiazole moiety is formed by a cyclisation reaction of the
diacylhydrazine derivatives 8 with phosphorous oxychloride. The unsymmetrical bishydrazides
8 are prepared by treatment of benzoyl chlorides 6 with the appropriate monohydrazides
7. The last reaction step is an esterification with 4-(5-hexenyloxy)benzoyl chloride.
The esterification reaction is essential in this case because of two reasons. First the rod-like
18

2.2 Liquid crystalline oxadiazoles and thiadiazoles
mesogen is formed, and second a functional group for the preparation of LC polymers is introduced.
Additionally the oxadiazole monomer 14 was synthesized, in which the ester group is
replaced by a biphenyl unit. By this it is possible to investigate the influence of the ester
function on the photoconductive properties of the liquid crystalline compounds.
14
The liquid crystalline behaviour and the phase transitions of the monomeric oxadiazole
derivatives 10–14 were determined by DSC and polarizing microscopy. In Tab. 2.1 the
phase transition temperatures are summarized:
Table 2.1: Phase behaviour of the monomeric oxadiazoles 10–14
Compound
Transition temperature in 8C
10 k 141 n 167 i
11 k 90 s a 116 i
12 k 162 n 179 i
13 k 127 (s a 111) i
14 k 72 s a 77 i
Except compound 13 all monomers are enantiotropic liquid crystalline and show nematic
or smectic mesophases. Only derivative 13 shows a monotropic s a phase.
2.2.3 Oligo and polysiloxanes with pendant oxadiazole groups
The next step was the synthesis of liquid crystalline polysiloxanes with oxadiazole groups in
the mesogenic unit. The polymers were prepared by a polymeranalogous reaction of the
monomers 10 and 11 with poly(hydrogenmethylsiloxane) 15.
Both polymers are liquid crystalline. The transition into the isotropic phase takes place
at about 200 8C and therefore at a much higher temperature than in the monomers. The high
clearing temperature makes the orientation of the polymers difficult which is preferably carried
out near the clearing point. Up to now it was not possible to prepare well-defined polymers
with the monomers 12–14. A possible reason is the inactivation of the Pt catalyst by
the amino groups.
If the cyclic tetrasiloxane 18, which is commercially available in high purity, is used
instead of 15, well-defined monodisperse model compounds are obtained [9]. These com-
19

2 Novel Photoconductive Polymers
10, 11 15
16 R=Ph-OC 6
H 13
17 R=C 7
H 15
Scheme 2.3: Synthesis of polysiloxanes with pendant oxadiazole groups.
pounds can be highly purified, for example by preparative gel permeation chromatography.
This is very important for photoconductors, because it is well-known that even traces of impurities
may reduce the carrier mobility.
18
So the monomeric compounds 10 and 12–14 have been reacted with 18 to yield the
following tetrameric derivatives:
19 R=Ph-OC 6
H 13
,n=1 21 R = Ph-N(CH 3
)C 6
H 13
,n=
20 R = Ph-N(CH 3
) 2
,n=1 22 R = Ph-N(CH 3
)C 6
H 13
,n=
Scheme 2.4: Tetrasiloxanes with pendant oxadiazole groups.
20

2.2 Liquid crystalline oxadiazoles and thiadiazoles
All cyclosiloxanes are liquid crystalline and show several advantages compared to the
polymers. So the cyclic siloxanes form stable glasses but their clearing points are much
lower. For example compounds 21 and 22 possess clearing temperatures at only 102 8C.
The phase behaviour of the tetramers are listed in Tab. 2.2.
Table 2.2: Phase behaviour of the tetrameric siloxanes 19–22.
Comp.
Transition temperature in 8C
19 g 54
a)
m 1 108
a)
m 2 117 s c 155 n 185 i
20 g 67 s a 138 n 181 i
21 g 34 k 66 n 102 i
22 g 38 s a 82 n 102 i
a) Mesophase not identified
2.2.4 Photoconductivity measurements
For the characterization of the novel photoconductive compounds two different experimental
techniques have been used. Some measurements were made by BASF AG using a steady-state
method. These experiments allow a quick characterization, but in contrast to the time-of-flight
(TOF) method no statements about transient photocurrents and carrier mobilities are possible.
The TOF measurements were carried out in the group of D. Haarer, Universität Bayreuth.
The tetrameric cyclosiloxanes are very suitable compounds for physical investigations.
Because of the low viscosity the preparation of samples is much easier than in the case of
the polymers. Additionally the dark currents of the tetramers are very low.
All the compounds are photoconductive, as shown by the steady-state method. For example,
the measurements of the thiadiazole 3 showed a distinct rise of the photocurrent at
the transition from the crystalline to the mesophase, as shown in Fig. 2.1. The decrease of
Figure 2.1: Temperature dependence of the photocurrent of the thiadiazole 3.
21

2 Novel Photoconductive Polymers
the current at the transition from the smectic to the nematic phase is directly correlated with
the loss of order. These results show that the basic idea of this work is correct.
Nevertheless, we were not able to carry out TOF measurements with calamitic monomers,
because the dark currents in the liquid crystalline phases were too high. As mentioned
before, such measurements are possible with the cyclic tetramers. The transient photocurrents
of the tetrasiloxanes 19, 21, 22 illustrate that the carrier transport is totally dispersive,
i. e. dominated by deep traps in which the charge carriers are captured. This is a typical behaviour
of amorphous polymers. No transit time could be detected and no statements about
the carrier mobility can be made for the rod-like mesogens.
In contrast it was shown in the past years that discotic liquid crystals like hexapentyloxytriphenylene
(HPT) 23 have carrier mobilities up to 10 –3 cm 2 /Vs in the liquid crystalline
D ho phase [10]. Here, the disc-shaped molecules are ideally stacked above each other
and therewith allow a fast carrier transport. The discotic hexathioether 24 with a highly ordered
helical columnar (H) phase exhibits mobilities up to 10 –1 cm 2 /Vs, which are almost as
high as in organic single crystals [11].
6
6
23 24
One important observation during the photoconductivity measurements of the tetramer
19 was, that it showed almost the same photocurrent for holes and electrons. This was interesting
because of the lack of electron transporting materials and led us synthesize a variety
of starburst oxadiazole compounds, which are described in the next Chapter.
2.3 Starburst oxadiazole compounds
2.3.1 Motivation
The synthesis of novel materials with high carrier mobilities is one of the major goals in the
field of photoconductive polymers. Within the last years different approaches have been pursued
to reach this goal. First, the photoconductive properties of conjugated polymers like
poly(phenylenevinylene) and poly(methyl phenylsilane) have been investigated [12]. Another
approach are liquid crystals which are the topic of the first Chapter. The third way to realize
22

2.3 Starburst oxadiazole compounds
the goal are glasses of large extended aromatic amines. The best investigated representatives
of this class of compounds are N,N'-diphenyl-N,N'-bis(3-methylphenyl)-[1,1'-biphenyl]-4,4'-
diamine (TPD) 25 from Xerox [13] and 1,1-bis(di-4-tolyl-aminophenyl)cyclohexane (TAPC)
26 from Kodak [14].
Both compounds are derivatives of triphenylamine, a well-known photoconductor. If
thin films of TPD and TAPC are prepared by vacuum evaporation both compounds form
metastable glasses. In such glasses carrier mobilities up to 10 –3 cm 2 /Vs for TPD [13] and
10 –2 cm 2 /Vs for TAPC [14] have been reported. But both TAPC and TPD glasses are metastable
and have a strong tendency to crystallize. If the molecules are imbedded in a polymer
matrix, e. g. polycarbonate or polystyrene, morphologically stable materials are formed, but
the mobilities decrease drastically [13].
25 26
Several attempts have been made to overcome the problems with metastable TPD and
TAPC glasses. Recently we described the synthesis of 3,6-bis[(9-hexyl-3-carbazolyl)ethynyl]-9-hexylcarbazole
27, a trimeric model compound of poly(carbazolylene ethynylene)
[15]. This material shows a glass transition temperature of 41 8C and forms glasses which
are stable for more than a year. Mobilities up to 10 –4 cm 2 /Vs (E =67 10 5 V/cm, T =308C)
have been measured by the time-of-flight technique.
27
Another interesting approach are starburst compounds with high glass transition temperatures.
So 4,4',4@-tris(N-carbazolyl)triphenylamine, which has been published recently
[16], shows a glass transition at 151 8C and forms a stable glass.
Beginning with the work of Thomalia [17] in 1986, starburst molecules (dendrimers)
have achieved enormous interest within the last years. Dendrimers are highly branched regular
molecules, which are usually prepared by stepwise reactions. In many cases the behaviour
of dendritic macromolecules are different from that of linear polymers, e. g. the former
23

2 Novel Photoconductive Polymers
usually show enhanced solubility. The reasons for these differences are the unique three-dimensional
structure and the large number of chain ends in dendrimers.
Two different methods have been developed for the stepwise synthesis of starburst
dendrimers:
a) the divergent approach, where the synthesis starts from a core molecule with two or
more reactive groups;
b) the convergent approach, in which the synthesis starts at the outer sphere of the dendrimer.
In the divergent approach, the reaction of the core molecule with two or more reagents
containing at least two protected branching sites is followed by removal of the protecting
groups and subsequent reaction of the liberated reactive groups which leads to the starburst
molecule of the 1st generation. The process is repeated until the desired size is reached. In
the convergent approach the synthesis starts at what will become the outer surface of the
dendrimer. Step by step large dendrimer arms are prepared and finally the completed arms
are coupled to the core. Both methods produce well-defined large dendritic molecules whose
structures are manifolds of the building blocks. They allow structural as well as functional
group variation.
On the other hand, hyperbranched polymers can be synthesized in a one-step reaction
using highly functionalized monomers of the type A x B, where x is 2 or larger. This method
does not yield polymers of such a well-defined structure, but has the advantage to provide
rapidly large quantities of material. The control of the degree of branching is difficult and
mainly depends on statistics, steric effects, and the reactivity of the functional groups.
The aim of our work is the synthesis of starburst compounds with oxadiazole moieties.
In contrast to the triphenylamine and carbazole derivatives oxadiazoles are strong electron
acceptors. Therefore their transport characteristics can be switched from hole transport materials
like 2,5-(4-diethylaminophenyl)-oxadiazole 1, in which the electron withdrawing effect
of the oxadiazole ring is overcompensated by the two electron donating amino groups,
to electron transport materials like 2-(biphenyl)-5-(4-tert.butylphenyl)-oxadiazole (PBD) 2.
Recently, Saito demonstrated that oxadiazoles in a polycarbonate matrix show electron transport
with mobilities up to 10 –5 cm 2 /Vs [18]. Such electron transport materials are attractive
for the use in copiers and in organic light-emitting diodes.
2.3.2 Synthesis of starburst oxadiazole compounds
Starburst oxadiazole compounds have been mentioned for the first time in a thermodynamic
study of the structure-property-relationship in low molecular weight organic glasses [19,
20]. Upon rapid cooling these compounds form glasses with T g s between 77 and 169 8C.
We have used three different approaches for the preparation of starburst oxadiazole
compounds, which are schematically shown in Fig. 2.2.
The methods A and B can be compared to the divergent synthesis of dendrimers.
Method A starts from a central core and involves the reaction of the acid chloride groups
and a subsequent cyclisation with phosphorous oxychloride to the oxadiazole ring (Chap-
24

O
O
Cl
C
C
Cl
O
C
Cl
Method A
Method B
O
H 2N NH C
O
O
O
O
C
NH
NH
C
C
NH NH C
O
O
C
NH
NH
C
C
N N
N NH
N
N
O
N N O
O
N
N
=core
= shell
Method C
X
Z
+
Y
X
X
Z
Z
X = OH, C CH Y = Br, F Z = O , C C
=core
=shell
Figure 2.2: The different approaches to starburst oxadiazoles.
25

2 Novel Photoconductive Polymers
ter 1). The main difficulty in this case is that three functional groups must react in one step.
Partially incomplete cyclisation causes problems, because the products contain one or two
bishydrazides and are difficult to separate from the target compounds, which have three oxadiazole
groups. Therefore we recently developed a totally different route (method B). The
compounds 31 a–f were prepared by the reaction of benzene tricarbonyl chloride 30 with
the tetrazoles 29. The latter were synthesized from benzonitriles 28 with potassium azide
(Scheme 2.5). With the loss of nitrogen the tetrazole ring is transferred to the oxadiazoles
[21]. This gives an easy access to unsymmetrically substituted oxadiazoles which we used
for the first time in the synthesis of starburst oxadiazole compounds.
In the third approach (method C) which is similar to the convergent synthesis a preformed
oxadiazole precursor is prepared by stepwise synthesis and finally coupled to the
core molecule.
R 1
R 2 CN + NaN 3
NH 4 Cl
R 1
R 2
R 3
R 3
28 29
C
N
N
NH
N
Cl
O
C
O
C
Cl
O
C
Cl
+ 3
R 2
R 1
R 3
C
N
N
NH
N
31a-f
pyridine
2h, reflux
31a-f
30 29
Scheme 2.5: Synthesis of starburst oxadiazoles via tetrazole intermediates.
For the preparation of the novel starburst molecules, different core molecules have
been used: 1,3,5-benzene tricarboxylic acid chloride, 1,3,5-tris(4-benzoyl)benzene, 1,3,5-
triethynylbenzene, and 4,4',4@trihydroxytriphenylamine.
Except benzenetricarboxylic acid which is commercially available, all core molecules
were synthesized following well-known literature procedures [22, 23].
The structures of the starburst oxadiazole compounds are summarized in Tab. 2.3.
31 c and 31g have been described before [19, 20], but no synthetisis procedure was given.
The oxadiazoles 31a–h have been prepared according to method B (Fig. 2.3), starting
from benzene tricarboxylic acid chloride.
Compound 31g with tert.-butyl substituents was synthesized by method A, too. So a
comparison of methods A and B is possible. The tetrazole route (method B) exhibits several
advantages compared to the ring closure with dehydrating agents (method A). One advantage
is the short reaction time. The reaction is finished within 2 hours whereas heating for
1–3 days is necessary if the ring closure is carried out with phosphorous oxychloride. Even
more important is the facile work up procedure. In the case of the dehydration with POCl 3
column chromatography and subsequent recrystallization is necessary to purify the product
which is finally obtained in 33% yield. In contrast only one filtration on a short silica gel
26

2.3 Starburst oxadiazole compounds
Table 2.3: Structures of the starburst oxadiazole compounds 31–34.
Comp. Core a) Shell a)
31
N
O
N
R 1
R 3
R 2
31a R 1 =R 3 =H,R 2 =CH 3
31b R 2 =R 3 =CH 3 ,R 1 =H
31c R 1 =R 3 =CF 3 ,R 2 =H
31d R 1 =R 3 =H,R 2 =C 2 H 5
31e R 1 =R 3 =H,R 2 =OC 2 H 5
31f R 1 =R 3 =H,R 2 = CH(CH 3 ) 2
31g R 1 =R 3 =H,R 2 = C(CH 3 ) 3
31h R 1 =R 3 =H,R 2 = N(C 2 H 5 ) 2
N N
32 R
R = C(CH 3 ) 3
O
C
C
C C
N N
33 R
R = C(CH 3 ) 3
C
C
O
N N
34 N
O
R R = C(CH 3 ) 3
O
a) cf. Figure 2.2.
column is sufficient for the purification of 31 g, prepared by the tetrazole route. In this case
the yield of the pure product is 69%. The starburst oxadiazole 32 has been prepared by
method B too, whereas for 33 and 34 method C (Fig. 2.2) was used. The identity of all compounds
was confirmed by NMR and FTIR spectroscopy.
2.3.3 Thermal properties
The thermal behaviour of the starburst molecules has been investigated by differential scanning
calorimetry and thermogravimetric measurements.
27

2 Novel Photoconductive Polymers
Table 2.4: Thermal properties of the starburst compounds 31–34.
a)
a)
Compound Molecular mass T g T m
[g/mol] [8C] [8C] [8C]
31a 553 – 334 348
31b 595 – 333 358
31c 919 – 250 336
31d 595 – 276 360
31e 643 108 259 337
31f 637 97 225 349
31g 679 142 257 366
31h 724 128 299 329
32 907 165 297 409
33 979 – 320 356
34 1122 137 – 391
a) 3rd heating, determined by DSC with 20 K/min
b) onset of decomposition in nitrogen, thermogravimetric measurement with 10 K/min
T dec
b)
Among the compounds 31 a–h with a small benzene core both crystalline and glass
forming materials exist. So 31 a–d with small methyl, ethyl, or trifluoromethyl substituents
show only melting points in the DSC experiment with cooling rates of 20 K/min. Naito and
Miura reported that it is possible to obtain glasses even with small methyl or trifluoromethyl
substituents if the compounds are rapidly cooled with liquid nitrogen [19]. In contrast, the
compounds 31 e with ethoxy, 31f with iso-propyl, 31g with tert-butyl substituents, and 31h
with diethylamino groups form glasses upon cooling with 20 K/min in the DSC equipment.
When the amorphous samples are heated again the glass transition appears first but on
further heating the samples start to recrystallize and consequently show a melting point at
higher temperatures. Compound 32 with the triphenylbenzene core behaves similar like
31e–h. The most stable glasses are formed by 34 with the triphenylamine core. The DSC
diagram of the novel glass forming oxadiazole compound is shown below. Upon both, heating
and cooling, only a glass transition is observed at 137 8C (Fig. 2.3). In our DSC experi-
Figure 2.3: DSC scan of the starburst oxadiazole compound 34, 3rd heating and cooling, with 20 K/min.
28

2.3 Starburst oxadiazole compounds
ments we never found any evidence for crystallization of 34. Even in the first heating no
melting point is observed up to 350 8C. Consequently, transparent amorphous films are obtained
from the starburst oxadiazole compound 34 by solvent casting.
The thermal behavior of compound 33 is somewhat different. No reproducible DSC
scans are obtained in subsequent heating-cooling cycles. We attribute this to thermal crosslinking
of the triple bonds [24].
The thermal stability has been monitored by thermogravimetric measurements. In
most cases the onset of decomposition is in the range from 330–370 8C. The oxadiazole
compound 32 with a triphenyl benzene core shows a somewhat higher thermal stability up
to 410 8C.
(CH 3 ) 3 C
N
O
N
O
C(CH 3 ) 3
N
N O
O
N
O
N
O
N
C(CH 3 ) 3
34
The starburst oxadiazole compounds are now being tested as electron injection and
transport layer in organic LEDs and as photoconductors. First tests of two layer LEDs with
PPV show that the novel materials possess properties comparable to 2 but have the great advantage
to show no recrystallization if thin films were made by spin-coating. We will report
on these measurements in the near future.
29

2 Novel Photoconductive Polymers
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H. Hoegl: J. Phys. Chem., 69, 755 (1965)
2. W. Wiedemann: Chem.-Ztg., 106, 275 (1982) and references therein
3. A.R. Brown, D.D.C. Bradley, J.H. Burroughes, R.H. Friend, N.C. Green-ham, P.L. Burn, A.B.
Holmes, A. Kraft: Appl. Phys. Lett., 61, 2793 (1992)
4. E. Buchwald, M. Meier, S. Karg, W. Rieß, M. Schwoerer, P. Pösch, H.-W. Schmidt, P. Strohriegl:
Adv. Mater., 7, 839 (1995), M. Greczmiel, P. Pösch, H.-W. Schmidt, P. Strohriegl, E. Buchwald,
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York, (1993)
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199, 345 (1991)
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Phys. Rev. Lett., 70, 457 (1993)
11. D. Adam, P. Schuhmacher, J. Simmerer, L. Häussling, K. Siemensmeyer, K.H. Etzbach, H. Ringsdorf,
D. Haarer: Nature, 371, 141 (1994)
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Phil. Mag. B, 61, 25 (1990)
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Carl Hanser Verlag, Munich, 137 (1994)
30

3 Theoretical Aspects of Anomalous Diffusion
in Complex Systems
Alexander Blumen
3.1 General aspects
Many relaxation phenomena in the solid phase depend on diffusion, which is usually treated
in the framework of statistical methods. Regular diffusion, known as Brownian motion, is
characterized by a linear increase of the mean-squared displacement with time. On the other
hand, for a whole series of phenomena this simple relation does not hold; their temporal
evolution of the mean-squared displacement is non-linear and thus obeys at long times:
r 2
…t† t
…1†
with g 0 1. Relation 1 is referred to as anomalous diffusion. In the case that g < 1 one denotes
the behaviour as subdiffusive. A subdiffusive pattern of motion often results from disorder
[1–4]. One has to note, however, that the asymptotic law, Eq. 1, emerges only when
the disorder influences the motion on all scales. In the case that g > 1 the motion is termed
superdiffusive. A classical example for superdiffusive behaviour is furnished by the motion
of particles in a turbulent flow. In this paper we focus on several models for anomalous diffusion
which involve polymeric systems.
Now, the mean-squared displacement is a basic characteristic feature for the motion
but, as an averaged quantity, it can provide only restricted information about the basic microscopic
mechanisms involved. In several works we have also studied the propagator P(r,t),
the probability to be at r at time t having started at the origin at t = 0. Space limitations prevent
us from going into details. Here we focus on Ar 2 (t)S and refer to Klafter et al. and Zumofen
et al. [4, 5] for in-depth analyses of the propagator.
From the beginning, the role of time-dependent aspects in disordered media has to be
emphasized. In usual random walk problems it is often assumed that the disorder is
quenched, so that the dynamics evolves over a static substrate, i. e. the geometrical or the energetical
disorder is frozen in. However, it is also possible that the dynamical processes are
directly under the influence of temporal (possibly medium induced) fluctuations.
First, in Section 3.2 we concentrate on photoconductivity, whose canonical description
involves the continuous time random walk (CTRW) approach [1, 4, 6–8]. Basic ingredients
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
31

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems
in CTRWs are waiting time distributions with long time-tails. For photoconductivity one
usually has g 1 shows up.
An interesting dynamical feature is observed when the object to move has itself an internal
structure; that is the case, for instance, for a polymer. In this case, because of the connections
which exist between different segments of the polymer, the motion gets a very rich
structure. The details of this situation will be discussed in Section 3.4.
Finally, we close in Section 3.5 with some conclusions.
3.2 Photoconductivity
A major field of application of scaling ideas in the time domain is photoconductivity [1–3,
7, 12]; here the use of polymeric photoconductors in printing and copying represents one of
the most sophisticated applications for organic materials. Such materials are nowadays
superior to their inorganic counterparts. Now, the issue at stake in amorphous photoconductors
is the appearance of dispersive transport, as contrasted to the familiar diffusive Gaussian
behaviour. One observes experimentally that the motion of the charge carriers becomes
slower and slower with the passage of time, a situation mirrored by Eq. 1 with g t T one finds
I t 1 : …4†
For instance measurements on polysiloxanes with pendant carbazole groups [13] follow
Eqs. 3 and 4 with g = 0.58 very well over four decades in time, see below. In fact this is
32

3.2 Photoconductivity
typical for g significantly lower than one. However, one should note that the behaviour of
I (t) is complex, since for g > 1 in the long-time limit the transport behaviour is non-dispersive.
This is related to the existence of a finite mean waiting-time t = g/(g – 1). It follows
that around g = 1 there is a crossover from dispersive to non-dispersive behaviour [13], as is
also confirmed experimentally.
From the preceding discussion it is obvious that the transport of charge carriers is
much influenced by the waiting-time distributions (WTD) between hops. Physically the
WTD arise from the local disorder in the sample. For large disorder the WTD are not exponential
but decay much more slowly. In line with the preceding arguments one takes for the
WTD expressions c (t) which behave at long times algebraically [7, 12]
…t† t 1 ; …5†
where 0 < g < 1. This choice for c (t) reproduces Eqs. 2–4, see below. Equation 5 is not valid
near the time-origin. Therefore in calculations one prefers to work with functions welldefined
for t 6 0. A suitable choice is for instance the series
…t† ˆ1 a
a
X 1
nˆ1
a n b n exp … b n t† …6†
with a < 1 and b < a –1 [8]. This function is everywhere continuous and finite, and for purely
imaginary t it turns into the Weierstrass function. One verifies readily that for large t Eq. 6
obeys Eq. 5 and that the corresponding g in Eq. 5 is g =lna/ln b. Equation 6 is very useful,
since it allows to vary g freely by a judicious choice of a and b. Another choice for c (t)
which is continuous for t 6 0is
…t† ˆ…1 ‡ t† :
…7†
Here again the long-time behaviour follows Eq. 5.
We now recall the basic ingredients of random walks in continuous time, the so-called
CTRW [6–8]. Let c (r, t) be the probability distribution of making a step of length r in the
time interval t to t +dt. The total transition probability in this time interval is
…t† ˆX
r
…r;t†:
…8†
Furthermore the survival probability at the initial site is
…t† ˆ1
R t
0
…† d;
…9†
so that, switching to the Laplace space (t ? u), one has
…u† ˆ‰1 …u†Š=u: …10†
33

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems
The probability density Z(r,t) of just arriving at r in the time interval t to t +dt obeys
the iterative relation
…r;t†ˆP
r 0
R t
0
…r 0 ;† …r r 0 ;t † d ‡ …t† r;0 ; …11†
in which the initial condition of starting at t = 0 from r = 0 is incorporated. One then has
for the probability P (r,t), that the particle is at r at time t,
P …r;t†ˆRt
…r;t † …† d: …12†
0
Now P (r, t) also obeys an iterative relation:
P …r;t†ˆX
r 0
R t
0
P …r 0 ;† …r r 0 ;t † d ‡ …t† r;0 …13†
as may be seen either by inspection or, more formally, by using Eqs. 11 and 12. Clearly, a
description of such convolutions, Eqs. 11–13, is more compact in Fourier-Laplace space.
For P (k, u) one has from Eq. 13:
P …k;u†ˆP …k;u† …k;u†‡ …u†
…14†
with the immediate solution
P …k;u†ˆ …u†=‰ 1 …k;u† Š ˆ 1 …u†
u
1
1 …k;u† : …15†
The last expression generalizes the usual diffusion relation to random media, in which
spatial and temporal aspects are coupled through c (k,u). The analysis is much simplified if
such aspects decouple, which is the case for instance when the disorder is mainly energetic
and the random walker moves over a rather regular lattice. In the decoupled case:
…r;t†ˆ …r† …t† :
…16†
From this it follows immediately that also c (k,u) =l(k) c (u) is decoupled. In the decoupled
scheme P (k, u) takes the form
P …k;u†ˆ1 …u†
u
1
1 …k† …u† ; …17†
where l (k) = P qp(q)e –ik7q is the structure function of the infinite lattice and p (q) denotes
the probability that a step extends over the distance q.
34

3.2 Photoconductivity
Coupling is very important for superlinear behaviour. Thus for the c (r,t) given by
Eq. 16 the mean-squared displacement is either divergent or increases sublinearly or at most
linearly in time. In order to obtain finite Ar 2 (t)S with a superlinear temporal behaviour,
coupled c (r,t) forms have to be used [4]. An example is the WTD
…r;t†ˆAr …r t †; …18†
in which the d-function couples r and t. This WTD leads to Lévy walks.
Let us now focus on the mean-squared displacement. Evidently, one has
r 2 R
…t† ˆ r 2 P …r;t† dr ˆ rk 2 P …k;t† j kˆ0
…19†
from which, in the decoupled scheme, using Eqs. 5 and 17 it follows that in the absence of
any bias Eq. 2 is fulfilled. The procedure is more readily examplified by calculating the current
from the mean displacement of the carrier Ar(t)S in a biasing field [8]. Setting L –1 for
the inverse Laplace transform it follows:
P
r…t† ˆ rP …r;t†ˆir k P …k;t† kˆ0 ˆ L 1 … ir k P …k;u† j kˆ0 †
r
ˆ L 1
@P…k;u†
@
ir k …k† j kˆ0
ˆ1
ˆ L 1 …u†
u ‰ 1 …u† Š
q ;
where in the last line we work in the decoupled scheme and AqS = P q qp (q) is the mean displacement
per hop. In the presence of a bias AqS 0 0. Thus, setting |AqS| = 1, we have for the
current I (t) in an infinite lattice of any dimension,
I …t† ˆ d
r…t†
dt ˆ L 1 …u†
: …21†
1 …u†
For a finite chain of N sites Eq. 21 takes in the Laplace space the form [3, 13]:
I…u† ˆXN
nˆ1
‰ …u† Š n ˆ
…u†
1 …u†
…20†
1 ‰ …u† Š N : …22†
It is precisely this function which was used in Refs. [13, 14] to analyse, together with
the WTD Eq. 6, the time-of-flight currents of photoconductive carriers in polysiloxanes with
pendant carbazole groups. In this work one achieved with g = 0.58 a good agreement with
the experimental findings. The short time behaviour indeed obeys Eq. 3, whereas at long
times the form of Eq. 4 is reproduced fairly well.
35

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems
3.3 The Matheron-de-Marsily model
As mentioned in the introduction, superdiffusive (enhanced) behaviour is often found in turbulent
flows. In this Section we adopt a picture different from CTRW. We focus on the influence
of randomly distributed biasing external fields and follow the description of Oshanin
and Blumen [11]. We take a three-dimensional (3D) solvent for which the flow is parallel to
the Y-axis. The direction and magnitude of the flow depend only on the X coordinate of the
position vector [9] so that V Y , the non-vanishing component of the flow field, obeys:
V Y …X;Y;Z†ˆVX ‰ Š: …23†
Here furthermore V[X] is a random function of X. Geometrically the system consists
of parallel layers perpendicular to the X-axis. In each layer the value of V Y is constant but
varies from layer to layer [9].
We assume the random function V[X] of Eq. 23 to be Gaussian with zero mean,
AV[X]S = 0, and with the covariance
VX ‰ 1 ŠVX ‰ 2 Š ˆ … j X1 X 2 j†: …24†
Here the brackets denote configurational averages, which are conveniently expressed
through Fourier integrals,
… jX 1 X 2 j† ˆ R1 1
dwQ…w† exp ‰ iw…X 1 X 2 † Š: …25†
Now many possibilities for Q(w) can be envisaged. For simplicity we take here only a
flat spectrum, Q(w) =W/2p, as in the original Matheron-de-Marsily (MdM) model [9] in
which the flows are delta-correlated,
… jX 1 X 2 j† ˆ W …X 1 X 2 †: …26†
We start from the Langevin dynamics of a single spherical bead subject to the MdM flow.
This allows us to display enhanced (superlinear) diffusion in a simple situation [9, 10, 22]. The
study of the dynamics of Rouse polymers in such flows is deferred to the next Section.
Let R(t) be the position of the center of mass of the bead at time t, and we assume
that R(0) = 0. The components X(t), Y(t) and Z(t) ofR(t) obey the following Langevin equations
m d2 X
dt 2 ˆ dX
dt ‡ f X…t† ;
m d2 Z
dt 2 ˆ dZ
dt ‡ f Z…t† ;
…27†
…28†
36

3.3 The Matheron-de-Marsily model
m d2 Y
dt 2 ˆ dY
dt
VX ‰ Š
‡ f Y …t† : …29†
Here m denotes the mass of the bead and z the friction constant. The terms f X (t), f Y (t),
and f Z (t) give the random (thermal-noise) forces exerted on the bead by the solvent molecules.
These forces are Gaussian, with the moments
f i …t† ˆ0
…30†
and
f i …t† f j …t 0 †ˆ2T i;j …t t 0 † ; …31†
where i,j B {X,Y,Z}. The dash stands for thermal averaging, d i,j is the Kronecker-delta and
the temperature T is measured in units of the Boltzmann constant k B .
Conventionally the acceleration terms in Eqs. 27–29 are neglected, since they are
small relative to the other terms [15]. This leads to
dX
dt ˆ f X…t† ;
dZ
dt ˆ f Z…t† ;
dY
dt ˆ V‰X…t†Š ‡ f Y…t† :
…32†
…33†
…34†
Note that in Eq. 34 the X and Y coordinates are coupled. Equations 32 and 33 are
readily solved,
X…t† ˆ 1 Rt
df X …† ;
Z…t† ˆ 1 Rt
df Z …† :
0
0
…35†
…36†
Thus the bead undergoes a conventional diffusive motion between the layers (along
the X-axis) and in the Z direction. One sees it readily by evaluating, say:
X 2 …t† ˆ 2Rt R
d t
1 d 2 f X … 1 † f X … 2 †ˆ2 1 T Rt R
d t
1 d 2 … 1 2 †ˆ2…T=† t; …37†
0 0
0 0
so that X 2 …t† ˆ2D 1 t, where D 1 = T/z is the diffusion coefficient of a single bead.
37

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems
A similar procedure can be performed with respect to the solution of Eq. 34,
Y…t† ˆRt
0
dV‰ X…† Š‡ 1 Rt
df Y …† :
0
…38†
The averaging involves now both, the thermal noise and the configurational disorder:
Y 2 …t† ˆ 2 Rt R t
d 1 d 2 f Y … 1 † f Y … 2 †‡ Rt R t
d 1
0
0
0
0
d 2 V‰X… 1 †Š V‰X… 2 †Š
ˆ 2D 1 t ‡…W=2† Rt R t R 1
d 1 d 2 dw exp ‰iw…X… 1 † X… 2 ††Š; …39†
1
0
0
where in the last line use was made of the representation Eqs. 25 and 26 with the flat spectrum
of the delta-function. The remaining average on the rhs. of Eq. 39 is readily evaluated
by remembering that it is the characteristic functional of the Brownian trajectory
… 1 ; 2 ; w† exp‰iw…X… 1 † X… 2 ††Š ˆ exp w 2 D 1 j 1 2 j
: …40†
Inserting Eq. 40 into Eq. 39 one recovers the following result for the average squared
displacement (ASD) in the direction of the flow field,
Y 2 …t† ˆ 2D1 t ‡ 4W 3
t 3 1=2
: …41†
p D 1
One should remark that at times greater than t c =9pD 1 3 /(4W 2 ) the superlinear growth
hY 2 …t†i t 3=2 in Eq. 41 dominates and the first, diffusive term can be neglected. One has
then a superdiffusive behaviour with an exponent of g = 3/2 in Eq. 1 [9].
We now turn to the analysis of the behaviour of polymers in MdM flow fields.
3.4 Polymer chains in MdM flow fields
The conformational properties and the dynamics of polymers in solutions under various
types of flows have been a subject of considerable interest within the last decades. Much
progress has been gained in the explanation of experimental data for systems in which the
flow velocities are given functions in space and time, see Refs. [16–19]. On the other hand,
the behaviour of polymers in random flows is less understood. In recent works [11] we
(Oshanin and Blumen) succeeded in establishing analytically the behaviour of Rouse polymers
[20] in MdM flow fields. The presentation here follows closely Ref. [11].
38

3.4 Polymer chains in MdM flow fields
In the Rouse model N monomers (beads) are coupled to each other via harmonic
springs [16, 17, 20]. As is well-known, the forces are of entropic origin. It is customary to
revert to a continuous picture in which n, the bead’s running number, takes real values. For a
detailed discussion see Doi and Edwards [17]. The Langevin equations of motion for such a
polymer in the MdM flow field are
@X n…t†
@t
@Z n…t†
@t
@Y n…t†
@t
ˆ K @2 X n …t†
@n 2 ‡ f x …n; t† ; …42†
ˆ K @2 Z n …t†
@n 2 ‡ f Z …n; t† ; …43†
ˆ K @2 Y n …t†
@n 2 ‡ V‰X N …t†Š ‡ f Y …n; t† ; …44†
see Ref. [11]. Equations 42–44 are the generalization of Eqs. 27–29 to polymers. They are to
be solved subject to the Rouse boundary conditions at the chain’s ends, n = 0 and n = N [17]:
@X n …t†
@n
ˆ @Y n…t†
@n
ˆ @Z n…t†
@n
ˆ 0 :
…45†
As before, the fluctuating forces on the rhs. of Eqs. 42–44 are Gaussian and also
delta-correlated with respect to the running index [11, 17]. For the X and Z components,
which are not subject to the flow, the procedure is standard [17]. Say, for the averaged X
component of the end-to-end vector one has
P 2 X …t† ˆ…X 0…t† X N …t†† 2 ˆ b2 N
; …46†
3
where b is the so-called persistence length.
Furthermore, the X component of the radius of gyration is
X 2 g ˆ 1
2N 2 Z N
0
Z N
0
dndm …X n …t†
X m …t†† 2 ˆ b2 N
18 : …47†
For isotropic situations (in the absence of flow fields) the end-to-end vector P R and
the radius of gyration R g of the polymer are related to P X and X g through P R 2 =3P X 2 and
R g 2 =3X g 2 . Because of the anisotropy one has to consider in the MdM model the different
components separately.
The dynamics of a flexible polymer chain is richer than that of a single bead. In the
Rouse model the dynamics of X n (t) depends essentially on the time of observation t and on
the Rouse time t R ,
R ˆ b2 N 2
3p 2 T : …48†
39

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems
Now t R is the largest internal relaxation time of the chain [17, 20]. Exemplarily for
t P t R one finds for the mean-squared displacement of a bead of the chain, say the zeroth one,
…X 0 …t† X 0 …0†† 2 ˆ 2b D 1=2
1t
: …49†
3p
Equation 49 is subdiffusive with g = 1/2 in Eq. 1. This is due to the fact that the trajectory
of a bead in a chain is spatially confined by its neighbours. In the limit t p t R the
chain diffuses as one entity and the bead’s trajectory follows mainly the motion of the
chain’s center of mass. The chain’s center of mass obeys
X 2 …t† ˆ2D R …N† t
…50†
with D R (N) =D 1 /N { T/(zN).
Let us now turn to the Y component of the center of mass for which Eq. 44 leads readily
to
Y…t† 1 N
so that we obtain
Z N
0
dnY n …t† ˆ 1 Z t
d
N
0
Z N
0
dn VX ‰ n …† Š‡ 1 f Y …n; † ; …51†
Y 2 …t† ˆ 2DR …N† t ‡ W Z t Z t Z 1
d 1 d 2
2N
0 0 1
dwg…w; 1 ; 2 †;
…52†
where g(w;t 1 ,t 2 ) denotes the dynamic structure factor of the chain,
g…w; 1 ; 2 †ˆ 1
N
Z N Z N
0
0
dn dm exp‰iw…X n … 1 † X m … 2 ††Š : …53†
It turns out [11], that all these integrations can be performed analytically. In the longtime
limit t p t R one finds
Y 2 …t† ˆ 2DR …N† t ‡ 4W 3
whereas for t P t R the short-time behaviour is given by
40
N 1=2
t 3=2
1 O…t 1=2
† ; …54†
pT
p
Y 2 …t† 2DR …N† t ‡ 3 W
bN 1=2 t2 1 O…t 1=4
† : …55†

3.4 Polymer chains in MdM flow fields
The interpretation of Eq. 54 is that at long times the t 3/2 dynamics dominates the picture;
the Rouse chain behaves like a compact bead. At short times the term t 2 may become
important. This g = 2 case in Eq. 1 is called ballistic; at very short times the center of mass
of the chain hardly moves and it practically does not change the flow pattern to which it is
subjected.
We remark that at short times the motion of the center of mass of the chain and
the motion of a tagged bead are characterized by different dependences on time. The
mean-squared displacement along the Y-axis of a tagged bead, say the zeroth one, grows
in time as
Y0 2…t†
e2DR …N† t ‡ W 1=4
D
b1=2 1
t 7=4 ; …56†
see Ref. [11]. This g = 7/4 dependence is, of course, related to the fact that the segmental
motion at short times is confined, the number of distinct flow layers visited by the bead
growing as t 1/4 . Results of such fractal-type behaviour may be formulated exactly [21, 22].
We now turn to the question of the elongation of the Rouse chain in the MdM flow
and sketch the evaluation of the end-to-end distance along the Y-axis [11].
The solution of Eq. 44 under the boundary conditions Eq. 45 has the form of a Fourier
series,
Y n …t† ˆY…0;t†‡2 X1
pˆ1
cos ppn
Y…p; t† ;
N
where the Y(p,t), p = 0, 1, …, denote the normal coordinates [17]. At t = 0 the chain is assumed
to be in thermal equilibrium, i. e. to have a Gaussian conformation. This can be accounted for
automatically by stipulating it to be subject to the thermal fluctuations since t =–?. Furthermore,
the MdM flow fields are switched on at t = 0. This leads to
…57†
Y…p; t† ˆ 1 Rt 1
d exp… p 2 …t †= R † ~ f Y …p; †
‡ 1 N
Z t
0
d
Z N
0
dn cos ppn
V‰X n …t †Š exp… p 2 = R † ; …58†
N
where the functions ~ f Y denote the Fourier components of the thermal fluctuations [17]. The
Y component of the end-to-end vector follows now from:
P Y …t† ˆY 0 …t†
Y N …t† ˆ2 P1
… 1 … 1† p †Y…p; t† : …59†
pˆ1
After performing the averaging over both, the thermal fluctuations and the realizations
of the flow fields, one obtains from Eqs. 58 and 59 in the long-time limit the equilibrium
value of the end-to-end vector for delta-correlated flows [11],
41

3 Theoretical Aspects of Anomalous Diffusion in Complex Systems
P 2 Y …1† b 2
N ˆ 1 ‡ CW2 bN 5=2
3
T 2 ; …60†
where C is a constant. Therefore, a Rouse polymer stretches along the Y-axis under MdM
flows, taking the form of a prolate ellipsoid. The correction term to the Gaussian behaviour
grows as N 7/2 .
It is now interesting to confront this finding with the situation in simple shear flows,
for which the result is [18, 19]:
P 2 Y …t† ˆb2 N
3
1 ‡ 3
6480
_ 2 2 b 4 N 4
T 2
: …61†
In Eq. 61 _ stands for the (constant) shear rate. Because all flow lines point now in
the same direction, the correction term shows a stronger N-dependence and obeys a N 5 law.
3.5 Conclusions
In this review several situations were presented, which lead naturally to the appearance of
anomalous diffusion. Besides the already well-discussed subdiffusive behaviour, which is often
seen in disordered media, we also considered superdiffusive dynamics, such as encountered
in layered random flows. Anomalous diffusion was analysed through several models
and we focussed on the behaviour of polymeric materials under such conditions. The basic
aspect underlying these phenomena is dynamical scaling, which is often encountered experimentally
and theoretically.
On the other hand not all systems scale with time, and care is required in applying the
models presented here; one has to be aware of intrinsic limitations of the scaling range (e. g.
size limitations, other temporal scales involved). Because of this, close cooperations between
experimentalists and theoreticians are much needed for the analysis of systems similar to the
ones described here. Furthermore, despite the success of the now-closing Collaborative Research
Center 213, much work still remains to be done.
Acknowledgements
The research collaboration with Prof. D. Haarer, Prof. J. Klafter, Dr. G. Oshanin, Dr. H.
Schnörer, and Dr. G. Zumofen in our joint work reviewed here was always very helpful and
pleasant. The support of the Deutsche Forschungsgemeinschaft through the Sonder-
42

References
forschungsbereich 213 and Sonderforschungsbereich 60 was fundamental for the whole project.
Additional help was provided by the Fonds der Chemischen Industrie and – in the later
stages – by the PROCOPE-Program of the DAAD.
References
1. A. Blumen, J. Klafter, G. Zumofen: in: I. Zschokke (ed.): Optical Spectroscopy of Glasses, Reidel,
Dordrecht, p. 199 (1986)
2. D. Haarer, A. Blumen: Angew. Chem. Int. Ed. Engl., 27, 1210 (1988)
3. A. Blumen, H. Schnörer: Angew. Chem. Int. Ed. Engl., 29, 113 (1990)
4. G. Zumofen, J. Klafter, A. Blumen: in: R. Richert, A. Blumen. (eds.): Disorder Effects on Relaxational
Processes: Glasses, Polymers, Proteins, Springer, Berlin, p. 251 (1994)
5. J. Klafter, G. Zumofen, A. Blumen: J. Phys., A24, 4835 (1991)
6. E.W. Montroll, G.H. Weiss: J. Math. Phys., 6, 167 (1965)
7. H. Scher, M. Lax: Phys. Rev. B, 7, 4491; 4502 (1973)
8. M.F. Shlesinger: J.Stat. Phys., 36, 639 (1984)
9. G. Matheron, G. de Marsily: Water Resour. Res., 16, 901 (1980)
10. G. Zumofen, J. Klafter, A. Blumen: Phys. Rev. A, 42, 4601 (1990)
11. G. Oshanin, A. Blumen: Macromol. Theory Simul. 4, 87 (1995); G. Oshanin, A. Blumen: Phys.
Rev. E, 49, 4185 (1994)
12. H. Scher, E.W. Montroll: Phys. Rev. B, 12, 2455 (1975)
13. H. Schnörer, H. Domes, A. Blumen, D. Haarer: Philos. Mag. Lett., 58, 101 (1988)
14. D. Haarer, H. Schnörer, A. Blumen: Dynamical Processes in Condensed Molecular Systems, in: J.
Klafter, J. Jortner, A. Blumen (eds.): World Scientific, Singapore, p. 107 (1989)
15. M. Fixman: J. Chem. Phys., 42, 3831 (1965)
16. P.G. de Gennes: in: Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, N.Y., (1979)
17. M. Doi, S.F. Edwards: in: The Theory of Polymer Dynamics, Oxford Univ. Press, Oxford, (1986)
18. R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager: in: Dynamics of Polymeric Liquids, Vol.2,
2nd Ed. Wiley, New York, (1987)
19. W. Carl, W. Bruns: Macromol. Theory Simul., 3, 295 (1994)
20. P.E. Rouse: J. Chem. Phys., 21, 1273 (1953)
21. J.-U. Sommer, A. Blumen: Croat. Chem. Acta, 69, 793 (1996)
22. G. Zumofen, J. Klafter, A. Blumen: J.Stat. Phys., 65, 991 (1991)
43

4 Low-Temperature Heat Release, Sound Velocity and
Attenuation, Specific Heat and Thermal Conductivity
in Polymers
Andreas Nittke, Michael Scherl, Pablo Esquinazi, Wolfgang Lorenz, Junyun Li,
and Frank Pobell
We have measured the long time (t =5 h to 200 h) heat release of polymethylmethacrylate
(PMMA) and polystyrene (PS) at 0.070 K ^ T ^ 0.300 K. After cooling from a temperature
(the charging temperature) of 80 K the heat release in PMMA shows a t –1 -dependence
in the measured time and temperature ranges in agreement with the tunneling model. In contrast,
for PS we observe strong deviations from a t –1 -dependence and a heat release smaller
than in PMMA in by a factor of ten, in apparent contradiction to specific heat and thermal
conductivity data for PS.
To compare the heat release with other low-temperature properties and to verify the consistency
of the tunneling model we have measured also the acoustical properties (sound velocity
and attenuation), the specific heat and the thermal conductivity of PMMA and PS in the
temperature ranges 0.070 K ^ T ^ 100 K, 0.070 K ^ T ^ 0.200 K and 0.3 K ^ T ^ 4K,
respectively. We show that the anomalous time dependence of the heat release of PS is due to
the thermally activated relaxation of energy states with excitation energies above 15 K.
4.1 Introduction
A disordered material releases heat after cooling it from an equilibrium or charging temperature
T 1 to a measuring temperature T 0 [1]. This heat release _Q (T 1 ,T 0 , t) depends on the
charging temperature T 1 as well as on the temperature T 0 , at which the measurement is performed,
and the elapsed time during and after cooling [2, 3]. The time-dependent heat release,
observed in several disordered systems, is a consequence of the long-time relaxation
of the low-energy excitations, identified as two-level tunneling systems (TS) [4–6]. As a
consequence of the finite relaxation time of the TS through their interaction with thermal
phonons [7] the specific heat depends also on the time scale of the experiment [5], as first
measured by Zimmermann and Weber [1].
In recent publications [5, 6] it was shown that within the tunneling model we can
quantitatively understand the observed temperature and time dependence of the specific heat
44 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

4.1 Introduction
and heat release of vitreous silica (SiO 2 ) over ten orders of magnitude in time. It has also
been pointed out that the measurements of the heat release at workable long times is probably
the only feasible method to obtain at least approximately the low-energy limit of the
tunnel splitting D 0 of the distribution function of TS. This low-energy limit is considered in
the literature by the cut-off parameter u min =(D 0 /E) min of the distribution function (E is the
energy splitting of the TS) and is introduced to keep the number of TS finite, avoiding divergences
in the calculated properties.
The interpretation of the heat release data in terms of the tunneling model is a difficult
task due to the not well-known:
a) temperature and time dependence of the specific heat due to TS at T >3K;
b) influence of relaxation processes of the TS at T > 3 K other than one-phonon tunneling
relaxation, i. e. high-order phonon tunneling and thermally activated relaxation;
c) influence of the cooling procedure.
In a recently published paper Parshin and Sahling [8] showed the complexity of the interpretation
of the heat release data when thermally activated relaxation is taken into account
within the framework of the soft potential model. Further theoretical work on the residual
properties of two-level systems and its dependence on the cooling procedure has been published
by Brey and Prados [9].
In this paper we present a further example of the complexity in the interpretation of
the heat release data and an experimental proof of the influence of the relaxation, probably
by thermally activated processes, of excited states that contribute to the heat release of the
TS at low temperatures. We have studied the long-time heat release of two amorphous polymers
with similar low-temperature specific heats and thermal conductivities, cooled under
similar conditions. In spite of those similarities we have found a large difference in the absolute
value and in the time dependence of the heat release between the two polymers when
cooled from temperatures above 15 K. The similarities and differences in the low-temperature
properties, their interpretation within the tunneling model, and the influence of thermally
activated relaxation are the main scope of this work. Preliminary results were published
in Ref. [10].
The paper is organized in five sections. In Section 4.2 we describe the phenomenological
theory for the heat release, based on the tunneling model and the influence of different
relaxation rates, and the cooling process. In Section 4.3 we briefly describe the experimental
procedures and samples. In Section 4.4 we show and discuss the experimental results. Conclusions
are drawn in Section 4.5.
45

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
4.2 Phenomenological theory for the heat release
4.2.1 Generalities
New theoretical work for the calculation of the heat release within the soft potential model
has been published recently [8]. In order to simplify the calculations, to minimize the number
of free parameters, and to assure a more transparent interpretation of the results we
decided, however, to interpret our results in terms of the standard tunneling model. According
to Ref. [8] and to our numerical calculations the differences between the soft potential
and standard tunneling model are not significant at the low temperatures of our measurements.
The standard tunneling model, including a thermally activated relaxation of the twolevel
systems, was used successfully to interpret the acoustical properties of vitreous silica
in eight orders of magnitude in phonon frequency from approximately 0.1 K up to room
temperature [11].
For a system of N two-level systems with energy difference E, the difference in the population
of the two levels at a given temperature T and at thermal equilibrium is given by
n 0 ˆ N tanh…E=2k B T† :
…1†
If the thermodynamic equilibrium of the system is slightly perturbed, e. g. the system
is rapidly cooled to a temperature T, the dynamical behaviour of the population difference
can be calculated according to the relaxation time approximation formula:
d…n…t†
n 0 …T††
dt
ˆ n…t† n 0 …T†
…E;T†
; …2†
where t(E,T) is the relaxation time of a tunneling system with energy E at a temperature T.
The heat released by the N two-level systems after the temperature change is given by
_Q ˆ _nE=2 :
…3†
The problem of calculating _Q simplifies to calculating n from Eq. 2. However, the
cooling process has to be taken into account. In this case Eq. 2 must be rewritten as
_n ˆ @n 0
@T
dT
dt
n n 0
: …4†
Equation 4 describes the response of two-level systems during a temperature change
given by the function T(t). In the general case Eq. 4 has to be solved numerically since n 0
and t are temperature-dependent variables.
46

4.2 Phenomenological theory for the heat release
4.2.2 The standard tunneling model with infinite cooling rate
If N two-level systems are in thermal equilibrium at a temperature T 1 and they are cooled to T 0
with infinite cooling rate, i. e. the time to cool the sample from T 1 to T 0 is zero, from Eq. 2 we
obtain
n…t† ˆ…n 0 …T 1 † n 0 …T 0 †† exp… t= …E;T 0 †† ‡ n 0 …T 0 † : …5†
The standard tunneling model assumes that at low temperatures (T < 2 K) the
one-phonon process is the dominating mechanism. The one-phonon relaxation rate is given
by
1
p ˆ 1
pm u2
…6†
with
1
pm ˆ AE3 coth…E=2K B T† ;
…7†
where A =(g 2 l /n 5 l +2g 2 t n 5 t )/2 pr –4 . The indices l, t refer to the longitudinal and transversal
phonon branches, r is the mass density, g l and g t are the coupling constants between phonons
and TS, and u=D 0 /E (D 0 is the tunneling splitting). For symmetrical TS, i. e. D 0 = E (u = 1),
t p reaches its minimum value t pm .
Furthermore, the standard tunneling model assumes that the distribution function of
TS is constant in terms of two independent variables, namely the asymmetry D and the tunneling
parameter l, i.e.
P…;†dd ˆ Pdd:
…8†
According Ref. [7] the distribution function can be written in terms of the variables E
and u as
P…E;u† ˆ P=u…1 u 2 † 1=2 : …9†
It is also convenient to have the distribution function in terms of the asymmetry and
the barrier height V between the potential wells. Following the work of Tielbürger et al. [11]
and assuming two well-defined harmonic potentials, it can be shown that in a first approximation
l = V/E 0 where E 0 represents the zero-point energy. In this case the distribution function
is
Pdd ˆ
P
E 0
ddV :
…10†
Replacing the total number N of TS with the integrals in E and u, the heat release is
given by
47

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
_Q ˆ PV Z 1 Z
E
E
1
dEE tanh tanh
E 0 2k B T 0 2k B T 1
0
u min
du
p
u 1 u 2
1 …T 0 † exp… t=…T 0 †† ;
(11)
where V is the sample volume and u min is the cut-off in the distribution function at u ?0.
Figure 4.1 shows the heat release at T 0 = 0.1 K as a function of time for different
charging temperatures T 1 following Eq. 11 and taking into account one-phonon relaxation
rate, Eqs. 6 and 7. The calculation was performed with parameters appropriated for vitreous
silica, Ak 3 B =4610 6 s –1 K –3 , P = 1.6610 38 J –1 g –1 and u min =5610 –8 [5]. We observe that
at short times and small T 1 the heat release _Q ! t –1 T 2 1. This dependence follows from the
logarithmic time dependence of the specific heat and holds for tPt m /u 2 min. In this limit the
heat release follows the often used approximation from Eq. 11 [1]:
_Q
p2
24 k2 B PV…T 2 1 T 2 0 † 1 t ; …12†
where _Q and P are measured in W and (Jg) –1 . We recognize in Fig. 4.1, however, that at
longer times the theory deviates from the t –1 -dependence. This deviation comes from the
exponential term in Eq. 11 that decreases strongly at long times and is determined by the
product Au 2 min. The smaller this product the larger is the time-range where the approximation
given by Eq. 12 holds. The influence of the cut-off u min can be recognized comparing the
calculated heat release in Fig. 4.1 with that in Fig. 4.2 (dashed lines) calculated with a
smaller u min .
In Fig. 4.1 we note also that at large times the heat release becomes independent of
T 1 . This feature was recognized experimentally in Refs. [2, 3] and was discussed in Ref. [6].
This T 1 -independence within the assumptions described above is a direct consequence of the
finite number of TS given by the cut-off u min . We should note, however, that in SiO 2 a saturation
of the heat release for large charging temperatures (T 1 > 20 K) is observed and still
a t –1 -dependence was measured [2]. This result cannot be explained within the standard tun-
Figure 4.1: Heat release as a function of time t (in s) according to the standard tunneling model and infinite
cooling rate with Ak 3 B =4610 6 s –1 K –3 , P = 1.6610 38 J –1 g –1 and u min =5610 –8 at a measuring
temperature of 0.1 K and at different charging temperatures T 1 , bottom: 5 K, top: 80 K.
48

4.2 Phenomenological theory for the heat release
Figure 4.2: Heat release as a function of time t (in s) within the standard tunneling model and infinite
cooling rate. The parameters are the same as in the previous figure but with u min =5610 –9 (dashed
lines) and an additional constant (energy-independent) relaxation time t TA =10 3 s (continuous lines).
neling model. The reason for this discrepancy is the relaxation rate, which is different from
the one-phonon process at higher temperatures and the influence of a finite cooling rate (see
below).
4.2.3 Influence of higher-order tunneling processes and a finite cooling rate
At temperatures above 3 K tunneling processes of higher-order are, in principle, possible. One
particular high-order process was studied theoretically in Ref. [12] and is similar to the optical
Raman process; the relaxation time of the TS is given by an interaction that involves two phonons.
Following Ref. [12] we added this Raman relaxation rate to the one-phonon relaxation
rate (Eq. 6). Following Ref. [12] a new free parameter, i. e. the coupling constant R for the Raman
process, is assumed. Due to its strong temperature dependence (T 7 ), the Raman process
can influence the population difference of the energy states of the TS mainly at high temperatures
(T > 5 K) during the cooling process. We should note that for infinite cooling rate and for
the temperature range of our measurements (T 0^1 K) the contribution of the Raman process
to the time and temperature dependence of the heat release is negligible. This follows from Eq. 5
where only the relaxation of the TS at the measurement temperature T 0 enters.
In order to take into account the cooling process we used the algorithm explained below.
The zero-time point is chosen as the time when the cooling process is started. The function
n (E,u,t) represents the difference in population of the TS energy states at any time t
and n 0 (E,T) means this difference in equilibrium (t ? ?) at a temperature T. Analog to
Eq. 11 the heat release can be written as:
_Q ˆ 1
2 PV
Z 1
0
dEE
Z 1
u min
du
p 1 …T 0 † e 1 …T 0 † t …n 0 …E;T 0 † n…E; u; t ˆ 0†† …13†
u 1 u 2
49

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Note that the population difference n depends on E and u because the relaxation rates,
that influence the relaxation of the TS, depend on these variables. As already described
above, to obtain the function n we need to solve the differential equation given by Eq. 4. In
general this is only possible by numerical methods.
The time dependence of the temperature during the cooling process performed in this
work can be well approximated by a linear function given by
T…t† ˆT 1 ‡ T 0 T 1
t A
: …14†
We assume that the sample is at t =0(t A )atT 1 (T 0 ); t A is the time needed to cool the
sample from T 1 to T 0 . The calculations have been done splitting the function T(t) inN steps
of time Dt (Fig. 4.3). During the time Dt the temperature remains constant and n (E,u,t)
shows an exponential behaviour given by Eq. 5; this is qualitatively shown in Fig. 4.3. To
obtain the population difference at the time Dt cooling from T 1 at t = 0 for a given E and u
we calculate iteratively the value
n…t ˆ 0 ‡ t† ˆn 0 …0 ‡ t† …n 0 …0 ‡ t† n…0† e 1 …T…0‡t††t : …15†
The calculations given in this work have been made taking into account 10 to 200
time-steps or iterations depending on the convergence of the numerical results.
Figure 4.4 shows the heat release as a function of time taking into account one-phonon
(dashed lines) and the Raman process (Rk 7
B = 100 s –1 K –7 ) with a cooling time
t A = 3000 s from T 1 to T 0 = 0.1 K. Taking into account our experimental time scale the
calculations were performed for the time interval 10 4 s ^ t ^ 10 6 s only. For one-phonon
Figure 4.3: Time dependence of the sample temperature during a cooling process and its splitting in N
steps (dashed lines) assumed for numerical calculations. Bottom: qualitative behaviour of the population
difference of the tunneling systems n 0 (T) as a function of time.
50

4.2 Phenomenological theory for the heat release
Figure 4.4: Heat release as a function of time for a finite cooling time T A = 3000 s and with (continuous
lines) or without (dashed lines) Raman processes.
process only, our calculations indicate that a finite cooling rate with t A = 3000 s to 5000 s
has a negligible influence (< 3%) on the heat release in our experimental time scale (compare
the dashed lines in Fig. 4.4 with the dashed lines in Fig. 4.2 obtained with an infinite
cooling rate).
The influence of the Raman process at T 0 = 0.1 K can be well observed if we take
into account a finite cooling rate. In comparison with the results using the one-phonon process
only and for T 1 >20K, _Q is smaller and shows a slightly different time dependence.
Note that the results for T 1 6 20 K resemble those obtained taking into account the one-phonon
process only but with a smaller charging temperature T 1 . We note also that at T 1^ 20 K
the heat release still shows a T 1 2 -dependence but reaches its saturation at a smaller T 1 in
comparison with the results with the one-phonon process only (Fig. 4.4).
We conclude that taking into account Raman processes, that influence the relaxation
rate of TS at temperatures larger than 2 K, it is possible to understand qualitatively the results
of, for example, SiO 2 where _Q reached a saturation at T & 10 K but still shows a t –1 -
dependence at t ^ 3610 5 s [2]. This result is only valid if the product Au 2 min in Eq. 7 remains
small enough.
Because of the uncertainty in the coupling constant R between TS and phonons a
quantitative comparison, however, of the heat release results with the theory taking into account
the contribution of higher-order processes is not useful. Phenomenologically and in order
to simplify the calculations, one can take into account the influence of higher-order processes
by choosing an appropriate smaller charging temperature than the real one. In that
sense charging temperatures T 1 6 10 K can be considered for comparison with theory as
free parameters. From our calculation we learn that it is practically impossible to obtain reliable
information on the density of states of TS from heat release experiments for charging
temperatures in the region of the T 1 -saturation of _Q.
51

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
4.2.4 The influence of a constant and thermally activated relaxation rate
There are different attempts to link the high-temperature (T > 10 K) with the low-temperature
properties of disordered solids [13, 14]. In particular, it has been proposed that the maximum
in the attenuation of phonons observed in amorphous materials at T > 10 K can be interpreted
assuming a thermally activated relaxation of the two-level systems. In this Section
we discuss the influence of thermally activated relaxation rate on the heat release for a finite
cooling rate.
For a single activation barrier V 0 between the two potential wells and at a measuring
temperature T 0 the thermally activated relaxation rate is given by
1
TA ˆ 1
0 e V 0=k B T 0
: …16†
Taking into account this relaxation rate in addition to quantum tunneling (Eq. 6) and
assuming that both processes are independent, the total relaxation rate can be written as
1 ˆ 1
P ‡ 1
TA :
…17†
Obviously, at a given temperature T 0 and for a single activation barrier we add a constant
relaxation rate to the tunneling process. Figure 4.2 shows the time dependence of the
heat release at different temperatures T 1 and using Eq. 17 with t TA =10 3 s. As expected, at
t P t TA the introduction of an additional constant relaxation rate does not influence the heat
release. At t & t TA the heat release is larger than taking only into account the one-phonon
process (Fig. 4.2). At longer times _Q decreases exponentially with time. This decrease can
be easily understood: the energy levels with long (tunneling) relaxation time have been depopulated
at t & t TA at a larger rate. Since the total amount of heat released by the TS
should be finite a decrease below the standard result (dashed lines in Fig. 4.2) is expected.
To take into account the distribution of potential barriers V and a thermally activated
rate we will follow the approach used by Tielbürger et al. [11] and transform Eq. 11 in a double
integral in the variables D and V using the approximation l = V/E 0 and D 0 & 2E 0 e –l /p:
_Q ˆ PV
2
Z 1
0
d
ZV max
0
dV E
E
tanh
E 0 2k B T 0
E
tanh 1 …T 0 † exp… t=…T 0 †† : …18†
2k B T 1
For the following discussion it is not relevant to introduce in Eq. 18 a specific distribution
of potential barriers V [11]. This is done below for comparison with experimental results
and for computing the acoustic properties. After finding the range of potential barriers
relevant for the heat release we divided the calculations in two regions: T 1 ^ 1 K and
T 1 > 1 K as described below.
In order to find the range of potential barriers relevant for the heat release in our measuring
time and temperature ranges we have calculated it following Eq. 18, splitting the V-
limits of the inner integral from V min = V max –10KtoV max , using only the one-phonon process,
and measuring temperature T 0 = 0.200 K. We have recognized that for V < 130 K and
52

4.2 Phenomenological theory for the heat release
at t&10 3 s the TS are mostly relaxed and do not contribute appreciably to the heat release
in our measured time range (10 4 s ^ t ^ 10 6 s) any more. The TS that relax through the
one-phonon process and are relevant to the heat release are found to be those with potential
barriers 130 K < V < 200 K. The contribution of tunneling systems with V > 200 K to the
heat release is negligible.
At T 1 < 1 K the contribution of a thermally activated rate for TS with potential barriers
130 K < V < 200 K is irrelevant, i. e. using t&10 –13 s (from acoustic measurements,
see below) we obtain a t TA (V = 130 K k B ) of several years. The potential barriers which are
relevant in our time and temperature range through a thermally activated rate are V/k B 1 K. Figure 4.5 shows the calculated heat release as a function of T 1 at a
given time t = 1.5610 5 s taking into account the cooling process and one-phonon process
as well as thermally activated relaxation rate. The numerical results (Fig. 4.5) show clearly a
saturation of _Q for T 1 > 5 K, i. e. at lower temperatures in comparison with the results without
thermally activated relaxation, in qualitative agreement with the results assuming Raman
processes and experimental results [2]. It is also interesting to note that a thermally activated
relaxation rate decreases slightly the exponent in the time dependence of the heat release,
i. e. _Q ! t –a with a < 1. This result is shown in Section 4.4.
Figure 4.5: Heat release at t = 1.5610 5 s as a function of charging temperature T 1 calculated with and
without thermally activated processes and a finite cooling rate.
53

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
4.3 Experimental details
The experimental setup for measuring the heat release and the specific heat consists of a calorimeter
on which the sample is mounted. The samples were cooled from 80 K to 0.050 K
in about (5–8)610 3 s. The calorimeter is attached to a holder through a thermal resistance
or a superconducting heat switch made of an Al strip. The holder is screwed to the mixing
chamber of a top loading dilution refrigerator.
Two sample holders were used, one from plastic and a second from Teflon. The first
one showed a heat release approximately proportional to t –1 whereas for the second one no
heat release was measured in agreement with previous measurements [2].
We used two different methods to measure the heat release. Firstly, after cooling to
low temperatures (T&0.060 K) the heat switch was opened and the time dependence of the
sample temperature (warmup rate) was measured till about 0.120 K. Then the sample was
cooled again to about 0.050 K before starting another warmup run. The heat release is obtained
from the slope of the temperature in the warmup run dT (t)/dt with the knowledge of
the specific heat of the sample plus calorimeter C(T), i. e. _Q = C(T) dT (t)/dt. The measurements
were completely automated with a personal computer. The background heat leak was
about (0.03–0.50) nW at 0.090 K and at t & 10 5 s depending on the sample holder and the
vibration of the connecting cables. To test the measuring setup and procedure we have produced
well-known heat leaks with a electrical heater, fixed to the sample.
The second method for measuring the heat release is based as before on the measurement
of the time dependence of the sample temperature at constant bath (mixing chamber)
temperature. The difference lies on the selection of a fixed thermal resistance R th between
sample and holder that enables the semiadiabatic measurement of the heat release in a time
scale larger than the intrinsic thermal relaxation time of the arrangement. Approximately
30 minutes after reaching a constant bath temperature T b the decrease of the sample temperature
T s with time provides directly the heat release, i. e. _Q =(T s – T b )/R th (T s ,T b ). The
measurement of the thermal resistance together with the background contribution to the heat
release is made in situ after reaching the time-independent minimum temperature. Within
experimental error (&10 %) both methods show the same results.
The specific heat of both polymers was measured with the heat pulse technique in semiadiabatic
fashion. The thermal conductivity was measured with a top loading 3 He refrigerator
using the standard procedure. The acoustic properties, sound velocity and attenuation, were
measured with the vibrating reed technique [16] in the frequency range (0.2–3) kHz.
Both polymers were prepared following standard procedures. The PMMA sample had
additionally 10 –2 mol% of tetra-4-tert.butyl-phthalocyamin (dye molecule) because it was
used in an early optical hole burning experiment [15]. For the specific heat and heat release
measurements the mass and density of the PMMA sample were determined as 11.97 g and
1.15 g/cm 3 , respectively. We have measured two PS samples prepared from different
batches. The densities of these samples were 1.05 g/cm 3 , and the masses 11.4 g (sample
PS1) and 38.0 g (sample PS2).
For the thermal conductivity measurements two slices were cut from the bulk samples
with length l = 1 cm, width w = 2 mm, and thickness d = 300 mm. For the acoustic measurements
the reeds had the geometry l = (0.7–1) cm, w = (0.1–0.3) cm and d = (100–300) mm.
54

4.4 Experimental results and discussion
4.4 Experimental results and discussion
4.4.1 Specific heat and thermal conductivity
Figure 4.6 shows for both polymers the specific heat devided by the temperature as a square
function of temperature. These measurements were performed only when the heat release is
negligible (t & 10 6 s). For t ?? the specific heat can be written in terms of two contributions:
c (T) =c 1 T + c 3 T 3 with coefficients for the tunneling system c 1 and phonon contribution
c 3 . From Fig. 4.6 we obtain the values c 1 = (3.0+0.3) mJ/gK 2 and c 3 = (93 + 18) mJ/gK 4
for PMMA, and c 1 = (4.6+0.5) mJ/gK 2 and c 3 = (77 +23) mJ/gK 4 for PS. The values of the
linear term are in fair agreement with those published by Stephens [17, 18] c 1 = 4.6 mJ/gK 2
for PMMA and c 1 = 5.1 mJ/gK 2 for [PS], but we obtained larger phonon contribution
(Stephens: c 3 =29mJ/gK 4 for PMMA and c 3 =45mJ/gK 4 for PS). In Figs. 4.7 and 4.8 we
compare our specific heat results with data from Ref. [18]. Within the scatter of the data it
is difficult to conclude whether the c values are really different.
Figure 4.6: Specific heat divided by the temperature as function of the squared temperature for PMMA
(a) and PS (b).
Figure 4.7: Comparison of our specific heat for PMMA with data from Ref. [18].
55

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Figure 4.8: Comparison of our specific heat for PS with data from Ref. [18].
According to the tunneling model and for t ?? the specific heat due to the TS is given
by [5]:
c…T† ˆp2
6 k2 2
BPV ln… †T ˆ c 1 T:
u min
…19†
If we assume (from heat release data, see below) u min =10 –10 for PMMA and
u min =6610 –9 for PS and from the measured c 1 we obtain a density of states P = 4.0610 38
(7.5610 38 ) 1/Jg for PMMA (PS) in reasonable agreement with the values obtained in earlier
specific heat measurements [17] and also from acoustic measurements for PMMA [24].
Recent optical hole burning experiments on PMMA and PS [15] indicate that the density of
states of TS for PS is about two times larger than for PMMA in reasonable agreement with
our specific heat measurements.
Figure 4.9: Thermal conductivity as a function of temperature for PMMA and PS.
56

4.4 Experimental results and discussion
Figure 4.9 shows the thermal conductivity for PMMA and PS as a function of temperature.
For PMMA and at T < 0.7 K we obtain k =28T 1.84 610 –3 W/mK 2.84 and for PS
k =19T 1.93 610 –3 W/mK 2.93 , in agreement with earlier measurements [17, 19–22], (Figs. 4.10
and 4.11), and with measurements in epoxies [23]. According to the tunneling model and for
T < 1 K the thermal conductivity is given by:
…T† ˆk3 B v
6p 2 … P 2 † 1 T 2 ; …20†
where v is the sound velocity and r the mass density. From Eq. 20 and with the sound velocity
from Ref. [17] we obtain Pg 2 = 9.2610 5 J/m 3 (11.5610 5 J/m 3 ) for PMMA (PS). If we
replace the density of states P from the specific heat results we obtain similar values for the
coupling constant between TS and phonons in both materials g = 0.28 eV (0.27 eV) for
PMMA (PS). The results for the thermal conductivity indicate that the density of states of
TS for the PS sample is about a factor of two larger than for the PMMA sample.
Figure 4.10: Comparison of our thermal conductivity data for PMMA with earlier publications. The
dashed line has been calculated within the tunneling model with parameters taken from our internal friction
data.
Figure 4.11: Comparison of our thermal conductivity data for PS with earlier publications. The dashed
line has been calculated within the tunneling model with parameters taken from our internal friction
data.
57

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
4.4.2 Internal friction and sound velocity
Figures 4.12 and 4.13 show the internal friction of PMMA and PS. Below 1 K the internal
friction of PMMA and PS is approximately temperature-independent. According to the tunneling
model the sound absorption should be temperature-independent in the region ot p P1
(fulfilled in our temperature range) where o =2pv is the phonon frequency and t p the relaxation
time of the TS (Eq. 6). In this temperature range a simple relationship is found for
the internal friction [11, 16]
Q 1 ˆ p
2 C …21†
Figure 4.12: Squares: PMMA internal friction as a function of temperature at a frequency of 535 Hz.
Full circles: indicate the attenuation data taken from the ultrasonic measurements at 15 MHz from
Ref. [24]. Solid and dotted lines: the calculated values for v=535 Hz and v=15 MHz following the
modified tunneling model considering a thermally activated relaxation rate. For more details see text.
Figure 4.13: Squares and Circles: PS internal friction as a function of temperature at two different frequencies.
Solid line: the calculated values for v=0.24 kHz following the modified tunneling model described
in the text.
58

4.4 Experimental results and discussion
with the constant C = Pg 2 /ru 2 and the sound velocity u. Equation 21 should hold even if
higher-order processes dominate the relaxation of the TS (if the relaxation rate is proportional
to u 2 !D 0 2 , Eq. 6). From the temperature-independent value of the internal friction
and assuming no background contribution (e. g. due to the clampling) we obtain for PMMA
(PS) C % 2.6610 –4 (8.3610 –4 ). Taking into account the mass density, sound velocity, and
the coupling constant obtained from the thermal conductivity and specific heat results described
above, the values for the parameters C from the internal friction indicate that the
density of states of TS for PS is larger than for PMMA by a factor &2.5. This is slightly
larger than that obtained from the specific heat (about 1.5). This might be attributed to the
unknown clamping contribution to the measured attenuation (which is always present) or to
different u min values (Eq. 19). In Figs. 4.10 and 4.11 we show the calculated thermal conductivity
using Eq. 20 with Pg 2 values from our internal friction at the plateau. Excellent agreement
is obtained for PMMA (Fig. 4.10) but for PS the thermal conductivity is by a factor
of two smaller. This difference might be attributed partially to the background contribution
which is not subtracted from the measurement before computing the parameter C.
The behaviour of the internal friction above about 3 K is qualitatively different for
both polymers. PMMA shows a decrease in the internal friction reaching a minimum at
about 30 K and increasing monotonously to the highest temperature of our experiment,
120 K (not shown in Fig. 4.12) in very good agreement with previous work at similar frequencies
[25]. On the contrary, the internal friction of PS shows the typical temperature dependence
measured for other amorphous materials, for example SiO 2 [16]. It increases,
reaching a frequency-dependent maximum at T&37 K (;40 K), (Fig. 4.13), and increases
again at T > 60 K (70 K) at the frequency 0.24 kHz (3.2 kHz). This internal friction increase
at T > 60 K for PS together with the increase at T > 30 K for PMMA will not be discussed
here. Instead, we will discuss the behaviour of the internal friction at lower temperatures in
terms of an extension of the tunneling model following the procedure described in Ref. [11].
We assume the simplified relation for a thermally activated relaxation rate given by
Eq. 16 and add it to the tunneling rate (Eq. 17). For two well-defined harmonic potentials
and within the approximations given by Eq. 10 and with l = V/E 0 , it can be shown that the
internal friction increases linear with temperature just above the temperature-independent region
(plateau) [11]:
Q 1 ˆ pCk BT
E 0
: …22†
This increase is observed for PS, (Fig. 4.13); applying Eq. 22 to the results below
20 K we obtain a zero point energy E 0 =13+ 2 K. In the same temperature region and due
to the influence of the thermally activated relaxation the relative change of the sound velocity
can be written as [11]:
uu
u ˆ Ck BT
E 0
ln…! 0 † : …23†
A nearly linear temperature dependence of uu/u is observed in both polymers at temperatures
about below 50 K (Figs. 4.14 and 4.15). At T ~ 1 K a crossover, due to tunneling
and due to thermally activated relaxation, from the linear T-dependence to the logarithmic
59

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Figure 4.14: Relative change of sound velocity at v=535 Hz for PMMA. The solid line represents the
linear temperature dependence of the sound velocity at temperatures below 20 K used for the numerical
calculations. Inset: the same data but below 10 K in a semilogarithmic scale.
Figure 4.15: Relative change of sound velocity at two different frequencies for PS. The solid line was
calculated according to the modified tunneling model described in the text. Inset: the same data but below
10 K in a semilogarithmic scale.
T-dependence can be observed for both polymers (insets in Figs. 4.14 and 4.15). Although
the tunneling model with the assumption of thermally activated relaxation of the TS provides
a reasonable fit for the linear temperature dependence, its origin is still controversial. We
note that a linear temperature dependence of the sound velocity above a few Kelvin is a
rather general behaviour observed in several amorphous [26, 27], disordered [28], and polycrystalline
metals [29]. Nava [28] argued recently against an interpretation in terms of thermally
activated relaxation of the TS for the linear T-dependence of the sound velocity. However,
new acoustical results in polycrystalline materials indicate a linear T-dependence of the
sound velocity comparable with those found in amorphous materials [29].
Applying Eq. 23 to the measurements at the two phonon frequencies for PS and with
the value of E 0 from the internal friction we obtain t 0 % (1 +0.5)610 –17 s. As discussed
in Ref. [11] the meaning of the prefactor t 0 is still unknown. The very small value of t 0
found for PS in comparison with the one for SiO 2 (t 0 %10 –13 s) should be taken as an ef-
60

4.4 Experimental results and discussion
fective value until a systematic comparison of the applied model to other experimental data
is available.
To obtain a maximum in the internal friction an upper limit of the potential height
must be assumed. Instead of a cut-off in the distribution P (D, V) and following Ref. [11], a
Gaussian distribution with a width s 0 will be assumed:
P…;V†ˆ
P
E 0
e … V2 =2 2 0 † : …24†
Figures 4.13 and 4.15 show the result of numerical calculations with only the width of
the distribution s as free parameter. The fits were obtained assuming s 0 = 1200 K and with
the values C = 8.3610 –4 , v = 0.24 kHz, t 0 =10 –17 s and E 0 = 13 K using the equations
given in Ref. [11]. A reasonable agreement with the experimental data is achieved for both
acoustic properties in the expected temperature region. The minimum obtained numerically
at T ~ 3 K (Fig. 4.13) has been discussed in Ref. [11] and is attributed to the difference in
the number of TS contributing to the relaxation process in the tunneling or thermally activated
regime.
As pointed out above, the internal friction of PMMA behaves differently from PS
above the plateau. It decreases at T > 3 K and reaches a minimum at T ~ 30 K. The predicted
linear temperature dependence of the internal friction (Eq. 22), at the measured frequency
(v = 535 Hz) is not observed for PMMA. It is tempting to interpret the decrease of
the internal friction results above 3 K assuming an upper bound of the TS density of states
around 15 K. This assumption has been indeed used to interpret the irreversible line broadening
of the optical hole burning experiments [30]. However, ultrasonic attenuation measurements
at 15 MHz for PMMA [24] show clearly a thermally activated maximum at 12 K.
Therefore we have decided to search for a set of parameters that might explain the low and
high frequency results under the assumptions described above and as done for PS.
From the sound velocity data below 20 K (Fig. 4.14) and assuming the validity of
Eq. 23 (ot P1) we obtain (E 0 /k B )/ln (ot 0 ) %C/(u ln (u)/uT) ~ 0.9 +0.2. If we assume
t 0 ~10 –17 s like PS, we obtain E 0 /k B ~ 30 K. With this value we are not able to obtain a reasonable
set of parameters that explain the data. If we take the value for SiO 2 from Ref. [11],
t 0 % 10 –13 s, we obtain E 0 /k B % (20 + 5) K, a value comparable to that for SiO 2 . After several
trials, eventually we have found a set of parameters that fits reasonably the low and
high frequency measurements without the assumption of an energy cut-off.
In Fig. 4.12 we compare the experimental results of the internal friction with the numerical
calculations for frequencies at 535 Hz and at 15 MHz using E 0 /k B =10K,t 0 =10 –13 s,
and a rather small distribution width s 0 = 150 K. The model reproduces fairly well the position
of the maximum in the attenuation at 15 MHz as well as its relative value in comparison
with the plateau [24]. It is worth pointing out, that for PMMA, since the changes produced by
the free parameters E 0 , t 0 and s 0 compete with each other, it is not possible to find another set
of values which can explain the results as well as we can. It is important to note that the internal
friction data for both polymers indicate a much smaller thermally activated contribution to
the phonon attenuation for PMMA than for PS, in spite of the fact that in the quantum tunneling
regime both samples show similar results. This difference may influence the low-temperature
heat release because the number of excited states that are available after the cooling process
will be different.
61

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
4.4.3 Heat release
Figure 4.16 shows the heat release as a function of time for the 12 g PMMA sample at
T = 0.090 K and after cooling from 80 K. The heat release follows very well the predicted
t –1 law from the tunneling model (Eq. 12). The same result was obtained in different runs
with slightly different cooling rates and measuring temperatures 0.070 K^T 0 ^0.110 K.
For finite cooling rates and for charging temperatures T 1 p 1 K this independence cooling
rate is to be expected (Section 4.2.3).
As discussed in Section 4.2 the unknown higher-order relaxation processes are taken
into account through an effective charging temperature T 1 . With the tunneling systems density
of states from the specific heat and using Eq. 12 we obtain T 1 % 26 K. As stated in
Section 4.2 for charging temperatures T > 10 K it is not possible to obtain reliable values of
the density of states for TS from heat release measurements.
From the observed t –1 -dependence and using the value for the coupling constant between
TS and phonons from Ref. [24], Ak B 3 = 1.6610 9 K –3 s –1 , we obtain an upper limit for
the parameter u min

4.4 Experimental results and discussion
and (5). Even worse, the computed curve (5) with thermal activation shows much smaller
values for the heat release by almost two orders of magnitude. This result indicates that too
many energy states were depopulated during the cooling process through thermal activation.
This disagreement might be ascribed to the assumed potential distribution function (Eq. 24)
or the approximation l = V/E 0 . Curve (2), which is calculated without thermally activated relaxation,
lies only a factor of two higher than the measured one (Fig. 4.16) and is in qualitative
agreement with the acoustical data where no influence from thermal activation were taken
into account.
For PS, according to the specific heat and acoustical data (at T < 3 K) and due to a larger
density of states of tunneling systems, we expected a larger heat release than for PMMA.
Surprisingly, the opposite is observed. After cooling from 80 K with the same cooling rate,
the heat release for PS measured at the same temperature as before was at least of a factor of
ten smaller than for PMMA (Figure 4.16), and it does not follow the t –1 -dependence.
In order to understand the observed deviations we have measured the heat release of
PS at lower charging temperatures. Figure 4.17 shows the heat release as a function of time
at charging temperatures 0.5 K^T 1 ^1 K. The heat release follows very well the t –1 -dependence
as well as an increase with T 1 2 (Eq. 12, Fig. 4.18). It is interesting to note that a
fair quantitative agreement with the prediction of the tunneling model is obtained. If we calculate
the density of states of tunneling systems P from these data with Eq. 12 (straight line
in Fig. 4.18) we obtain P = 9.0610 –38 J –1 g –1 which is similar to the one obtained from the
specific heat assuming u min =6610 –9 .
Figure 4.17: Heat release as a function of time (in hours) for Polystyrene at a measuring temperature
T 0 = 0.200 K cooling from different T 1 . The straight lines have a t –1 -dependence.
At higher temperatures T 1 we observe the expected saturation of the heat release
(Fig. 4.18), however, no deviation from the t –1 -dependence has been measured within experimental
error. Deviations are observed if we charge the sample at temperatures T 1 >15K
(Fig. 4.16). In this case we have cooled the sample from 80 K and measured the heat release
at 0.090 K (sample PS1) and 0.300 K (sample PS2). According to theoretical estimates the
observed difference in the heat release between the two PS samples is not attributed to the
difference in measuring temperatures. The rather abrupt saturation of the heat release at
T 1 6 7 K is attributed to the depopulation of the excited states through thermally activated
relaxation in the cooling process (Figs. 4.5 and 4.18), as it will become clear below.
63

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
Figure 4.18: Heat release multiplied by the time t as a function of the difference between the squares of
charging temperature T 1 and measuring temperature T 0 for PS. The solid line follows the theoretical
prediction with a density of states of tunneling systems 1.45 larger than the one obtained from the specific
heat results.
The following experiment provides further verification that the non-simple time dependence
of the heat release for PS is due to energy states excited above 15 K and depopulation
during the cooling process, very likely through thermally activated relaxation. We have
charged the sample at 80 K for several hours and cooled it then to 1 K. We have left the sample
for 22 h at 1 K and later continued to 0.3 K, the temperature at which the heat release
was measured (Curve (1) in Fig. 4.19). We observe now a clear deviation from the t –1 law.
That means that even after 22 h at 1 K the energy states excited above 15 K have not relaxed
completely. After 60 h at 0.3 K we warmed the sample to 1 K for 17 h, cooled it to 0.3 K
and measured the heat release (Curve (2) in Fig. 4.19). The heat release follows the theoretical
t –1 -dependence very well.
We have calculated the heat release taking one-phonon,thermally activated processes,
and the experimental cooling process into account. We used the same parameters obtained
from the fit to the acoustical data (Figs. 4.13 and 4.15). With only one free parameter, u min ,
Figure 4.19: Heat release as a function of time for PS. (1): cooled from 80 K to 1 K and leaving the
sample 17 h at this temperature, T 0 = 0.300 K. (2): the sample was warmed to a charging temperature
T 1 = 1 K for 22 h and cooled again to T 0 = 0.300 K. The solid line in (1) is only a guide and the solid
line (2) has a t –1 -dependence.
64

4.5 Conclusions
we can reproduce the measured heat release of PS reasonably well (Fig. 4.16). In this figure
we present the two curves (3) and (4) using the parameter u min =10 –10 and 6610 –9 . This
fit indicates that we can observe the influence of a finite u min through the steeper decrease
of the heat release at long times. Curve (1) in Fig. 4.16 was calculated with parameters for
PS but without thermally activated relaxation.
4.5 Conclusions
We conclude that thermal conductivity, specific heat and acoustical properties at low temperatures
(T < 1 K) can be quantitatively interpreted within the tunneling model for PMMA
and PS. At higher temperatures we have measured very different temperature dependence of
the internal friction for the two polymers. At the used frequencies and at 0.1 K^T^120 K
PMMA shows no thermally activated dissipation peak. PMMA shows a minimum in the internal
friction at T ~ 30 K where PS shows a maximum. Nevertheless, within the tunneling
model and introducing a thermally activated relaxation rate as well as a Gaussian distributon
density of states of tunneling systems, it is possible to understand the acoustical properties.
From the fits we may conclude that the density of states of tunneling systems for PMMA is
restricted to smaller energies, i. e. a smaller width s 0 , than for PS.
Although the thermal conductivity, specific heat, and sound velocity temperature dependence
for both samples are similar, the heat release shows different time dependences as
well as different values in apparent contradiction with the other measured low-temperature
properties. Taking into account the internal friction results, it is tempting to correlate the differences
in the heat release between both samples to the difference in the contribution of
thermally activated processes in the relaxation of the tunneling states during the cooling process.
We have demonstrated that the deviations of the time dependence of the heat release
from the t –1 -dependence predicted by the tunneling model is due to the relaxation of energy
states excited above 15 K. The smaller values of the measured heat release of PS in comparison
with the standard tunneling model can be explained by the depopulation of energy
states relaxing mainly by thermally activated processes during the cooling process.
Without detailed knowledge of the short relaxation times at T 1 > 1 K, a quantitative
fit or even a qualitative understanding of the heat release _Q(T 1 ,T 0 ,t) data for high charging
temperatures seems to be impossible. Therefore, the heat release with high charging temperatures
can be hardly used to obtain information on the density of states of tunneling systems.
Heat release experiments at high measuring temperatures T 0 > 1 K are necessary to
obtain information on the dynamics of the tunneling systems with a relaxation rate not given
by the tunneling one-phonon process. These experiments may clarify the influence of the
two different relaxation mechanisms discussed nowadays in the literature to interpret the
low-temperature properties of amorphous solids above 1 K, namely incoherent tunneling
65

4 Low-Temperature Heat Release, Sound Velocity and Attenuation, …
[31] and thermal activation within the soft potential model [8]. Both approaches have been
used with impressive success in the last years. It is, however, not yet clear if incoherent tunneling,
a mixture of the two processes (incoherent and thermal activation), or only thermal
activation is the main relaxation mechanism of the TS above 1 K.
Acknowledgements
We wish to acknowledge W. Joy for the sample preparation, and S. Hunklinger, D. Haarer
and J. Friedrich for valuable discussions. This work was supported by the Deutsche Forschungsgemeinschaft
through the Sonderforschungsbereich 213. A. Nittke was supported by
the “Graduiertenkolleg Po 88/13” of the Deutsche Forschungsgemeinschaft.
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67

5 Spectral Diffusion due to Tunneling Processes
at very low Temperatures
Hans Maier, Karl-Peter Müller, Siegbert Jahn, and Dietrich Haarer
5.1 Introduction
Pioneered by the work of Zeller and Pohl [1] it was discovered in the early 1970s that amorphous
solids show low-temperature thermal and acoustic properties which are very different
from those observed in crystals. For reviews see for example Refs. [2, 3]. The most wellknown
of these anomalous properties is the specific heat, which is in general considerably
larger than would be expected from the Debye model and varies linear with temperature in
contrast to the Debye T 3 -dependence. Other anomalies are the temperature dependence of
the thermal conductivity, the properties of phonon echoes, and ultrasonic absorption. These
features seem to be quite universal for all kinds of amorphous solids, irrespective of their
chemical composition and structure, i. e. inorganic as well as organic or polymeric.
Very soon after the first experimental evidence for these anomalous low-temperature
excitations Anderson et al. [4] and independently Phillips [5] developed a theoretical description,
called the tunneling model [4, 5]. It is based on the assumption that the low energy
degrees of freedom in amorphous solids arise from tunneling motions of atoms or groups of
atoms between local energetic minima which should exist in any disordered solid. If only
the lowest energy level of each minimum is considered, this leads to two eigenstates. For
this reason, these degrees of freedom are often referred to as two-level system (TLS). An
important consequence of this model is the formal analogy to a system of particles with
spin 1/2, which for example immediately explains the existence of phonon echoes. Another
very striking feature of the tunneling model was the prediction of a time-dependent specific
heat [4]. This was confirmed experimentally several years later [6]. Furthermore, the model
can explain the heat release [7] of amorphous solids, which is a time-dependent non-equilibrium
phenomenon. From these kinds of experiments, it could be concluded that TLS dynamics
occur on time scales extending over many orders of magnitude.
The sensitivity of optical experiments on amorphous solids was hindered, in many instances,
by the large inhomogeneous broadening which arises from the distribution of local
environments of the involved optical transitions. Therefore a very important step was the
discovery of persistent spectral hole burning in 1974 [8, 9]. This method of high-resolution
laser spectroscopy eliminates the inhomogeneous effects induced by the static disorder in
68 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

5.2 The optical cryostat
amorphous hosts. For this purpose dye molecules are embedded in the solid which change
their absorption spectra when irradiated with resonant narrow band laser light. This produces
a dip in the inhomogeneous absorption band, called a spectral hole. A narrow spectral hole
can be regarded as a highly sensitive spectral probe for any kind of distortion of the matrix,
which induces small spectral changes [10]. In this way variations of strain fields caused by
pressure changes of almost some 10 hPa can be detected [11]. Spectral holes are also sensitive
to small changes of other external parameters including electric fields [12, 13].
After observing quite a few anomalous properties of optical transitions in glasses and
attributing them to the dynamics of TLS [14], the tunneling model was adopted by Reinecke
[15] to explain the low-temperature line widths of optical transitions in amorphous solids
using the concept of spectral diffusion. This concept had originally been developed for the
description of spin resonance experiments [16] and had already been applied to the theoretical
treatment of the above mentioned ultrasonic properties of glasses [17]. Soon after this
step, the possibility of a connection between thermal and optical properties of amorphous
solids was supported by the observation of time dependence of spectral hole widths [18].
The application of the tunneling model to the description of spectroscopic properties
proposed an important link suggesting a correlation between the specific heat and the optical
line width. This implies that, besides the mentioned calorimetric and acoustic methods,
optical spectroscopy yields information about the dynamics of the same low energy excitations
of amorphous solids, which dominates the calorimetric experiments. A crucial point in
establishing this connection is the temperature dependence of these physical solid state parameters.
The tunneling model in its original form predicts linear temperature dependence for
the specific heat as well as for the optical line width. This prediction, however, is only valid
at temperatures where the contribution of phonons is of minor importance. Earlier hole burning
measurements on the temperature dependence of optical line widths [19] indicated the
necessity of expanding the temperature range accessible to optical absorption spectroscopy
to values far below 1 Kelvin in order to exclude any other mechanisms except TLS dynamics.
For this reason we have constructed a 3 He/ 4 He dilution refrigerator with optical
windows allowing transmission spectroscopy at temperatures down to 0.025 K, a temperature
regime in which only very few data from optical spectroscopy existed before [20, 21].
In addition, cooling with a 3 He/ 4 He mixture can be performed continuously for very long
times (in principle unlimited) if the necessary care in cryostat design and operation is taken.
In our experiments we have reached operation times up to three months.
5.2 The optical cryostat
In the design of a cryostat for optical spectroscopy two main problems have to be overcome.
In an absorption experiment the sample has to dissipate the absorbed energy in order to
avoid overheat during the measurement. Therefore the best possible thermal contact to the
cold stage is of extreme importance. The second problem is how to guide the laser light to
the sample without creating an intolerably large additional heat leak, for example by infrared
69

5 Spectral Diffusion due to Tunneling Processes at very low Temperatures
irradiation from the windows. Since various construction methods for optical cryostats in the
temperature range below 0.500 K did not yield the desired results, special care had to be taken
in the design of our dilution refrigerator. For example it had been reported [22] that for
samples located outside the mixing chamber temperatures below 0.300 K could not be
reached, even if the best thermal contact is realized by pressing the samples to the wall of
the mixing chamber with indium metal as contact material. In another kind of experiment a
glass fibre winding around the cold finger of a dilution refrigerator shows a significant temperature
gradient with respect to the mixing chamber below 0.100 K [20]. For these reasons,
we decided to place the sample directly into the dilute phase of the mixing chamber. This
way optimal thermal contact and a constant temperature of the sample was achieved via the
superfluid 4 He. For performing optical experiments the mixing chamber was designed as a
glass cylinder.
The optical path, used in our cryostat, is shown in Figure 5.1 [23]. Three sets of windows
are mounted on cold copper shields to eliminate most of the room temperature radiation
in three steps: at liquid nitrogen temperature (77 K), liquid helium temperature (4.2 K),
and at approximately 1 K, the latter being achieved by cooling with the pumped 3 He of the
distillation chamber. In contrast to other constructions, the vessel containing the dilution refrigerator
part of the cryostat is in our design not surrounded by the helium tank which has
the advantage that the laser beam is not scattered by boiling liquid helium. We estimate that
the total heat leak caused by room temperature radiation is below 50 nW. The minimum
temperature reached is 0.024 K. The cooling power is about 6 mW at 0.100 K. Although this
is a very low value, it is sufficient since the power of the absorbed laser light was always far
below 1 mW. The light powers used for hole burning in our experiments were on the order of
several nanowatts to several tens of nanowatts depending on the temperature. For hole detection
the power is reduced to the picowatt range. The cooling power of our cryostat and the
thermal conductivity of the involved polymers are high enough to guarantee a uniform temperature
distribution with no significant sample heating during the hole burning process.
4
He - shield mixing - chamber
LN 2 - shield 1-K - shield
sample
Figure 5.1: Alignment of optical windows and sample in the cryostat. The sample is placed inside the
mixing chamber of the dilution refrigerator. The shields are connected to the distillation chamber, the
helium tank, and the nitrogen tank, respectively. The liquid N 2 windows possess an infrared reflective
coating.
70

5.3 Theoretical considerations
The temperature is measured with a RuO 2 thick film resistor (TFR) supplied by Phillips
(type RC-01). The heat capacity and thermal relaxation time of the TFR are equivalent
to those of a carbon resistor but its thermal reproducibility during temperature cycles is better
[24]. The resistor is calibrated against several primary and secondary thermometers including
a NBS fixed point device, CMN and Ge, thermometers. Its accuracy is better than
2% in the temperature range between 4.2 K and 0.025 K. The thermometer is placed into
the dilute phase of the mixing chamber in direct thermal contact to the sample. The resistance
is measured using a four wire ac resistance bridge (AVS-46 RV Elektroniikka, Finland).
At the minimum temperature, the energy dissipated in the resistor is less than 0.5 pW.
5.3 Theoretical considerations
In the tunneling model [4] the TLS was described by two parameters (Fig. 5.2), the asymmetry
parameter D and the overlap parameter l, which contains the barrier height, the distance
of the two minima, and the mass of the tunneling particle. They are related with the total energy
splitting E of the two levels and the tunneling matrix element D 0 by the relations
q
E ˆ 2 ‡ 2 0
and 0 ˆ k exp… † ; …1†
where kO is a typical zero point energy. One of the most important ingredients of the TLS
model is the distribution function characterizing the two parameters. The originally used assumption
for this function, which has also been applied to explain spectral diffusion [15,
17], is a flat distribution in both parameters, namely
P…;†dd ˆ Pdd:
…2†
Figure 5.2: Double minimum potential as a model for a TLS. All relevant parameters are shown in the
figure.
71

5 Spectral Diffusion due to Tunneling Processes at very low Temperatures
It is convenient, however, to use experimentally accessible quantities as variables, i. e.
the energy E and a dimensionless relaxation rate R, which can be defined as: R = r/r max =
D 2 0 /E 2 . The transformation to the new variables yields for the distribution function [25]:
1
P…E;R† ˆ P p : …3†
2R 1 R
A very important feature of this distribution is the fact that P (E, R) is independent of
E. In order to keep the total number of TLS finite, some cut-off value for R ? 0 has to be
introduced [3], which is usually denoted by R min .
It was shown in Ref. [15] that for optical transitions in glasses the TLS dynamics results
in spectral diffusion, which shows up in the experiment as a time and temperature-dependent
Lorentzian line broadening. The width of this Lorentzian line must be calculated by
averaging over the distribution of energies and relaxation rates P(E, R). It can be written as:
…t; T† ˆ2p2
3k C ij
Z Emax
E min
Z 1
dEn…T†
R min
dR E P…E;R†…1 exp‰ tr max RŠ† : …4†
Here AC ij S represents an average coupling constant between the TLS and the optical
transition, n (T) is a thermal occupation factor of the TLS states. For the system which is investigated
in this work, the parameter r max can be estimated from experimental data on ultrasonic
attenuation [26] to be on the order of 10 7 s –1 at a temperature of 0.100 K and even larger
at higher temperatures. Therefore, the condition t 7r max p 1 is well fulfilled on all time
scales exceeding several microseconds. In this limit the rate integration in Eq. 4 can be performed
analytically [27], which leads to the well-known logarithmic time dependence of
spectral diffusion:
!…t; T† ˆ p2
3k C
ij P k B T ln…r max t† : …5†
Equation 5 represents the theoretical prediction of the tunneling model for the time
and temperature-dependent broadening of spectral holes. This result, however, is the result
of a particular form of the density of tunneling states P (E, R), which is based on the a
priori assumption of Eq. 2. A uniform density of states in D is a physically reasonable
choice; the independence of l, however, is difficult to justify, since l consists of several
parameters [28]. The temperature dependence of spectral diffusion is dominated by D the
time evolution stems mainly from l. Therefore, these two predictions from Eq. 5 do not
have the same validity. In our experiments we have investigated time and temperature dependence
separately.
72

5.4 Temperature dependence
5.4 Temperature dependence
In these experiments, we investigated polystyrene (PS) doped with phthalocyanine and polymethylmethacrylate
(PMMA) doped with tetra-4-tert-butyl phthalocyanine. Both sample
materials have also been investigated by heat release and specific heat measurements [29].
The samples had optical densities of about 0.4 at a typical thickness of 3 mm. The samples
were prepared by bulk polymerisation of the solution of the dye in the monomer.
The observed hole shapes are Lorentzian with no detectable deviations in all cases. To
eliminate saturation effects at least 10 holes with different energies were burned at each temperature
and the homogeneous line width was determined by extrapolation to zero burning
fluence [30]. Due to their low relative depths the extrapolation was done assuming a linear
dependence of the hole area on energy [10]. A weak variation of the line width with light
power was observed at our minimum temperature of 0.025 K for PMMA. We attribute this
variation to sample heating.
The temperature dependence of the hole width is shown in Fig. 5.3 [23]. The results are:
a) both systems display a linear temperature dependence over the whole temperature range
from 0.500 K down to 0.025 K and no crossover to a constant line width is seen. The linearity
of the data plot shows that the density of states P (E, R) is indeed independent of the
TLS energy;
b) the relative magnitude of the line width in the two systems correlates with the respective
specific heat [31];
c) the extrapolated line width for T ? 0 is nearly the same for both systems, confirming
that in an amorphous system the line width reaches the lifetime limited value G h = 1/2 pT 1
(Heisenberg limit) of the electronic transition only when T ?0. This limit is reached in our
experiment within 10%!
Figure 5.3: Temperature dependence of the hole burning line width for PS and PMMA between
0.025 K and 0.500 K.
73

5 Spectral Diffusion due to Tunneling Processes at very low Temperatures
In the case of PS, where unsubstituted phthalocyanine was used as dye, the result for
the extrapolated line width G (T ? 0) is in very good agreement with the fluorescence lifetime
measurements of this molecule [32]. We are not aware of measurements of the fluorescence
lifetime of substituted phthalocyanine, but our results show that there is no major difference
between both dyes as far as their excited state lifetimes are concerned.
These results imply that in amorphous solids there are dynamical processes of twolevel
systems with an energy spectrum extending down to values as low as k B 70.025 K.
These low-energy excitations dominate the line widths of optical transitions even at low
temperatures and the lifetime limited value can only be reached by extrapolating to zero
temperature.
5.5 Time dependence
After the system has been cooled down spectral holes are burned immediately and after
some delay times, while the sample was at constant temperature, we observe a hole broadening
which depends on this delay time. This behaviour is demonstrated in Fig. 5.4. Dataset A
is the broadening of a hole which is burned immediately after the sample (phthalocyanine in
PMMA) has been cooled from liquid nitrogen temperature to 0.300 K. The curves show the
evolution of holes burned about one day, three days, and one week after cooling down. The
amount of spectral diffusion in a given time interval decreases continuously until a small residual
effect becomes observable, which is independent of the time delay after reaching the
final temperature. This is represented by dataset B in Fig. 5.4.
A
B
Figure 5.4: Time evolution of spectral holes burned at different times after sample cooling. Datasets
corresponding to A and B are investigated in the following.
74

5.5 Time dependence
Investigating the temperature dependence of the time evolution of the hole width [33],
we performed experiments on a PMMA sample doped with phthalocyanine at different temperatures
T, varying from 0.100 K to 1 K. For each of the temperatures T = 0.100, 0.300,
0.500, and 0.700 K, the respective experiments were carried out in a separate run. It took
about one hour to cool the system from 77 K to 4 K. The time for cooling down from 4 K
to the final temperature T depends slightly on T, but is in the range of about (6+1) hours.
Immediately after reaching T, holes are burned and their subsequent broadening is observed
for about one week. These experiments yield data corresponding to set A in Fig. 5.4.
The results of these measurements are shown in Fig. 5.5 [33]. In this plot the time origin for
each run is identical, namely the time at which the cooling down procedure from 77 K was
started. For better visibility, however, the data corresponding to T = 0.500, 0.300, and
0.100 K are shifted by 0.5, 1.0, and 1.5 orders of magnitude along the logarithmic time axis,
respectively. Except for the data taken at 0.500 K the total observation time was 6–10 days.
Figure 5.5: Broadening of holes burned immediately after reaching T. Data shifted for better visibility
(see text).
Keeping the samples at constant temperature new holes are burned after 1–2 weeks
and their broadening is observed for several hours, corresponding to set B in Fig. 5.4. These
data are shown in Fig. 5.6 [33]. Here the observation time is 2–16 hours. Note that the range
of the DG-axis is about 10 times less as compared with Fig. 5.5.
Comparing our data of Figs. 5.5 and 5.6, it is obvious that the measurements which are
started immediately after cooling down show no variation of the observed broadening with the
phonon bath temperature, while the residual behaviour observed many days later exhibits a
rather pronounced dependence on the experimental temperature. It has to be emphasized that
the data shown in Fig. 5.5 and the corresponding data in Fig. 5.6 for the same values of Twere
measured without any change of temperature in between the two subsequent experiments.
In spite of the variation of the final temperature over almost one order of magnitude,
all four datasets in Fig. 5.5 show the same hole broadening behaviour.
We attribute this to the existence of a non-equilibrium state of the TLS ensemble due
to the fast cooling procedure: The large amount of spectral diffusion observed after fast
75

5 Spectral Diffusion due to Tunneling Processes at very low Temperatures
Figure 5.6: Broadening of holes burnt after 1–2 weeks at low temperature.
cooling is due to the relaxation of TLS to their thermal equilibrium [34]. This relaxation
process is known to produce the thermal phenomenon of heat release [7]. Thus, it can be
concluded that we have established an important connection between optical and calorimetric
phenomena.
The data of Fig. 5.6 exhibit a pronounced dependence on temperature, as can be expected
from theoretical considerations outlined in Section 5.3 and from the non-time resolved
experiments of Section 5.4. The time evolution, however, shows a distinct non-logarithmic
behaviour in contradiction to the prediction of the tunneling model. For short times
the Fig. 5.6 shows clearly that the hole broadening starts fairly logarithmic. We have performed
transient hole burning experiments on the millisecond time scale, which also yielded
a logarithmic behaviour [35]. At longer times, however, the diffusional broadening increases
faster than logarithmic. This increase occurs later at lower temperatures, which is to be expected
from the temperature dependence of the relaxation rates. These new features are in
qualitative agreement with theoretical work that invokes strong coupling of TLS with phonons
[28, 36].
We believe that our experimental work shows that the method of spectral hole burning
spectroscopy is suitable for studying the low-temperature properties of amorphous solids.
Especially for the investigation of the long-time behaviour of TLS equilibrium dynamics,
where other methods cease to function, it is a very powerful experimental technique.
References
1. R.C. Zeller, R.O. Pohl: Phys. Rev. B, 4, 2029 (1971)
2. Amorphous Solids. Low Temperature Properties, in:W.A. Phillips (ed.): Topics in current Physics,
Vol. 24, Springer, Berlin, (1981)
76

References
3. S. Hunklinger, A. K. Raychaudhuri: in: D. F. Brewer (ed.), Progress in Low Temp. Physics,Vol. IX,
Elsevier Science, Amsterdam, p. 265 (1986)
4. P.W. Anderson, B.I. Halperin, C.M. Varma: Philos. Mag., 25, 1 (1972)
5. W.A. Phillips: J. Low Temp. Phys., 7, 351 (1972)
6. M.T. Loponen, R.C. Dynes,V. Narayanamurti, J.P. Garno: Phys. Rev. Lett., 45, 457 (1980)
7. J. Zimmermann, G. Weber: Phys. Rev. Lett., 46, 661 (1981)
8. B.M. Kharlamov, R.I. Personov, L.A. Bykovskaya: Opt. Commun., 12, 191 (1974)
9. A.A. Gorokhovskii, R.K. Kaarli, L.A. Rebane: JETP Lett., 20, 216 (1974)
10. J. Friedrich, D. Haarer: Angew. Chem., 96, 96 (1984); Angew. Chem. Int. Ed. Engl., 23, 113
(1984)
11. Th. Sesselmann, W. Richter, D. Haarer, H. Morawitz: Phys. Rev. B, 36(14), 7601 (1987)
12. A.P. Marchetti, M. Scozzafara, R.H. Young: Chem. Phys. Lett., 51(3), 424 (1977)
13. V.D. Samoilenko, N.V. Rasumova, R.I. Personov: Opt. Spectr., 52(4), 580 (in Russian) (1982)
14. P.M. Selzer, D.L. Huber, D.S. Hamilton, W.M. Yen, M.J. Weber: Phys. Rev. Lett., 36, 813 (1976)
15. T.L. Reinecke: Solid State Commun., 32, 1103 (1979)
16. J.R. Klauder, P.W. Anderson: Phys. Rev., 125(3), 912 (1962)
17. J.L. Black, B. I. Halperin: Phys. Rev. B, 16, 2879 (1977)
18. W. Breinl, J. Friedrich, D. Haarer: J. Chem. Phys., 81(9), 3915 (1984)
19. G. Schulte, W. Grond, D. Haarer, R. Silbey: J. Chem. Phys., 88(1), 679 (1988)
20. M.M. Broer, B. Golding, W.H. Haemmerle, J.R. Simpson, D.L. Huber: Phys. Rev. B, 33, 4160
(1986)
21. A. Gorokhovskii,V. Korrovits, V. Palm, M. Trummal: Chem. Phys. Lett., 125, 355 (1986)
22. Korrovits, M. Trummal: Proceedings of the Academy of Sciences of the Estonian SSR, 35(2), 198
(1986)
23. K.-P. Müller, D. Haarer: Phys. Rev. Lett., 66, 2344 (1991)
24. W.A. Bosch, F. Mathu, H.C. Meijer, R.W. Willekers: Cryogenics, 26, 3 (1985)
25. J. Jäckle: Z. Phys., 257, 212 (1972)
26. G. Federle: PhD thesis, Max-Planck-Institut für Festkörperforschung Stuttgart (1983)
27. S. Hunklinger, M. Schmidt: Z. Phys. B, 54, 93 (1984)
28. K. Kassner: Z. Phys. B, 81, 245 (1990)
29. A. Nittke, M. Scherl, P. Esquinazi, W. Lorenz, Junyun Li, F. Pobell: J. Low Temp. Phys., 98(5/6),
517 (1995)
30. L. Kador, G. Schulte, D. Haarer: J. Phys. Chem., 90, 1264 (1986)
31. K.P. Müller: PhD thesis, Universität Bayreuth (1991)
32. W.H. Chen, K.E. Rieckhoff, E.M. Voigt, L.W. Thewalt: Mol. Phys., 67(6), 1439 (1989)
33. H. Maier, D. Haarer: J. Lum., 64, 87 (1995)
34. S. Jahn, K.-P. Müller, D. Haarer: J. Opt. Soc. Am. B, 9, 925 (1992)
35. S. Jahn, D. Haarer, B.M. Kharlamov: Chem. Phys. Lett., 181, 31 (1991)
36. K. Kassner, R. Silbey: J. Phys. Condens. Matter, 1, 4599 (1989)
77

6 Optically Induced Spectral Diffusion in Polymers
Containing Water Molecules: A TLS Model System
Klaus Barth, Dietrich Haarer, and Wolfgang Richter
6.1 Introduction
In the past decade much experimental and theoretical work was done on spectral diffusion in
amorphous solids. It turned out that at low temperatures, optical dephasing phenomena [1]
as well as the numerous caloric data [2] can be well described by assuming the presence of
low-energy excitations, the so-called two-level systems (TLS). Photochemical hole burning
[3] and optical echo experiments [4] have provided experimental evidence for different dynamics
of optical transitions in glassy systems as compared to crystalline matrices. For
glassy organic solids the TLS concept was first proposed by Small and co-workers [5, 6]
and was later used by Reinecke [7] to explain low-temperature optical line widths. Especially
the photochemical hole burning data over long observation periods (observation times
longer than three months) at temperatures down to 0.050 K allowed a critical test of the theoretical
approach within the tunneling model [8]. Theoretical and experimental results to
this topic are also given in this book by H. Maier et al.
At temperatures above 1 K tunneling transitions and localized vibrations influence the
physical properties [9]. Both types of mechanisms have recently been incorporated in the
soft potential model [10] which contains the well-known tunneling model as a special case.
A microscopic interpretation of optical line broadening phenomena in terms of e. g.
local excitations of the matrix or switching of optically addressed chemical groups suffered
from the lack of experimental data. In the past, several experimental techniques have been
developed to shed light on the microscopic mechanisms of the phenomena. These techniques
are all based on the generation of phonons in the matrix material, especially in the neighbourhood
of the spectral probe. Heat pulses from external heaters [11] produce a broad distribution
of phonon frequencies inside the chromophore host system and permit the investigation
of the dynamics of barrier crossings and the coupling between the TLS and the dye
molecules [12]. Light-induced phonon generation in hole burning systems can in principle
be achieved either by exciting non-radiative decay processes in appropriate chromophores
[13] or by direct absorption of IR light in the matrix material [13, 14]. The non-radiative decay
method, which has been used to study transient and irreversible spectral diffusion processes,
yields also a broad distribution of phonon frequencies and gives rise to similar experimental
processes as e. g. the heat pulse technique. On the other hand, the IR absorption
78 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

6.2 Experimental setup for burning and detecting spectral holes
method where the sample is illuminated with narrow band IR light allows us to generate
phonons of high density and of a comparatively small energy distribution in the direct neighbourhood
of the optical probe. The contribution of the broad band black-body background
radiation can be minimized by using an appropriate intensity of a narrow band IR source.
For the optical investigation of amorphous systems, photochemical hole burning (PHB)
is a high-resolution technique and a powerful tool to examine local sites in polymers at low
temperatures. Dye molecules embedded in polymers at low concentration exhibit strongly inhomogeneously
broadened absorption bands in a conventional spectroscopic experiment. Selecting
an ensemble of these dye molecules with narrow band laser light, which gives rise to a
photoreaction, leads to a sharp spectral dip at the laser frequency, the so-called spectral hole.
Due to the high resolution, this method is appropriate for detecting small perturbations in the
matrix such as a transient change of the phonon distribution or permanent matrix rearrangements
in the vicinity of the dye molecules. Such rearrangements can be due to a configurational
change of parts of the polymer main chain or due to a reorientation of small molecules
(water), which may be embedded in the matrix in addition to the dye molecules.
In the following, two different experimental conditions are discussed which are based
on the absorption of IR light. Both experimental situations lead to a change of the local interaction
of the optical probe with its local matrix environment. In the first part the phonons
are treated as a time-dependent temperature bath. In the second part experiments are discussed
where local groups of the matrix are selectively addressed by IR radiation of very
low intensity. The observed enhanced spectral diffusion shows a characteristic dependence
on the IR frequency and coincides with only a few of the numerous IR absorption bands. A
quantitative description of this new process together with the identification of the resonant
vibrations is given within a simple kinetic model.
6.2 Experimental setup for burning and detecting spectral holes
The experimental setup for burning and detecting spectral holes with a narrow band laser is
described elsewhere [16]. The experiments were performed at 1.8 K in a bath cryostat in
which the sample was immersed in superfluid helium. As samples we used different polymeric
matrices such as polymethylmethacrylate (PMMA), polyamide (PA), polyethylene, polystyrene,
etc. doped with free base phthalocyanine (H 2 Pc) at low concentration (10 –2 mol%). Their
preparation is described in Ref. [16]. The water content of the samples has been controlled by
heating them under vacuum conditions. Because of its strong hydrophilic nature PMMA has
a high capacity of water absorption at atmospheric conditions. In order to investigate the influence
of the water molecules the experiments, described in Section 6.4, were carried out either
with a nearly water-free PMMA sample or with a PMMA sample containing water molecules
under equilibrium conditions. The optical windows of the cryostat consisted of crystalline
BaF 2 which allowed the study of hole burning spectra in the lowest vibrational band of the
electronic S 0 ? S 1 transition of H 2 Pc at about 0.69 mm as well as under IR illumination between
2 mm and 11 mm. In this setup, the broad band background radiation with an emission
79

6 Optically Induced Spectral Diffusion in Polymers Containing Water Molecules
maximum at 10 mm has an integrated intensity of about 300 mW/cm 2 at the location of the
sample. As IR radiation sources we used a Globar with a maximum emission intensity near
3 mm and different CO 2 laser lines around 10 mm.
An acousto-optic modulator was used for reducing the intensity of the CO 2 laser down
to 1 W/cm 2 and for switching with times some microseconds. The radiation of the Globar
was dispersed in a monochromator and with different dielectric filters to a bandwidth between
0.05 mm and 0.2 mm with typical intensities around 100 mW/cm 2 .
6.3 Reversible line broadening phenomena
In this Section the mechanisms of reversible line broadening phenomena under continuous
or pulsed irradiation conditions are discussed. Phonons are generated by illuminating the
polymeric sample with the light of a single CO 2 laser line. The wavelength of the CO 2 laser
was selected to generate high frequency phonons only within a small penetration depth of
less than 10 mm. The dye molecules, which are homogeneously distributed throughout the
sample thickness of some 100 mm, are a local probe for the increase in the phonon density
during the IR irradiation. Due to the thermalisation processes, phonons show a broad energy
distribution. This change of the phonon distribution gives rise to an increased effective temperature
under cw irradiation conditions. Because of the high cooling rate of the surrounding
superfluid helium, typical irradiation intensities of about 1 W/cm 2 were used to obtain a
significant increase in the phonon density. The influence on the homogeneous line width of
the dye molecules can be investigated in a zero burning fluence experiment, i. e. the limit of
the hole width is given by extrapolating to zero burning fluence. The lower curve in Fig. 6.1,
which was measured without any IR irradiation, yields an extrapolated value for the line
Figure 6.1: Zero burning fluence experiment under different phonon density conditions in polyamide
doped with H 2 Pc (see text).
80

6.3 Reversible line broadening phenomena
width of about 375 MHz. The two upper curves show increased hole widths due to an enhanced
phonon generation either by IR irradiation during the burning process only(triangles)
or during the detection process only(circles).
The profile of a spectral hole can be written as
Z 1 1
A…! ! L †/
nz…! 0 ! L †z…! ! 0 †d! 0 ; …1†
where nz(o' – o L ) is the site distribution function of the molecular line contributing to the
hole spectrum and z(o – o') is the absorption profile of a single molecule.
On the time scale of this experiment (up to some minutes) the spectral hole profiles
are mainly reversible (see also Fig. 6.2 and Fig. 6.3). Therefore, the homogeneous line profile
function z(o) is much more affected by the altered phonon density than the site distribution
function nz(o). Thus, it is obvious that the two upper curves in Fig. 6.1 are similar and
give for the hole width nearly the same zero burning fluence value of 440 MHz. Both curves
are well separated from the lower curve which was obtained without any additional phonon
generation. Applying an effective temperature model the difference between these two zero
burning fluence values corresponds to a temperature change of about 0.3 K within the sample.
Because of the strongly inhomogeneous experimental conditions during phonon generation
and phonon diffusion through the sample, however, each different subensemble of dye
molecules experiences a different distribution in phonon energies and, as a consequence, a
bath description in terms of one single temperature is not appropriate. By using different op-
Figure 6.2: Lower part: Spectral profile of a photochemical hole with and without phonon generation
during the detection process. Upper part: Directly measured difference between these two profiles as
obtained by modulating the IR light at a frequency of 150 Hz.
81

6 Optically Induced Spectral Diffusion in Polymers Containing Water Molecules
Figure 6.3: Decay of the transmission maximum during a laser pulse.
tical energies for the phonon generation, it is in principle possible to study the contribution
of different matrix vibrations on the dephasing of a single absorber function z(o).
The time scale of the hole burning experiments discussed above is several minutes. It
is about 3 orders of magnitude larger than the typical thermal relaxation time of the sample
as given by
ˆ d2 c
; …2†
where d is the dimension (thickness) of the sample, c the specific heat, r the mass density,
and k the thermal conductivity. For the temperature of our experiment (1.8 K) and a typical
sample dimension of 1 mm the thermal relaxation time t is calculated to 10 ms.
A consequence of this short relaxation time is shown in Fig. 6.2 and Fig. 6.3. The two
spectral holes in the lower part of Fig. 6.2 were obtained with and without phonon generation
during the hole detection process. A modulation of the IR light during the detection
process results in a continuous switching between these two hole profiles. In the upper part
of Fig. 6.2, a modulation frequency of 150 Hz was used during the scan of the optical frequency.
The switching time between the two hole profiles is of the same order of magnitude
as the thermal relaxation time. The continuous switching between the two hole profiles is
detected with lock-in technique. The result shows that the phonon generation and relaxation
process is sufficiently fast to follow the 150 Hz modulation of the IR light.
A time-resolved experiment is shown in Fig. 6.3. The optical transmission at the peak
of a hole spectrum was monitored while the sample was being illuminated with a single
CO 2 laser pulse of 600 ms duration. The decay of the transmission signal cannot be described
with a single exponential function. It is well represented by the superposition of two
exponentials with a fast contribution t 1 & 10 ms and a slow contribution t 2 & 150 ms. The
fast contribution is close to the thermal relaxation time at the temperature of the helium
bath. The slow contribution contains all the inhomogeneous experimental conditions as mentioned
above, in particular a temperature gradient inside the sample that gives rise to a variation
of the thermal relaxation time. Since the amplitude of the fast contribution is larger by
a factor of 4, the switching time between the two curves in Fig. 6.2 is mainly determined by
the fast contribution t 1 .
82

6.4 Induced spectral diffusion
The experimental results of this Section show that optically generated phonons can be
used to study the transient broadening of the optical line shape of a single absorber. In principle,
the dependence on the phonon frequency can be studied in such an experiment. In the
same experiment the time-resolved dynamics of phonon diffusion and phonon decay in polymeric
systems can be investigated via the dephasing mechanism of the optical probe.
6.4 Induced spectral diffusion
In this Section we will demonstrate that the resonant absorption of IR light in a polymeric
system can also lead to irreversible line broadening phenomena. Even at a very low irradiation
intensity (a typical value is 100 mW/cm 2 ) the investigation of specific types of relaxation
processes is possible. Such low irradiation intensities do not increase the temperature of
the matrix by more than 0.01 K and hence yield no measurable dephasing effect via a
change of the phonon distribution. Because of the irreversible nature of the optical line
broadening, such phenomena are usually ascribed to spectral diffusion processes. The IR induced
spectral diffusion is caused by addressing specific moieties within the matrix via the
excitation of local vibrations. In the following weakly bound water molecules are studied.
When a change in the spatial configuration takes place during the relaxation process of a
water molecule the optical transition of a neighbouring dye molecule will be affected. Optical
absorbers in glasses are therefore very sensitive probes for local rearrangements and are
suited for a microscopic study of this topic.
Due to the hydrophilic nature of the matrix the investigated guest host system, PMMA
doped with H 2 Pc, contains a comparatively large number of water molecules (up to one volume
percent). The experimental procedure is the following: A spectral hole is burned and its
spectrum is repeatedly recorded while the sample is being exposed to IR radiation of a certain
wavelength at a constant irradiation intensity. Figure 6.4 shows the typical increase of the hole
Figure 6.4: Time evolution of the hole broadening induced by different IR wavelengths. From top to
bottom: 2.80 mm, 2.86 mm, 2.90 mm, 2.96 mm, and 3.04 mm.
83

6 Optically Induced Spectral Diffusion in Polymers Containing Water Molecules
Figure 6.5: Transmission spectra of PMMA without (curve A) and with (curve B) natural water content.
The quotient C = B/A shows the H 2 O absorption lines. Dependence of the hole broadening on the
IR wavelength after t 0 = 35 min (curve D) due to induced spectral diffusion.
width and the strong dependence of this behaviour on the IR wavelength. To illustrate the pronounced
spectral selectivity, one value of each hole broadening curve at t 0 = 35 min is plotted
in Fig. 6.5 (curve D) versus the corresponding IR wavelength. The data points do not simply
reflect the absorption behaviour of the sample. Although the PMMA matrix has many absorption
bands between 2 mm and 11 mm (curve B in Fig. 6.5), there are only two distinct wavelengths
which give rise to induced spectral diffusion processes.
The small concentration of water molecules in the matrix gives rise to non-saturated
absorption bands near 2.8 mm and 6.1 mm. They correspond to the fundamental stretching
and bending vibrations of H 2 O, respectively. The difference spectrum (curve C in Fig. 6.5)
of water free and water saturated PMMA yields the exact position of the H 2 O absorption
bands, which are in good agreement with the resonances found in our experiment. On an expanded
scale in Fig. 6.5, even the asymmetric shape of the inhomogeneous H 2 O band is reflected
by the experimental data. Replacing protonated with deuterated water yields a red
shift of the IR resonances which also agrees very well with the corresponding absorption
spectra of heavy water.
In analogy to the tunneling model, which is based on the assumption of two potential
minima, we assume two stable sites for each water molecule (Fig. 6.6). Since the experimental
results indicate that a permanent change in the matrix occurs transitions between the two
sites are only allowed via the first excited vibrational states. Using such a simple four-level
model for the underlying kinetics, we are able to explain the temporal behaviour of the induced
spectral diffusion in a quantitative fashion.
84

6.4 Induced spectral diffusion
Figure 6.6: Energy level scheme of a H 2 O molecule in a two-site model.
The two ground states are labelled 1 and 2, the first vibrationally excited states 3 and
4, respectively. The IR induced and the spontaneous transition rate for each site are denoted
by the Einstein coefficients B and A, respectively. In our notation, B is proportional to the irradiated
IR intensity. Transitions between the two sites are denoted by the conversion rate k.
The transitions shown in Fig. 6.6 lead to a system of 4 coupled linear rate equations.
Since we used in our experiments very low intensities of the IR light, we consider only the
limit B P A where the population of the excited levels 3 and 4 is negligible at all times. The
result is a time-dependent number of flips between the two ground states n f = n 1?2 + n 2?1 .
Only those flips are taken into account which contribute to a change in the total configuration
of all water molecules with respect to the time t = 0 of the hole burning process.
According to Reinecke [7] the width of the spectral diffusion kernel Do is proportional
to the number of flips n f . Though using the approximation that each flip yields the
same contribution to Do, when taking into account the decrease of the IR intensity across
the sample thickness as well as a random orientation of the H 2 O molecules, the result for
Do can only be calculated numerically. Using a Taylor expansion, however, a very good analytical
approximation of the width of the induced spectral diffusion is given by
!…t† ˆG N
0
4pk 1 1 ‡ 2Bp !
f t b
: …3†
b
The difference between this formula and the numerical result is less than 5%. p f is the
flip probability of a single H 2 O molecule as defined by p f = k /(A + k), b is a dimensionless
geometry parameter of about 2, N 0 is the density of the H 2 O molecules and G is the coupling
constant between dye and water molecules. Equation 3 describes very well the dependence
of the broadening of a spectral hole on the number of absorbed IR photons (Fig. 6.4).
It is possible to perform a test of the model. Equation 3 contains only two fitting parameters,
the flip probability p f , and the coupling constant G. These two parameters can be obtained
from a single experiment.
An example is given in Fig. 6.7. One set of the experimental data (filled circles) is obtained
at an IR intensity of 80 µW/cm 2 . The solid line is a fit according to Eq. 3 and yields
a flip probability of 18% and a coupling constant of G = 3.6 7 10 –44 Jcm 3 .(Note: the slight
difference with respect to the data given in reference [17] is due to the extension of the
model by including the random orientation of the water molecules and the decrease of the
IR intensity according to Beer’s law.) With these two values it is possible to calculate the
85

6 Optically Induced Spectral Diffusion in Polymers Containing Water Molecules
Time (s)
Figure 6.7: Test of the model. Hole broadening versus IR irradiation time for three different IR intensities.
The solid line is a fit, the dashed lines are calculated according to Equation 3.
time dependence of the induced spectral diffusion for any other irradiation intensity. The
dotted lines in Fig. 6.7 are calculated with Eq. 3 for the obtained values and the intensities
230 µW/cm 2 and 20 µW/cm 2 . The experimental results under these conditions, represented
by the squares and triangles, are in very good agreement with the theoretical predictions.
The high quantum yield of 18% for the flip of a single water molecule becomes obvious
by a comparison of the number of absorbed IR photons (typically 10 14 s –1 ) with the
total number of water molecules (about 10 18 ) in the sample volume. Since a strong induced
spectral diffusion is observed within several minutes, a local reorientation process with a
high quantum yield must be involved.
The coupling strength between water and dye molecules can be estimated by taking
into account the static dipole moment of the H 2 O molecule. A reorientation of the H 2 O molecules
affects the dye molecules by the concomitant change in the local electric field at the
location of the chromophores. Assuming a mean distance of 15 Å and inserting data from
Stark effect experiments [18], one obtains a value of approximately 2710 –44 Jcm 3 for the
coupling constant which is very close to our fitted value. Together with the resonant phenomena
discussed above, this yields a detailed microscopic understanding of this special
kind of interaction.
We can conclude that spectral diffusion induced by the absorption of IR photons in
the frequency range 2–11 µm is a resonant process. Its microscopic origin is a spatial rearrangement
of weakly bound water molecules after vibrational excitation. Assuming a configurational
model with only two possible sites for each water molecule and applying the theoretical
model of spectral diffusion by Reinecke [7], an analytical description of the resonant
hole broadening behaviour can be obtained. A variation of the IR irradiation intensity shows
that the model description yields excellent agreement with the experimental data.
86

References
References
1. S. Jahn, K. P. Müller, D. Haarer: J. Opt. Soc. Am. B, 9, 925 (1992)
2. M. Deye, P. Esquinazi: Zeitschr. für Phys. B, 39, 283 (1989)
3. J. Friedrich, D. Haarer: Angew. Chem., Int. Ed. Engl,. 23, 113 (1984)
4. C. A. Walsh, M. Berg, L. R. Narashiman, M. D. Fayer: Chem. Phys. Lett., 130, 6 (1986)
5. J. M. Hayes G. J. Small: Chem. Phys., 27, 151 (1978)
6. G. J. Small: Persistent nonphotochemical hole burning and the dephasing of impurity electronic
transitions in organic glasses, in: V. M. Agranovich R. M. Hochstrasser (eds.): Spectroscopy and
Excitation Dynamics of Condensed Molecular Systems, North-Holland, Amsterdam, p. 515 (1983)
7. T. L. Reinecke: Sol. Stat. Com., 32, 1103 (1979)
8. H. Maier, D. Haarer: J. Lumin., 64, 87 (1995)
9. R. Greenfield, Y. S. Bai, M. D. Fayer: Chem. Phys. Lett., 170, 133 (1990)
10. D. A. Parshin: Phys. Sol. State, 36, 991 (1994)
11. U. Bogner: Phys. Rev. Lett., 37, 909 (1976)
12. T. Attenberger, K. Beck, U. Bogner: in: S. Hunklinger, W. Ludwig, G. Weiss (eds.): Proc. of the
3rd Int. Conf. on Phonon Physics, p. 555
13. A. A. Gorokhovskii, G. S. Zavt, V. V. Palm: JETP Lett., 48, 369 (1988)
14. W. Richter, Th. Sesselmann, D. Haarer: Chem. Phys. Lett., 159, 235 (1989)
15. W. Richter, M. Lieberth, D. Haarer: JOSA B, 9, 715 (1992)
16. G. Schulte, W. Grond, D. Haarer, R. Silbey: J. Chem. Phys., 88, 679 (1988)
17. K. Barth W. Richter: J. of Lumin., 64, 63 (1995)
18. R. B. Altmann, I. Renge, L. Kador, D. Haarer: J. Chem. Phys., 97, 5316 (1992)
87

7 Slave-Boson Approach to Strongly Correlated Electron
Systems
Holger Fehske, Martin Deeg, and Helmut Büttner
7.1 Introduction
The problem of understanding high-temperature superconductivity has been a challenge to
theoreticians from a wide variety of fields. Many theoretical investigations have been carried
out in order to identify the mechanism of this fascinating phenomenon as well as to establish
the canonical model itself [1]. Although excellent progress is being made in deducing
a consistent description of the high-temperature superconductivity systems from a
first-principles theory, at present no microscopic theory can account for their unconventional
normal-phase data in its entirety. The interplay of charge and spin dynamics in the
normal state seems to hold the key to the understanding of the physical mechanism behind
high-temperature superconductivity in the cuprates. Both macroscopic measurements of
transport and magnetic properties as well as microscopic measurements probing the charge
and spin excitation spectra are fundamental in establishing the anomalous normal-state
properties of these materials [1].
The nature of spin excitations of the high-temperature superconductivity cuprates has
been experimentally studied by means of nuclear magnetic/quadrupole resonance (NMR/
NQR) and inelastic neutron scattering (INS) techniques clarifying the persistence of strong
antiferromagnetic (AFM) correlations in the normal and superconducting states [2]. Detailed
investigations of the wave-vector dependence of the low-frequency spin fluctuation spectrum
have revealed remarkable differences between the YBa 2 Cu 3 O 6+x and La 2–x Sr x CuO 4 families
at low doping level x; the dynamic structure factor S (~q, o) keeps its maximum at (p,p) in
the YBCO system, while in LSCO, the peaks are displaced from the commensurate position
to the four incommensurate wave-vectors p(1 +q 0 , 1), p (1,1+q 0 ), where q 0 F2x. On the
other hand, the analysis of the NMR data shows that the relaxation rates on the planar Cu
sites are similar in all materials.
Two striking features are associated with 63 T 1 –1 :
a) as a result of strong local spin fluctuations on 63 Cu sites it is enhanced by one order of
magnitude over the oxygen rate 17 T 1 –1 ; and
b) in sharp contrast to the Korringa-like behaviour at the planar 17 O sites, its temperature
dependence does not follow the Korringa law [2].
88 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

7.1 Introduction
The physical origin of the contrasting ~q-dependence of the spin fluctuation spectrum
is still under discussion. Millis and Monien [3] have argued that the spin dynamics and, in
particular, the temperature dependence of the spin susceptibility w s (T) in LSCO are caused
by a spin density wave instability, whereas in the YBCO family they are due to in-plane
AFM fluctuations and a novel non-Fermi liquid spin singlet pairing of electrons in adjacent
planes. On the other hand, the magnetic properties are intimately related to the energy band
dispersion of the non-interacting system within a Fermi liquid based framework, i. e., in this
way the observed spin dynamics can be attributed to different Fermi surface (FS) geometry
of LSCO-type and YBCO-type, respectively. Along this line, details of the spin fluctuation
spectrum are studied using a nearly antiferromagnetic Fermi liquid approach by Monthoux
and Pines [4]. From a more microscopic point of view, the important effects of FS shape on
the magnetic properties were confirmed by Si et al. [5] within a large Coulomb-U auxiliary
boson scheme and by Fukuyama and co-workers [6] on the basis of a resonating valence
bond (RVB) slave-boson mean-field approach to the extended t-J model. Furthermore, Ito et
al. [7] have recently reported that the charge transport in the CuO 2 plane is determined by
dominant spin scattering, i. e., the spin dynamics are manifest in the extraordinary transport
properties of the high-temperature superconductivity cuprates as well.
Encouraged by these findings it is the aim of this report to study magnetic and transport
phenomena of high-temperature superconductors probably in terms of the most simple
effective one-band model describing both correlation and band structure effects, the socalled
t-t'-J model:
H t t 0 J ˆ t X hi;ji;
~c y i ~c j t 0 X
h
hi;jii;
~c y i ~c j ‡ J X ij h i
~n i ~n j
~S i
~S j
4
: …1†
H t–t'–J acts in a projected Hilbert space without double occupancy, where ~c …y†
c …y†
i …1 ~n i † is the electron annihilation (creation) operator, ~S i ˆ 1
P
0 ~cy i ~ 0 ~c i
i =
0;
2
and
~n i ˆ P ~cy i ~c i. J measures the AFM exchange interaction, t and t' denotes hopping processes
between nearest-neighbour (NN; Ai,jS) and next nearest-neighbour (NNN; AAi,jSS) sites
on a square lattice. Compared to the original t-J model the t'-term incorporates several important
effects near half-filling. Starting from a rather complex three-band Hubbard or Emery
model [8] for the CuO 2 planes, quantum cluster calculations [9] have revealed that the
relative large direct transfer between NN oxygen sites (t pp < t pd /2) leads to a sizeable NNN
hopping t' in the context of an effective one-band description. More recently the t'-term has
been introduced to reproduce the FS geometry observed in ARPES experiments [6, 10–12].
Fitting the quasi-particle dispersion relation
" ~k ˆ 2t…cos k x ‡ cos k y † 4t 0 cos k x cos k y ; …2†
involved in Eq. 1 to experimental and band theory results yields t in the order of 0.3 eV and,
e. g., for the case of YBCO, t'&–0.4 t [4, 5, 13]. Moreover, Tohyama and Maekawa [12]
have emphasized that a t-t'-J model with t' > 0 can be used to describe the electron-doped
systems, e. g. Nd 2–x Ce x CuO 4 (NCCO). In this case one has to shift the momentum ~ k ? ~ k +
(p,p) [12], i. e., within a band-filling scenario one obtains a hole pocket-like FS centred at
(p,p)-point which shrinks with increasing doping [14]. Finally, as pointed out by Lee [15],
89

7 Slave-Boson Approach to Strongly Correlated Electron Systems
in a locally AFM environment doped holes can propagate coherently only on the same sublattice
without disturbing spins. Therefore, the t'-term coupling the same sublattice becomes
crucial for the low-lying magnetic excitations. This clearly is a correlation effect related to
the NNN hopping processes.
7.2 Slave-boson theory for the t-t'-J model
Apart from numerical techniques, the slave-particle methods have been employed extensively
in studying the effects of electron-electron correlation in the (extended) t-J model [6, 16–21].
The two commonly used approaches, the NZA slave-boson [16, 17] and slave-fermion [18]
schemes, however, yield quite different results concerning the (mean-field) ground-state phase
diagram and spin/charge excitations for this model [22]. As yet, the relationship between both
types of slave-field theories is not well understood. Within the NZA formulation, for example,
the Hamiltonian can be solved by a mean-field approximation with the (uniform) RVB order
parameter. A serious difficulty of this approach is the absence of AFM correlations [18, 22],
e. g., in the half-filled case, the lowest energy state, the energy of which is considerable higher
than numerical estimates indicate, does not satisfy the Marshall sign rule and fails to show the
expected long-range Néel order [23]. On the other hand, the mean-field slave-fermion schemes
[18] are known to give reasonable results for the spin susceptibility as well as for the spin correlation
length [21, 22], but also suffer from neglecting important correlation effects, especially
the fermion charge degrees of freedom are not described sufficiently well. In contrast, the fourfield
slave-boson (SB) technique, introduced by Kotliar and Ruckenstein (KR) [24] in the context
of the Hubbard model, has the advantage of treating spin and charge degrees of freedom on
an equal footing. Starting from the scalar KR SB representation of the Hubbard model, one may
generate an intersite exchange interaction via a loop expansion in the coherent state functional
integral [19]. However, the effective t-J Lagrangian, derived in this way, contains only an Ising
interaction term, i. e., important spin-flip exchange processes are neglected. As we shall see below,
to bosonize the complete exchange interaction term one has to use the spin-rotation-invariant
(SRI) extension of the KR SB theory from the beginning [25, 26].
7.2.1 SU(2)-invariant slave-particle representation
For the sake of definiteness, we return to the extended t-J Hamiltonian (Eq. 1), that may be
cast into the form
H t t 0 J ˆ X
t ij
~ y i
i;j
~ j
J
4
X
hiji
~ y i
~ i ~ y
j ~
j : …3†
90

7.2 Slave-boson theory for the t-t'J model
In order to bring out the SU(2) symmetry of the system, we have used in Eq. 3 a spinor
representation, where the one-row [one-column] matrices ~ y i ˆ…~ y i ; ~ i †‰ ~ i Š are built up
by the projected fermion creation [annihilation] operators ~ y i ~cy i ‰ ~ i ~c i Š: ‰ Š denotes
the contravariant [covariant] four-component vector (m =0,x, y, z) of Pauli’s matrices.
To preserve SRI, we apply to H t-J the manifest [SU(2) 6 U(1)] invariant SB scheme [26,
27] based on the SRI SB approach developed for the Hubbard model by Li et al. [28]. Accordingly,
we define scalar boson fields e ({) i and bosonic matrix operators p ({) i (representing
empty and singly occupied sites, respectively) and pseudofermion spinor fields C ({) i in the
following way:
j0 i i ˆ e y i jvaci;
j i i ˆ X y
i py i vac
j i: …4†
The unphysical states in the extended Fock space of pseudofermionic and bosonic
states are eliminated by imposing two sets of local constraints:
C …1†
i
ˆ e y i e i ‡ 2Trp y i p i
1 ˆ 0 ; …5†
expressing the completeness of the bosonic projectors, and
C …2†
i
ˆ i
y
i ‡ 2p y i p i
o ˆ 0 ;
…6†
relating the pseudofermion number to the number of p-type slave-bosons (hereafter underbars
denote a 262 matrix in the spin variables). Obviously, double occupancy has been projected
out. In the transformation of the fermionic spinor fields analogous to [26, 28],
~ i ! z i i ; …7†
the non-linear bosonic hopping operators
z i ˆ‰ o 2p y i p i Š 1=2 e y i ‰1 ‡ ey i e i ‡ 2Trpy i p i Š1=2 p i
‰…1 e y i e i † o 2 ~p y i ~p i Š 1=2 …8†
yield a correlation-induced band renormalization where ~p ({)
irr' = rr'p ({)
i,–r',r. Exploiting the
SU(2),O(3) homomorphism, the matrix operators p i
may be decomposed into scalar (singlet)
p o ({) and vector (triplet) ~p i =(p ix , p iy , p iz ) components as
p …y†
i
ˆ 1
2
X
p …y†
i ;
…9†
where the p im obey the usual Bose commutation rules ‰p i; p y j 0Šˆ ij 0. Consequently, the pseudofermions
i ˆ… i; i † T satisfy anticommutation relations of the form f i ; y
j 0gˆ ij 0.
91

7 Slave-Boson Approach to Strongly Correlated Electron Systems
This way the interaction term is converted due to
~ y i ~ i ! 2Trp y i p i
; …10†
where, more explicitly, the locally defined particle number and spin operators are just
~n i ! ~n i …p i †ˆ2Trp y p ˆ P p y p ;
i i i i
~S i ! ~S i …p i †ˆTrp y i
~p i ˆ 1
2 …py io ~p i ‡ ~p y i p io i~p y i ~p i † : …11†
Actually, the components of the bosonized SB spin operator act as generators of rotations
in spin space, i. e., ~S i …p i † satisfies the spin algebra. Since C (1) i and C …2†
i commute with
the SB t-J Hamiltonian, the constraints in Eq. 5 and Eq. 6 can be ensured by introducing the
time-independent Lagrange multipliers …1†
i and …2†
i
ˆ P
…2†
i .1 As a consequence, the
Hamiltonian of the t-J model (Eq. 3) in terms of the slave-boson and pseudofermion operators
has to be replaced by
H SB
t t 0 J ˆ P
t ij
i;j
‡ P i
y
i zy i z j j
… …1†
i C …1†
i
J P
Trp y i p i
Trp y j p j
hiji
‡ Tr …2†
i C …2†
i † : …12†
In the physical subspace, H SB
t–J possesses the same matrix elements for the basis states
(Eq. 4) as the original t-J Hamiltonian (Eq. 3) for the purely fermionic states. To verify this
directly, special attention has to be paid to the bosonization of AFM exchange interaction
term
~ y
i ~ i
~ y j ~
j ˆ 2 P …~c y i ~cy j ~c j ~c i
~c y i ~cy j ~c j ~c i † ; …13†
which includes besides the Ising exchange contributions # i " j $
#i " j the spin-flip processes
# i " j $ "i # j . Let us emphasize that the matrix elements of the spin-flip terms
are not reproduced in the scalar KR SB approach [19]. By contrast, within the SRI SB approach
it is a straightforward exercise to show that these contributions can be expressed in
terms of the p ({) irr' :
X
p y i py j 0 p j p
1
0 i i j ˆ
4
i j :
0
…14†
Thus our SRI SB scheme (Eqs. 4–12) provides a consistent bosonization of the extended
t-J model.
1 Note that additional constraints do not exist. Especially, if the constraint ~p i ~p i ˆ 0 is added [25] the
spin algebra is not satisfied.
92

7.2 Slave-boson theory for the t-t'J model
7.2.2 Functional integral formulation
To proceed further, it is convenient to represent the grand canonical partition function for
the redefined SB Hamiltonian (Eq. 12) in terms of a coherent-state functional integral [29]
over Grassmann fermionic and complex bosonic fields as
Z
R
Z ˆ D‰ ; ŠD‰e ;eŠD‰p ; p Š d‰ …1† Š d‰ …2†
dL…†
Š e 0 ; …15†
L…† ˆP
e i …@ ‡ …1†
i †e i ‡ 2Trp T ‰…@ ‡ …1†
i † o …2†T
i
‡ P
Trp ~p i i Trp~p Trp p j j i i Trpp j j
hi;ji
‡ P
i …@ † o ‡ …2†
i i ‡ P t ij
i z i z j j :
i
i;j
i
i
Šp T i
…1†
i
…16†
Apart from the above symmetry considerations the SB functional-integral formalism
reveals additional global and local gauge invariances [25, 30–32].
The action S ˆ R
0
dL…† is invariant under the following site and time-dependent
phase transformations (y i (t)w i
(t)), i. e., under the local symmetry group SU(2) 6U(1),
i ! e ii e i i i ;
e i ! e i e i i
; …17†
p i
! p i
e i i ;
provided that the (five) Lagrange parameters act as time-dependent gauge fields, l (1) i (t) and
…t†, absorbing the time derivatives of the phase factors:
l …2†
i
…1†
i
! …1†
i ‡ i _ i ;
…2†
i
! e i i …2†
i e i i i _ i
‡ i _ i o : …18†
Next, in the continuum limit, the radial gauge is introduced by representing the Bose
fields by modulus and phase,
e i ˆ je i je i'…ei† ;
p i ˆ 1
2
X
p i
e
i …p † i ;
…19†
93

7 Slave-Boson Approach to Strongly Correlated Electron Systems
(for a time-discretized version of this gauge fixing, see [31]). Exploiting the gauge freedom
of the action, we can now fix the five real-valued coefficients y i (t), w im (t) to remove five
phases (j i (t), f im (t)) of the Bose fields e i , p im in the radial gauge. As a consequence, all
the Bose fields {e i , p im } become real, in contrast to the Hubbard model, where one SB field
remains complex [25, 32]. At this point one should notice that the particle number and spin
operators are changed into ~n i …p i †ˆP p2 i ; ~S i …p i †ˆp io ~p i (the previous notations e i , p io
and ~p i now denote the radial parts of the corresponding Bose fields). Then, using the familiar
identity for Gaussian integrals over Grassmann fields [29],
R
D‰ ; Š e ‰ G 1 Š 0
ˆ e Tr ln‰ G 1Š ; …20†
the fermionic degrees of freedom can be integrated out and we obtain the following exact representation
of the grand canonical partition function
Z ˆ R D‰Š e S ef f
…21†
in terms of the real-valued bosonic fields f ia ,
i …† ˆ… i …†† ˆ …e i ;p io ; …2†
io ;…1† i
; p ix ; …2†
ix ; p iy; …2†
iy ; p iz; …2†
iz † :
…22†
In Eq. 21 the effective bosonic action S eff takes the form
(
S ef f ˆ R d P
0 i
…1†
i
e 2 i ‡ P
…1†
i
…2†
io
p 2 i
2p io ~p i ~ …2†
i
…1†
i
‡J P )
1
p io ~p i ~p j p jo
hiji
4 …p2 io ‡ ~p2 i †…p2 jo ‡ ~p2 j †
h
i
Tr ij; 0 ; 0 ln G ij; 1 † ; …23†
where the (inverse) SB Green propagator is given by
h
Gij; 1 0…; 0 †ˆ @ ‡ …2†
io
0
i
~ …2†
i ~ 0 ij … 0 †
t ij …z y i z j† 0 ; 0…1 ij† : …24†
It may be remarked that since the bosons are taken to be real, their kinetic terms,
being proportional to the time derivatives in Eq. 16, drop out due to the periodic boundary
conditions imposed on Bose fields (f ia (b) =f ia (0)). Strictly speaking it follows from this
property that all the Bose fields do no longer have dynamics of their own [25].
94

7.2 Slave-boson theory for the t-t'J model
7.2.3 Saddle-point approximation
The evaluation of Eq. 23 proceeds via the saddle-point expansion 2 , where at the first level
of approximation we look for an extremum S … i † of the bosonized action S eff with respect
to the Bose and Lagrange multiplier fields f ia :
@S ef f
@ i
ˆ!
0 ) S ˆ S ef f
iˆ i
: …25†
The physically relevant saddle-point F i is determined to give the lowest free energy
(per site)
f t t 0 J ˆ =N ‡ n ; …26†
where at given mean electron density n =1–d, the chemical potential m is fixed by the requirement
n ˆ 1 @
N @
…27†
( O ˆ S= denotes the grand canonical potential). Obviously, an unrestricted minimization of
the free energy functional is impossible for an infinite system. To keep the problem tractable,
we use the ansatz
~m i ˆ m~u i ; ~u i ˆ…cos ~Q ~R i ; sin ~Q ~R i ; 0† …28†
for the local magnetization ~m i ˆ 2 ~S i ˆ 2p io ~p i . Following earlier analyses of spiral states
for the Hubbard model [34] the unit vector ~u i is chosen as a local spin quantization axis
pointing in opposite directions on different sublattices. Thereby, the order parameter wavevector
~Q is introduced as a new variational parameter to describe several magnetic ordered
states: PM, FM ( ~Q = 0), AFM ( ~Q =(p,p)), and incommensurate (1,1)-spiral ( ~Q =(Q, Q)),
(1,p)-spiral ( ~Q =(Q,p)) and (0,1)-spiral ( ~Q = (0,Q)) states. Note that since fluctuations of
the charge density are not incorporated the scalar Bose fields are homogeneous: e i = e, p io =
p o , l (1) i = l (1) , and l (2) io = l (2) o . The vector fields exhibit the same spatial variation as the magnetization:
~p i ˆ p~u i and l ~ …2†
i ˆ l …2† ~u i . Then transforming Eq. 23 in the ( k; ~ o)-representation
the trace in S eff can be easily performed to give the free energy functional
2 Mean-field approximations to the functional integral are achieved by replacing the bosonic fields by
their time averaged values. For the Hubbard model, a comparison of the paramagnetic and antiferromagnetic
SB solutions with quantum Monte Carlo results shows that such a mean-field like approach
yields an excellent quantitative agreement for local observables [33] and therefore can give a qualitative
correct picture with relatively little effort.
95

7 Slave-Boson Approach to Strongly Correlated Electron Systems
where
f t t 0 J ˆ …1† …e 2 ‡ p 2 o ‡ p2 1† o …2† …p2 o ‡ p2 † 2 …2† p o p ‡ n
‡J p 2 1
o p2 …cos Q x ‡ cos Q y †
2 …p2 o ‡ p2 † 2
‡ 1
N
X
ln‰1 n ~k Š ; …29†
~k
n ~k ˆ‰expf…E ~k
†g ‡ 1Š
1
…30†
holds. The renormalized single-particle energies E ~k … ˆ†are obtained by diagonalizing
the kinetic part of Eq. 23 [26]. Requiring that f t-J be stationary with respect to the variation of
the magnetic order vector, the wave-vector ~Q can be obtained from the extremal condition
sin Q x;y ˆ 1
Jp 2 o p2 1
N
X
~k
n ~k
@E ~k
@Q x;y
:
…31†
If one substitutes Eq. 31 together with f a (obtained from the solution of the coupled
self-consistency equations (Eq. 25)) into Eq. 29, the free energy of the t-t'-J model is obtained
as
f t t 0 J ˆ J
4 ‰m2 …cos Q x ‡ cos Q y † 2n 2 Š‡ …2† m ‡… o …2† †n
1 X
ln 1 ‡ e …E k ~ ; …32†
N
~k
where the quasi-particle energy takes on the form
E ~k ˆ …1 ‡ †…" ~k ‡ " ~k
†=y ‡ …2†
~Q o
h i 1=2
2 …" ~k " ~k ~Q †2 =y ‡‰m…" ~k ‡ " ~k ~Q †=y ‡ …2† Š 2
…33†
with m=2 p o p and y=(1+d) 2 – m 2 .
In particular, at the spatially uniform paramagnetic saddle-point, F …PM† =(e, p o , l o (2) ,
l (1) ; 0, 0; 0,0; 0,0) the remaining bosonic fields
96
e 2 ˆ ;
p 2 o ˆ 1 ;
…1† ˆ
…2†
o
ˆ
2 ‡ 3 2
1 2 ~"…0† ;
2
~"…0† …1 †J; …34†
2
…1 ‡ †

7.2 Slave-boson theory for the t-t'J model
are explicitly given in terms of J, d and a single energy parameter ~" (0) defined by
~"…~q† ˆ 2
N
X
" ~k ~q
… E ~k † …35†
~k
(at T=0). Here, the quasi-particle energy E ~k is
E ~k ˆ 2" ~k =…1 ‡ †‡ …2†
0 ; …36†
and m can be determined from d =1– 2 N
free energy becomes simply
P
~k … E ~ k
†. Finally, in this approximation, the
f …PM†
t t 0 J ˆ z2 ~"…0†
1
2 J …1 †2 ‡ …1 † : …37†
7.2.4 Magnetic phase diagram of the t-t'-J model
In the numerical evaluation of the self-consistency loop we proceed as follows: at given
model parameters J and d, we solve the remaining saddle-point equations for m, l (1) , l o
(2)
and l (2) together with the integral equation for m using an iteration technique. Then, in an
outer loop, the order parameter wave-vector ~Q is obtained from the extremal Eq. 31 by
means of a secant method. Convergence is achieved if all quantities are determined with relative
error less than 10 –6 . Note that our numerical procedure allows for the investigation of
different metastable symmetry-broken states corresponding to local minima of the variational
free energy functional (Eq. 32).
At first, let us consider the case t' = 0, i. e., the pure t-J model. The resulting groundstate
phase diagram in the J-d plane is shown in Fig. 7.1. For the case d = 0 (Heisenberg
model), we obtain an AFM ground state. At J=0, the FM is lowest in energy up to a hole concentration
of 0.327, where a first-order transition to the (1, p)-spiral takes place; above
d = 0.39, we find a degenerate ground state with wave-vector (0,p). At d = 0.63 the PM becomes
the lowest in energy state (second-order transition). This coincides with the U ?? SB
results of the Hubbard model (J=4t 2 /U) [35]. The PM-FM instability occurs at d PM-FM = 0.33
[32]. In contrast, the slave-fermion phase diagram [20] exhibits a much larger FM region.
This can be taken as an indication that correlation effects are treated less accurate within the
slave-fermion mean-field scheme [20, 36]. For finite exchange interaction J the (1,1)-spiral
is the ground state at small doping. With increasing d a transition to the FM takes place,
which becomes unstable against the (1,p)-spiral at larger doping concentrations. For
J/t > 0.08, we find a transition from (1,1)-spirals to (1,p)-spirals at about d & 0.2. In
Figure 7.1, the dotted line separates the (1, p)-spiral state from the region, where the (1, p)
and (0, p) states are degenerate. If we admit only homogeneous phases, the phase boundaries
(1,1)-spiral ⇔ (1,p)-spiral at d & 0.2 and (1,p)/(0, p)-phase ⇔ PM at d &0.63 remain
nearly unchanged at larger values of J. However, analyzing the thermodynamic stability of
97

7 Slave-Boson Approach to Strongly Correlated Electron Systems
Figure 7.1: SB ground-state phase diagram of the t-J model.
the saddle-point solutions (i. e. the curvature of f t J …d), one observes a tendency towards
phase separation into hole-rich and AFM regions above the ‘diagonal’ solid curve in the left
upper part of the J-d phase diagram (see below).
Figure 7.2 displays the variation of the extremal spiral wave-vector ~Q as function of
doping. At d =0, ~Q =(p, p) indicates the AFM order. At low doping the (1,1)-spiral order vector
decreases approximately linear. With decreasing J the AFM exchange is weakened, consequently
the deviation of the order vector from (p, p) increases. The discontinuities reflect the
first-order transition from (1,1)-spirals to (1, p)-spirals. For J/t =0.05 the transition to the FM
state takes place, with Q x jumping down to zero. The (1,p)-spiral wave-vector shows a monotonous
decrease of Q x until at d = 0.63 the transition to the PM (Q x = 0) occurs. Comparing
the magnitude of the theoretical order vector of the spiral solutions to results from inelastic
neutron scattering experiments on La 2–x Sr x CuO 4 [37], we find good agreement for an exchange
interaction strength of J/t =0.4 (which seems to be a reasonable value with respect to
the strong electron correlations observed in the high-temperature superconductors).
Figure 7.2: The x-component of the spiral wave-vector ~Q as a function of doping d (compared with experiments
[_] on LSCO [37]).
98

7.2 Slave-boson theory for the t-t'J model
Next, we investigate the ground-state properties of the t-t'-J model, where we fix
J/t =0.4. Comparing the free energies of several (homogeneous) symmetry-broken states,
we obtain the SB phase diagram shown in the t'/t-d plane in Fig. 7.3. Obviously, we can distinguish
two regions. In the parameter region |t'/t| ^ 0.2, Figure 7.3 resembles the groundstate
phase diagram of the pure t-J model (Fig. 7.1), i. e., we found large regions with incommensurate
spiral order. However, compared to the pure t-J model, the t'-term stabilizes Néel
order in a finite d region near half-filling. The increasing stability of AFM configurations
can be intuitively understood because the t'-term moves electrons without disturbing the
Néel-like background [12]. For larger ratios |t'/t| we have a completely different situation. In
this parameter regime only commensurate states (AFM, FM, PM) occur, where for
t' < t' c = –1.4 t we obtain the AFM state for all d. Note the rather large differences to the value
of t' c obtained within a semiclassical (1/N)-expansion [38]. By varying the exchange coupling
J, the phase boundaries in the t'/t-d plane are not much affected, e. g. for J/t =1 and
t'/t =–0.4, the transitions AFM ⇔ (1,1)-spiral and (1,1)-spiral ⇔ FM take place at d = 0.17
and d = 0.6, respectively. We would like to point out that the main qualitative features of our
SB phase diagram do confirm recent studies of magnetic long-range order in the t-t'-J
model [38, 39].
Figure 7.3: Restricted SB ground-state phase diagram of the 2D t-t'-J model at J/t =0.4.
In Fig. 7.4 we plot the order parameter wave-vector as a function of doping at t'/t
= +0.16 and t'/t =–0.4. The behaviour of Q x reflects a series of transitions AFM ⇔ (1,1)-
spiral ⇔ (1, p)-spiral ⇔ PM. The corresponding (sublattice) magnetization abruptly changes
at the (1,1)-spiral ⇔ (1, p)-spiral first-order transition. From Fig. 7.4 the asymmetry between
hole (t' < 0) and electron doping (t' > 0) becomes evident. In contrast to recent Hartree-
Fock results for the Hubbard model [40] we found the AFM phase near half-filling for both
electron-doped and hole-doped cases (provided t'( 0). In the absence of t ' hopping, for arbitrarily
small doping the AFM is found to be unstable against the (1,1)-spiral phase (cf.
Fig. 7.3). Obviously, the AFM correlations are strongly enhanced by a positive t'-term,
which is also in qualitative agreement with exact diagonalization studies of the t-t'-J model
[12] and confirms the experimental findings for the electron-doped system NCCO [41, 42].
Note that the stability region of the AFM phase agrees surprisingly well with the combined
99

7 Slave-Boson Approach to Strongly Correlated Electron Systems
Figure 7.4: The x-component of the SB spiral wave-vector ~Q away from half-filling for the negative values
t'/t =–0.16 (solid) and t'/t =–0.4 (long-dashed), i. e., hole doping (d > 0), and for the positive one
t'/t =0.16 (dashed), i. e., electron doping.
phase diagram for La 2–x Sr x CuO 4 and Nd 2–x Sr x CuO 4 obtained from neutron scattering [43]
and muon spin relaxation measurements [44], respectively. For the YBCO parameter t'/t =
–0.4, the AFM disappears around d = 0.1 whereas in the phase diagram of YBCO, determined
by neutron diffraction [45], this transition takes place at about x = 0.4 oxygen content.
However, there exits strong evidence that at least up to x = 0.2 no holes are transferred
from CuO chains to CO 2 planes.
As we have already noted, the phase diagrams in Fig. 7.1 and Fig. 7.3 result from the
relative stability of various homogeneous states. On the other hand, there are arguments for
the existence of inhomogeneous, e. g., phase-separated states in the t-J and related models
[46]. Using very different methods, it was realized by several groups [47–51], that at large
J/t the ground state of the t-J model separates into hole-poor (AFM) and hole-rich regions.
Unfortunately, in the physically interesting regime of small exchange coupling (J/t ~ 0.2–
0.4) and low doping level this point is still controversial. To gain more insight into the phenomenon
of phase separation in t-J-type models of strongly correlated electrons it seems to
be important to investigate the effect of an additional NNN hopping term t' as well. Therefore
we study the free energy as a function of hole density d, where a (concave) convex curvature
indicates local thermodynamic (in)stability implying a (negative) positive inverse isothermal
compressibility k –1 = n 2 @2 f
If k –1 < 0, the domain of the two-phase regime is determined
performing a Maxwell construction for the anomalous increase of the chemical poten-
@n 2 :
tial m by doping.
The results of our analysis of thermodynamic stability are depicted in Figs. 7.5a and
7.5 b for the t-J and t-t'-J model, respectively. The boundary of phase separation for the t-J
model is given by the solid curve in the J-d plane shown in Fig. 7.5 a. As can be seen from
Fig. 7.1, the different phase separated domains are built up by the (AFM) states at half-filling
(d = 0) and the corresponding hole-rich state on the right boundary of the respective region.
At J=0, where we recover the U ?? result of the Hubbard model [32], the free energy
is a convex function Vd, i. e., our SRI SB theory does not support recent arguments
[47] for phase separation in this limit. In the opposite limit of large J, complete charge se-
100

7.2 Slave-boson theory for the t-t'J model
Figure 7.5: Phase diagram of the t-(t')-J model including phase separated states. The phase separation
boundary (solid curve) for the t-J model is shown in the J-d plane (a). We include the transition lines of
Refs. [49] (dashed), [50] (dotted), and [51] (chain dashed). The triangles are the Lanczos results of
Ref. [47]. The phase diagram of the t-t'-J is calculated in the t'/t-d plane at J/t =0.4 (b). Here the twophase
region consists of AFM and (1,1)-spiral states. For further explanation see text.
paration takes place for J/t > J PS /t=4.0, which seems to be an essentially classical result.
From exact diagonalization (ED) we obtain J PS /t=4.1 +0.1 [52] compared with 3.8 derived
by means of a high-temperature expansion [51]. We note that the homogeneous magnetic
phases are always unstable close to half-filling (provided J/t > 0). This is in qualitative
agreement with results obtained from ED studies [47] as well as from semiclassical [49] or
renormalization-group calculations [50]. Also plotted in Fig. 7.5 a are the results of the hightemperature
expansion method [51], where phase separation may occur only above a critical
exchange J/t =1.2 as d?0, contrary to all the other approaches. The line separating the
two-phase region from the stable states was determined by Marder et al. [49] within a semiclassical
theory to vary as J/t =4 d 2 whereas our theory yields an approximately linear dependence
at small d. We believe that, due to an improved treatment of spin correlations in
our approach, the region of incomplete phase separation is reduced. The instability towards
phase separation at small J can be taken as an indication that charge correlations may play
an important role as well.
The dotted lines of zero inverse compressibility in Fig. 7.5b show that also for |t'/t| >0
there is a finite range of d over which the (1,1)-spiral is locally unstable. Similar results were
recently obtained by Psaltakis and Papanicolaou [38]. But it is important to stress that in our
theory the AFM state is locally stable for both signs of t'. In addition, based on the Maxwell
construction, we can show that near half-filling the AFM state remains also globally stable
against phase separation for the t-t'-J model (cf. Fig. 7.5 b, where the two-phase region is
bounded by the solid lines). At larger values of |t'/t| 6 0.5, the phase-separated region is due
to the first-order nature of the transition AFM ⇔ (1,1)-spiral (dashed curve) [38].
Finally to demonstrate the quality of our SRI SB approach, for the t-J model the expectation
value of the kinetic energy is compared with the results from ED for a finite 16-
site [48] (36-site [52]) lattice in Fig. 7.6. We find an excellent agreement between SB results
and ED data. Obviously, this result does not depend on the interaction strength J, i. e. the SB
101

7 Slave-Boson Approach to Strongly Correlated Electron Systems
Figure 7.6: Expectation value of the kinetic energy as a function of doping at several interaction
strengths J in comparison to ED results for the 16 (36) site lattice [48, 52].
theory well describes important correlation effects. Note that AH t S t–J /t is directly related to
the effective transfer amplitude of the renormalized quasi-particle band, which is taken as input
for the calculation of transport coefficients in the following section.
7.3 Comparison with experiments
7.3.1 Normal-state transport properties
As one of the main normal-state puzzles of the CuO 2 based high-temperature superconductors,
the anomalous transport properties, in particular the temperature and doping dependence
of the Hall resistivity R H (T,d) [53–56], has been under extensive experimental and
theoretical study. Quasi-particle transport measurements suggest a small density of charge
carriers and hence a small pocket-like Fermi surface (FS) [57]. On the other hand, direct angle-resolved
photoemission (ARPES) probes of the FS [14, 58] yield a large FS which satisfies
Luttinger’s theorem and might be well described by (LDA) band structure calculations
[13, 59]. In principle, this contrasting behaviour is found for hole-doped (La 2–x Sr x CuO 4 ,
YBa 2 Cu 3 O 6+x ) and electron-doped (Nd 2–x Ce x CuO 4 ) copper oxides as well.
Adopting the hypothesis of Trugman [60], most of the normal-state properties of the
cuprates may be explained by the dressing of quasi-particles due to magnetic interactions
and the subsequent modification of their dispersion relation. Then, once the quasi-particle
band E ~k has been obtained, the Hall resistivity R H = s xyz /s xx s yy can be calculated in the relaxation
time approximation, using standard formulas for the transport coefficients:
102

7.3 Comparison with experiments
ˆ
ˆ
e 2 X @n ~k
k 2 u
V k ~ u k ~ ;
@E ~k
~k
e 3 2 X
k 4 cV
~k
u ~ k
" u ~ k
@u ~ k
@k
@n ~k
@E ~k
:
(38)
Here n ~k is given by Eq. 30,V denotes the volume of the unit cell, e lkg is the completely
antisymmetric tensor, and u ~ k ˆ @E ~k =@k . Note that R H does not depend on the relaxation
time t.
To make the discussion more quantitative, let us now consider the doping dependence of
R H (d,T) in terms of the t-t'-J model using the saddle-point and relaxation time approximations,
where FS and correlation effects are involved via the renormalized SB band E ~k (Eq. 33). As we
have pointed out above, in our approach the SB quasi-particle band dispersion E ~k has to be determined
in a self-consistent way at each doping level d. This should be in contrast to the NZA
SB mean-field approach to the t-t'-J model of Chi and Nagi [61] where, in the J ? 0 limit, the
calculation of transport properties is based on the simple replacement " ~k ! ~" ~k =
–2td [(cos k x + cos k y )+2(t'/t)cos k x cos k y ] of the non-interacting band dispersion (Eq. 2).
Figure 7.7 shows the theoretical Hall resistivity as a function of carrier density in
comparison to experiments on LSCO [53], YBCO [54] and NCCO [55, 56]. In the LSCO
and NCCO systems, the concentration of chemically doped charge carriers in the CuO 2
planes (d) definitely agrees with the composition (x) of the substitutes Sr and Ce. This simple
relation, however, no longer holds for YBCO, i. e., the number of holes transferred into
the planes does not increase linearly with the oxygen content. Indeed, the magnetic properties
indicate d & 0uptox=0.2 [45]. In order to compare our theoretical model with the
R H (x) data found on oxygen-doped YBCO, we use the relation d =(x – 0.2)/2 [62].
Figure 7.7: Doping dependence of the Hall resistivity for hole-doped (left panel) and electron-doped
(right panel) systems. The slave-boson results for J/t =0.4 and different ratios t'/t = 0 (solid), –0.16
(dashed), –0.4 (chain dotted) and t'/t =0.16 (dotted) are compared with experiments on LSCO (_) [53]
(at 80 K), YBCO (O) [54] (at 100 K) and NCCO (Z) [55, 56] (at 80 K), respectively. The inset shows
the temperature dependence of R H for t'/t =0atd = 0.1 (short dashed) and d = 0.15 (dotted).
103

7 Slave-Boson Approach to Strongly Correlated Electron Systems
Figure 7.7 clearly demonstrates the importance of the NNN transfer term t' for a consistent
theoretical description of the experimental Hall data. For J/t =0.4, an excellent agreement
with experiments on LSCO and NCCO, including the sign change of R H (d) at a very similar
value, can be achieved using the parameter values t'/t =0 and t'/t = 0.16, respectively. It
should be noted that in the case of LSCO we obtain t'/t =0 from R H (d) while LDA calculations
yield a ratio t'/t =–0.16 [13, 59]. For YBa 2 Cu 3 O 6+x , where the experiments [54] give
R H > 0 up to x=1, a negative t'-term suffices to give the correct tendency to R H (d). Using
t'/t =–0.4, our theory yields a sign change of R H at d & 0.7. The strong increase (decrease)
of the positive (negative) Hall coefficient as d?0 can be attributed to the formation of
small hole (electron) pockets in the FS, which is a correlation effect. A recent analysis of resistivity
saturation in LSCO [57], based on Boltzmann transport, has been taken as an indication
of a small FS as well, however, the existence of a pocket-like FS is still a subtle and
unresolved issue. We found that the temperature dependence of R H (d,T) is at least in qualitative
agreement with experiments on LSCO (see inset Fig. 7.7 (left panel)). Note that the
quasi-particle dispersion E ~k exhibits extremely flat minima implying the presence of a new
small energy scale D. Therefore, when the temperature becomes comparable to D the hole
pockets are washed out and a sign change of R H occurs (cf. Fig. 7.7 at fixed d (inset)). 3
The FS of the interacting system are shown in Fig. 7.8 for typical ratios t'/t at J/t =0.4
and d = +0.1, where the diagonal (1,1)-spiral phase is lowest in energy. As observed for t-J
and Hubbard models as well [64], we obtain small hole (or electron) pockets with a volume
! |d|. The calculated FS are very anisotropic. As |d| increases, the pockets grow, until the
FS topology changes completely at a critical doping value d c (R H (d c ) = 0 (cf. Fig 7.7), reflecting
the transition from hole to electron carriers for t'/t < 0 and vice versa for t'/t >0.
Figure 7.8: Quasi-particle Fermi surface in the (1,1)-spiral phase at J/t =0.4 for d = 0.1 (hole-doped
system) and d = –0.1 (electron-doped system).
3 We recently learned of a related exact diagonalization study of Dagotto et al. [63], where, similar in
conclusion, the doping and temperature dependence of R H was calculated using a strongly renormalized
flat quasi-particle dispersion.
104

7.3 Comparison with experiments
We want to point out that the renormalization of the quasi-particle band E ~k strongly depends
on both interaction strength J and doping level d [62] which, in fact, calls into question
the frequently used rigid band approximation. Due to the strong coupling of spin and
charge dynamics the characteristic energy scale for the coherent motion of the charge carriers
is J and not t (provided t > J).
7.3.2 Magnetic correlations and spin dynamics
In this Section we try to understand the INS and NMR experiments on the basis of the SRI
SB mean-field theory. Using a generalized RPA expression for the spin susceptibility and assuming
that the AFM correlations are spatially filtered by various ~q-dependent hyperfine
form factors, we focus, in particular, on the spin dynamical properties in the paraphase of
the t-t'-J model. Our starting point is an RPA-like form for the exchange-enhanced spin susceptibility
[5, 6, 65]
o …~q; !†
s …~q; !† ˆ
1 ‡ J 2 …cos q x ‡ cos q y † o …~q; !†
: …39†
The irreducible part 4
o …~q; i! m †ˆ
2 X
N
~k;n
G… ~ k; i! n † G… ~ k ‡ ~q; i! n‡m †
…40†
contains the (dressed) SB Green propagators G… k; ~ i! n †ˆ‰i! n E ~k ‡ ~Š 1 describing noninteracting
electrons with the renormalized band structure E ~k …d† (Eq. 26).
Once the dynamic spin susceptibility has been obtained both, INS measurements and
NMR experiments, can be explored. Probed by INS from the fluctuation-dissipation theorem
the q-dependent and o-dependent spin structure factor is related to the dynamical susceptibility
by
S …~q; !† ˆ 1 1
p 1 exp… k!† Im s…~q; !† …41†
On the other hand, the nuclear spin-lattice relaxation rate a T –1 1a (a =k,k), e. g. for a
field H a applied parallel to the c-axis, given by [66]
a T 1k T 1 k B
1ˆ
2 2 B k lim 1
!!0 N
X
~q
a F 2 ? …~q† Im s…~q; !†
k!
…42†
4 o n = 2np/b [o m = (2m +1)p/b] denote the fermionic [bosonic] Matsubara frequencies.
105

7 Slave-Boson Approach to Strongly Correlated Electron Systems
and the transverse spin-spin relaxation rate, T –1 2G , for the RKKY coupling of the nuclear Cu
spins, given by [67]
T 2
2G ˆ
2
0
123
c 1 X 2
8k 2 …2 B † 4 Fk 2 N
…~q† 1 X
4
s…~q† @ Fk 2 N
…~q† s…~q† A 5 ; …43†
~q
provide local, atomic site (a = {63, 17} specific information. Here m B denotes the Bohr
magneton and the constant c=0.69 is the natural-abundance fraction of the 63 Cu isotope.
The form factors are given for 63 Cu and the planar 17 O nucleus as [68]
63 F …~q† ˆA ‡ 2B…cos q x ‡ cos q y † ; …44†
~q
17 F …~q† ˆ2C cos q x
2 : …45†
Together with the anisotropy of the Cu relaxation rates the measurements of the
Knight shift, a K ˆ 2
a B n k lim ~q!0 a F …~q† s …~q; ! ˆ 0†, have been used to determine the hyperfine
coupling constants A a , B and C on the basis of the Mila-Rice Hamiltonian [69]. Following
Ref. [66] we take A || & –4B, A k & 0.84B, C & 0.87B and B & 3.3610 –7 eV, where
63 gk = 7.5610 –24 erg/G and 17 gk = 3.8610 –24 erg/G.
As we know, the RPA susceptibility contains an unphysical instability of the paramagnetic
phase at some particular wave-vector ~q below a critical doping d c as signaled by the
zero in the denominator of Eq. 39 at o = 0 (Stoner condition). Therefore the use of Eq. 39
only makes sense if the system is far from the magnetic instability, i. e., d > d c , where for
J=0.4 we have d c (t'/t =–0.16) H 0.27 and d c (–0.4) H d c (0) H 0.17.
7.3.3 Inelastic neutron scattering measurements
We begin with a discussion of the RPA dynamical spin structure factor S (~q; o†. The ~qdependence
of S (~q; o† along the main symmetry axis of the Brillouin zone is shown in
Fig. 7.9 at J=0.4 and hole density d = 0.3 for different ratios t'/t. Here the temperature
is T=35 K and the frequency ko = 0.010 eV. For LSCO-type parameters (t'/t =0,
–0.16) we found four pronounced incommensurate peaks located at the points
p (1+q o , 1), p (1,1+q o ). The incommensurate modulation wave-vectors move with increasing
doping level d away from the corner of the Brillouin zone along the directions
(1, p) or(p, 1) (square lattice notation). Note, that the incommensurate peak position obtained
from a three-band RPA calculation of S (~q; o† [5] can be parametrized consistent
with the experimental observation that q 0 H 2 x [37], while all the effective one-band
RPA approaches [6, 10] yield an incommensurability scaling rather as q 0 H d. A more
detailed investigation of the LSCO-type ~q-scans show that the ~q-variation of S(~q; o† is
mainly governed by that of w o …~q; o† and, in accordance with experiments [70], the in-
106

7.3 Comparison with experiments
commensurate peaks considerably broaden when temperature or energy transfer are increased
[71]. By contrast, the same plot for YBCO-type (t'/t =–0.4) parameters shows a
broad and nearly T-independent [71] maximum around the (p, p)-point [72] (Fig. 7.9)
which, due to the flat topology of w o , mainly reflects the ~q-dependence
J (~q) =J (cosq x + cosq y ) (cf. Eq. 39). In this way, our calculations confirm recent arguments
[5, 6, 73] for the importance of band structure (Fermi surface geometry) effects
in explaining the difference between observed LSCO and YBCO spin dynamics. Nevertheless,
whether the incommensurate signals arise from an intrinsic magnetic structure or
whether they result from the formation of domains (charge superstructures) in the LSCO
system remains unanswered by INS [2, 68].
Figure 7.9: Dynamic magnetic structure factor S(~q; o† is plotted along the (1,1) and (1, p)-directions of
the Brillouin zone for different ratios t'/t.
In a next step, we calculate the longitudinal or spin-lattice relaxation rate, T –1 1 , using the
hyperfine form factors (Eq. 45). In Figure 7.10 the temperature dependences of 63 T –1 1k and
17 T –1 1k (inset) are shown for d = 0.35 and t'/t =–0.4 in comparison to experiments [74] on
fully oxygenated YBCO materials (x =1). Although our theory does not succeed in giving
the correct amplitude of a T –1 1k the qualitative features of the NMR data are described surprisingly
well. Obviously, the broad magnetic peak in S (~q; o† at the AFM wave-vector
~Q AFM =(p,p) strongly enhances the relaxation rate on Cu sites while, due to a geometrical
cancellation ( 17 F a (~q ~Q AFM † 0†, the corresponding oxygen rate is rather insensitive to
nearly commensurate AFM fluctuations and therefore is governed by the long wavelength
part ~q 0 of the spin susceptibility [2]. For 63 Cu the nominal Korringa ratio S : (1/T 1 TK 2 S )
(K S denotes the spin part of the Knight shift) is at least one order of magnitude larger. As
can be seen from Fig. 7.10, for YBa 2 Cu 3 O 7 -type parameters, a Korringa (1/T 1 ! T) dependence
(dotted line) holds at both Cu and O sites below T* ~ 120K, demonstrating the existence
of a characteristic temperature T* as well as in all other near-optimum T c compounds
[2, 75]. T* is in good agreement with the coherence energy scale suggested experimentally
[75]. Above T* the 17 O NMR relaxation remains linear whereas the 63 Cu relaxation time
does not follow the Korringa law.
107

7 Slave-Boson Approach to Strongly Correlated Electron Systems
Figure 7.10: Spin-lattice relaxation rates 63 T –1 1k and 17 T –1 1k (inset) as a function of temperature. SB results
((+); t'/t =–0.4, d = 0.35) are compared with experiments (D) on YBa 2 Cu 3 O 7 [74].
In the oxygen-deficient compound YBa 2 Cu 3 O 6.6 , 1/( 63 T 1k T) shows a broad maximum
at about 150 K (Fig. 7.11), which reflects a strong deviation from the canonical Korringa behaviour.
In the normal-state regime the theoretical results agree even quantitatively with the
experimental 63 Cu NMR data [76]. At this point it is important to stress that the present theory
incorporates considerable band renormalization effects already via w 0 …~q; o†, especially
at low doping level. Thus a rather moderate strength J = 0.4 of the AFM exchange interaction
yields the experimentally observed enhancement of 63 T 1k . In striking contrast to the optimally
doped YBCO system, the Korringa relation is no longer satisfied for the planar 17 O
nucleus sites in the underdoped material (cf. inset Fig. 7.11). Instead, a different behaviour
17 T 1k T 17 K S = const was suggested to hold down to T c [76]. The unconventional T-scaling of
17 T 1k has been taken as a signature for another important feature of the normal-state spin dynamics,
the so-called spin-gap behaviour [2].
Figure 7.11: 63 Cu and planar 17 O relaxation data (^) for underdoped YBa 2 Cu 3 O 6.6 [76] are plotted vs
temperature. Theoretical results (6) are given at t'/t =–0.4, and d = 0.2.
108

7.4 Summary
Complementary measurements of the transverse spin-spin relaxation rate, T –1 2G ,have
provided further insights into the drastic change in the magnetic properties when passing
from the overdoped to the underdoped regime [67, 77]. As experimentally observed, we
found that T –1 2G increases (decreases) with increasing (decreasing) hole doping (temperature)
[78]. In order to detect the opening of a spin-pseudogap as a function of T, a powerful technique
is to measure the ratio T 2G /T 1 T [79] which is nearly constant above ~ 200 K for deoxygenated
YBa 2 Cu 3 O 6.6 [67]. The calculated temperature dependence of this quantity is
shown in Fig. 7.12 together with recent experimental results [67]. Most notably, the opening
of a spin-pseudo gap at T* ~ 135 K [79], i. e. well above T c , is clearly seen as a decrease of
below T*. Note that for YBa 2 Cu 3 O 7 , as predicted by Fermi liquid theories, the ratio T 2 2G/T 1 T
is approximately constant above 150 K [2].
Figure 7.12: T 2G /( 63 T 1k T) in YBa 2 Cu 3 O 6.6 (^) as measured by Takigawa [67] compared with the SB
data (6) att'/t =–0.4, and d = 0.2.
7.4 Summary
In this work we have used a spin-rotation-invariant SB approach to investigate magnetic and
transport properties of the 2D t-t'-J model. Our main results are the following (see also [81]):
a) We present a detailed magnetic ground-state phase diagram of the 2D t-(t')-Jmodel, including
incommensurate magnetic structures and phase separated states. At finite t', a main
feature of the phase diagram, we would like to emphasize, is the existence of an AFM state
away from half-filling, which is locally and also globally stable against phase separation.
This result agrees with the experimentally observed AFM long-range order in the weakly
doped LSCO and YBCO compounds. In contrast, for the simple t-J model we observe no
AFM long-range order at any finite doping due to phase separation.
109

7 Slave-Boson Approach to Strongly Correlated Electron Systems
b) The next nearest-neighbour hopping process (t') incorporates important correlation and
band structure effects near half-filling. In particular, the t'-term can be used to reproduce
the FS geometry of LSCO, YBCO, and NCCO. Also the NNN hopping provides a possible
origin for the experimentally observed asymmetry in the persistence of AFM order of holedoped
and electron-doped systems.
c) The quality of the SRI SB approach was demonstrated in comparison with exact diagonalization
results available for the t-J model on finite square lattices with up to 36 sites. In
this case, the SB method yields an excellent estimate for the quasi-particle band renormalization.
d) Within the saddle-point and relaxation time approximation, our SB calculation of the
Hall resistivity in the t-t'-J model provides a reasonable explanation of the experimentally
observed doping dependence of R H on both hole-doped (La 2–x Sr x CuO 4 , YBa 2 Cu 3 O 6+x ) and
electron-doped (Nd 2–x Ce x CuO 4 ) copper oxides.
e) Using a generalized RPA expression for the spin susceptibility and assuming that the
AFM correlations are spatially filtered by the hyperfine form factors, we have calculated the
temperature dependences of spin-lattice and spin-spin relaxation rates for planar copper and
oxygen sites. The results agree qualitatively well with various NMR experiments on YBa 2-
Cu 3 O 6+x . In addition, we can attribute the contrasting ~q-dependence of the magnetic structure
factor S (~q; o† seen in INS experiments for LSCO-type and YBCO-type systems to differences
in their fermiology.
Recently, our theory was improved to include (Gaussian) fluctuations beyond the paramagnetic
saddle-point approximation [27, 80]. We derived a concise expression for the spin
susceptibility w s (~q; o† of the t-t '-J model which does not have the standard RPA form. Then
we were able to show that the instability line obtained from a divergence of w s (~q; 0† is in
agreement with the PM ⇔ spiral state phase boundary in the saddle-point phase diagram,
which in fact proves the consistency of both approaches [27].
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112

8 Non-Linear Excitations and the Electronic Structure
of Conjugated Polymers
Klaus Fesser
8.1 Introduction
Conjugated polymers have attracted quite substantial research activities during the last 15
years [1]. On one hand these materials promise interesting technological applications most
of which are related to the possibility of a reversible charging and decharging of these systems.
These include various battery designs as well as storage of (charged) pharmaceuticals
which in turn can be released in a controlled way by application of an electrical current. For
the design of non-linear optical components they play an important role as organic materials
due to their processibility and fine tuning of their physical properties via suitable side
groups. Recently light-emitting diodes made from these systems have made conjugated polymers
to possible candidates for the construction of thin displays [2].
On the other hand a theoretical description of the whole class of these materials poses
interesting questions which are worth to study on their own right. As essentially quasi onedimensional
systems they give rise to the hope that many of these questions might be answered
analytically. The main problems are the nature of the insulator-metal transition observed
during doping and the origin of the intragap states, which are mainly responsible for
the relaxation processes relevant for the light-emitting properties. We have addressed both
aspects within this project and this article is organized accordingly. In Section 8.2 we present
the theoretical model stressing the relevant physics and discuss the related fundamental symmetries.
Then in Section 8.3 we adopt the view that the doping process mainly introduces
disorder into these systems and calculate various properties (density of states, optical absorption)
from this assumption. In Section 8.4 we investigate the non-linear excitations responsible
for the intragap states in more detail and close in Section 8.5 with an outlook on
still open problems.
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
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8 Non-Linear Excitations and the Electronic Structure of Conjugated Polymers
8.2 Models
Common to all conjugated polymers is the existence of a carbon backbone with an alternating
sequence of (short) double and (long) single bonds. The various systems, as polyacetylene,
polyparaphenylene, polypyrrole, polythiophene, polyparaphenylenevinylene, etc.,
differ only – from a physcist’s point of view – in the side groups and other chemical structures
which are energetically far away from the Fermi energy. Emphasizing the general aspects
of these systems, these chemical details are safely neglected if one restricts to properties
which are mainly due to the electrons around the Fermi energy. Thus the parameters in
the simple model presented below should be regarded as effective parameters including
some aspects of the interactions which are otherwise neglected.
Beside some biological systems such as b-retinol where a finite chain of a conjugated
polymer is present (and thus the properties of this polymer might be of relevance for the
biological functions of this material), there may be other, related systems where similar theoretical
concepts are or can be applied. These include the metal-halogen (MX) chains where
instead of one, as in the conjugated polymers, now two electron bands govern the essential
physics. On the same level the polyanilines can be modelled where the phonons of the conjugated
polymers have to be replaced by the librons, i. e. oscillations of the quinoid/benzoid
rings. Finally, a true one-dimensional modification of carbon, carbene with alternating single
and triple bonds, can also be understood along these lines.
So far we have only mentioned the construction of adequate models for the investigation
of physical properties. However, there are competing approaches to address the same
set of questions. One of these approaches uses sophisticated quantum-chemical codes [1c,d]
to calculate structure and electronic states of these materials. Although such a procedure
may be able to reproduce the observed properties of a specific material quite well it is very
difficult to obtain information about general trends and physical mechanisms. Therefore we
did not follow this route. Another well-established method for calculating ground-state properties,
namely the local density functional, has been used to some extent [3]. The drawback
of this method, however, is the finite size of the system which can be calculated within a
reasonable amount of computer resources. Therefore only a few questions have been addressed
via this approach.
Therefore, we shall argue in favor of a simple model which is capable to include the
most essential physics. Assuming the sp 2 -hybridisation of the carbon atom leaves one p-electron
per atom the others being incorporated into the bonds as s-electrons. It is the physical
behaviour of this single p-electron which is responsible for all the interesting effects.
Since there is only this one electron per site the polymer would be metal-like in its
ground state, in contrast to nature where one finds an energy gap of the order of 1 eV. In
one dimension, however, these electrons are unstable when they are coupled to the lattice.
Due to the Peierls effect, scattering off 2 k F phonons, a gap opens right at the Fermi energy
in accordance with the observations. In addition, electron-electron interaction contribute to
the size of this gap [4] stabilizing the semiconductor ground state even further. A single-particle
model along these lines has been put forward by Su et al. [5] in the early stages of conjugate
polymer research. Its parameters although, derived from a simple picture, should
114

8.2 Models
nevertheless be interpreted as effective parameters including parts of the electron-electron
interaction which is otherwise neglected. It has been shown [6] that for some ground-state
properties such redefinition can indeed be performed.
It turns out that there is a typical length scale in this model v F /D (with the Fermi velocity
v F and the electronic gap 2 D) which is considerably larger than the interatomic spacing
a. Thus for most physical properties of interest a continuum approximation is valid.
This stresses the generic aspects of the whole class of these materials even more.
In rescaled variables this model now reads
H ˆ X Z
s
dx ‡ s …x† f i 3@ x ‡ 1 …x†g s …x†‡ 1 Z
2
dx 2 …x†
…1†
Here c (x) is a two-component spinor describing electrons moving to the left (right)
along the one-dimensional polymer, s is the spin index which, except for external fields, is
not relevant here. The s 3 term is the kinetic energy originating from the hopping of electrons
between neighbouring sites. D (x) is the lattice order parameter where a constant
D (x) =D 0 describes a uniform dimerization of the lattice. The s 1 term is the electron-phonon
coupling responsible for the Peierls distortion. Finally we have an elastic energy for the
lattice. The kinetic energy is neglected within an adiabatic approximation. All quantities
have been scaled in order to have a single coupling constant (l & 0.2).
At this stage we postpone the non-linear excitations of this model to a subsequent
Chapter, we only discuss some of the symmetries of this model which are of relevance also
for these excitations. First we note that the form of the electronic part of the Hamiltonian
(Eq. 1) has exactly the form of a (relativistic invariant) Dirac operator in one dimension.
This correspondence has been exploited [7] successfully in obtaining solutions of this model
from results known in models of elementary particles. We remark here that similar analogies
can be made for specific forms of the Fermi surface also in higher dimensions. Thus the
connection between solid state physics and quantum field theory models can be used for a
better understanding of both.
In addition there are two symmetries which can be directly related to observable quantities.
The charge-conjugation symmetry, which can be expressed as H being invariant under
c?s 2 Kc (K complex conjugate operator), relates particle and hole states and thus is responsible
for the single particle spectrum being symmetric with respect to the Fermi energy,
which has been set to zero in H (Eq. 1). A more hidden property is the supersymmetry of
the electronic part H el (Eq. 1),
0 ˆ H el ;H ‰ el ; 3 Š : …2†
We have shown [8] that this more formal property is responsible for the asymmetry of
the optical absorption peaks of transitions involving intragap states. Both symmetries are absent
in the real systems under consideration, but for most questions this breaking of symmetries
is merely a question of quantity rather than of importance for the existence of these localized
states.
Finally we mention that Eq. 1 has been derived as a model for polyacetylene where
the ground-state order parameters D o and –D o yield the same energy. This is not the case for
115

8 Non-Linear Excitations and the Electronic Structure of Conjugated Polymers
the other polymers of this class. A simple extension of this model, introduced by Brazovski
and Kirova [9], corrects this. As a result all the solutions of Eq. 1 can be carried over [10]
to this more general model where only the location of the intragap states are now parameters
which can be adjusted to a specific material of interest. In consequence the model (Eq. 1)
can be considered as the most simple generic model of conjugated polymers.
8.3 Disorder
A major focus of theoretical investigations is the experimental observation that the electronic
structure and consequently the physical properties change drastically upon doping. Here
we restrict ourselves to the question how these properties are affected through a random
force introduced by these impurities. For simplicity we only consider isoelectronic doping,
i. e. the total number of carriers remains unchanged and the dopants introduce only scattering
centers for the electrons.
In consequence two different types (site or bond impurities) of scatterers can be identified.
The first gives rise to a random contribution to the on-site energy of an electron
whereas the latter modifies the hopping integral between neighbouring sites. In the spirit of
the continuum approximation we are thus lead to consider only backward (Eq. 3) or forward
(Eq. 4) scattering.
P R
H imp ˆ U b dx ‡ …x† 1 …x†…x x j † ; …3†
j
P R
H imp ˆ U s dx ‡ …x†1 …x†…x x j † ; …4†
j
x j denotes the (random) position of an impurity. Since this problem cannot be solved exactly
we have to redraw to approximate methods.
We have used three main routes.
In the first method we employ the first Born approximation for the impurity self-energy
[11]. This enables us to formulate equations of motion for the full space-dependent
Green functions and thus consider the influence of disorder on the non-linear excitations as
well [12] (see next Chapter). Furthermore, the replacement of Eq. 3 and Eq. 4 by random
(Gaussian) fields
Z n
H imp ˆ dx ‡ …x†
1
1
o
V b=s …x† …x† ;
which is correct within the Born approximation, allows the determination of the Green function
and higher correlation functions via a functional integral technique [13]. The averaging
procedure is formulated through the introduction of additional Grassmann variables in a
116
…5†

8.4 Non-linear excitations
supersymmetric way. We note that this supersymmetry is not related to the one discussed
earlier. Using the algebraic properties of these variables the average can be performed and
the resulting functional integral can be calculated via a transfer method in one dimension.
We note that this procedure still works when two coupled chains are considered but cannot
be done exactly in any higher dimension. As result we obtain the density of states and also
information about the extension of the (in principle) localized wave functions via the Thouless
formula. It turns out [14] that for moderate disorder a typical realistic chain length of
approximately 200 units is smaller than the localization length of the band states not too
close to the band edgesunverständlich. Thus for such systems these states can indeed be considered
as extended states even in the presence of disorder putting various models which
treat the propagation of electrons along one chain as metal-like on a firmer basis. This
method can easily be extended to higher dimensional systems which means that also the
coupling of polymer chains can be taken into account. Thus we are able to calculate, within
a saddle-point approximation, the optical absorption coefficient for a two-dimensional film.
For an orientation along the chains we find [15] a typical disorder induced broadening of the
absorption edge together with a shoulder on the low-energy side due to neighbouring chains
coupling. Both features agree quite well with experimental findings. Concomitantly, the absorption
perpendicular to the chain direction is featureless.
In the second method, going beyond Born, we examined the density of states within
the coherent potential approximation (CPA) which takes into account multiple scattering processes.
One might think that on this level impurity states are introduced in the gap. However,
we find [16] that the existence of such localized impurity states strongly depends on the relative
strength of site vs. bond impurity. Only states in the gap due to disorder can be found
if the site amplitude |U s | is stronger than the bond amplitude |U b |. Since CPA is an effective
medium theory this result might be questionable in one dimension.
In the third method, in order to check the CPA result, we performed a numerical simulation
[17] where for a given random distribution the electronic eigenvalues have been calculated
numerically and the results been averaged over a large number of realizations. It turned
out that the CPA results could be reproduced quite satisfactorily thus establishing the different
role of both types of impurities.
8.4 Non-linear excitations
Given the single particle states according to Eq. 1 with a uniform order parameter D o one
might expect that the lowest excited state corresponds to the lowest level in the conduction
band being occupied thus requiring the amount 2D o in energy, since the half-filled case considered
here the valence band with E |D o |is
empty. The model (2.1), however, exhibits the property that an additional carrier modifies
the order parameter locally, yielding a non-homogeneous D (x), and at the same time creates
through a rearrangement of the band statesadditional state(s) deep in the gap. Mathematically
this behaviour is due to the fact that Eq. 1 is a non-linear model, the non-linearity re-
117

8 Non-Linear Excitations and the Electronic Structure of Conjugated Polymers
sulting from the requirement that the total energy functional AH{D (x)}S has to be stationary
with respect to a variation of the order parameter D (x). This gives rise to a self-consistency
equation where this order parameter is governed by the occupied electronic states,
…x† ˆ
P ‡ …x† 1 …x† : …6†
occ
Various exact solutions to this problem are known. The most prominent one is the
kink (soliton) for the case of polyacetylene being,
p
…x† ˆ o thx= 2 :
…7†
As already mentioned, this kink does not exist for the other polymers of this class because
the ground state is non-degenerate in D o . Therefore the simplest non-linear excitation
in a more general sense is the bound kink-antikink pair (polaron) which is characterized by
two localized electronic states in the gap at +o o (e. g. o o /D o & 0.5 for polythiophene). In
addition there exist periodic solutions, e. g. the kink lattice with a periodicity determined by
the concentration of excess charges. Physically these states lower the total energy of the system
through an inhomogeneous order parameter D (x), which raises the energy (cf. Eq. 1)
and a much larger compensation through the intragap states, which altogether give a smaller
value of the total energy than the simple single-particle picture. For the polaron this gain in
energy amounts to E p /2D o = 0.98 < 1 for polythiophene. For trans-polyacetylene this number
is 0.90.
One can now envisage the processes which are involved in generating visible light. A
sufficiently strong electric field can promote a single electron into the lowest unoccupied
conduction band state. This state is unstable and relaxes on a fast time scale (femtosecond)
into a polaron-like state which then can recombine to the ground state under the emission of
radiation. A full microscopic understanding of all the processes involved is only possible if
for the dynamics of the lattice the degrees of freedom are fully taken into account as well as
the residual Coulomb interactions.
One step in this direction has been made within this project by Bronold [18]. In his
doctoral thesis he treats on an equal footing electron-electron (exciton) and electron-phonon
(polaron) interactions. The coupling of this system to short laser pulses gives rise to characteristic
changes of position and shapes of absorption/emission lines in optically stimulated
emission and inverse Raman scattering experiments. As these effects have a very short time
scale (femtoseconds), experiments are difficult to perform and a comparison with existing
theoretical predictions is not convincing. Nevertheless, one expects that this line of approach
will finally give a detailed understanding of the functioning of organic light-emitting diodes.
Having gained some insight into the non-linear mechanisms giving rise to localized
electronic (intragap) states, how dopants, mentioned in the previous Chapter, might influence
these states. Two alternatives are feasible:
a) the non-linear aspect dominates, i. e. the picture developed so far is still valid but only
some details are modified due to the disorder. On the basis of the Born approximation we
have indeed calculated [12] the electronic structure of the kink solution in the presence of
impurities and found that the spatial extension of this structure is enlarged when the doping
118

8.5 Perspective
concentration is raised. In accordance with the closure of the gap at a critical concentration
we find that the width of the kink tends to infinity at this value. The more interesting case
of the polaron, however, could not be solved satisfactorily due to numerical instabilities,
which could not be avoided. For details see Ref. [12].
b) the more subtle case of the competition between this non-linear mechanism and the
multiple scattering processes off the impurities, which is treated in terms of the T-matrix or
the CPA gives also rise to localized states. The experimental observation that the formation
of these states is independent of microscopic details leads to the conclusion that the non-linear
aspect is the dominant one. A fully self-consistent treatment of this problem with an impurity
located at x o and (for simplicity) a kink at x 1 gives complicated coupled integral equations
[19] which have not been solved. A recent investigation for the case of a kink lattice
shows both mechanisms working quite independently, however, the method employedwelche
Methode ? does not give a full self-consistent solution..
Summing up this Chapter we note that the non-linear excitations play a dominant role
in various physical applications of conjugated polymers. But a full understanding of the interplay
of various mechanisms giving rise to these localized states has not been reached yet.
8.5 Perspective
The potential technical applications have stimulated a myriad of experimental and theoretical
studies. It is obvious that similar investigations have also been performed, mostly along
different lines, by various groups. The disorder aspect, responsible for the observed metalinsulator
(or semiconductor) transition, in conjunction with a kink (soliton) or polaron lattice
has been treated by many authors [20] in all kinds of approximate approaches. All these studies
resulted in the same prediction that such an M-I transition would occur at the experimentally
observed dopant concentration level. It is now clear from the foregoing Chapter
that only a fully self-consistent treatment will give a satisfactory answer to this question.
But since the applications for conjugated polymers envisaged at present focus on optical
properties rather than electronic transport this question has been lost out of sight. Still, the
nature and dynamics of localized electronic states in these materials must be fully understood.
In addition, from an application point of view, these polymers appear to be similar to
conventional semiconductors. The recent proposal [21] that the non-linear excitations actually
modify the conventional picture, e. g. at the interface between a polymer and a metal (or
conventional semiconductor) space charge regions (depletion layers) differ from an inorganic
semiconductor, finds renewed interest because transistors made from conjugated polymers
are feasible and of interest for the integration of optical and electronic components.
From a theoretical point of view the functional integral techniques promise interesting
opportunities to make contacts to other areas of research. Universal properties have been dis-
119

8 Non-Linear Excitations and the Electronic Structure of Conjugated Polymers
covered [22] in the absorption spectrum, discussed earlier, as well as in the distribution of
energy levels in certain disordered systems [23]. We propose that there is a close connection
between both aspects. A careful treatment of the underlying correlation functions, including
a more general type of disorder than discussed here, has to be performed. We expect that
the result will lead to new universality classes which technically spoken will show up in different
supersymmetric non-linear s models. Work along this line is in progress.
In summary, conjugated polymers pose interesting problems, both for applications and
pure theoretical studies. Both aspects have matured during the past 15 years but still questions
of a more general nature are left unanswered.
Acknowledgements
The author is indebted to all his collaborators for enlightening and stimulating discussions
as well as fruitful collaborations. Thanks to A.R. Bishop, F. Bronold, H. Büttner, D.K.
Campbell, K. Harigaya, U. Sum, Y. Wada, and M. Wolf.
References
1. a) A.J. Heeger, S. Kivelson, J.R. Schrieffer, W.P. Su: Rev. Mod. Phys. 60, 781 (1988)
b) T.A. Skotheim (Ed.): Handbook of Conducting Polymers. Dekker, New York (1986)
c) J.L. Brédas, R. Silbey (Eds.): Conjugated Polymers. Kluwer, Dordrecht (1991)
d) W.R. Salaneck, I. Lundström, B. Ranby (Eds.): Conjugated Polymers and Related Materials. Oxford
University Press, Oxford (1993)
e) Proc. Int. Conf. Science Technology of Synthetic Metals: ICSM ’90, Synth. Met. 41–43 (1991);
ICSM ’92, Synth. Met. 55–57 (1993); ICSM ’94, Synth. Met. 69–71 (1995)
2. J.H. Burroughes et al.: Nature 347, 539 (1990)
3. P. Vogl and D.K. Campbell: Phys. Rev. Lett. 62, 2012 (1989); Phys. Rev. B 41, 12 797 (1990)
4. For a review see: D. Baeriswyl, D.K. Campbell, S. Mazumdar, in: Conjugated Conducting Polymers,
H. Spiess (Ed.), Springer Series in Solid State Sciences 102, 7 (1992)
5. W.P. Su, J.R. Schrieffer, A.J. Heeger: Phys. Rev. B 22, 2099 (1980)
6. D. Baeriswyl, E. Jeckelmann, in: Electronic Propteries of Polymers, H. Kuzmany, M. Mehring, S.
Roth (Eds.), Springer Series in Solid State Sciences 107, 16 (1992)
7. D.K. Campbell and A.R. Bishop: Nucl. Phys. B 200, 297 (1982)
8. U. Sum, K. Fesser, H. Büttner: Ber. Bunsenges. Phys. Chem. 91, 957 (1987)
9. S. Brazovskii and N. Kirova: Pis’ma Zh. Eksp. Teor. Fiz 33, 6 (1981) [JETP Lett. 33, 4 (1981)]
10. K. Fesser, A.R. Bishop, D.K. Campbell: Phys. Rev. B 27, 4804 (1983)
11. a) K. Fesser: J. Phys. C 21, 5361 (1988)
b) K. Iwano and Y. Wada: J. Phys. Soc. Jpn. 58, 602 (1989)
120

References
12. F. Bronold and K. Fesser, in: Nonlinear Coherent Structures in Physics and Biology, M. Remoissenet
and M. Peyrard (Eds.), Springer Lecture Notes in Physics 393, 118 (1991)
13. K.B. Efetov: Adv. in Phys. 32, 53 (1983)
14. M. Wolf and K. Fesser: Ann. Physik 1, 288 (1992)
15. M. Wolf and K. Fesser: J. Phys. Cond. Matter 5, 7577 (1993)
16. K. Harigaya, Y. Wada, K. Fesser: Phys. Rev. Lett. 63. 2401 (1989);
Phys. Rev. B 42, 1268 and 1276 (1990)
17. K. Harigaya, Y. Wada, K. Fesser: Phys. Rev. B 43, 4141 (1991)
18. F. Bronold: Doct. Thesis, Univ. Bayreuth (1995)
19. K. Fesser: Prog. Theor. Phys. Suppl. 113, 39 (1993)
20. a) E.J. Mele and M.J. Rice: Phys. Rev. B 23, 5397 (1981)
b) G.W. Ryant and A.J. Glick: Phys. Rev. B 26, 5855 (1982)
c) S.R. Philpott et al.: Phys. Rev. B 35, 7533 (1987)
d) E.M. Conwell, S. Jeyadev: Phys. Rev. Lett. 61, 361 (1988)
21. a) S.A. Brazovskii, N. Kirova: Synth. Met. 55–57, 4385 (1993)
b) G. Paasch and T.P.H. Nguyen: unpublished (1995)
22. K. Kim, R.H. Mckenzie, J.W. Wilkins: Phys. Rev. Lett. 71, 4015 (1993)
23. B.D. Simons and B.L. Altshuler: Phys. Rev. B 48, 5422 (1993)
121

9 Diacetylene Single Crystals
Markus Schwoerer, Elmar Dormann, Thomas Vogtmann, and Andreas Feldner
9.1 Introduction
Polydiacetylenes can be grown as macroscopic polymer single crystals [1–3]. This property
is unique. They comprise one linear polymer axis and can have spatial extensions of up to
several millimetres or more in all three spatial directions (Fig. 9.1). The covalent chemical
bonds along the polymer axis make them mechanically strong along the corresponding crystallographic
axis. Their Young’s modulus is about a quarter of the Young’s modulus of steel
[4, 5] and their tensile strength along the polymer axis has been reported to exceed that of
steel [6]. Weak bonds of van der Waals-type perpendicular to the polymer axis are responsible
for extremely low dimensional – generally one-dimensional – macroscopic electronic
properties of these polydiacetylene single crystals. They are insulators, they can become
pyro- or ferroelectric, and they show large optical non-linearities. These electronic and optical
properties are primarily determined by the p-electron system along the polymer axis.
Most polydiacetylenes differ only in the substituents R and R' (Scheme 9.1). But the electro-
Figure 9.1: Photo of macroscopic paratoluylsulfonyloximethylene-diacetylene (TS6) single crystals under
polarized light. Monomer (top), polymer (bottom).
122 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

9.1 Introduction
Scheme 9.1: Polydiacetylene in its isomeric structures: acetylene-type (a) and butatriene-type (b). R
and R' are the substituents for different diacetylenes (see also Tab. 9.1).
nic structure of their all-trans planar carbon chain is at first approximation of acetylene-type
(a) rather than of butatriene-type (b). Both isomeric structures contain a non-interrupted
p-electron system along the carbon chain.
The term polydiacetylene is somewhat puzzling, at least for a physicist. However, it
becomes quite clear if one takes into account the structure of the diacetylene monomer
(Fig. 9.2). A typical substituent R is paratoluylsulfonyloximethylene (Scheme 9.2).
The diacetylene with R = R' = paratoluylsulfonyloximethylene is termed TS6 (sometimes
TS). It was shown by G. Wegner and his co-workers in a series of works, published in
the early 1970s [1, 7], that large molecular crystals can be grown from a solution of TS6,
e. g. in acetone, and that these monomer diacetylene crystals can be converted by a topochemical
(or solid state) 1,4-addition reaction to the polydiacetylene single crystals
(Fig. 9.2). The crystal structures of TS before and after the reaction have been investigated
in detail by Kobelt and Paulus [8], Bloor et al. [8], and Enkelmann [9] and are sketched in
Fig. 9.3 and in Tab. 9.1.
Figure 9.2: The monomer diacetylene crystal is converted by a topochemical or solid state reaction to
the polydiacetylene single crystal.
123

9 Diacetylene Single Crystals
Scheme 9.2: Diacetylene monomer with the substituents R = R' = paratoluylsulfonyloximethylene (TS6).
Figure 9.3: Monomer and polymer crystal structure of TS6 deduced from X-ray data [9]. Note the reactive
carbene at the chain end.
Both crystal structures are monoclinic with two monomers or monomer units of different
orientation per unit cell (Fig. 9.3 sketches only one of these two.). The chemical bond
between two carbons of nearest neighbour diacetylenes (1,4-addition) results in the linear
polymer chain and the small change in the lattice parameter along the b-axis prevents the
destruction of the macroscopic single crystal during the topochemical reaction. Several surfaces
of diacetylene crystals have been studied by atomic force microscopy (AFM) in order
to investigate both, the single crystal surface structure and solid state reactions at the surface
[129, 131].
124

9.1 Introduction
Table 9.1: TS monomer and polymer crystals (monoclinic, space group P2 1 /c) [9].
T/K a/Å b/Å c/Å b/ 8 D x
g/cm 3
TS monomer 120 14.61(1) 5.11(1) 25.56(5) 92.0(5) 1.46
TS monomer 295 14.60 5.15 15.02 118.4 1.40
TS monomer 295 14.65(1) 5.178(2) 14.94(1) 118.81(3) 1.40
TS polymer 295 14.993(8) 4.910(3) 14.936(10) 118.14(4) 1.483
TS polymer 120 14.77(1) 4.91(1) 25.34(2) 92.0(5) 1.51
The topochemical reaction can be induced thermally and/or photochemically and/or
by electron-beam irradiation. For TS the thermal conversion versus time (Fig. 9.4) is
strongly temperature dependent and highly non-linear. The thermodynamics of the integral
reaction has been investigated extensively by Bloor et al. [10], Eckhardt et al. [11], Chance
et al. [11], and others. The reaction diagram (Fig. 9.5) for TS shows that the dark reaction is
thermally activated and has an activation energy of 1 eV per monomer. It is exothermic with
a polymerization enthalpy of 1.6 eV per addition of one monomer. The entire reaction is irreversible
and the TS6-polydiacetylene (PTS) crystals are not solvable in ordinary solvents.
During the solid state reaction almost all properties of the diacetylene crystals change
drastically, e. g. the transparent monomer crystals are converted to polymer crystals with
highly dichroic, strongly reflecting surfaces which contain the b-axis. In transmission the
polydiacetylene crystals can only be investigated as thin films (Fig. 9.6). Their absorption
spectrum, e. g. for TS, clearly shows the vibronic spectrum due to the single, double and triple
bonds of the polymer chain [12]. Batchelder et al. [13] investigated extensively the optical
absorption, reflection, and Raman spectra of TS single crystals.
While these experiments were directed towards the study of the electronic excitations
of the bulk and their coupling to the vibrations, Sebastian and Weiser [14] investigated the
Figure 9.4: Time conversion curves for the thermal polymerization of PTS at 60 8C (.), 70 8C (#), and
80 8C (d) [9].
125

9 Diacetylene Single Crystals
Figure 9.5: Reaction diagram for the thermal polymerization of TS (solid curve) and photopolymerization
of 4BCMU (dashed curve) [10, 11].
Figure 9.6: Absorption spectra of a thin TS diacetylene single crystal for light polarized parallel and
perpendicular to the polymer axis b (T = 300 K). Monomer (M) and polymer (P) absorption.
defects by electroabsorption. As an example Fig. 9.7 shows the absorption and the electroabsorption
spectra, which they have analyzed in great detail.
Since about 1980 and especially during our work for the Sonderforschungsbereich 213
we have synthesized several new diacetylenes, the substituents of which are shown in
Tab. 9.2 [15]. For selected diacetylene single crystals we have investigated:
126

9.1 Introduction
Figure 9.7: Absorption a and electroabsorption Da of photoproducts in PTS-monomer as a function of
ko. Full curve: experimental spectra, dashed curves: fit by Lorentzian and a charge transfer model for
Da [14].
a) in Section 9.2 the elementary steps and structures during the solid state photopolymerization
by transient optical spectroscopy, by electron spin resonance (ESR), and by electron
nuclear double resonance (ENDOR);
b) in Section 9.3 the application of the photopolymerization of thick diacetylene single
crystals as a very effective holographic storage process;
c) in Section 9.4 the tailoring of diacetylenes as ferro or pyroelectric crystals, which do
not demand considerable efforts for the poling processes and which show good thermal stability;
d) in Section 9.5 the optical non-linearity of second and of third order and their application
for an optical device with femtosecond time resolution.
The present paper is a review of our work with diacetylene single crystals.
127

9 Diacetylene Single Crystals
Table 9.2: A survey of the investigated diacetylenes [15].
Survey of the investigated diacetylenes
R 1 C C C C R 2
DNP
TS
PD-TS
CD -TS
2
FBS
O 2 N
CH 2 O NO
CH 2 O SO 2
D
CD 2 O SO2
D
H
CD2 O SO2
H
CH
2
O SO
2
2
CH 3
D
CD 3
D
H
CH 3
H
F
R 2
ability to polymerize
therm. γ
= R 1
= R 1
= R 1
= R 1
= R 1
+++
+++
+++
+++
+++
-
+++
+++
+++
+++
IPUDO
Name R 1
CH O
O CH 3
(CH 2 ) 4 O C NH CH
= R 1
-
+++
NP/PU
CH 2
O NO2
CH 3
O
2 C NH
-
+
O
NP/4-MPU CH2 O NO
2 CH2 O C NH
CH 3 - +
NP/MBU
DNP/MNP
O 2
O H
CH2 O NO2 CH2 O C NH C + +
CH 3
(-)(S) or (+)(R)
N
O 2 N
CH
NO CH O CH
+++
2 O 2
2
3
DNP/PU
CH
2
N
O
O 2
O
NO2 CH2 O C NH
- -
DNP/4-MPU
O2
N
CH
2 O
O
NO2 CH2 O C NH CH 3
- -
O 2 N
DNP/DMPU CH2 O
NO2
CH
O CH3
O NH
2 C
CH 3
- -
DNP/MPU
O 2 N
CH O
2
O H
NO2 CH2 O C NH C
CH
(-)(S) or (+)(R)
3
- -
TS/FBS CH2 O SO2
CH 3 CH2
O SO 2 F +++
FBS/TFMBS CH2 O SO 2 F CH2 O SO2 CF 3 +++
128

9.2 Photopolymerization
9.2 Photopolymerization
9.2.1 Carbenes
The aim of this Chapter is to review our spectroscopic work towards the analysis of both,
the electronic structures and the dynamics of the intermediate reaction products (Fig. 9.2),
during the photopolymerization, i. e. after the excitation of the solid state reaction by light.
For the entirely thermally activated solid state polymerization detailed spectroscopy of the
intermediate states turned out to be difficult or not very efficient. One exception was the
identification of carbenes as reactive species during the thermal solid state polymerization
of TS6 (Fig. 9.8).
In contrast to radicals with one non-bonded electron carbenes have two non-bonded
electrons. It has been shown for the first time by Wassermann et al. [16] that the electronic
ground state of pure methylene (:CH 2 ) is a triplet state, where the total spin quantum number
S is 1. Because of their non-centrosymmetry most molecular triplet states show a splitting
into three components even in the absence of an external field. This splitting is due to the
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
2hν
R
C
C
C
C
R
R
C
C
C
C
R R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R C R
C
C
R
C
C
R
C
C
C
R
kT
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R R
C
C
C
C
R C R
C
R
C
C
C
kT
R
C
C
C
R C
R
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R
R
C
C
C
C
R C R
C
C
C
R C R
C
C
R C
C R
C
C
R
C
C R
C
C
C
R
kT
R
C
C
C
C
R R
C
C
C
C
R
R
C
C
C
C
R C R
R C
C
C
C
R C
R
C
C
C
R
Figure 9.8: UV photopolymerization of TS6: The monomer crystal is irradiated with an UV-flash. The
dimer is formed from the monomer by a photoreaction, then a series of thermally activated monomer
addition reactions leads via the diradicals (DR) and dicarbenes (DC) to the polymer.
R
C
C
C
C
C
C
C
C
C
R
R
R C C R
kT
R
C
C
C
C
R
R
C
C
C
C
R C R
R
R
C
C
C
C
C
R C
C
C
C
C
C
C
C
C
C
C
R
R
R
R C C R
kT
C
R C
R
C
C
C
R
R
C
C
C
C
R C R
R
R
R
C
C
C
C
C
C
C
C
R C R
C
C
R C C
C
C
C
C
C
C
C
C
C
R C
C
R
R
R
R
R
C
C
C
R
kT
R
R
R
R
R
R
R
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
R
R
R
R
R
R
C
R C
R
C
C
C
R
129

9 Diacetylene Single Crystals
magnetic dipole-dipole coupling of the two unpaired electrons and is called zero-field splitting.
In an external magnetic field the resulting fine structure of the ESR spectra is highly anisotropic,
i. e. it strongly depends on the direction of the external magnetic field with respect
to the orientation of the tensor describing the magnetic dipole-dipole interaction. This interaction
is of course an intramolecular property and therefore single crystals are ideal candidates
for the measurement and the quantitative analysis of molecular triplet states by ESR. As originally
shown by the work of Hutchison and Mangum [17], and van der Waals [18] the ESR
spectrum of a molecular triplet state is described by the spin Hamiltonian T H s [19]
T H s ˆ B B o g S^
‡D S^ 2
z ‡ E…S^ 2
x S^ 2
y † ; …1†
where m B is Bohr’s magneton, B 0 the external magnetic field, g the spectroscopic splitting
factor, ^S the spin operator for a spin with total spin quantum number S = 1, ^S u with u = x, y, z
are the components along the principal axes of the magnetic dipole-dipole interaction tensor.
D and E represent the two independent values of this tensor, the trace of which is zero. They
usually are called zero-field splitting parameters:
D ˆ o
4p
E ˆ o
4p
3
4 g2 2 z 2
B hr2 r 5 i ; …2†
3
4 g2 2 x 2
B hy2 r 5 i ; …3†
r is the distance of the two unpaired electrons and x, y, and z are the components of r along
the principal axes of the magnetic dipole-dipole interaction tensor of these two electrons.
Only for systems of cylindrical symmetry (around z) the zero-field splitting parameter E
vanishes (E = 0). And only for spherical symmetry (Ax 2 S = Ay 2 S = Az 2 S = 1/3 Ar 2 S) the entire
fine structure is zero. The values of D and E for the triplet carbenes as detected during the
purely thermal solid state polymerization [20] are shown in Tab. 9.3 in comparison with typical
values of different molecular triplet states. By these values and especially by the strong
anisotropy of the fine structure in the ESR spectrum these carbenes (as sketched in Fig. 9.8)
are clearly identified as reactive species during the pure thermal solid state polymerization
of TS. They will play a major role in the electronic structures of the intermediate products
during the photopolymerization as described in the following paragraphs.
Table 9.3: Zero-field splitting parameters for the reactive species during the thermal solid state polymerization
of TS [20], pure methylene [16], diphenylmethylene [21], and benzene [22] in their first excited triplet
state.
D
hc =cm–1
E
hc =cm–1
TS 0.2731 –0.0048
:CH 2 0.6636 0.0003
:C(C 6 H 5 ) 2 0.39644 –0.01516
C 6 H 6 0.1581 –0.0046
130

9.2 Photopolymerization
9.2.2 Intermediate photoproducts
Further experiments for analyzing the electronic structure of intermediate states were not
very successful, until Sixl et al. [23] and Bubeck et al. [24] published their first low-temperature
spectroscopic experiments on partially photopolymerized TS crystals. Thereby, the
monomer crystal is cooled to 4.2 K in the dark. Then it is irradiated with UV light
(l ^ 310 nm) for a short period. After this procedure a large number of different species
show up in both, the optical absorption and the ESR spectrum. They persist at helium temperature.
Subsequent annealing in the dark produces further intermediate reaction products
which are also identified by their optical or ESR spectra. Finally, further irradiation with
visible light, which is absorbed only by the intermediate reaction products, produces still
more and different reaction products. In a comprehensive series of investigations [25–47]
most of these intermediate products have been identified and classified. Moreover, the mechanisms
of their production and their reaction kinetics have been analyzed. According to
Sixl, there exist three different series of intermediate products:
1. The diradical-dicarbene series: DR 2 , DR 3 , DR 4 , DR 5 , DR 6 , DC 7 , DC 8 , DC 9 , DC 10 ,
DC 11 … polymer;
2. The asymmetric carbene series (AC);
3. The stable oligomer series (SO).
The DR-DC series leads directly from the monomer to the polymer. It is initiated and
processed in the following simple and clear way. The monomer crystal is irradiated with an
UV flash and subsequently rests in the dark. The flash excites the monomers and produces
dimers (DR 2 ). These react by a thermally activated step by step addition of monomers, as illustrated
in Fig. 9.8. In the following paragraphs we will show for a few selected examples
how these results have been achieved and we will present details of both, the electronic
structures and the dynamics.
The AC and the SO series are produced by additional irradiation with light, i. e. these
are photoproducts which do not necessarily arise during the solid state polymerization of
TS6. Although they are an important part of the entire variety of structures in partially polymerized
TS6 crystals, we will not treat them in this review. They have been described extensively
by Sixl in his papers cited above and in Ref. [48].
9.2.3 Electronic structure of dicarbenes
9.2.3.1 Electron spin resonance of quintet states ( 5 DC n )
As an example for the electronic structure analysis of the intermediate states with ESR, we
will review below the ESR spectra of the dicarbenes DC 7 …DC 13 [28, 32, 42]. They are
characterized by their fine structure and temperature dependence. Figure 9.9 shows the ESR
131

9 Diacetylene Single Crystals
ESR-Signal
T
Q
T
T
Q
Q Q T
Q
Q
Q
T
Q
T=10 K
ν =9,46 GHz
T
0 200 400 600
Magnetic Field B 0/mT
Figure 9.9: The ESR spectrum of perdeuterated TS after irradiation for 1000 s. The signals marked
with T arise from triplet states and those with Q from quintets. The T-lines are microwave saturated. The
magnetic field B 0 is oriented parallel to the z-axis of the quintet fine structure tensor [28].
spectrum of a perdeuterated TS crystal which has been irradiated with UV light (313 nm) of
a mercury high-pressure arc lamp (HBO 200) at 4.2 K for about 1000 s. All ESR lines in
Fig. 9.9 labelled with Q are due to dicarbenes in their quintet state (S = 2). Prior to the UV
irradiation no ESR signal is observed.
The temperature dependences of the ESR signals (Fig. 9.10) show one common feature:
the ESR intensities vanish for T ? 0, i. e. the ESR signals are thermally activated and
the ground state is spinless. But the temperatures for the maximum intensities are different
for each ESR signal, indicating different activation energies.
1
10
5
T/K
2
Signal Intensity
0,5
0
0
0,2 0,4 0,6 0,8
1/T/K -1
Figure 9.10: Temperature dependence of the ESR intensities of dicarbenes DC 9 ,DC 10 , and DC 11 . The
calculated lines have been fitted by Eq. 9 with the activation energies De* SQ for the quintet states [33, 47].
132

9.2 Photopolymerization
The resonance fields for fixed microwave frequencies are strongly anisotropic.
Figure 9.11 shows as an example this anisotropy for the dicarbenes DC 9 ,DC 10 ,DC 11 , and
DC 12 , respectively. The external field has been rotated to the polymer backbone plane. Each
anisotropy belongs to one distinct temperature dependence.
The following model (Fig. 9.12 and inset Fig. 9.13) describes quantitatively the anisotropic
fine structure and the temperature dependence. The ground state (|S>) of each dicarbene
in first order is a singlet state with the total spin quantum number S = 0. The excited
state (|Q>) in first order is a quintet state with S = 2. The excitation energy is De SQ . The
quintet state is built up by the electronic coupling of two triplet carbenes at both ends of the
oligomer via exchange interaction. R 12 is the distance between the two triplet carbenes. If
the singlet-quintet splitting De SQ is large as compared to the magnetic dipole-dipole coupling
(zero field splitting) within the triplet carbenes, the total spin quantum numbers S = 0
and S = 2, respectively, are good quantum numbers, i. e. the quintet state is pure. The spin
Hamiltonian for this case has been analyzed by Schwoerer et al. [34].
Figure 9.11: Angular dependence of the resonance fields B 0 of the 4 dicarbene structures DC 9 ,DC 10 ,
DC 11 , and DC 12 in perdeuterated TS-diacetylene crystals. The crystal is rotated so that the external
magnetic field B 0 is in the plane of the polymer backbone. The b-axis is the direction of the polymer
chain, y and z are the principal axes of the fine structure tensor. The curves are fitted to the experimental
points by computer calculations. A and B indicate the two magnetically equivalent directions of the
molecular orientation within the monoclinic unit cell. Dots: experimental values; lines: calculated by
Q H S 0 (Eqs. 4–7) [32, 42].
Figure 9.12: Dicarbene configuration. The dicarbene molecule consists of n diacetylene units with two
identical triplet carbene chain ends. D t and E t are the triplet fine structure parameters of the S = 1 carbene
species. j gives the orientation of the triplet fine structure z-axis with respects to the crystal b-
axis.
133

9 Diacetylene Single Crystals
Figure 9.13: Experimental values for the singlet-quintet splitting De SQ for seven different dicarbenes
DC 7 …DC 13 [42].
If De SQ is smaller than or somewhere in the order of the magnetic dipole-dipole coupling
then the singlet and quintet states are mixed. The spin Hamiltonian Q H S for this general
case has been derived in the elegant work of Benk and Sixl [35]:
Q H S ˆ g B B 0 …^S 1 ‡ ^S 2 †‡ 1 6 " SQ …^S 1 ‡ ^S 2 † 2 ‡H DD
S
…4†
The first term of Eq. 4 represents the electronic Zeeman term, ^S 1 and ^S 2 the spin operators
for two triplet carbenes, and the second term represents the electronic exchange interaction.
If ^S 1 and ^S 2 couple to a quintet state (S = 2), (^S 1 + ^S 2 ) 2 = S 2 = S(S + 1) = 6. If
they couple to a singlet, S = 0. Therefore, this term directly results in the energy level
scheme, indicated in the inset of Fig. 9.13. The pure singlet and the pure quintet states are
split by De SQ which turns out to be the characteristic property of each dicarbene. The third
term of Eq. 4 represents the magnetic dipole-dipole coupling of the two triplet carbenes:
H DD
S
^
ˆ D …S 2 1z
^
1
3 S2 1 †‡E…S2 1x
X S^
1x S^
2x ‡ XAS^
1y S^
2y
^
S 2 ^
1y†‡D…S 2 2z
1
^
3 S2 2
^
†‡E…S 2 2x
^
S 2 2y†‡
X…1 ‡ A† S^
1z S^
2z ‡ Xa…S^
1z S^
2y ‡ S^
1y S^
2z† …5†
D and E are the fine structure parameters of the identical triplet carbene chain ends;
x, y, and z are the principal axes of the corresponding fine structure tensors. The intercarbene
magnetic dipolar interaction is represented by the parameter
134

9.2 Photopolymerization
X ˆ g 2 2 B … 0=4p†R 3
12 : …6†
The geometrical factors A and a are only dependent on the orientation j of the fine
structure tensor z-axis with respect to the b-axis of the crystal.
A ˆ 1 3 sin 2 '; ˆ 3 sin 2': …7†
2
The diagonalization Q H S of yields the allowed ESR transitions, their resonance fields,
and their dependence on the orientation of the external magnetic field B 0 . At a first sight,
the number of fit parameters seems to be high: D, E, g, the orientation of the triplet fine
structure tensor, R 12 , and De SQ . But besides R 12 and De SQ these parameters are well-known
from earlier ESR experiments on triplet carbenes [20, 49, 50]. Therefore, R 12 and De SQ are
the only free to fit parameters. Figure 9.11 shows the result of the fit. The ESR anisotropy
was calculated by exact diagonalization of Q H S with the fitting parameters R 12 and De SQ .In
all cases the fit is almost perfect. Furthermore, it turns out that the intertriplet magnetic dipole-dipole
interaction does not influence the results if R 12 exceeds 12 Å. Therefore, only
the parameter De SQ remains to fit for each dicarbene. The obvious differences between the
anisotropies of the different dicarbenes (Fig. 9.11) are due to De SQ only! The result is shown
in Fig. 9.13 where De SQ is decreasing exponentially with increasing number n of monomer
units:
" SQ ˆ " 0 SQ e R=R 0
; …8†
with R = n1 m (1 m is the length of the monomer unit within the oligomer). The slope in
Fig. 9.13 yields R 0 = 5.4 Å&10 a 0 (a 0 = Bohr radius). The origin of the abscissa in Fig. 9.13
was deduced by Neumann [47]. But even without the knowledge of this origin the value of R 0
as compared to a 0 shows that the triplet carbene is highly delocalized and not at all restricted
to the end of the oligomer, as one might think because of Fig. 9.12. The model also gives the
temperature dependencies of the dicarbene signals. Because in first order the ground state is
spinless, the ESR intensities I of the quintet states (beyond Curie’s law) must be thermally activated:
I / 1 T ‰5 ‡ exp…" SQ =kT†Š :
…9†
The lines in Fig. 9.10 are calculated with Eq. 9 by fitting De* SQ . For the dicarbenes
DC 8 ,DC 9 ,DC 10 , and DC 11 , both values, De* SQ and De SQ , are determined respectively. Within
the experimental uncertainty they are identical [32, 42]. This latter result is an excellent
proof for the dicarbene model because De* SQ and De SQ were determined from two completely
different properties: the anisotropy of the fine structure and the intensity of the ESR spectra.
We therefore have no doubt that we really did observe dicarbenes (as sketched in Figs. 9.8
and 9.12). They are an ideal modelling substance for short linear oligomers of diacetylenes
which are perfectly oriented in the single-crystal lattice. The longest dicarbene (DC 13 ) observed
has according to our model a length of (R =1364,9 Å) 64 Å!
135

9 Diacetylene Single Crystals
9.2.3.2 ENDOR of quintet states
The most important result of the preceding Chapter is the delocalization of the spins S = 1
of the two triplet carbenes which is necessary for their coupling over a distance of up to
64 Å (!) to the well defined quintet states. It was therefore attractive to measure and analyze
the ENDOR spectrum of at least one quintet dicarbene. ENDOR should detect at
least the protons of the CH 2 groups of the substituents if the nuclear spins of these protons
are hyperfine coupled with the electron spin of the triplet carbene. As compared to
ENDOR with an electron spin of S = 1/2, the complication is the high anisotropy of the
five electronic Zeeman levels Q u (u = 1 … 5) of the quintet state (Fig. 9.14). Not only
their energy separation, i. e. the ESR transition fields (Fig. 9.11), are strongly dependent
on the direction of the external field. Also their effective spin S eff , i. e. the expectation
value of the spin, is dependent on the direction and on the strength of the external field
B 0 . Therefore in this case the orientation quantum number m s is an unsuitable quantum
number not only because of the large zero field splitting, as expressed by D, but also because
of the singlet-triplet mixing, as expressed by De SQ . As described in the preceding
paragraph this problem has been solved with high accuracy, Hartl et al. [36] measured
the ENDOR spectra and their anisotropies for one quintet dicarbene ( 5 DC 10 ), which is accessible
most comfortably at T = 4.2 K (Fig. 9.10) and for which the total spin quantum
number (S = 2) is a good quantum number. The aim of these experiments was to determine
the hyperfine coupling constants with the above-mentioned protons and subsequently
to extract the electron spin density from these values, i. e. the delocalization of
the triplet carbene quantitatively.
Figure 9.14: Quintet state with an external field B o interacting with one proton (I = 1/2). All allowed
ESR transitions (a–d) and NMR transitions (1–5) are shown. By ENDOR, in first order, only the two
NMR transitions directly connected to the observed ESR line are detectable [36].
136

9.2 Photopolymerization
The spin Hamiltonian Q H S;i of a quintet dicarbene coupled to one individual proton,
numbered i, is
Q H s;i ˆ QH 0 s ‡ g I K B 0 ^I i ‡ ^SA i I i :
…10†
The first term in Eq. 10 is given by Eq. 4 and the second is the nuclear Zeeman energy.
^I i is the nuclear spin operator for nuclear spin 1/2. The third term is the hyperfine interaction
of the individual proton i, as defined by the hyperfine tensor A i . ^S is the total electron
spin operator, ^S ˆ ^S 1 ‡ ^S 2 . As nuclear dipole-dipole interaction can be neglected we
will omit the index i in the following.
The nuclear terms in Eq. 10 are small as compared to the electronic terms. Therefore,
we treat them in first order perturbation theory, taking the solutions of Q H 0 S as basis. Q H 0 S
has five quintet eigenstates |Q u S , u = 1, 2, 3, 4, 5. For very high fields they become the
high field states |Q m S, for which ^S z |Q m s
S = km s |Q m s
S,m s = +2, +1, 0, –1, –2. But for the
fields used in ordinary ESR spectrometers the electronic Zeeman energy is in comparison to
the fine structure not large and therefore the electron spin is not quantized along the external
field B 0 [39]. This results in a strong |Q u S-dependence on the direction of B 0 with respect to
the crystal axes.
From Q H 0 S the effective spin
S u eff ˆ
D E
Qu j ^S jQ u
…11†
can be calculated exactly [34]. It is this electron spin which interacts via A with the proton
spin.
The first order perturbation theory of the nuclear terms calculates the shift Dn u of the
individual proton Larmor frequency with respect to the free proton Larmor frequency n F
(ENDOR shift):
hDn u = hn u – g I m K |B 0 | , (12)
(g I m K |B 0 |/h = n F ). The result of the calculation of the Larmor frequencies n u of the hyperfine
coupled protons is [36, 43]:
h u ˆjS u eff A g I K B 0 j : …13†
For a quintet state one should observe 5 different ENDOR lines per proton. This is illustrated
in Fig. 9.14 where the observed NMR transitions in the ENDOR experiment are indicated
by the ciphers (n) = (1), (2), (3), (4), and (5); the first order ESR transitions are indicated
by the lower case letters (a), (b), (c), and (d).
Fig. 9.15 shows four ENDOR spectra as detected via the ESR transitions (a), (b), (c),
and (d), respectively. Four protons i = 1, 2, 3, and 4, respectively, are clearly separated from
the free proton frequency n F . In the vicinity of n F a large number of weakly coupled protons
are visible. They also have been resolved by expansion of the NMR frequency scale. For a
few strongly coupled protons we are able to detect all five NMR transitions (1) to (5). This
is a further and definite proof that we do observe quintet states.
137

9 Diacetylene Single Crystals
3(2)
2(2)
1(2)
(a)
Bo II X
= 9.570 MHz
ν F
ν F
(c)
Bo II Y
ν F = 14.193 MHz
2(1)
1(4) 2(4) 3(4)
4(4)
0
ν F
3(2) 2(2)
10 20 30
40
0 10 20 30
(b)
Bo II Y
= 10.911 MHz
ν F
2(4)
ν F
ν F
(d)
Bo II X
= 15.553 MHz
ν F
3(4)
4(4)
4(2)
1(2)
0 10 20 30 0 8 12 16 20
NMR Frequency ν/MHz
NMR Frequency ν/MHz
NMR FREQUENCY ν / MHz
Figure 9.15: ENDOR spectra as detected by the ESR transitions (a–d); n F is the free proton frequency.
(1) to (5) label the NMR transitions illustrated in Fig. 9.14, and 1, 2, 3 … number the individual protons.
The external field B o is oriented along the y or x-axis of the fine structure tensor [36].
The ENDOR shift anisotropy is shown in Fig. 9.16 for the strongly coupled protons
i = 1, 2, 3, and 4. This anisotropy is mainly due to the anisotropy of the effective spin S u eff.
The lines were calculated (via Eqs. 12 and 13) by fitting A i . In total we have analyzed
22 protons, the hyperfine tensors A i of which are presented in Tab. 9.4. Two features can be
Figure 9.16: ENDOR shift anisotropies for the four strongest coupled protons i = 1, 2, 3 and 4, detected
via the ESR transition (b). x, y and z are the principal axes of the fine structure tensor. The experimental
values were taken for the rotation of B 0 in the yz-plane and in the zx-plane, respectively. The ENDOR
shifts Dn are calculated for (b)i(2) transitions (drawn out) and the (b)i(3) transitions (dashed), respectively
[36].
138

9.2 Photopolymerization
Table 9.4: Complete hyperfine tensors for 22 protons, calculated by fitting the experimental anisotropy
in a least-squares method. The A ij are diagonalized principal values of the fitted tensor, a is the isotropic
coupling constant, B jj the dipolar anisotropic tensor, F, Y, and c are Euler angles of the hyperfine tensor
axes relative to the fine structure axes [36].
i A xx A yy A zz a B xx B yy B zz F Y C Assign-
MHz MHz MHz MHz MHz MHz MHz degr. degr. degr. ment
1 16.298 18.953 16.747 17.333 –1.035 1.620 –0.586 13.6 –51.8 2.8 CH2
2 11.870 14.290 10.683 12.281 –0.411 2.009 –1.598 –39.2 76.4 –12.8 CH2
3 2.348 3.737 5.667 3.917 –1.569 –0.180 1.749 28.0 80.5 –6.9 CH2
4 1.905 1.797 1.253 1.652 0.253 0.146 –0.399 74.9 101.2 4.6 CH2
5 1.788 0.710 –0.391 0.702 1.086 0.007 –1.093 56.0 –16.4 –21.8 CH2
6 0.407 0.804 0.703 0.638 –0.231 0.166 0.065 10.3 10.1 –36.9 CH2
7 1.456 0.089 –0.119 0.475 –0.981 –0.387 –0.594 26.3 –25.4 –4.4 CH2
8 0.700 0.620 0.076 0.465 0.234 0.155 –0.389 57.4 –15.3 8.0 CH2
9 –0.791 0.981 0.520 0.237 –1.028 0.744 0.284 –23.4 –80.7 –15.4 CH2
10 –0.992 1.424 0.148 0.194 –1.185 1.230 –0.045 –23.2 1.6 –67.2
11 0.219 –0.079 0.400 0.180 0.039 –0.259 0.220 –67.5 111.6 41.8
12 0.098 0.161 0.271 0.177 –0.079 –0.016 0.094 21.5 90.6 –25.7
13 –0.819 1.321 –0.275 0.076 –0.895 1.245 –0.351 –6.7 –5.7 –6.0 CH2
14 0.075 –0.120 0.239 0.065 0.011 –0.185 0.174 12.6 7.3 –38.9
15 1.969 –1.045 –0.889 0.012 1.958 –1.057 –0.901 –68.2 –7.4 49.5 ARYL
16 –0.066 –0.165 0.235 0.002 –0.067 –0.166 0.234 –24.1 –5.2 38.5
17 –0.456 –0.051 –0.203 –0.203 –0.254 0.254 0.000 91.1 114.5 27.5
18 2.350 –2.533 –0.473 –0.219 2.569 –2.314 –0.254 –20.4 9.5 –53.6 ARYL
19 –0.311 –0.237 –0.442 –0.330 0.019 0.093 –0.112 –23.5 91.3 89.0
20 –0.777 –0.289 –0.143 –0.403 –0.374 0.114 0.260 –36.0 –30.2 18.1 ARYL
21 –2.070 1.303 –1.288 –0.685 –1.385 1.988 –0.603 6.0 –49.3 6.5 CH2
22 –1.072 –0.735 –0.652 –0.820 –0.252 0.085 0.168 4.7 18.6 –27.0 CH2
extracted from Tab. 9.4 immediately: first, the anisotropic part of the hyperfine tensor is
small as compared to the isotropic part for almost all protons, and second, several protons
(i = 17, 18, 19, 20, 21, 22) show a negative value of the isotropic coupling constant a.
We are able to assign 6 protons unambiguously: i = 1, 2, 3, 13, 21, and 22 (Fig. 9.17).
The analysis of the hyperfine data shows that the spin density at C 1 (Fig. 9.18) is only 11%,
–2,2% at C 2 , 17% at C 3 , –6% at C 4 , 7,9% at C' 1 and –2% at C' 2 , i. e. the spin is highly delocalized
within the oligomer (Fig. 9.18). This corresponds with the above-described exchange
coupling of the two carbenes and is to our opinion a very impressive demonstration
of the power of ENDOR.
9.2.3.3 ESR and ENDOR of triplet dicarbenes 3 DC n
For a long time it was unclear whether the triplet state (S = 1) of the dicarbenes ( 3 DC) does
exist, and if it does whether its energy is higher or lower than the energy De SQ of the dicarbene
quintet state ( 5 DC). Müller-Nawrath et al. [51] have shown theoretically and experimentally
that the ESR transitions of the triplet states of carbenes ( 3 C) and of dicarbenes ( 3 DC),
respectively, are mutually degenerated in diacetylene oligomers of diacetylene crystals if the
139

9 Diacetylene Single Crystals
Figure 9.17: Assignment of 6 hyperfine tensors to methylene protons at the polymerization head. The
arrows indicate the directions of the strongest main value of the respective hyperfine tensors. These 6
tensors have been fitted simultaneously yielding the spin density distribution shown in Tab. 9.4 [36].
20
17
15
10
7,9
11
5
0
-5
-2
-2,2
-6
-10
C2' C1' C4 C3 C2 C1
Figure 9.18: Carbene spin density at the reactive polymer chain end (Fig. 9.17).
oligomers are long, i. e. if the number of added monomers n is more than 7. Therefore 3 C
and 3 DC cannot be discriminated by ESR. Their ENDOR spectrum, however, is completely
different. The ENDOR shifts Dn are related by Dn DC = –1/2 Dn C . The existence of several
ENDOR lines, which fulfills the characteristic factor –1/2 in the above-mentioned relation,
unambiguously shows the existence of the triplet state of dicarbenes. Its excitation energy is
lower than the excitation energy De SQ of the quintet states of the same dicarbene [51].
140

9.2 Photopolymerization
9.2.4 Flash photolysis and reaction dynamics of diradicals
A single UV laser flash with wavelength l = 308 nm, pulsewidth 15 ns, and flash energy
1 mJ initiates photopolymerization by the production of the diradical DR 2 [52]. At low temperature,
i. e. 4.2 K, this photoproduct is stable. It is detected by its absorption spectrum. In
the spectral range the 0-0 transition peaks at 422 nm between monomer and polymer absorption
(Fig. 9.20a). Annealing the crystal containing photoproduct DR 2 , prepared as described
above, produces the diradicals DR 3 ,DR 4 ,DR 5 , and DR 6 respectively by addition of one
monomer per step (Fig. 9.8). All these diradicals can be detected by optical absorption spectroscopy,
for example by cooling the crystals to low temperature in order to slow down the
dark reaction (Fig. 9.20b). Sixl et al. [48] were the first who detected these optical absorption
spectra [131]. The dark reaction at low annealing temperatures has been investigated extensively
by Gross [38, 46].
If the reaction is photoinitiated by a single UV laser flash at high temperatures
(T > 180 K) the entire time-dependent reaction series, DR 2 ? DR 3 ? DR 4 ?DR 5 ?DR 6 ,
can be observed by monitoring the transient optical absorption of each of the products DR 2
to DR 6 . By this experiment we were able to analyze the reaction kinetics of these thermally
activated steps separately [37, 44]. As an example Fig. 9.19 shows the transient absorption
0.23
T=270 K
DR 422nm
2
0
0.23
DR 514nm
3
∆OD
0.29 0
DR 578nm
4
0.5 0
DR 664nm
5
0
0
10
t/ µ s
20
Figure 9.19: Time sequence of the intermediate products DR 2 ,DR 3 ,DR 4 , and DR 5 at T = 270 K after
the UV flash which is indicated by an arrow. DOD is the change of the optical density after the UV
flash [37, 44].
141

9 Diacetylene Single Crystals
Figure 9.20a: Difference of the optical absorption spectra (DOD) of TS6 after and before one single
UV pulse (l = 308 nm, t = 15 ns, E = 0.1 mJ, T = 80 K). The peak at 422 nm is due to the absorption
of the Dimer DR 2 .
Figure 9.20b: Appearance of DR n reaction intermediates after a single pulse irradiation at 308 nm and
additional annealing. (a): 5 K, 0 min; (b): 100 K, 6 min; (c): 100 K, additional 30 min plus 120 K,
36 min; (d): 130 K, 36 min plus 140 K, 16 min [131].
at T = 270 K as detected by the change of the optical density (DOD) after the flash vs. time.
Each intermediate product is detected at the maximum of its optical absorption: DR 2 at
422 nm, DR 3 at 514 nm, DR 4 at 578 nm, and DR 5 at 664 nm. The delay in the production
of a subsequent intermediate is clearly demonstrated. The whole reaction passes in a 10 ms
time scale at room temperature.
Assuming that DR 2 is produced by the flash promptly, DR 3 by a dark reaction from
DR 2 ,DR 4 by a subsequent dark reaction from DR 3 , and so on, we used a simple kinetic
142

9.2 Photopolymerization
model for the quantitative analysis. This model is described by the following equations for
the concentrations n i and the rate constants K i for the DR i , i =1,2,3,4,5:
dn i
dt ˆ K i 1 n i 1 K i n i ; …14 a†
K i ˆ K 0 e …E i=kT† : …14 b†
As an example, the fit of this model to the transient DR 4 at 200 K is shown in
Fig. 9.21. The fit does not show any significant deviation from the experimental curve.
Figure 9.21: Experimental transient and model curve according to Eq. 14a for DR 4 in perdeuterated TS
at T = 200 K. The difference between experiment and model is plotted around the baseline [37, 44].
One result of these experiments is shown in Fig. 9.22. The addition reactions are thermally
activated, the activation energies being about 0.25 eV per monomer, almost identical
for each step (Tab. 9.5). Figure 9.22 also includes values for a product labelled V which is
presumably due to long polymer chains with reactive carbene chain ends.
A second result of these experiments is an estimation of the polymer yield for the
photoreaction as defined by the number of polymerized monomer molecules per absorbed
UV photon. Q is the quantum yield for the initiation process. Thus, the polymer yield P is
the product of Q and the kinetic chain length L, i. e. the number of monomer molecules
which are added to the chain after one initiation process:
P = Q7L = 0.07 ± 0.02 (15)
The polymer yield was determined from the increase of the polymer absorption DOD
due to UV irradiation [53].
If we assume a kinetic chain length of 100, which is a reasonable value, we get a quantum
yield for chain initiation of 7610 –4 . P increases with decreasing temperature from
300 K to 180 K by nearly a factor of five. If TS is perdeuterated P also increases by a factor
of 2.5 [44, 54].
Both, the temperature effect and the isotope effect on the polymer, can be explained
qualitatively by an increase of the kinetic chain length due to a longer carbene chain end
lifetime [37, 44, 54].
143

9 Diacetylene Single Crystals
Figure 9.22: Temperature dependencies of the rate constants for the decays of DR 2 ,DR 5 , and V. They
can be described by the Arrhenius law in a range of three orders of magnitude [37].
Table 9.5: Activation energies DE i and frequency factors K o for the reaction rates of DR i , evaluated from
their Arrhenius plots [37].
K 2 K 3 K 4 K 5
DE i / eV 0.25 ± 0.03 0.26 ± 0.03 0.30 ± 0.03 0.30 ± 0.03
K o /s –1 10 10 ± 1 10 11 ± 1 10 11 ± 1 10 11 ± 1
9.3 Holography
Diacetylenes have been subject to intense work due to their unique ability to undergo topochemical
solid state polymerization, resulting in macroscopic polymer single crystals [1–3,
25, 28, 37]. Whether this reaction takes place depends on the monomer stacking distance
and the tilt angle (Fig. 9.3). Both can be influenced by varying the rest groups R. The polymerization
can be initiated by heat, UV radiation, X rays, or g rays and is irreversible. The
optical properties, especially the absorption coefficient a and the refractive index n, are
known to change dramatically during polymerization.
By means of UV photopolymerization high-efficiency holographic grating on diacetylene
crystals can be recorded, as was first shown by Richter et al. [55, 56]. Utilizing a fre-
144

9.3 Holography
quency-doubled argon laser (l w = 257 nm) Richter et al. obtained surface phase gratings,
due to the low penetration depth of this UV wave (Fig. 9.6). At high exposures higher diffraction
orders up to five has been observed. Because of the 5% stacking distance mismatch
between monomers and polymers, Richter et al. often observed a destructive surface peel
off. This problem has been shown to become less important by using longer UV wavelengths
and, therefore, higher penetration depths [57].
The first aim within the Collaborative Research Centre 213 was to find a suitable technique
for recording such gratings in an effective and reproducible way. Using this technique
we have investigated the most important diffraction characteristics of these gratings: efficiency,
thickness, angular selectivity and their dependence on exposure, sample thickness,
and prepolymerization. The second aim, however, was to explain these characteristics within
appropriate theoretical approach. Using this knowledge, the chain length of the polymers during
UV polymerization and subsequent thermal treatment can be estimated. In the final investigation
we have shown that images and even a holographic trick film can be recorded in the
diacetylene crystals at room temperature. The peculiarity of the method is the difference of
recording (UV) and reading (VIS) wavelengths. This difference allows prompt readout without
a developing process and without perturbation of the hologram by the readout laser.
9.3.1 Theory
Figure 9.23 shows the writing and reading beams for recording and replay of the simple holographic
grating, respectively. Provided that they are of equal intensity two coherent UV
Figure 9.23: Schematic geometry of writing and reading of a holographic grating. Two coherent UV
waves (dashed) form the grating of a spatial periodicity L. A VIS wave (solid) will generally be diffracted
into different orders j.
145

9 Diacetylene Single Crystals
waves, impinging symmetrically the photoactive medium, form an intensity pattern on the
surface of the photoactive material,
I…x† ˆI 0 cos 2 …px=L† :
…16†
L is the grating distance given by
L ˆ w =2 sin w ;
…17†
where l w is the vacuum wavelength and y w the angle of incidence outside the medium. This
intensity pattern results in a photoproduct distribution of the same periodicity, represented
by a grating vector K,
K ˆ 2p ^x=L ;
K ˆ jKj ˆ 2p=L ; …18†
and results in a UV photopolymerization pattern, the refractive index n, and the absorption
coefficient a of which can be described as
n…x† ˆn 0 ‡ P1
n h cos …Khx† ;
hˆ0
a…x† ˆa 0 ‡ P1
a h cos …Khx† :
hˆ0
…19†
The importance of the Fourier coefficients besides h = 1 generally depends on reaction
kinetics, exposure, and saturation effects. In general, n and a can also vary with the z coordinate.
In principle a readout light beam (vacuum wavelength l r ) will be diffracted by the
grating resulting in different orders j as indicated in Fig. 9.23. Each of them have an amplitude
S j and a propagation vector r j . The total electric field inside the medium is a superposition
of all theses waves:
E…z† ˆ
jˆ1 P
jˆ 1
S j …z† ^s j exp… ir j r† : …20†
Here ^s j are the polarization vectors. The wave vectors r j are coupled to the grating
vector K via
r j ˆ r 0 ‡ jK :
…21†
This treatment was used first by Magnusson and Gaylord [58] and leads to a system
of coupled differential equations for the complex amplitudes S j :
146
uS j
uz ‡ cos …a 0 ‡ i# j †S j ‡
1 X1
i cos S j h …2pn h = r ia h † ^s j h ^s j ‡ S j‡h …2pn h = r ia h † ^s j‡h ^s j ˆ 0 …22†
2
hˆ1

9.3 Holography
where is given by
# j ˆ jK cos… '† …jK† 2 r =4pn 0 …23†
with a slant angle j between K and the x direction; j =908 for the symmetric case shown
in Fig. 9.23. This so-called coupled wave approach is a generalization of the work of Kogelnik
[59], who assumed pure sine gratings read under the Bragg condition where only one
transmitted and one diffracted wave is present. Kogelnik [59] gives analytical solutions for
the amplitudes of the reference and the signal wave (zeroth and first order respectively) –
only for the first ascent period of a growth curve, extending earlier work for the transparency
region. For the efficiency Z j of the transmission grating with the thickness d
j …d† ˆS j …d†S j …d†
…24†
he gets for j = 1 the well-known formula
…d† ˆ sin 2 …pn 1 d=cos 0 †‡sinh 2
…a 1 d=2 cos 0 † exp… 2ad=cos 0 † ; …25†
with the replay wavelength l and the Bragg angle y 0 . It should be pointed out that this formula
is only valid for the special case of thick (volume) gratings, which show pure Bragg behaviour.
That is they possess neither higher Fourier coefficients nor a modulation amplitude n 1 nor large
enough a 1 , to produce higher diffraction orders, assuming n 1 and a 1 do not vary with z.
The thickness of a holographic grating is often described in terms of the Q factor, defined
by
Q ˆ 2pd=L 2 n 0 :
…26†
Gratings with Q^1 are regarded as thin those with Q 610 as thick.
9.3.2 Experimental setup
We used diacetylene single crystal platelets, approx. 20 mm to200mm thick and some mm 2
in area, cleaved from a parent TS6 crystal parallel to the (100) surface.
The experimental setup is shown in Fig. 9.24. For writing holographic gratings we utilized
a cw helium-cadmium laser (l w = 325 nm) or a xenon chloride excimer laser (l w = 308 nm).
We gave preference to the recording geometry suggested by Bor et al. [62] and not to the wellknown
beamsplitter geometry. The first order diffracted beams of a reflection grating R are reflected
by a pair of parallel mirrors M1 and M2 (or pass a biprism instead) and are superimposed
on the sample S. This geometry yields four main advantages:
a) Given the fringe distance of the reflection grating, the resulting fringe distance on the
sample is L = D/2 and wavelength-independent;
147

9 Diacetylene Single Crystals
b) Both writing beams are superimposed correctly because they have passed the same
number of reflections. This is especially important if non-Gaussian beams are used;
c) The intensities of both beams are equal and wavelength-independent;
d) The setup is realizable compactly and can easily be adjusted for the use of short coherence
lengths.
Control of exposure is possible using an UV enhanced photodiode (De3). To read the
gratings we used a 3 mW helium-neon laser (l r = 633 nm). At this wavelength the increase
in refractive index can be expected to be high, whereas the absorption should not become
too strong during the induction period of the polymerisation reaction (Fig. 9.4). The sample
holder was mounted on the axis of a stepping motor for varying the angle of incidence. For
analyzing transmitted or diffracted light a preamplifed large-area photodiode (De1) was
mounted on a radius level of a second stepping motor being coaxial to the first one. Both
motors include a reduction gear, giving an angular resolution of 0.06 mrad. Signal improvement
was achieved by chopping the readout beam and using a second photodiode (De2) as
an intensity reference. Both photodiode outputs were led to lock-in amplifiers. Two polarizers,
Pol1 and Pol2, were used to attain both UV and VIS laser polarizations parallel to the
sample’s b-axis.
Figure 9.24: Experimental setup. De1, De2, De3: large area photodiodes, R: reflection grating, S: sample
on sample holder, Sh: beam shutter, BS: beam splitter, M: mirrors, Pol1, Pol2: polarizers (l/2
plates), Ch: chopper.
Two sample orientations relative to the grating fringes were investigated:
a) UV polarization,VIS polarization, and polymer axis b lying in the plane of incidence (E
mode), henceforth denoted as b k orientation;
b) UV polarization, VIS polarization, and polymer axis b were perpendicular to the plane
of incidence (H mode), called b || orientation.
148

9.3 Holography
The high dichroism of the crystals and their well-formed habit can be used to align
their orientation under a polarizing microscope. Of course, the optical anisotropy of the crystals
as well as anisotropic reaction kinetics give rise to birefringence effects.
9.3.3 General characterization
Figure 9.25 shows for typical samples the maximum efficiency Z(y 0 )=Z as a function of
the exposure. These growth curves show that an optimum can be reached between 0.5 and
1.0 J/cm 2 for l w = 308 nm and between 5 and 10 J/cm 2 for l w = 325 nm. For TS6 the penetration
depth at 308 nm is about 110 mm, whereas at 325 nm it is about 420 mm. This difference
and in addition a presumably lower quantum yield at 325 nm results in much slower
growth curves for this wavelength. The decrease in efficiency is not only an effect of coupling
back intensity from first to zeroth order, but also produced by an increase of absorption.
Sample quality and thickness can influence these curves. A destruction of the surface
is only observed if the optimal exposure is exceeded by a factor 2 to 3. This is due to the
lattice mismatch of monomer and polymer stacking distance.
The energy needed to reach the maximum is approximately by a factor 5 larger than
at 257 nm [55, 56]. From the growth curves, i. e. from the exposure E which was needed to
produce a certain efficiency Z, the holographic sensitivity S of the material given by
S ˆ
p
=E
…27†
can be estimated. For TS6 and depending on sample quality this value can range from 0.4 to
1.4 cm 2 /J. This is in the order of typical sensitivities for photorefractive crystals but far less
0 12.5 25
0.5 0.13
Efficiency η
0.25
0.065
0 0
0
1.5
3.0
Exposure E / Jcm -2
Figure 9.25: Holographic growth curves of TS6 (solid curve: l w = 308 nm, d = 130 mm; dotted curve:
l w = 325 nm, d = 270 mm) and IPUDO (dashed curve: l w = 308 nm, d = 260 mm). Orientation b k ,
L = 3.3 mm. Z(y 0 )=Z: efficiency for readout Bragg condition.
149

9 Diacetylene Single Crystals
than for common silver halide materials [63]. Nevertheless, the resolution of TS6 is comparable
to that of common photographic materials. From the experiments of Richter et al. [55]
it is known that a resolution of 2500 lines/mm is achievable. In our electron beam lithography
experiments with a resolution of 5000 lines/mm has been achieved [64].
Supposing the thickness of the sample is equal to the grating thickness, we can calculate
the factor Q for our gratings. At a grating distance of L = 3.3 mm this factor
ranges from 2.5 to 60 for samples of thickness 25–500 mm, whereas at L = 0.8 mm Q
factors of 800 can be achieved. These values are typical for thick phase gratings. Nevertheless,
it should be pointed out that for example a grating with Q = 40 already shows
higher diffraction orders. This confirms the arguments brought by Moharam and Young
[60] that the r factor (r = L 2 /L 2 n 1 n 0 ) should be preferred when discussing grating diffraction.
9.3.4 Angular selectivity
The angular selectivity is a measure for diffracted intensity when readout is performed under
off-Bragg conditions or, quantitatively speaking, it is the half-width Dy of the function Z (y),
where y is the angle of incidence. This property is of central interest, when discussing holographic
storage media, because it limits the number of holograms/holographic gratings
which can be stored simultaneously.
Only as a rule of thumb, Kogelnik [59] gives Dy & Ln 0 /d, whereas Magnusson and
Gaylord [58] do not give any analytical expression for this quantity. Both approaches can easily
be implemented on a computer to simulate diffraction properties. Assuming n h = 0 for
h 62 and only for the first ascent period of a growth curve we find a Dy-dependence of the
following simple form
ˆ n 0 =d
…28†
valid only for the first ascent period of a growth curve, because the angular selectivity
curves get split beyond the first maximum [61].
The dependence of Dy on sample thickness d proved to be very well reproducible with
respect to adjustment and sample quality. Figure 9.26 shows two typical measurements of
Z (y) representing samples of different thickness. In contrast to the early experiments of
Richter et al. [55] and of Niederwald et al. [56] these values represent an improvement of
Dy by up to two orders of magnitude.
Figure 9.27 shows Dy values versus sample thickness for TS6. No significant difference
could be observed with respect to exposure or orientation for this grating distance.
There is also no significant difference between TS6 and IPUDO with respect to angular selectivity.
Measurements at L = 0.8 mm are summarized in Fig. 9.28. Within the series the minimum
angular selectivity of 0.188 arose, using a 380 mm thick TS6 platelet. The hyperbolas
plotted in Figs. 9.27 and 9.28 are fits of the function Dy = Ln 0 /d to the data points. Averaged
over all exposures n 0 ranges from 1.3 to 1.7.
150

9.3 Holography
1
Rel. Efficiency
0.5
0
-30 0
30
Readout Angle
/ Deg.
Figure 9.26: Dependence of relative efficiency Z as a function of the incidence angle y for two TS6
samples of different thickness d.
Figure 9.27: Angular selectivity Dy as a function of sample thickness d for TS6 (open circles) and
IPUDO (full circles). L = 3.3 mm, l w = 308 nm [65].
Angular Selectivity ∆Θ / Deg.
2
1
0
0 250
500
Sample Thickness d / µ m
Figure 9.28: Angular selectivity Dy as a function of sample thickness d for TS6. L = 0.8 mm,
l w = 325 nm for both writing geometries, b || (full circles) and b k (open circles) [65].
151

9 Diacetylene Single Crystals
9.3.5 Prepolymerized samples
Until now we were assuming that sample thickness and hologram thickness are equal. In
fact, for large thickness values d one can observe slight deviations from the hyperbola function
due to the finite penetration depth of the UV light. These deviations should become
more obvious for smaller penetration depths. To observe this some Dy measurements were
carried out using thermally prepolymerized (up to 5 h at 70 8C before recording gratings)
TS6 samples. Penetration depths for these samples varies between 18 to 80 mm at 308 nm
and from 16 to 420 mm at 325 nm in the b || orientation.
The effect of prepolymerization on angular selectivity is shown in Fig. 9.29. The values
for samples prepolymerized for 2 h still show for thin samples a weak thickness dependence.
For thick samples Dy does not tend to decrease further. Samples prepolymerized for
5 h do not show any dependence on d, indicating a penetration depth distinctly smaller than
the smallest sample thickness used. Thus, for an increasing polymer concentration the holographic
gratings get thinner and thinner. The possibility of producing high efficiency volume
phase gratings is restricted to fresh monomer crystals.
Figure 9.29: Angular selectivity of prepolymerized TS6 samples: 5h (upper open circles), 2 h (full circles),
and fresh crystals (lower open circles) [66].
9.3.6 Chain length, polymer profile, and grating profiles
A model [66, 68] describing the spatially inhomogeneous reaction kinetics of diacetylenes
must take into account a kinetic chain length L, which depends on the polymer conversion P.
The monomolecular reaction in a simple homogeneous situation then is given by
dP=dt L…P†…1 P† : …27†
For the experiments described so far, L can be assumed constant during the exposure
time, because only very little conversion takes place. Furthermore, for all experiments de-
152

9.3 Holography
scribed, the factor (1 – P) can be dropped, because the experiments were carried out in the
low conversion regime of the induction period and because saturation effects affect development
experiments in both orientations. For the two orientations, different situations must be
investigated corresponding to the different interaction of the two characteristic lengths L
and L (Fig. 9.30),
b jj : dP=dt L…P† cos 2 …px=L† : …28†
Here a polymer molecule grows parallel to the fringes contributing its whole chain
length L to the polymer growth at point x, where it has been initiated. In the other case, a chain
initiated in x contributes to the polymer growth in the whole interval [x – L/2, x + L/2]:
x‡L=2
b ? : dP=dt R
cos 2 …px 0 =L†dx 0 :
x L=2
…29†
Integrating both Equations we see that in the second case the chain growth smears out
the photoproduct distribution for a certain amount. This effect should be stronger if L/L approaches
1. For the quotient of both polymer modulations we get the ratio d,
ˆ P ?
P jj
ˆ sin…pL=L†
pL=L : …30†
The resulting d values range for fresh samples from 0.85 to 0.95 and tend to decrease
with increasing prepolymer content. For samples prepolymerized (5–6 hours, 3–4% polymer)
we find that d is between 0.65 and 0.80. These values correspond to chain lengths (L)
of 0.15 to 0.4 mm or 300 to 800 repeat units [66].
Figure 9.30: Microscopic model to understand the interaction of the two characteristic lengths L and L:
The two geometries investigated change the angle between chain growth direction and grating fringes.
153

9 Diacetylene Single Crystals
9.3.7 Multrecording
In additional experiments, we succeeded in recording more than only one grating into one
thick TS6 crystal. After each exposure interval, we rotated the sample for a certain amount
to slant the gratings relatively to each other. Figure 9.31 shows the first order diffraction versus
the angle of incidence for a 310 mm thick sample containing 42 gratings tilted stepwise
by 1.08 8. We observed that gratings already present are almost not influenced by the subsequent
recording processes except by the increase of total absorption, which causes a loss of
efficiency less than 10 %. Thus, we did choose an exposure of only 0.07 J/cm 2 for each grating.
As can be seen from Fig. 9.31, the signal-to-noise ratio is bad for the last written gratings.
It will get worse by decreasing the angular distance of the individual gratings but will
hardly improve by increasing it [67]. A serious diffraction of the UV light by the gratings already
recorded in the sample does not take place. UV diffraction efficiencies are extremely
small in diacetylene crystals.
-3
Efficiency η / 10
8
4
1
2
4
0
-10 20
50
Readout Angle Θ/ Deg.
Figure 9.31: Efficiency of the first order of a multihologramm consisting of 42 single gratings successively
recorded in a 310 mm thick TS6 crystal. The gratings were recorded in the order of the numbers
indicated [69].
9.3.8 Holography
For the reconstruction of a real hologram with different writing and reading wavelengths, it
was necessary to use a divergent reference wave. With a test platelet as object a resolution
of 300 lines/mm was achieved in excellent TS6 samples (Fig. 9.32). Holograms up to 32
pictures were written (l w = 325 nm) in one crystal using the angular selectivity of the thick
phase. When rotating the sample the pictures can be reconstructed (l r = 633 nm) successively
(holographic trick film).
The storage density of about 6.7610 9 cm –3 , estimated from the real resolution and
angular selectivity, is about one to two orders of magnitude lower than the storage density
calculated from the grating distance [70].
154

9.4 Di-, pyro-, and ferroelectricity
Figure 9.32: Reconstructed image from one of the 32 pictures of a holographic trick film [70].
9.4 Di-, pyro-, and ferroelectricity
The dielectric properties of diacetylene monomer single crystals are not strikingly different
from those of other organic materials. Typically, they have permittivity values (earlier called
dielectric constant e r ) of about 4–6. Solid state polymerization, however, results in a pronounced
anisotropy of the electric permittivity with maximum values parallel to the polymer
chain direction, due to the large polarizability of the extended p-orbitals. During our TOPO-
MAK activities the change of e r , accompanying solid state polymerization, was analyzed
quantitatively (Section 9.4.1.1). The results of the e r analysis can be applied for the in situ
monitoring of the polymer content (Section 9.4.1.2).
The tailoring of pyro or ferroelectric properties of diacetylenes (Section 9.4.2) is less
straightforward than might be surmised from the well-ordered arrangement of the R, R' substituents
in Scheme 1, and from the fact that polar side groups can easily be introduced as
substituents R and R' (Tab. 9.2). The difficulties originate from the packing of the individual
polar diacetylene monomer molecules in the elementary cell of the solution-grown single
crystal. This, generally, gives rise to a center of inversion symmetry for a pair of molecules
thus compensating the individual molecular electric dipole moments in an antiferroelectric
arrangement. During solid state polymerization spurious electric polarization has been observed
[71] resulting from intramolecular distortion of originally centrosymmetric monomer
units. Here we want to emphasize that our systematic TOPOMAK investigations have realized
pyroelectric (Section 9.4.2) as well as ferroelectric properties (Section 9.4.3) for appropriately
substituted diacetylenes.
155

9 Diacetylene Single Crystals
9.4.1 Dielectric properties of diacetylenes
Typically, the electric permittivity of monomer single crystals of substituted diacetylenes
ranges from 4 to 6 at room temperature. These e r values are therefore larger than values of
simple non-polar organic polymers, like polytetrafluoroethylene, in agreement with the existence
of polar side groups. As is exemplified in Fig. 9.33, monoclinic crystals of 2,4-hexadiynylene
di-p-toluenesulfonate (TS, see Tab. 9.2) shows generally a weak but non-negligible
anisotropy for e r [72, 75].
ε r (h)
8
7
6
5
TS
T = 60°C
(1)
(2)
4
3
0
10 20 30
polymerization time t(h)
Figure 9.33: Electric permittivity e r (t) as a function of the polymerization time for TS crystals at
60 8C. The electric field was applied parallel to the chain direction of the monoclinic crystals (1) and in
the two orthogonal directions (2) and (3). The permittivity was measured at 1 kHz for three different
thin parallel-plate single crystal capacitors [75].
(3)
40
9.4.1.1 Correlation of polymer content and electric permittivity
Generally, a sigmoid time-conversion curve is observed for thermal solid state polymerization
of diacetylenes. This behaviour was shown for TS in Fig. 9.4 and is reported for the unsymmetrically
substituted 6-( p-toluenesulfonyloxy)-2,4-hexadiynyl-p-fluorobenzenesulfonate
(TS/FBS, Tab. 9.2). In Fig. 9.34a the slow conversion of the initial induction period of the
solid state polymerization lasts until a polymer content of about 10 % is achieved. For higher
up to complete conversion, this is followed by an autocatalytic reaction enhancement. The
standard technique for the derivation of time-conversion curves is the gravimetrical analysis
of a large number of crystals in a point-by-point procedure. After thermal polymerization
for a well-defined period both, the soluble monomer and the insoluble polymer portions, are
determined gravimetrically (Fig. 9.34 a). This technique consumes a considerable number of
crystals and a substantial amount of time. Furthermore, only averaged time-conversion
curves are obtained.
Figure 9.34b shows that during solid state polymerization of TS/FBS the electric permittivity
parallel to the chain direction, e r|| , increases by a factor of about 2. The permittivity
e r|| is almost a linear function of the polymer content, as is exemplified in Fig. 9.34 c. We
have found comparable behaviour for the substituted diacetylenes TS, FBS, FBS/TFMBS,
and DNP [72–75]. Because the reorientation of the side groups during the polymerization is
weak, it does not influence substantially the increase of e r parallel to the chain direction.
156

9.4 Di-, pyro-, and ferroelectricity
Figure 9.34: Thermal solid state polymerization of TS/FBS (PTS is another acronym for TS).
(a): Time-conversion curve derived by gravimetrical analysis. (b): Time-permittivity curve derived at
1 kHz for a thin parallel-plate single-crystal capacitor oriented with the polymer chain direction (b-axis)
parallel to the electric field. (c): Correlation of conversion and electric permittivity (with time as implicit
parameter) obtained by combination of (a) and (b) [73].
This proves that only the extended p-electron system of the diacetylene backbone is responsible
for the enhanced polarizability. This conclusion is supported by the experimental analysis
for three orthogonal directions in TS single crystals shown in Fig. 9.33 [75]. The minor
variations for the two orthogonal directions (2) and (3) can be explained by the changes of
the lattice parameters. In contrast, the change of the respective lattice parameters with polymer
content does not suffice to explain the change of De r|| .
The linear relation between permittivity and polymer content is in principle surprising
because there is a distribution of chain lengths of solid state polymerization, differing between
induction period and autocatalytic range. The linearity, shown in Fig. 9.34 c, reveals
that the polarizability of the p-electron system saturates already at chain lengths below the
shortest ones occurring during thermal solid state polymerization.
The experimental range for e r is 1.4 to 2.2 derived for different substituted diacetylenes
by Gruner-Bauer [72–75], which agrees with the observations of other groups [76–
78]. These values compare favourably with the estimate De r & 1.6 obtained for TS by simplified
model calculations [75]. For these theoretical estimates the method of Genkin and
157

9 Diacetylene Single Crystals
Mednis [79] has been modified by Gruner-Bauer, extending earlier work for the transparency
region [80].
9.4.1.2 Application to topospecifically modified diacetylenes
The linear relation of electric permittivity parallel to the chain direction vs. polymer content
thus established can be used for the control of the polymer content of prepolymerized samples
as well as for the derivation of time-conversion curves of individual single crystals in
situ. We have used this technique to study the solid state polymerization of topospecifically
and fully deuterated TS.
Striking differences of their reactivities were reported before by Ch. Kröhnke [54].
The limited amount of samples available was sufficient for the permittivity analysis.
Figure 9.35 shows the behaviour of different topospecifically deuterated derivatives of TS
at T =608C [73]. It should be stressed that the induction periods of single crystals with
nominally the same history did not differ by more than 10% at the same polymerization
temperature. The toposelective modification of these substituted diacetylenes by the deuteration
of the methylene groups close to the triple bonds of the diacetylene monomer
(that are engaged in the crankshaft-type motion of the monomer molecule around its center
of mass during solid state polymerization) evidently has a drastic influence on the solid
state polymerization.
8
7
PTS
T = 60°C
6
ε (t)
5
8
7
6
5
8
7
CD - PTS
T = 60°C
CD - PTS
T = 60°C
2
6
5
0
10
Figure 9.35: Electric permittivity as function of time for solid state polymerization at T =608C for TS,
PD-TS (fully deuterated TS), and CD 2 -TS (where only the methylene groups close to the triple bond of
the diacetylene monomer unit are deuterated, see Tab. 9.2 a) [73].
158
20
30
polymerization time t/h
40

9.4 Di-, pyro-, and ferroelectricity
9.4.1.3 Additional applications
Structural phase transitions accompanying solid state polymerization influence the behaviour
of e r (t). Thus they can not be observed but additional hints concerning the type of structural
changes can be obtained. For example, a structural phase transition for DNP (Tab. 9.2) occurs
at a polymer content above 95% resulting in the loss of order perpendicular to the polymer
chains [81]. The accompanying increase of the side-group mobility results in a distinct
increase of the electric permittivity [75]. Even more dramatic changes of e r were observed
at the transition to a fibrillar structure in polar crystals of DNP/MNP (Tab. 9.2) [74, 75].
Thinking of applications outside fundamental research, the correlation of permittivity
and polymer content can also be used for different kinds of ageing control. Since solid state
polymerization is an activated process, the temperature-weighted time at elevated temperature
– like a low-temperature radiation dose – influences the capacity of a diacetylene crystal-plate
capacitor according to a well-defined characteristic history-capacity. Evidently, this
can be adapted to an appropriate electronic ageing control.
9.4.2 Pyroelectric diacetylenes
One fascinating goal of diacetylene materials research is the realization of polar polymer single
crystals without complicated poling procedures. Substituted diacetylenes R 1 –C:–C:C–R 2
incorporating polar side groups R 1 and R 2 with large but different electric dipole moments
have to be synthesized. Since solid state polymerization depends on the ability of individual
molecule side groups, which can perform intramolecular torsion and giving rise to an overall
crankshaft-like motion, it is difficult – and was impossible for us – to predict the packing arrangement
of individual monomer molecules in the solid. Frequently, the unit cell of substituted
diacetylenes was observed to accommodate pairs of formula units in a centrosymmetric
arrangement, thus compensating the net electric polarization. Polymorphism turned out to be
an obstacle to systematic tailoring of dielectric properties of diacetylene single crystals, because
different crystal structures of the same diacetylene derivative could be obtained from
different, and occasionally even from the same solvent [82]. Nevertheless, Strohriegl synthesized
during our TOPOMAK activities several non-centrosymmetric diacetylenes, which crystallized
also in a polar phase. Typically, their permittivities were relatively large and anisotropic
[83]. We restrict this report to three examples.
9.4.2.1 IPUDO
IPUDO (for the molecular structure see Tab. 9.2) is an example of a symmetrically substituted
diacetylene. For the monomer as well as polymer crystals, non-centrosymmetrical
orthorhombic crystal structures can be found already at room temperature [84]. The c-axis
is the polar axis of the monomer crystal. IPUDO can only be polymerized by g radiation
( 60 Co). For 85% polymerized crystals pyroelectric properties were observed only in b direction.
The permittivity of IPUDO is highly anisotropic, with maximum values of 8.5 for the
159

9 Diacetylene Single Crystals
monomer (parallel to a), or 11.6 for the 85% polymer (parallel to b) and with minimum values
(parallel to c) smaller by a factor of 2 (3.5) for the monomer (polymer) [83]. The distortion
of the long side groups by the development of hydrogen bonds between neighbouring
–CO–NH- groups is supposed to be responsible for the non-centrosymmetry [83, 85, 86].
Thus it is not surprising that the variation of the electric polarization of about
3610 –8 Ccm –2 between room temperature and 4 K amounts to an unbalancing of only
about 1% of the compensation of oppositely oriented C=O … HN dipole moments.
9.4.2.2 NP/4-MPU
NP/4-MPU (for the molecular structure see Tab. 9.2) is an example of a non-centrosymmetric
diacetylene that forms polar monomer single crystals only, if it is grown from appropriate
solvents, here from 2-propanol [82]. The resulting modification I is orthorhombic
with the polar space group Fdd2 (Z = 16) and c as the polar axis [82]. This modification is
not reactive thermally or under X-ray irradiation, because the monomer packing is outside
the favourable range for solid state polymerization. The diacetylene rods make an angle of
678 with the stacking axis c , and the stacking distance d = 4.61 Å. The permittivity of this
diacetylene is highly anisotropic and shows the largest value of about e r &23 for an electric
field applied in the direction of the polar axis [82, 87].
Figure 9.36 shows the variation of the spontaneous electric polarization DP of NP/4-
MPU with temperatures between 10 K and the melting point. DP amounts to about 15% of
the electric polarization, which can be estimated from the volume density of molecular electric
dipole moments of about 3 Debye (10 –29 Cm).
The pyroelectric coefficient p(T) =dP S /dT =8.8610 –10 Ccm –2 K –1 is of the same
order of magnitude (smaller by 1/3) as that of the well-known and commercially used pyro
and ferroelectric polyvinylidenefluoride. Therefore NP/4-MPU single crystals can be used
for the detection of radiation [89], which we showed by using chopped low-power laser light
as radiation source. The pyroelectric current and the total variation of the surface charge
Figure 9.36: Temperature-dependent change of the spontaneous polarization for a NP/4-MPU single
crystal (modification I) along the polar c-axis for temperatures below the melting point T m . The pyroelectric
coefficient P at 300 K is also given [87].
160

9.4 Di-, pyro-, and ferroelectricity
have been used for the detection. Furthermore, a transversal piezoelectric coefficient of
1018 fCN –1 for NP/4-MPU was derived at room temperature. It is comparable with the value
of a-quartz [88].
9.4.2.3 DNP/MNP
DNP/MNP (for molecular structure see Tab. 9.2) was the most successful one of Strohriegl’s
syntheses [74]. It has a polar crystal structure for monomer and polymer crystals (space
group P2 1 ). DNP/MNP polymerizes – thermally or exposed to UV radiation – extremely
fast, because during solid state polymerization the molecules are packed optimally.
The solid obtained by thermal polymerization of monomer crystals exhibits a fibrous
texture, probably due to the large changes in lateral packing of the side groups. The temperature
influence on the spontaneous electric polarization perpendicular to the chain axis c
is shown in Fig. 9.37 for the monomer as well as the polymer crystal. The polarization varies
up to room temperature by 7610 –8 Ccm –2 for the monomer crystal, with a pyroelectric
coefficient of about 3.2610 –10 Ccm –2 K –1 at room temperature.
0.0
-2
∆P(T) / 10 C cm
-7
-0.2
-0.4
-0.6
-0.8
-1.0
0
monomer
DNP / MNP
parallel to polar axis
100 200
T/K
polymer
Figure 9.37: Variation of the spontaneous electric polarization parallel to the polar b axis of DNP/MNP
crystals with temperature. The data of DP (T) =P (T)–P (5 K) were derived by charge integration during
a temperature cycle [74].
300
9.4.2.4 Spurious piezo and pyroelectricity of diacetylenes
Sample defects can be another origin of piezo and pyroelectric phenomena in substituted
diacetylenes, which generally can be identified via sample dependence and smaller size of
these effects [88]. Bloor et al. discussed such spurious pyroelectric effects, which seemed to
be correlated with the occurrence of macroscopic deformations, such as screw dislocations
in TS single crystals [90]. Similarly the analysis of weak polarization, caused by molecular
distortion during solid state polymerization of TS, was reported by Bertault et al. [91]. We
have observed comparable weak pyroelectric phenomena for TS/FBS [87].
161

9 Diacetylene Single Crystals
9.4.3 The ferroelectric diacetylene DNP
Whereas several pyroelectric diacetylenes were identified, uniform ferroelectric phases seem
to be rather the exception, according to our experience with many new substitutions of diacetylenes
[92]. The ferroelectric low-temperature phase of the symmetrically disubstituted
diacetylene DNP, i. e. 1,6-bis(2,4-dinitrophenoxy)-2,4-hexadiyne (Tab. 9.2), turned out to be
one of the rare and interesting exceptions [93–98].
On both ends DNP carries polar dinitrophenoxy groups, with an electric dipole moment
of 10 –29 Cm, whose mutual twisting gives rise to the spontaneous electric polarization
of the non-centrosymmetric low-temperature phase (space group P2 1 ) (Fig. 9.38). The DNP
monomer crystal has besides a pyroelectric [93] low-temperature phase a ferroelectric one
for T < T c = 46 K. According to our investigations, the direction of the spontaneous polarization
(Fig. 9.39) can be influenced by an external electric field [97]. Structural defects, occurring
in all DNP crystals, give rise to a distribution of transition temperatures and to the
existence of domains, whose polarization can not be inverted with the accessible external
fields – thus behaving like pyroelectrics. Only the domains with the highest transition temperatures
could be poled [97].
b
a
b
a
b
a
a)
b)
c)
Figure 9.38: Packing arrangement of DNP molecules in the monomer crystal viewed along the c-axis
at three temperatures (a): T = 296 K, (b): T = 145 K, (c): T = 5 K [96].
The temperature dependence of the spontaneous polarization (Fig. 9.39) of the monomer
crystal can be described in the framework of Landau’s phenomenological theory assuming
a tricritical phase transition [97]. The maximum experimental value of the electric polarization
of 2.4610 –7 Ccm –2 compares favourably with the polarization that was calculated
from the intramolecular twisting angle of the polar dinitrophenoxy groups of 5.18 determined
by X-ray structural analysis at 5 K [96].
The electric permittivity (Fig. 9.40) reaches values of about e r & 150 at the transition
temperature. Its temperature dependence is strongly influenced by defects [97]. Thus, it is
less appropriate for a comparison with theoretical predictions.
In the early days of TOPOMAK, H. Schultes has already observed that the phase transition
of DNP shifts to lower temperatures (Fig. 9.41–43) with increasing polymer content
162

9.4 Di-, pyro-, and ferroelectricity
3
DNP
-2
P(T) / 10 -7 Ccm
2
1
0
0
10 20 30 40 50 60
T/K
Figure 9.39: Temperature dependence of the spontaneous polarization parallel to the polar b-axis for
different DNP monomer single crystals [97].
75 0.3
DNP
ε r (T)
50
25
0.2
0.1
-1
(χ ferro
(T))
0
0
20
40
Figure 9.40: Temperature dependence of the low-frequency electric permittivity (left axis) and the inverse
of the ferroelectric part of the susceptibility (right axis) for a DNP monomer crystal [97].
T/K
60
80 0
-7
polarization / 10 C cm
-2
1,0
0,5
10 h
5h
2h
0h
0
25
Figure 9.41: Variation of the zero-field electric polarization in crystallographic b-direction of DNP single
crystal with duration of thermal polymerization at 130 8C [94] (Fig. 9.34 for conversion curve).
50
T/K
163

9 Diacetylene Single Crystals
ε (T) / ε (293 K)
r
r
8
4
10 h
5h
13 h
14 h
16 h
0
0 25 50 75 100
T/K
Figure 9.42: Influence of duration of thermal solid state polymerization at 130 8C on temperature dependence
of electric permittivity of DNP [94].
50
T c / K
40
30
1,6
1,2
-1
∆S /Jmol K
-1
20
10
0
0
X/%
80
40
0
0
8 16
t/h
4 8
16 0,0
Figure 9.43: Variation of the transition temperature T c (e r maximum) and the conversion entropy (transition
entropy change) DS with the duration t p of the thermal solid state polymerization of DNP at
129 8C [83, 87]. The inset shows the typical variation of the polymer content X with t p .
12
t p/h
0,8
0,4
(thermal polymerization) and can not be observed in the polymer crystal [94, 98]. This was
a puzzle for our early attempts to understand the ferroelectric phase transition of DNP.
In the microscopic picture of the phase transition, nuclear magnetic resonance spectroscopy
and relaxation of the DNP protons gave important additional information [95, 96].
The proton NMR spectrum reflects the orientation of the proton-proton axis of the methylene
groups close to the central diacetylene unit via the nuclear-spin magnetic dipole interaction
in a rather clear-cut way (Fig. 9.44). For fixed crystal orientation and varied temperature
the spectrum of the methylene group protons of the DNP monomer proved that both DNP
moieties are twisted around the central C–C single bond of the diacetylene below T c . Since
this degree of freedom is lost during the solid state polymerization the ferroelectric phase
transition is suppressed.
164

9.4 Di-, pyro-, and ferroelectricity
r
H0IIb
r
H 0
b
-40
0 40
-40 0 40
Rel. Frequency (kHz)
Figure 9.44: Simulated (left) and experimental (right) H NMR spectra as a function of the crystal orientation
with respect to the external magnetic field, recorded at room temperature (n p = 200 MHz). The
crystal was rotated in steps of 58 around its long axis (a-axis), which was oriented perpendicular to the
external field [96].
Additional information on the phase transition was obtained from proton-spin-lattice
relaxation measured as function of the Larmor frequency, temperature, and orientation of
the single crystals (Fig. 9.45). The analysis indicated the slowing down of a molecular motion
on approaching the ferroelectric phase transition with the activation energy of about
0.020 eV, which is in the range of known librational and torsional modes of diacetylenes
[96].
The phase transition of DNP could further be characterized by specific heat measurements
for monomer and thermally polymerized single crystals (Fig. 9.46) [98]. These data
support the description of the phase transition of the monomer crystals as a tricritical transition.
This means it is a borderline case between a first-order and second-order phase transition,
with a distribution of transition temperatures. The transition enthalpy was much lower
than the corresponding order-disorder transition, in agreement with results obtained by Bertault
et al. via Raman spectroscopy, which proved the importance of displacive contributions
to the DNP phase transition [99].
165

9 Diacetylene Single Crystals
31 MHz
-3 -1
Spin-lattice relaxation rate (10 s )
90 MHz
200 MHz
Temperature (Kelvin)
Figure 9.45: Spin-lattice relaxation rate of the protons in a DNP monomer single crystal for three Larmor
frequencies. Two rate maxima can be discerned with temperature dependence explained by the
model of Bloembergen, Purcell, and Pound (solid line) [96].
30
-1
∆/C/J mol K -1
20
2h
2 p =0h
10
6h
0
30
Figure 9.46: The ferroelectric contribution to the molar heat capacity of solid state polymerized DNP
single crystals for different annealing times t p at 129 8C for [98].
40
T/K
50
9.4.4 Summary
The increase of the electric permittivity for electric fields parallel to the polymer chain direction
during solid state polymerization of diacetylenes can be used for in situ monitoring
the monomer to polymer conversion of individual single crystals. The large librational am-
166

9.5 Non-linear optical properties
plitudes of the diacetylene moiety, required for solid state polymerization, are also the basis
for the occurrence of interesting dielectric phase transitions. The tailoring of diacetylenes as
ferro or pyroelectric crystals, which show good thermal stability and do not demand considerable
efforts for the poling process, is a trial-and-error process, but has been realized during
our TOPOMAK activities in a number of cases. Thus, material properties useful for applications
of pyro- or piezoelectricity thus have been obtained.
9.5 Non-linear optical properties
9.5.1 Aims of investigation
Polydiacetylenes are polymers showing a one-dimensional semiconducting behaviour. This
one-dimensional structure causes exceptionally high third order non-linearities (w (3) ) [100],
also in off-resonant wavelength regions [101], with extremely short sub-picosecond switching
times [102]. After this discovery it was believed that an optical amplifying switch (optical
transistor) or even an optical computer was close at hand.
At the start of our SFB 213 project the initial optimism about the application of the
huge non-linearity of polydiacetylenes in optical switching was already somewhat damped.
It was becoming clear, that the polydiacetylenes’ non-linear optical coefficients, despite belonging
to the largest non-resonant non-linearities, were not sufficient for cascadable, intensity
amplifying switches [103]. There was not much known about the mechanisms leading to
the large non-linearities and different, sometimes contradicting theoretical models were proposed
to describe them. On the other side, there was a demand for tailor-made materials
with predictable non-linearities, absorption bands, and good optical quality.
In this scenario, polydiacetylenes were nevertheless interesting, as the mechanisms of
their large non-linearity form a good basis to build on. Only a few modifications were well
characterised and there was a general lack of measurements with single crystals due to problems
with sample preparation. Therefore, the aim was to characterise systematically the influence
of side groups on the optical properties, preferably in the macroscopically ordered,
stable, and reproducible framework of good and – later on – thin crystals.
9.5.2 Experimental setup
Mainly two kinds of experiments will be described here, two-beam pump-probe and degenerate
four wave mixing (DFWM) measurements.
In a pump-probe experiment, an intense pulsed pump beam and a weaker equally
pulsed probe beam of the same wavelength are focused into the sample, and the transmission
167

9 Diacetylene Single Crystals
of the probe beam is measured. The timing of the two pulses can be shifted, to make it possible
to probe the decay of excitations generated by the pump beam. Two contrary effects
may occur, bleaching and induced absorption.
Near-resonant pump-probe lifetime measurements were performed by W. Schmid
[105, 123], using the same picosecond dye laser system as for DFWM, described below.
Th. Fehn further on investigated the subject in the off-resonant wavelength region between
720 nm and 820 nm, using a commercial Titan-Sapphire laser system (Coherent) with pulse
lengths of ca. 120 fs, pumped by an argon ion laser (Coherent Mira).
DFWM measurements can be done in a variety of geometries. Here, the forward mixing
geometry is used (Fig. 9.47), where three beams, forming a right angle, are focused into the
sample. This setup allows time-resolved measurements and, in contrast to third harmonic generation
(THG) measurements, yields the w (3) (o;–o,o,–o) tensor which is related to an intensity-dependent
refractive index, the interesting quantity for optical switching applications.
Figure 9.47: Beam geometry for DFWM measurements.
The interference pattern of the pump beams 1 and 2 forms horizontal stripes in the
medium. As w (3) is directly related to an intensity-dependent refractive index, this interference
pattern generates a refractive index pattern. The third probe beam is partially reflected
on these horizontal planes, generating the signal beam 4. The efficiency of diffraction is related
to |w (3) |. The pulse timing is adjustable, so the decay of the refractive index pattern can
be probed. When the delay between the beams 2 and 3 is in the range of the laser’s coherence
length, these pulses generate a diffraction grating for beam 1, resulting in an artificial
raise of the signal, called coherence peak.
All DFWM measurements were done with a commercially available synchronously
pumped dye laser with a cavity dumper (Spectra Physics model 3500). When using Pyridine-1
as radiant dye, the wavelength can be tuned between 670 nm and 750 nm. The pulse
width is about 1 ps; the repetition rate can be adjusted between single shot and 8 MHz. This
is important to reduce the average heat load absorbed by the crystals.
To suppress stray light signals a well-known modulation technique was used, the modulation
of two pump beams with different frequencies using a chopper blade with a set of
different divisions. As the DFWM signal depends on the product of the input intensities, the
signal can be detected at the sum or difference of the two modulation frequencies [104],
168

9.5 Non-linear optical properties
I 3 sig / I 1I 2 I 3 ˆ I 1 …0†…cos ! 1 t ‡ 1†I 2 …0†…cos ! 1 t ‡ 1†I 3 ˆ
1
ˆ I 1 …0†I 2 …0†I 3 ‰
2 cos…! 1 ‡ ! 2 †t ‡ cos…! 1 ! 2 †t ‡ 2 cos ! 1 t ‡ 2 cos ! 2 tŠ: …31†
W. Schmid proposed the separated detection of w (5) effects by an extension of this
modulation method. A w (5) signal without w (3) contributions can be detected at twice the sum
or difference frequency [105], as the w (5) signal contains terms depending on the square of
the product of the modulated beam intensities,
I sig / I 1 I 2 I 3 s k 3 ‡ k 35 … I 1 ‡ I 2 ‡ I 3 †‡k 5 … I 1 ‡ I 2 ‡ I 3 † 2 ; …32†
where k 3 is a function of w (3) , k 5 of w (5) , and k 35 depends on both non-linearities and their
relative phase.
Although this modulation method permits qualitative measurements of w (5) effects,
especially determination of relaxation times, quantitative evaluation of w (5) is not possible,
due to the infinite number of modulation harmonics with ill-defined relative amplitudes
caused by the trapezoidal modulation form provided by a chopper blade [106].
This problem has been solved by A. Feldner. He developed a new type of modulator,
working with two independently rotating polarizing foils (Fig 9.48). This type of modulator
generates two harmonics in the intensity modulation spectrum with a fixed ratio of 4 : 1
(cos 4 ot) [106].
Figure 9.48: Scheme of intensity modulation by rotating polarizing foils (a). The vertical polarizer (b)
ensures that an anisotropy of the material has no effect on the signal or the modulation spectrum.
For time-resolved pump-probe measurements of shorter relaxation times, a Titan-sapphire
laser pumped by an argon ion laser is employed. In the current configuration, the wavelength
can be tuned between 720 nm and 820 nm; the pulse width is about 120 fs.
169

9 Diacetylene Single Crystals
9.5.3 Theoretical approaches
The simplest approach is the free electron model [107, 108]. The electrons are treated to
move freely in a one-dimensional box, subject to a potential V 0 cos (px/d) by the ion cores
(d denotes the bond length). The polarizability a and hyperpolarizability g is derived from the
2nd and 4th order perturbation energies caused by an electrical field along the chain. This
rough model does not account for local field effects nor for the alternating bond length found
along the axis in PDAs. The model yields a static w (3) = 3.0610 –11 esu along the axis.
Agrawal, Cojan et al. [109–111] treated explicitly the linear and non-linear optical
properties of PDAs. They computed the electronic energy levels using a Hückel formalism.
They computed a static w (3) = 0.7610 –10 esu for PTS and w (3) = 0.25610 –10 esu for
TCDU. This model neglects electron-electron interaction and therefore excitonic effects.
The model yields the correct order of magnitude for w (3) , but the wrong sign.
The phase space filling (PSF) model was developed to describe non-resonant NLO
properties in semiconductors [112, 113], especially in quantum well structures [114]. Greene
et al. adapted this model to one-dimensional polymer chains [115, 116]. The model is only
applicable to systems were the low energy absorption band is excitonic, as is the case with
PDAs [117]. Formation of excitons is limited by the number of available electron states that
are necessary to form the exciton. With an increasing number of excitons, the dipole momentum
for forming a new exciton is reduced. The exciton band bleaches.
As the following measurements will show, the PSF model seems to describe best the
non-linear optical behaviour of polydiacetylenes. A very important prediction of this model
is the proportionality of w (3) and a in the near-resonant frequency regime [115]. This behaviour
was found in polydiacetylenes, strongly supporting the PSF theory [118].
9.5.4 Sample preparation
For measurements well off the resonance, i. e. with wavelengths larger than 720 nm, p-TS6
crystals were prepared by thermal polymerisation of monomer crystals, grown out of a saturated
solution and manually cut using a shaver blade. Thickness of these crystals varies between
40–100 µm.
Later, for measurements closer to resonance, a method has been developed to grow
thin mono-crystalline layers of TS6 and 4BCMU between glass substrates. To avoid crystal
strains the monomer crystals have to be removed from the substrate before polymerising.
The resulting crystal thickness can be made as low as 300 nm.
9.5.5 Value and phase of the third order susceptibility w (3)
From DFWM measurements |w (3) | was determined for several polydiacetylenes in the nearresonant
to off-resonant wavelength region, 680 nm to 750 nm (Tab. 9.6). Concurrently, the
170

9.5 Non-linear optical properties
Table 9.6: w (3) values for some different Polydiacetylenes [118, 119].
Material Modification |w (3) |/esu at wavelength
PTS crystal 2610 –10 720 nm
FBS crystal 2610 –10 720 nm
4BCMU amorphous film 4610 –11 720 nm
4BCMU thin monocrystalline film 3610 –10 720 nm
4BCMU thin monocrystalline film 2610 –9 670 nm
imaginary part of w (3) can be computed from the non-linear absorption coefficient b, obtained
from measurements of the intensity dependence of the sample transmission. It was
found that the real part of w (3) was dominating the imaginary part by a factor of 3 [118].
This is predicted by the PSF model, from the bleaching of the exciton band.
The value of w (3) is nearly identical for p-TS6 and p-4BCMU, and 4 times lower for
p-IPUDO. These values were reproducible within 30%. For p-FBS, the reproducibility was
only within an order of magnitude. No difference was found between thermally and x-ray
polymerised crystals [118].
9.5.6 Relaxation of the singlet exciton
In p-TS6 energy relaxation times could be resolved at wavelengths below 700 nm. As the relaxation
times are of the same order of magnitude as the laser pulse width (1 ps), a model
function has to be fitted to the measured curve to obtain the relaxation (Fig. 9.49).
Figure 9.49: w (3) relaxation in p-TS6 at a wavelength of 680 nm and temperature of 200 K. For comparison,
the dashed line gives the 3 rd order autocorrelation of the laser pulse. The relaxation time of the
singlet exciton can be seen as a significant broadening effect [120].
171

9 Diacetylene Single Crystals
The DFWM signal decays with half the time constant of the exciton density. For measurements
of the relaxation time, the pump-probe experiments described below are to be preferred,
because the intensity-dependent transmission directly follows the exciton lifetime
[120].
In the off-resonant regime at 760 nm, the lifetime is about 0.5 ps, rising to more than 2 ps
at 680 nm (Fig. 9.50). As with DFWM measurements, the lifetime can not be seen as an exponential
signal decay, but as a broadening of the laser pulse autocorrelation signal. The lifetime
Figure 9.50: Wavelength dependence of the lifetime of the singlet exciton in p-TS6, derived from the
broadening of the autocorrelation function of picosecond pulses. Different symbols denote different
crystals [123].
can be estimated by fitting a model function accounting for the laser pulse shape.
Th. Fehn investigated the subject, using a Titan-sapphire laser system with pulse
lengths of 120 fs between 720 nm and 820 nm. Here, the exponential signal decay can be directly
resolved, because T 1 of the singlet exciton is much longer than the laser pulse
(Fig. 9.51). Accordingly, the relaxation time can be determined more accurately. In comparison,
W. Schmid’s fits tend to yield a slightly longer relaxation time, but within the error
ranges the results are consistent.
9.5.7 The w (3) tensor components
The w (3) tensor component parallel to the polymer chain of about 2–3610 –10 esu (off resonant)
is the well dominating source of the non-linear response. The DFWM signal was found
to be polarized parallel to the polymer chain and the signal decreased below the detection
threshold if any or all of the pump beams were polarized perpendicular to the chain. From
these observations w (3)
perp can be estimated to be lower than 1610 –12 esu, i. e. at least a factor
172

9.5 Non-linear optical properties
Figure 9.51: The energy relaxation time of the singlet exciton is clearly resolvable with Ti-sapphire laser
driven pump-probe experiments.
of 200 lower than w (3)
par [121, 122]. The behaviour of the different PDAs under investigation
(p-TS6, p-4BCMU, p-FBS) was very similar [118].
In the near-resonant frequency regime, a constant ratio of w (3) and the linear absorption
coefficient a has been found, as predicted by the PSF model [115]. This model provides
the best description of the experimental results [120].
9.5.8 Signal saturation
With high pump intensities saturation of the w (3) signal was observed (Fig. 9.52). Although this
general behaviour was found in all PDA samples, the finer details, e. g. the dependence on the
mean intensity, varied between different measurements. It turned out that mainly three effects
are responsible for the damping of the signal, whose relative influence depends on the wavelength,
the crystal thickness, and the general quality of the crystals. A thermo-optical effect
widens the focus diameter thus decreasing the intensity of the pump/probe beams. Induced absorption
and two-photon absorption decrease this intensity too. Finally, polydiacetylenes show
a significant influence of w (5) on the non-linear susceptibility, which is contrary to w (3) .
The induced absorption was measured concurrently to the DFWM signal by the transmission
of one of the pump beams. Thus, one can easily account for this effect in the measurements.
Our measurements on manually cleaved thick (50–150 µm) crystals [118] showed a
stronger damping with higher repetition rates of the laser system, i. e. higher heat load but
lower peak intensities. No damping was observed at off-resonant frequencies, e. g. at 750 nm.
Therefore, the damping is clearly related to the heat load absorbed by the crystal, an indica-
173

9 Diacetylene Single Crystals
Figure 9.52: The flattening of the intensity dependence of the DFWM signal can successfully be described
by a significant value of the fifth order susceptibility w (5) (inset solid curve: fit with assumed
contributions of both w (3) and w (5) ) [123].
tion of a thermo-optical effect. An amplification of the effect by inclusions of the solvent
was proposed.
In samples prepared later on with better optical quality W. Schmid observed a different
behaviour [120]. The damping became stronger with lower repetition rates, i. e. higher
peak intensity. No thermo-optic effect was observed. This can be well explained by the advances
in crystal preparation: solvent inclusions were reduced and the crystal quality improved,
resulting in a lower absorption coefficient at the same, near-resonant wavelength
(e. g. 720 nm). So, the absorbed heat load and the thermo-optical coefficients were reduced
concurrently.
The damping effect observed by W. Schmid has to be addressed to a w (5) effect contrary
to w (3) [123]. Later measurements on thin crystalline films, using an improved measurement
technique, sustained this interpretation [106].
9.5.9 Spectral dispersion, phase, and relaxation of w (5)
The value of w (5) has first been evaluated by Schmid [123], by fitting a polynomial to the
measured curve of signal intensity versus pump laser intensity. At 748 nm, this fit yields a
value of w (5) = 2.8610 –34 (m/V) 4 , with a phase angle of 1658 between w (3) and w (5) .
With the improved modulation method, a w (5) value of (1.69 ± 0.79)610 –33 (m/V) 4
(1.1610 –16 esu) was evaluated at 729 nm [106]. This is in reasonable agreement with
Schmid’s measurements, considering the different wavelengths, different measurement techniques,
and different samples.
After successful growth of thin (a few micrometers) p-TS6 crystals, it was possible to
study w (5) signals at near-resonant wavelengths without an interfering thermo-optical effect
174

9.5 Non-linear optical properties
Figure 9.53: Wavelength dependence of w (3) and w (5) in p-TS6 single crystals. The isolated dots are experimental
values obtained from a thin crystal (about 5 µm), all other dots are from a thick crystal
(about 50 µm) [106].
[106]. The spectral dispersion of w (3) and w (5) has been measured with the improved modulation
technique in the near resonant regime (Fig. 9.53). The w (5) amplitude increases roughly
proportional to w (3) , as expected from the PSF model. In this total regime, a phase between
w (5) and w (3) of about 1608 was found.
In the near resonant regime, the w (5) relaxation shows the same behaviour as w (3) .No
energy relaxation can be resolved in the off-resonant regime around ca. 720 nm, but the relaxation
time increases with shorter wavelengths (Fig. 9.54).
However, with even larger wavelengths, the w (5) relaxation times rise again (e. g. 2.3 ps
at 748 nm) [120]. This can be successfully described proposing a three level energy scheme
Figure 9.54: Relaxation time of the w (5) response in p-TS6. At 720 nm, only an 5 th order autocorrelation
can be seen. At 685 nm a finite lifetime is clearly resolvable [106].
175

9 Diacetylene Single Crystals
Figure 9.55: At wavelengths between 720 nm and 750 nm, the w (5) signal relaxes in a two-step process [120].
with an even state at 3.2 eV [123]. In addition to the fast dominant process, a very slow weak
relaxation has been found, with relaxation times between 20 ps and 100 ps (Fig. 9.55) [120].
9.5.10 Conclusion
The polydiacetylenes’ non-linear optical behaviour was found to be in accordance with the
predictions of the PSF model that assigns the non-linearity to the bleaching of the exciton
band. Theoretical approaches neglecting excitionic effects clearly fail. The exciton dominates
the non-linear optics of polydiacetylenes [118].
With increasing laser intensity, the w (3) signal shows significant deviations from the
ideal I 3 -dependence. In thick crystals at near resonant wavelengths, a thermo-optic effect
can be observed, which is amplified by crystal impurities [118]. However, also thin crystals
show a similar damping of the w (3) signal. In contrast to the thermo-optical effect, this does
not depend on the average heat load absorbed by the crystal, but on the peak intensity of the
pump pulses. This effect can be described by a w (5) effect with a phase shift of 1608 against
w (3) , by analysis of the intensity dependence of the net signal of both effects [120] as well as
by direct measurements using a new modulation technique [106].
Polydiacetylenes with different symmetrically substituted side groups differ mainly in
terms of crystal quality, stability, and solubility; the influence on the third order susceptibility
|w (3) | is rather small [118]. Interesting effects where found, however, in the unsymmetrically
substituted diacetylene NP/4-MPU. This diacetylene forms a non-centrosymetric crystal
with a significant w (2) [124, 125]. The second harmonic generation with this crystal can
176

References
be phase matched, making the material very interesting for commercial applications, e. g. as
a multiplying medium for a laser pulse autocorrelator [126].
In terms of optical quality, stability, and the value of w (3) polydiacetylenes still are
superior to other polymers with a linear p-electron system, e. g. the polyparaphenylenevinylene
investigated at our institute [127, 128, 130]. This is mainly due to their unique ability
to form macroscopically ordered single crystals with aligned polymer chains.
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180

10 Matrix-Molecule Interaction in Dye-Doped
Rare Gas Solids
Thomas Giering, Peter Geissinger, Wolfgang Richter, and Dietrich Haarer
10.1 Introduction
The investigation of the electronic states of polyatomic organic dye molecules is enhanced
considerably when they are doped into suitable solid host matrices [1] at low concentrations.
If the interaction between host and guest molecules is weak, the guest molecule will exhibit
its characteristic electronic and vibronic signature except for a possible shift of the total
spectrum, starting with the zero-phonon origin. The hosts are chosen according to their absorption
because they must not overlap with the guest states under investigation. This can be
achieved for a variety of polymers, n-alkanes, and rare gases, so that the optical absorption
spectra of these guest-host systems are dominated by the guest molecules.
A natural extension of these investigations was the incorporation of guest molecules into
disordered systems to serve as probes of the host properties. In contrast to n-alkane hosts, in disordered
host materials the optical absorption spectra of these guest-host systems are often characterized
by broad and featureless bands, the so-called inhomogeneous broadening. When a
dye molecule is incorporated into a host matrix its transition energy will experience a shift due
to the dye-matrix interaction. In a (hypothetic) perfect crystal all guest molecules will experience
exactly the same shift, whereas in disordered hosts a distribution of local environments
and therefore a distribution of transition energies is observed, which can be as large as several
100 cm –1 . The inhomogeneous broadening prevents access to the homogeneous absorption
lines, whose widths are related to energy and phase relaxation processes in the system.
The fact that the inhomogeneous broadening is a consequence of statistically distributed
local environments of the guest molecules means that it can be described by a stochastic
model which dates back to Markoff [2]. For a review see Stoneham [3]. In this model, for
details see next Section, the host material is dissected into matrix units, which in the case of
rare gas hosts are identical with the atoms. For polymeric hosts the matrix units correspond
to the respective monomer units. The inhomogeneous distribution of absorption frequencies
is then given by averaging over all possible arrangements of matrix units around a cavity
containing the dye molecule, which are weighted by the line shift caused by each respective
configuration. The overall inhomogeneous shift is calculated by adding up the contributions
of each matrix unit to the total line shift. Their contributions depend on their respective positions
to the dye molecule. A modification of the interatomic distances, for example
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
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10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids
through the application of external pressure, leads to a change of the interaction and therefore
to a shift of the line. This suggests that pressure effects can also be taken into account
within the framework of the stochastic model [4].
However, in order to generate measurable changes of the entire inhomogeneous band,
pressure changes in the range of several gigapascals are required. These high pressures alter
the structure of the investigated sample significantly which also affects the homogeneous
width [5]. To observe the effects of small pressure changes the spectral resolution has to be
increased significantly. This can be accomplished by the experimental technique of hole burning
spectroscopy, which was introduced in 1974 [6, 7]. The hole burning method, for a description
see Ref. [8] and Chapter 5, allows access to homogeneous line widths which are
masked by the inhomogeneous broadening. Furthermore, within the inhomogeneous band,
spectral holes can serve as persistent narrow frequency markers. The changes to these frequency
markers are due to: structural relaxations of disordered host materials (Chapter 5), IR
induced spectral diffusion (Chapter 6), external perturbations – like electric fields – [9, 10],
and pressure (Section 10.6) and can be monitored accurately over extended time periods.
In the ideal case the hole width is twice the homogeneous width. Therefore the increase
in spectral resolution is roughly given by the ratio of inhomogeneous to homogeneous
width. This ratio, also referred to as the multiplexing factor, which is due to the possible application
of the hole burning technique in data storage devices, is typically in the range of
10 3 to 10 5 [8]. This means that for producing detectable changes in a spectral hole due to
external pressure the magnitude of the external perturbation can be reduced by approximately
the multiplexing factor. This in turn allows sample investigations near equilibrium
conditions, meaning that the displacements of the matrix units from their zero-pressure positions
are small. The results of the first pressure tuning experiments reveal a red shift of the
hole center, when the pressure is raised, and a blue shift, when the pressure is lowered after
the burning of the hole [11, 12]. Furthermore, in both cases a hole broadening is observed.
Laird and Skinner’s extension of the stochastic approach [4] for describing pressure effects
was first applied to dye molecules in various polymer hosts like polyethylene (PE),
polystyrene (PS), and polymethylmethacrylate (PMMA). Their prediction of a frequencydependent
pressure shift was duly verified. Within 20% the predicted value agreed with the
experimental results. The apparent success of the stochastic model was surprising, because
polymers meet the basic requirements of the model quite poorly. Usually it is assumed that
one type of interaction, e. g. dispersive forces, between dye and matrix molecules predominates.
Also, the matrix units, which in this case are the monomer units, are considered to be
spherical and independent of each other yielding additive contributions to the solvent shift
of the doped dye molecules, which are also assumed to be spherical. Additionally, the matrix
units are assumed to be able to arrange themselves independently around the dye molecule.
In the case of polymer hosts, the matrix units clearly are unable to arrange themselves independently,
because they are connected by strong directional bonds. Moreover, the monomer
units are often slightly polar, like PMMA.
Furthermore, the validity range of the results of the stochastic model remained unclear,
since in the course of the calculation two conflicting approximations with regard to
the number density r of the matrix units within the interaction range of the dye molecule
were made: the Gaussian approximation (valid for r??) and the continuum approximation
(valid for r?0). The latter is neglecting correlations between matrix units, which arise
from their mutual steric exclusion.
182

10.2 Stochastic theory
To clarify these inconsistencies we set out to systematically investigate the stochastic
model, using model systems that come closest to its basic assumptions. For this reason we
chose solid rare gases as host matrices. For these systems additional information is readily
available [13, 14]. Furthermore, the interaction potential and its parameters are well-known for
pure rare gases. Matrix parameters can be varied systematically by using different rare gases,
while still retaining a structural similarity. Some rare gas mixtures show a different structure
than pure rare gases (Sections 10.3 and 10.6). Through a systematic variation of the mixing ratio
the impact of the involved structural transition onto the optical data can be investigated.
10.2 Stochastic theory
The above-mentioned stochastic model description considers an amorphous system of N matrix
units, containing a small concentration of dye molecules. Each matrix unit will shift the
electronic absorption line of the dye molecule by some amount n( ~R i †, where ~R i is the position
of the i th matrix unit with respect to the dye molecule. The contributions of all matrix
units are assumed to be additive. The total inhomogeneous distribution of absorption lines
can then be written as [2–4]:
I…† ˆ 1
V N Z
d ~R 1 ...d ~R N P… ~R 1 ; ...; ~R N †
X N
iˆ1
!
… ~R i † : …1†
V is the volume of the sample. In order to further evaluate this expression, the function
P ( ~R 1 ; ...; ~R N ), which is the probability of finding N matrix units at positions ~R 1 ; ...; ~R N ,
has to be specified. If correlations between the matrix units are neglected, the (N+1) particle
solute-solvent distribution function can be factorized into a product of two-particle distribution
functions g( ~R):
P… ~R 1 ; ...; ~R N †ˆ QN
nˆ1
g… ~R n † :
…2†
This is the so-called continuum approximation, which is valid in the case of small
number densities r of matrix units.
Furthermore, considering only a non-polar solute and solvent, the perturbation function
n( ~R) of the transition frequency is of the familiar Lennard-Jones type
( "
s 12 #
s 6
… ~R† ˆ 4"
R R 0 R R 0
if R R 0 ;
(3)
1 if R

10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids
with the origin shifted by R 0 to account for the large size difference between solute and solvent
[4]. In this representation R 0 + s/2 is the solute radius, while
6p
2s can be taken as the
solvent diameter. This means that a matrix unit located at the position of the potential minimum
R 0 +
6p
2s will shift the electronic absorption line of the dye by the potential depth e,
as given in Eq. 3. In the case of r??, the inhomogeneous distribution is Gaussian, which
is a consequence of the Central Limit Theorem [15].
For the spatial distribution of the matrix units a simple step function was chosen [4],
meaning that a matrix unit can be found anywhere outside a spherical cavity of radius
R 0 + R c , but not inside. The parameter R c determines how close a matrix unit can come to
the dye molecule. With the above conventions about solvent and solute dimensions, we have
R c = s/2 (1 +
6p
2 ) &1.061 s.
Inserting Eqs. 2 and 3, the inhomogeneous distribution (Eq. 1) can now be evaluated,
resulting in expressions relating the experimentally accessible parameters, namely the solvent
shift n s (spectral position of the band maximum with respect to its gas phase position)
and the full width at half maximum G s to the local number density r and to the microscopic
parameters e, R 0 , s, and R c [15].
We can now turn to the question of matrix correlations, which arises because of the
application of two conflicting approximations in the course of the calculation: the Gaussian
approximation for large and the continuum approximation for small number densities
of the matrix units. A reasonable way out of this dilemma is to retain the Gaussian approximation
and to introduce matrix correlations. Here the only correlation effect considered
is the principle that two matrix units cannot be located at the same position. This
means that the factorization used in Eq. 2 cannot be applied. Matrix correlations can be
accounted for, within the framework of the stochastic model, by introducing a three-particle
distribution function g 3 ( ~R; ~R 0 † [16–18]. Applying the Kirkwood superposition approximation
[19],
g 3 … ~R; ~R 0 †g… ~R†g… ~R 0 †g S ~R ~R 0
; …4†
we introduce a solvent-solvent distribution g S (| ~R ~R 0 |). In analogy to the solute-solvent distribution,
we insert a simple step function for g S (| ~R ~R 0 |) with the cut-off radius given by
6p
2s . As in the case of neglected correlations it is possible to derive expressions that allow
the calculation of the potential depth e and the local number density of the matrix units r
from spectroscopic data (Section 10.5).
As mentioned in the introduction, the stochastic approach can also provide a description
of the effects of external pressure on spectral holes. In our experiments, however, the
pressure change is always accompanied by a simultaneous change in temperature, see
Ref. [20]. Therefore, the observed hole shifts and broadenings will not only be due to pressure
changes, but also to the thermal expansion of the matrix. There may also be dynamical
effects such as phonon scattering and (fast) tunneling systems (TLS) relaxations, which we
will not treat in this contribution.
Pure pressure effects have been accounted for by the Laird and Skinner theory [4].
This theory can be conveniently expanded to also include the thermal expansion of the matrix
and its influence on the dye molecule, for details see Refs. [20, 21]. In analogy to the
inhomogeneous distribution (Eq. 1), the temperature-pressure kernel, i. e. the probability that
184

10.2 Stochastic theory
a guest molecule with the original solvent shift n will have a new transition frequency n'
after a pressure change Dp and a temperature change DT, can be written as
f … 0 ; p; T† ˆ
Z
1
I…†V N
P N
iˆ1
d ~R 1 ...d ~R N P… ~R 1 ; ...;R * N†
… ~R i † 0 P N
0 … ~R i ;p;T† : …5†
The observed temperature-pressure shift was found to be linear, while the concomitant
hole broadening can be described by a power law. The latter is clearly dominated by dynamical
processes which are affected by the change in temperature. It is now assumed that, in analogy
to pure pressure effects, the hole shift and the hole broadening due to the thermal volume
expansion are linear functions of the temperature change. The function n' ( ~R i ; Dp; DT)
in Eq. 5 can then be linearized to
0 … ~R;p;T† ˆ… ~R† k @… ~R†
@R
iˆ1
!
R
p ‡ g
3
@… ~R†
@R
!
R
T :
3
k is the compressibility and g the volume thermal expansion coefficient of the matrix.
It is important to note that the temperature and pressure effects have opposite signs, see
Eq. 6 and Refs. [11, 22].
The further evaluation follows the lines of the calculation given in Ref. [21]. Within
the above-mentioned approximations, the pressure-temperature kernel is found to be Gaussian.
At this point we restrict our considerations to the evaluation of the hole shift, which is
calculated as
" Z ( )( )#
…; p; T† ˆ d 3 R @… ~R†
Rg…R†
1 ‡… S † … ~R†
3 @R
s 2 …gT kp† :
S
(7)
s S is related to the full width at half maximum of the inhomogeneous line shape G S by
G
s S = p S
, while n is the burning frequency of the spectral hole with respect to the gas-phase
2 2 ln 2
position of the transition frequency of the dye molecule.
For the modified Lennard-Jones potential (Eq. 3) Eq. 7 has to be evaluated numerically.
It can, however, be simplified by considering only a purely attractive van der Waals
potential. This approximation seems reasonable, since only matrix units located in the attractive
part of the intermolecular potential can cause the observed red shift in pure pressure
tuning experiments [11]. We obtain for the temperature-pressure shift:
…; p; T† ˆ2 ‰ kp gTŠ: …8†
With this equation, we found a simple way to extract the pure pressure shift from our
pressure-temperature data. Due to the opposite sign of pressure and temperature shifts
(Eq. 8), the pure pressure shift will be even larger with the inclusion of the temperature ef-
…6†
185

10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids
fects, leading to a larger compressibility after including the correction. The evaluation of
our experimental data leads to an additional complication. The volume thermal expansion
coefficient depends, contrary to the compressibility, strongly on the temperature, even in the
small temperature interval from 1.8 K to 5 K. Fortunately, for crystalline samples experimental
data are available [23] which allow us to substitute gDT ? R TDT
T
dT'g(T') in Eq. 8.
With the help of this equation it can be shown that the pressure effects should clearly dominate
the experiment [20], which is confirmed by the observed red shift in pure pressure-tuning
experiments [11].
The stochastic description of pressure effects can also be modified to include correlations
between matrix units [20, 21]. It is interesting to note that for the purely attractive van
der Waals potential, Eq. 8 will not acquire any additional terms suggesting that for the determination
of the compressibility k matrix correlations will also play a minor role when the
Lennard-Jones potential Eq. 3 is valid (Section 10.7).
10.3 Rare gases
Due to their simplicity rare gases have served as model systems for a long time. This means
that a large data basis [13, 14, 24] is available. Of special importance for our investigations
is the knowledge of the microscopic interaction potential between rare gas atoms. For its attractive
region it was already calculated by F. London [25], to depend as 1/R 6 upon the interatomic
distance. In the well-known Lennard-Jones (6,12) potential the exponential R-dependence
for the repulsive contribution is approximated by an algebraic 1/R 12 term:
s
12
s
6
…R† ˆ4~"
: …9†
R R
The potential depths ~e, which are directly connected with the polarizability a and the
parameters s, which determine the distance of the potential minimum, are listed in Tab. 10.1.
Table 10.1: Lennard-Jones parameters ~e and s [26, 27] and polarizability a [28].
Gas ~e [cm –1 ] s [Å] a [10 –25 cm 3 ]
Argon 83.7 3.405 16.3
Krypton 113 3.65 24.8
Xenon 161 3.98 40.1
Another aspect is the structure of solid rare gases, which can be produced either by
slow freezing the liquid or by condensation onto a sufficiently cold substrate. While the for-
186

10.4 Experimental
mer method yields single crystals with macroscopic dimensions, condensed rare gases are
polycrystalline. Electron and X-ray diffraction [29, 30] experiments show the crystals and
crystallites to have a face-centred cubic structure, although theoretical simulations predicted
the hexagonal close packing to be energetically favoured by about 0.01%.
The situation is different for certain quench condensed rare gas mixtures, where computer
simulations [31] predicted the possibility of an amorphous structure for rare gases
whose atomic radii differ by at least 10%. X-ray diffraction measurements [30] indeed revealed
an amorphous structure for Ar 1–x Xe x matrices with mixing ratios 0.2 < x < 0.7. This
opens up the possibility to study the transition between the polycrystalline and amorphous
state by varying the mixing ratio in a single system.
10.4 Experimental
The dye-doped rare gas matrices had to be prepared in situ. For this purpose a gas handling
system was installed which provided alternatively the pure rare gases (Linde, 99.998% purity
for argon, 99.990 % purity for krypton and xenon) and rare gas mixtures. The concentrations
of the mixture components were determined by monitoring the pressure in the mixing
chamber during the composition process. Commercially available phthalocyanine (Aldrich,
used without further purification) was sublimated and subsequently mixed with the desired
matrix gas in a Knudsen effusion furnace. Following Bajema et al. it consisted of a sublimation
chamber (T&600 K) and a super-heating chamber (T&700 K). The gaseous matrix
passed through a nozzle onto a sapphire substrate being in thermal contact with the cold finger
of a continuous flow cryostat.
Two different cryo-systems were used, which opened a wide temperature range for investigations.
In system I temperatures down to 4 K could be produced using a commercially
available continuous flow cryostat (Oxford Instruments). A home-built special outer vacuum
container enabled either sample deposition or optical access through the sample. The system II
is our own design. It consisted of a continuous flow cryostat which was immersed into liquid
helium after the sample preparation. Temperatures down to 1.5 K could be reached by pumping
the helium. Furthermore, hydrostatic pressure could be exerted by sealing off the helium
bath. A typical dye concentration was approximately 2610 –3 mol/l with a sample thickness
of about 10 mm. To characterize the samples, absorption spectra were recorded with a monochromator
(Jobin Yvon THR 1500, used resolution about 0.2 cm –1 ). Hole burning studies
were performed using a single mode dye laser (Coherent 599, bandwidth about 3 MHz). The
optical setup is described in detail in Refs. [11, 32].
187

10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids
10.5 Inhomogeneous absorption lines
The visible absorption spectra of phthalocyanines show two regions with strong and characteristic
absorption bands. The lowest electronic energy transition, called Q band, is degenerated
in metallophthalocyanines due to a molecular D 4h symmetry. In free base phthalocyanines
(H 2 Pc), however, the reduction of molecular symmetry from D 4h to D 2h , caused by
the two H atoms in the inner ring, lifts the degeneracy of the Q band. It splits into two
bands, which are conventionally referred to as Q x (for the lower energy component) and Q y
(for the higher energy component). These two absorption bands are shown in Fig. 10.1 for a
H 2 Pc-doped krypton matrix. At higher frequencies a number of vibronic lines appear.
Figure 10.1: Absorption spectrum of H 2 Pc in a krypton matrix.
Our spectra agree quite well with spectra recorded previously by Bajema et al. [33].
Due to the inherent local disorder in vapour condensed matrices [30, 34] the absorption
spectra are inhomogeneously broadened. For the investigated systems Gaussian line shapes
are expected from theoretical considerations [2, 3]. However, this prediction is difficult to
test due to a slight asymmetry of the absorption spectra towards lower frequencies. This observation
indicates a second preferential position of H 2 Pc molecules in the rare gas matrix
and confirms earlier measurements [35]. In samples with a sufficiently high dye concentration
this second site leads to a weaker set of spectral transitions shifted about 70 cm –1 towards
lower energies [35, 36].
Table 10.2 compiles the spectral positions of the Q x and Q y bands as well as the
widths of the Q x bands and their solvent shifts n S , i. e. the frequency shifts of the respective
absorption maxima with respect to the absorption of free H 2 Pc molecules, as measured in a
supersonic free jet [37]. From argon to xenon increasing solvent shifts and inhomogeneous
widths can be observed.
188

10.5 Inhomogeneous absorption lines
Table 10.2: Spectroscopic data (positions n, solvent shift n S , width G S ) of the absorption bands of H 2 Pcdoped
rare gas matrices. Units are in cm –1 .
Matrix n (Q x ) G S (Q x ) n S (Q x ) n (Q y )
Argon 14764 44 364 15731
Krypton 14664 58 464 15605
Xenon 14540 89 593 15466
The solvent shift measures the strength of the dye-matrix interaction which is purely
dispersive for the investigated systems. When going from argon to xenon the increasing solvent
shift can therefore be explained by the increasing matrix polarizability. The latter is
even effective enough to compensate the reduction of the number of interacting matrix
atoms. Due to their increasing size the solvent shift is produced by fewer interacting units
which, according to basic statistics, enlarges the relative fluctuation of the line shifts and
manifests itself in the observed increasing inhomogeneous width.
However a quantitative analysis of the absorption bands requires the formalism of the
stochastic theory, outlined in Section 10.2, which is able to connect the measured solvent
shifts and inhomogeneous bandwidths to two microscopic parameters of the system, namely
the respective number densities r of matrix units around a dye molecule and the depths e of
the dye-matrix interaction potentials. While for polymer matrices the stochastic theory was
to be used to determine geometric parameters [4], they are already known for our rare gas
model systems from independent investigations. This enabled us to reduce the number of fit
factors and calculate the r and e values as listed in Tab. 10.3.
Table 10.3: Potential parameters e and number densities r calculated with (^r;^e) and without (r, e) matrix
correlations. r cryst : number density for pure rare gas crystals.
^e [cm –1 ] e [cm –1 ] ^r [Å –3 ] r [Å –3 ] r cryst [Å –3 ]
Argon 23.1 1.57 0.00975 0.145 0.0267
Krypton 33.4 2.45 0.00787 0.104 0.0222
Xenon 47.5 4.58 0.00594 0.062 0.0173
The widely used continuum approximation turned out to lead to the unreasonable result
that r exceeds the number densities measured in bulk rare gas crystals! The physical
reason for this discrepancy is that the continuum approximation does not prevent matrix
units from occupying identical positions, since their mutual steric exclusion has been neglected!
Therefore the calculation is based upon too many matrix units – reflected in the unrealistically
large number density – giving insufficient weight to the individual matrix unit
[20]. Taking matrix correlations into account, as sketched in Section 10.2, the depth of the
dye-matrix interaction potential ^e is increased and the number density ^r reduced to appropriate
values (Table 10.3).
189

10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids
10.6 Pressure effects
In Fig. 10.2 the effect of hydrostatic pressure on a spectral hole, as measured with our system
II, in an argon matrix is shown. A pressure increase causes a red shift and broadening
of the initial hole. It has to be emphasized, however, that due to the construction of our cryostat,
pressure changes were always connected with temperature changes following the helium
vapour pressure curve. Therefore the measured broadening and shift of spectral holes do
have thermal and dynamical contributions and are not only due to the pure pressure effect.
These two (thermal and dynamical) contributions to the hole shift are separated in
Fig. 10.3 for the case of argon. The measured hole shift is plotted versus the pressure increase
Dp. The pure pressure contribution to the hole shift was extracted by taking into account
the thermal expansion of the matrix (Section 10.2). Figure 10.3 also shows that the
Figure 10.2: Broadening and pressure shift of a spectral hole (H 2 Pc in an argon matrix). Trace (1): Original
hole profile; Trace (2): Spectral hole after a pressure increase of 88 kPa.
Figure 10.3: Shift of hole minimum vs. pressure change: Raw data (squares) and data after temperature
correction (bullets).
190

10.7 Rare gas mixtures
pure pressure shift depends linearly on the pressure increase and is larger than the total observed
pressure shift due to the opposite signs of temperature and pressure effect (Eq. 8).
Using Eq. 7 it is now possible to calculate the matrix compressibility k. The observed
linear pressure shift towards lower energies suggests that the interacting matrix atoms are located
in the attractive part of the Lennard-Jones potential. The van der Waals potential is
therefore expected to be a good approximation, which was employed in the derivation of the
analytical expression for the pressure dependence (Eq. 8). Indeed, Tab. 4 shows that the results
obtained, using the Lennard-Jones and the van der Waals potential respectively, are
identical within the accuracy of our experiment.
Table 10.4: Matrix compressibilities in units of GPa –1 determined with (^k) and without (k) correlations.
Calculation using Lennard-Jones (LJ) and van der Waals (vdW) potentials.
^k LJ k LJ k vdW
Argon 0.2500 0.2429 0.2758
Krypton 0.2899 0.2653 0.2895
Xenon 0.3768 0.3919 0.4191
Of further interest is the question whether matrix correlations are of the same importance
for the calculation of the matrix compressibility as they are for the potential parameter
and the number density. A detailed analysis, however, reveals that for our model systems as
well as for H 2 Pc-doped polymers PE and PS the corrections to the simple equations due to
matrix correlations are smaller than the experimental error. This explains the good agreement
between optically determined compressibilities, calculated conventionally without consideration
of matrix correlations, and mechanically determined bulk values for polymeric systems.
10.7 Rare gas mixtures
So far experiments with H 2 Pc-doped pure rare gas matrices have been reported. As discussed
above, the transition from a polycrystalline to an amorphous solid can be studied in a
single system using condensed rare gas mixtures, on which we will focus now.
The Q x absorption bands of H 2 Pc-doped matrices with various Ar-Xe compositions are
plotted in Fig. 10.4 together with the spectra for the pure argon and xenon matrices. In contrast
to the slight asymmetry of the spectra in the pure rare gas matrices, which was already mentioned
above, the absorption bands in rare gas mixtures can be well-described by a Gaussian line
shape. For all composition ratios the Q x bands are located within the spectral range given by the
absorption lines of the two pure rare gas hosts. However, it is not possible to describe the spectra
of the mixed matrices as a linear combination of the pure constituents spectra. This demonstrates
that the gases are homogeneously mixed before condensation as well as in the solid matrix.
191

10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids
Figure 10.4: Normalized absorption spectra of the Q x band of H 2 Pc in solid Ar 1–x Xe x matrices. From
left to right: x = 1 (pure Xe), x = 0.6, 0.4, 0.3, 0.15, x = 0 (pure Ar).
More insight into the dependence of the spectra upon the composition ratio can be
gained if the frequency of the absorption maximum is plotted against the Xe concentration.
This is done in Fig. 10.5, where the solvent shift is also given. The latter rises continuously
with increasing Xe concentration.
The solid line in Fig. 10.5 was calculated with Eq. 1 extended to mixed systems. The
density of the matrix units is interpolated from the values of the pure components assuming
a constant atomic volume, as described in [39]. Using this simple approach, the calculated
Figure 10.5: Position of the maximum of the Q x band (left-hand scale) and solvent shift n S (right-hand
scale) of H 2 Pc in Ar 1–x Xe x matrices versus Xe concentration. The solid line corresponds to the interpolation
of the mass density on the basis of constant atomic volume.
192

10.7 Rare gas mixtures
curve agreed quite satisfactory with the experimental data. The slight deviation from the theoretical
curve may have its origin in uncertainties of the respective matrix densities [40] and
compositions due to different sublimation temperatures of argon and xenon.
The quasi-homogeneous line width is obtained by extrapolating the spectral hole width
to zero hole area, assuming that the latter depends linearly on the burning fluence [8]. Thus,
saturation effects may be excluded [41].The quasi-homogeneous line widths are determined
for all mixed Ar-Xe matrices with a sufficient optical density at a temperature of 1.8 K. The
results are plotted in Fig. 10.6 versus the Xe concentration. Since the line width depends on
the spectral position in the absorption band [40] the values are interpolated to the frequency
of the absorption maximum of each sample. The quasi-homogeneous line width in pure argon
matrices is significantly lower than in pure xenon matrices. From argon to xenon concentrations
up to 50 % only minor changes for H 2 Pc-doped rare gas mixtures are observed
with no recognizable discontinuity at the expected transition from a polycrystalline to an
amorphous matrix.
Figure 10.6: Quasi-homogeneous width, as measured at the maximum of the Q x band, of H 2 Pc in
Ar 1–x Xe x matrices versus Xe concentration.
It is well-known that at liquid helium temperatures the fluorescence lifetime plays a
minor role only for the quasi-homogeneous width whereas the dominating contributions
come from fast relaxations of TLS and coupling to local modes. The experimentally observed
smaller widths in argon matrices as compared to xenon matrices can therefore be attributed
to the smaller TLS density, which was determined by specific heat investigations
[30], and to a higher local mode frequency, measured via hole burning [20]. These two relaxation
mechanisms are thought to be characteristic for disordered systems. Thus, TLS and
local modes are not only present in amorphous solids but also in polycrystalline samples, in
which these excitations can exist at grain boundaries as well as in the neighbourhood of im-
193

10 Matrix-Molecule Interaction in Dye-Doped Rare Gas Solids
purities inside the crystallites. The line width behaviour at the transition to the amorphous
state can therefore be attributed to the high disorder and to the porous structure that is already
present in the pure matrices.
10.8 Summary
The present investigation centred on a stochastic model, which was developed to describe
the inhomogeneous broadening of absorption lines of dye molecules doped into a disordered
host. Furthermore, the model seems to account well for the effects of external pressure on
spectral holes burned into these bands, making it possible to determine the host compressibility
by performing a purely optical experiment. This model has also been widely used for
dye-doped polymer hosts although they hardly satisfy the basic assumptions, making its application
questionable. The aim of the present work was therefore to put the model to the
test by investigating dye-doped rare gas solid systems that come closest to satisfying these
requirements.
In its original form the theory treats the matrix units in a continuum approximation by
neglecting correlations between them. This leads to unreasonable results for microscopic
parameters, which could be demonstrated for the first time in our model systems. Therefore,
the theory was extended to take into account steric exclusion of matrix units. For the depth
of the dye-matrix interaction potential and the local matrix density the modified theory produces
physically realistic results.
The behaviour is different with respect to pressure effects on spectral holes. Our investigations
verified that the local matrix compressibility, measured in our experiments, is
mainly sensitive to the dispersive part of the dye-matrix potential. Therefore, the details of
the repulsive region of the dye-matrix potential as well as the consideration of matrix-matrix
correlations cause only minor changes to the matrix compressibility, which are beyond the
experimental accuracy. This results justifies previous determinations of the matrix compressibility
using the original model.
In order to investigate the influence of the matrix structure our experiments were extended
to dye-doped rare gas mixtures of argon and xenon. In these matrices the transition,
from polycrystalline pure rare gas matrices to amorphous mixed rare gas solids, can be monitored
by varying the composition ratio. At the transition to the amorphous state the quasihomogeneous
line width revealed no increase, which can be attributed to the high degree of
structural disorder already present in the condensed pure rare gas matrices.
Structural transitions can also be studied in annealing experiments. Therefore the next
step in our investigations will focus on the changes of the inhomogeneous and quasi-homogeneous
widths, occurring during thermal cycling experiments in rare gases and – turning to
more complex systems – in dye-doped amorphous water [42]. The investigation of water
matrices promises, in addition to the structural aspects, to shed some light on the importance
of electrostatic interactions due to the polarity of the water molecules.
194

References
Acknowledgements
The authors gratefully acknowledge many elucidating discussions with Dr. L. Kador.
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196

II
Mainly Micelles, Polymers, and Liquid Crystals
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

11 The Micellar Structures and the Macroscopic
Properties of Surfactant Solutions
Heinz Hoffmann
11.1 General behaviour of surfactants
Surfactants consists of molecules with a hydrophobic and a hydrophilic part. Due to this amphiphilic
nature these molecules adsorb from aqueous solutions onto interfaces, as is expressed
in their name [1]. The molecules are densely packed in these adsorbed monolayers.
In the aqueous bulk phase the surfactant molecules assemble above a characteristic concentration,
called cmc, into micellar structures, which can be understood as interfaces in the
bulk solution. The driving force for the adsorption and the aggregation is the same for both
processes and is given by the hydrophobic interaction [2].
The molecular packing of the surfactant molecules in films and micelles is mainly determined
by the area a which a surfactant molecule requires at the interface. In both, films
and micelles, the molecules will occupy about the same area [3]. If the area a is larger than
the cross section a 0 of the hydrocarbon chain in its equilibrium conformation the interface
of the micelle will be curved towards the hydrocarbon core. If it is the same the interface
will be flat on a local scale and if it is smaller the interface will be curved the other way
around. The importance of the area a for the structures of micelles was recognized by Tanford
[2] and later by Ninham and co-workers [4].
From simple geometrical considerations it follows that the shape of a micelle can be
expressed by the packing parameter P = ar/v, where r is the length of the hydrocarbon chain
and v its volume. The packing parameter varies from 3 for globules to 1 for bilayers. As a
result of their packing parameter single-chain surfactants often form globular micelles and
double-chain surfactants tend to form bilayer structures.
In a film at a macroscopic interface the area a of the molecule controls the thickness of
the film and the order parameter of the hydrocarbon chains inside the film. If a>a 0 the film
is thinner than the length of the surfactant chain and water borders directly on the hydrocarbon
chains of the surfactant resulting in a large interfacial tension. If a &a 0 the film looks like
one half of a bilayer of a real membrane. In this case the water is not in contact with the hydrocarbon
chains and the interface has a low interfacial tension. In this qualitative model the
parameter a controls the curvature of a micelle and the interfacial tension at an interface.
Note that the differently shaped micelles are present only in surfactant solutions.
When the micellar solutions are in contact with hydrocarbon the micelles will solubilize hy-
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
199

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
drocarbon and will be transformed into microemulsion droplets. Then the curvature of the
droplets correlates with the interfacial tension [5]. The minimum of this interfacial tension
corresponds to the maximum of the solubilization capacity of the surfactants [6, 7]. This relation
has been used extensively to optimize surfactant systems for tertiary oil recovery. On
the basis of theoretical work the values of interfacial tension at the minimum seems to depend
on the bending constants of the films [8]. A high (low) minimum of interfacial tension
would mean a high (low) bending constant.
11.2 From globular micelles towards bilayers
Amphiphilic substances with an extremely small hydrophilic group, for example aliphatic alcohols
with intermediate chain lengths, are called cosurfactants. Normally cosurfactants do
not form micelles in aqueous solutions but they are surface active and make up mixed micelles
with normal surfactants. In cosurfactant/surfactant mixtures the mean area a per head
group must vary smoothly with the mole fraction of the cosurfactant, between the value for
the pure surfactant a s and the value for the cosurfactant a c . For most uncharged surfactants
the value a s corresponds to a situation where normal globular micelles are formed while a c
corresponds to a situation where inverse micellar structures of the L 2 phase exist. In principle,
by varying the mole fraction X c of the cosurfactant in aqueous solutions, we should encounter
all the types of micellar structures and mesophases that can possibly exist in surfactant
solutions. While the area per head group can vary smoothly with X c , it is evident that
the curvature of the micellar interface cannot vary continuously from strongly convex to
strongly concave for micelles of the same type. The system has to switch to differently
shaped micelles and mesophases if X c is varied.
There is evidence that the systems use defects in mesophases to adjust the mean curvature
that is set by X c . Strain in a mixed system can also be avoided by concentration fluctuations.
The system can form structures with two different X c values that are in equilibrium,
one with a lower X c and one with a higher X c than the average value. For all these reasons
we find a rich variety of micellar structures and phases if X c is varied and the total concentration
is kept constant. The diversity which is encountered in such situations is shown in
Fig. 11.1 which represents the phase diagram of the ternary system tetradecyldimethylaminoxide/hexanol/water
(C 14 DMAO/C 6 OH/H 2 O) at the water-rich corner. With increasing X c
we observe the six visibly different single-phase regions L 1 (micellar solution), L 1
*,L al ,L 3
*
(vesicle phases), L ah (lamellar phase with flat lamellae), and L 3 (bicontinuous sponge phase)
[9]. All these phases have distinct properties. Furthermore there is a change in shape of the
micelles in the L 1 phase from globules to rods. This transition is, however, not visible to the
unaided eye. The phases L 1 ,L 1
*,L 3
*, and L 3 are optically isotropic while the phases L al and
L ah are birefringent (the L al phase is only weekly birefringent in most cases).
The general features of this diagram are typical for such ternary systems and are practically
independent of the chain lengths of the surfactant and the cosurfactant. Figure 11.1
200

11.2 From globular micelles towards bilayers
c C6 OH mM
400
2Φ
300
L 3
2Φ
200
L αh
L αl-h
L
100
*
αl
L 3
2Φ
* L
L 1
1
0
0 50 100 150 200
Figure 11.1: Section of the phase diagram of the ternary system C 14 DMAO/C 6 OH/H 2 O in the waterrich
corner at 25 8C. For details see Section 11.4.1 and Fig. 11.11.
c C 14 DMAO /mM 201
shows that the phases with lamellar structures are already formed at total surfactant concentrations
around 1 wt%. These phases, except the L al and L ah phase, which probably do not
develop a phase boundary, are separated from each other by two-phase regions. Three phase
regions can also exist in the phase diagram although all phases consist of about 99 wt%
water. Therefore and because of the very similar densities of the different phases the phase
separation takes a long time and hence the determination of such phase diagrams is dificult.
Furthermore the refractive indices of the phases are almost the same, which makes it difficult
to distinguish one and two-phase regions unambiguously as the single phases are often
slightly turbid. Inspite of that most two-phase systems separate macroscopically into two
phases after a sufficiently long time.
The phase diagram of Fig. 11.1 serves as an example of what happens if the radius of
curvature varies continuously with the mixing ratio X c at a constant surfactant concentration.
For the classic non-ionic alkyl polyglycolethers (C 12 E 5 ) this can be done by varying the temperature
[10]. For ionic surfactants this can be done by increasing the salinity [11], by mixing
a cationic with an anionic surfactant, or vice versa [12]. In all these different situations
we can expect to observe the same sequence of phases. The phase diagram in Fig. 11.1 was
established on the basis of visual observation of samples. To check for birefringence the
samples were viewed between crossed polarizers.
Before we come to the microstructures there are several facts that are worth emphasizing.
Note that the phase boundaries between the different phases are given by more or less
straight lines, which means that the mixing ratio for the micelles is constant at the phase
boundaries and does not depend on the concentration. Notice also that the equimolar mixtures
of surfactant and cosurfactant are in the middle of the wide L a region. The combination
therefore acts like a real double-chain surfactant. The phases from L 1
* to L ah are actually
subphases of L a , as has been demonstrated by freeze fracture electron micrographs. They
consist of bilayer-type structures [13] but the topology of the bilayers is quite different.

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
11.3 Viscoelastic solutions with entangled rods
11.3.1 General behaviour
Surfactant solutions with globular micelles are generally Newtonian liquids with a low viscosity
which increases linearly with the volume fraction F of the particles according to Einstein’s
law
ˆ s …1 ‡ 2:5F† :
…1†
Here Z s is the viscosity of the pure solvent. F is an effective volume fraction which
also takes into account the hydration of the molecules. It can be up to three times the true
volume fraction. But also in this case the viscosity of a 10 wt% surfactant solution is only
about twice as high as the solvent viscosity. The same is true if anisometric micelles are present
in the solutions as long as their rotational volumes do not overlap.
On the other hand, many surfactant solutions are highly viscous even at low concentrations
in the range of 1 wt%. From this observation it can be concluded that the micellar
aggregates in these solutions must organize themselves into some kind of a supermolecular
network. The viscosity of such systems strongly depends on parameters like surfactant concentration,
ionic strength, temperature, or concentration of additives. The solutions usually
also have elastic properties because the zero shear viscosity Z 0 is the result of a transient
network of entangled rods that is characterized by a shear modulus G 0 and a structural relaxation
time t according to
0 ˆ G 0 t :
…2†
In this case the shear modulus is determined by the particle density n of entanglement
points
G 0 ˆ n kT :
…3†
The networks of entangled cylindrical micelles could be made visible by cryo-electron
microscopy by Talmon et al. [14]. These pictures clearly show the shape and the persistence
length of the rods but they do not reveal their dynamic behaviour. According to Eq. 2 the
viscosity is the result of structure and dynamic behaviour, i. e. the structural relaxation time
constant t which strongly depends on many parameters and can vary by many orders of
magnitude for the same surfactant, if for instance the counterion concentration is changed
[15]. This is shown in Fig. 11.2 where Z 0 for several cetylpyridiniumchloride (CPyCl) concentrations
is plotted vs. the sodiumsalicylate (NaSal) concentrations. With increasing
amounts of NaSal the viscosity passes a maximum, then a minimum, and finally a second
maximum. This behaviour is due to a corresponding dependence of t on the NaSal concentration
while G 0 is independent of this parameter for a constant CPyCl concentration. Simi-
202

11.3 Viscoelastic solutions with entangled rods
10 6 30mM CPyCl
10 5
60mM CPyCl
100mM CPyCl
η o /mPas
10 4
10 3
10 2
10 1
10 1 10 2 10 3
c NaSal /mM
Figure 11.2: Double logarithmic plot of the zero shear viscosity Z 0 for three different CPyCl concentrations
vs. the concentration of added NaSal at 25 8C.
lar results have been observed for many systems by various groups. The viscosity of a 1 wt%
surfactant solution varies from the solvent viscosity and up to a 10 6 times higher value.
Figure 11.3 shows a schematic sketch of a network of rod-like micelles. It is generally
assumed that the crosslinks of the network that cause the elastic behaviour are entanglements
[16]. However, this is not always true. Adhesive contacts between the micelles or a
transient branching point, like a many armed disc-like micelle, can act as crosslinks [17,
18]. Some experimental evidence for both possibilities have recently been observed. The entangled
thread or worm-like micelles have typical persistence lengths between some 100 to
1000 Å and they may, or may not, be fused together at the entanglement points.
Figure 11.3: Schematic drawing of an entanglement network of long cylindrical micelles. Note the different
length scales: k is the mesh size, l the mean distance between two knots, and m the contour length
between two knots.
The cylindrical micelles have an equilibrium network conformation. They constantly
undergo translational and rotational diffusion processes. They also break and reform. If the
network is deformed or the equilibrium conditions are suddenly changed it will take some
time until the system reaches equilibrium again. If a shear stress p 21 is applied to a network
203

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
solution in a much shorter time than the equilibration time, the solution behaves like a soft
material that obeys Hooke’s law,
p 21 ˆ G 0 g ;
…4†
with the spring constant G 0 and the deformation g. On the other hand, if the stress is applied
for a longer time the system flows like a Newtonian liquid,
p 21 ˆ 0 _g ;
…5†
with the zero shear viscosity Z 0 and the shear rate _g. A mechanical model for a viscoelastic
fluid is the so-called Maxwell model which consists of a spring with the constant G 0 and a
dashpot with the viscosity Z 0 . The zero shear viscosity of such a system can be expressed by
the product of G 0 and t according to Eq. 2.
Both quantities can be determined by oscillating rheological measurements [19].
Many viscoelastic surfactant solutions can be described in a large frequency range by the
Maxwell model with a single shear modulus G 0 and a single structural relaxation time constant
t [20]. This is shown in Fig. 11.4 a. However, there are surfactant solutions which behave
in a completely different manner as can be seen from Fig. 11.4b. The systems do not
show a frequency-independent plateau value of the modulus and the viscosity cannot be expressed
by a single G 0 or a single t value. In such situations the shear stress after a rapid deformation
relaxes exponentially [21],
t
a
p 21 ˆ ^p 21 exp : …6†
t
Figure 11.4a shows that the loss modulus G@ increases again with the frequency f.
This increase can be related to Rouse modes of the cylindrical micelles. On the basis of a
10 3
10 1
10 2
G´,G´´ / Pa
10 2
10 0
G´
10 -1
G´´
|η * |
10 -2
10 -3 10 -2 10 -1 10 0 10 1
f/Hz
Figure 11.4a: Double logarithmic plot of storage modulus G', loss modulus G@, and complex viscosity
|Z*| vs. frequency f for a solution with 100 mM CPyCl and 60 mM NaSal at 25 8C. The solution behaves
like a Maxwell fluid with a single shear modulus G 0 and a single structural relaxation time t.
204
10 1
10 0
10 -1
|η * |/Pas

11.3 Viscoelastic solutions with entangled rods
G´, G´´ / Pa
10 2
10 1
10 0
G´
G´´
|η * |
10 -1
10 -3 10 -2 10 -1 10 0 10 1
f/Hz
Figure 11.4b: The same plot for a solution with 80 mM C 14 DMAO, 20mMSDS , and 55 mM C 6 OH.
The solution does not behave like a Maxwell fluid. Note the differences to Fig. 11.4a: G@ does not pass
over a maximum, G' does not show a plateau value but increases with f after the intersection with G@
with a constant slope of 0.25.
10 2
10 1
10 0
10 -1
|η * |/Pas
theoretical model [22] the minimum value of G@ can be expressed by the storage modulus
G', the entanglement length l e and the contour length l k of the cylindrical micelles according
to
G 00 min ˆ G0 l e
l k
:
…7†
11.3.2 Viscoelastic systems
Rod-like micelles from ionic surfactants are usually formed at high ionic strengths with
strongly binding, or hydrophobic counterions, or with large hydrophobic groups, like double-chain
or perfluoro surfactants [4]. Solutions of such surfactants become highly viscous
and viscoelastic with increasing concentration. In Fig. 11.5 some results are shown as double
logarithmic plots of viscosity vs. concentration [23–25] in spite of the different chemistry
of the surfactants. All these surfactants show a concentration region where the slopes
are the same and in the range of 8.5, which is very high. The viscosity starts to rise
abruptly at an overlap concentration c* and follows the scaling law within a small transition
concentration
0 /
c x
; …8†
c
where x is about 8.5 ± 0.5. The exponent must therefore be controlled by the electrostatic interaction
in the solutions and is much bigger than the value of 4.5 ± 0.5 expected for large
polymer molecules, which do not change their size with increasing concentration. From the
large exponent it can be concluded that the rod-like micelles continue to grow with concentration
above c*. This was postulated by McKintosh et al. [26].
205

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
η ο /mPas
10 6 Lec / C 14 DMAO / SDS
10 5
10 4
10 3
10 2
10 1
C 8 F 17 SO 3 NEt 4
C 16 TMASal
C 16 PyCl + NaSal
C 16 C 8 DMABr
10 0
10 0 10 1 10 2
c/mM
Figure 11.5: Double logarithmic plot of zero shear viscosity Z 0 vs. concentration c for several solutions
of charged surfactants. Note that all different systems show the same power law exponent within a limited
concentration range above c*.
Figure 11.6 shows logarithmic plots of Z 0 vs. c for the system CPyCl + NaSal [27]. It
has been shown that this system can be described by the Maxwell model with one G 0 and
one t value above the first viscosity maximum. In this range the complex behaviour is due
to the concentration dependence of t, while G 0 steadily increases with concentration.
Furthermore, it was found that t is controlled by the kinetics of breaking and reforming of
the micelles [28]. These processes are faster than the reptation of the rods under the given
experimental conditions.
As already mentioned, the same is true for the system with a constant CPyCl concentration
with increasing amounts of NaSal, which is shown in Fig. 11.2. The dependence of
the viscosity on the counterion concentration is controlled by a corresponding behaviour of
t, while G 0 is independent of the NaSal concentration. The viscosity is therefore a result of
the dynamics of the system and not of its structure. This can be proved by cryo-electron microscopy
[14]. The electron micrographs show no differences between the structure of the
10 5 CPyCl + NaSal
CPyCl + 0.3MNaSal
10 4
uncharged
η ο /mPas
10 3
10 2
10 1
10 0 10 1 10 2 10 3
c/mM
Figure 11.6: Double logarithmic plot of zero shear viscosity Z 0 vs. concentration c of CPyCl + NaSal
for solutions at the first viscosity maximum (o), at the second maximum (p), and at the minimum (_)
at 25 8C (Fig. 11.2).
206

11.3 Viscoelastic solutions with entangled rods
micelles for all four concentration regions. This is very remarkable because the micelles are
differently charged in the concentration regions. Below the first maximum of the viscosity
they are highly and positively charged, at the minimum they are completely neutral, and at
the second maximum they carry a negative charge.
The power law behaviour in the different concentration regions is also completely different,
as can be seen from Fig. 11.6. The exponent at the first maximum is 8, at the minimum
1.3 and at the second maximum 2.5. No theoretical explanation is available for the extremely
low exponent of 1.3 which has also been observed for systems which completely
differ in chemistry and conditions. Therefore the exponent seems to represent a general behaviour
determined by fundamental physics.
Figure 11.7 shows logarithmic plots of Z 0 vs. c for zwitterionic alkyldimethylaminoxide
surfactants (C x DMAO) [17]. The data again show a power law behaviour over extended
concentration regions. Some curves show a break indicating that also uncharged systems can
switch the relaxation mechanism. At the lowest concentration region, where a power law behaviour
is observed, the slope is the highest and almost the same as for the observed polymers.
This is somewhat surprising because it could be expected that the length of the micellar
rods increases with increasing concentration, which should lead to a higher exponent of
the power law. It is therefore likely that the dynamics of the systems is already influenced
by kinetic processes under these conditions.
η o /mPas
10 6 C 14 DMAO
10 5 C 16 DMAO
OleylDMAO
10 4
10 3
10 2
10 1
10 0
10 0 10 1 10 2 10 3
c/mM
Figure 11.7: Double logarithmic plots of zero shear viscosity Z 0 vs. concentration c for solutions of alkyldimethylaminoxide
surfactants of various chain lengths at 25 8C.
For higher concentrations a smaller exponent is observed. Obviously a new mechanism
is operating under these conditions which is more effective in reducing a stress than the mechanism
in the low concentration region. Generally a mechanism can only become dominant
with increasing concentrations if it is faster than the one at low concentrations. The moduli
increase with the same exponent in the various concentration regions. Furthermore, Fig. 11.7
shows that the absolute value of Z 0 for C 16 DMAO and ODMAO (Oleyl) differ by an order
of magnitude even though the slope is the same in the high concentration region. This is due
to the fact that in the kinetically controlled region the breaking of the micelles depends very
much on the chain length of the surfactant.
207

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
According to the theory of micelle formation, addition of cosurfactants to surfactant
solutions leads to a transition of spherical micelles to rods or to a growth of rod-like micelles
[29]. As a consequence the viscosity of a surfactant solution increases with increasing
cosurfactant concentration. This is shown in Fig. 11.8 for the system of 100 mM C 14 DMAO
with various cosurfactants. The viscosity first increases and then passes a maximum. The situation
is similar as shown in Fig. 11.2. It is likely that the micelles still grow steadily with
increasing cosurfactant concentration but the system switches from one mechanism on the
left side of the maximum to a faster mechanism on the right side. The reason for the switch
is probably that the rods become more flexible with increasing cosurfactant concentration.
The different mechanisms become obvious in Fig. 11.9 where log(Z 0 ) is plotted vs. log (c)
for C 14 DMAO with different amounts of decanol (C 10 OH), which is completely solubilized
in the micelles due to its poor solubility in water. The plot shows that the slopes of mixtures
at the left side of the maximum are the same while the mixture with the highest cosurfactant/surfactant
ratio has the lowest slope of 1.3. This value is equal to the one for CPySal at
the viscosity minimum. Systems with similar low slopes from the literature [30] are shown
η o /mPas
10 3 hexanol
octanol
decanol
10 2
lecithin
10 1
10 0
0 10 20 30 40 50 60
c/mM
Figure 11.8: Semilogarithmic plot of zero shear viscosity Z 0 of a 100 mM solution of C 14 DMAO vs.
the concentration c of added cosurfactants at 25 8C. Note that all curves pass a maximum.
10 4 TDMAO/C 10 OH=5:1
TDMAO/C 10 OH=6.6:1
10 3 TDMAO/C 10 OH=10:1
TDMAO/C 10 OH=20:1
TDMAO
η o /mPas
10 2
10 1
10 0
10 1 10 2 10 3
C 14 DMAO / mM
Figure 11.9: Double logarithmic plot of zero shear viscosity Z 0 of a mixture of C 14 DMAO and C 10 OH
with different molar ratios of cosurfactant/surfactant vs. the concentration of C 14 DMAO at 25 8C.
208

11.3 Viscoelastic solutions with entangled rods
η o /mPas
10 3
10 2
CPyCl+NaSal (T=20 o C)
C 14 DMAO/C 10 OH=5.1(T=25 o C)
C 16 EO 7 (T=45 o C)
CTAB+ NaClO 3
10 1
10 1 10 2 10 3
c/mM
Figure 11.10: Double logarithmic plot of zero shear viscosity Z 0 vs. the total surfactant concentration
for several surfactant systems with the same power law exponent of 1.3.
in Fig. 11.10. The chemistry and also the viscosity values for these systems are completely
different, yet the slope is the same. The shear moduli for these systems are very similar for
given surfactant concentrations and they also scale with the same exponent. The low exponent
for the viscosity therefore comes about by the structural relaxation time which scales
with an exponent of –1 according to
t /
c 1
: …9†
c
11.3.3 Mechanisms for the different scaling behaviour
All studied surfactant systems show the same qualitative behaviour. The viscosity rises
abruptly at a characteristic concentration c* which is the lower the longer the chain length
of the surfactant is. At the concentration c* the rotational volumes of rod-like micelles start
to overlap and form a network. This network can be an entanglement network, as in polymer
solutions, or the micelles can be fused together, or can be held together by adhesive contacts.
All these types have been proposed and it is conceivable that they can really exist
[31]. Theoretical treatments assume that the cylindrical micelles are worm-like and flexible.
This can be the case for some of the presented systems, but certainly not for the binary surfactant
systems. For example, C 16 DMAO and ODMAO have very low c* values and if the
micelles would be flexible they would have to be coiled below c*. But both, electric birefringence
and dynamic light scattering determinations, show that at c 7c* the lengths of the rods
are comparable with the mean distances between them. Hence, the rods must be rather stiff
with persistence lengths of some 1000 Å. Similar results have been obtained by electron micrographs.
The abrupt increase of Z 0 at c* is difficult to understand for stiff rods even taking
into account further growth of the rods with increasing concentration. Many systems with
rod-like particles show that the rotational time constant for the rods is very little affected
around c* and the solutions do not become viscoelastic above c*. We therefore have to assume
that other interactions than just hard core repulsion between the rods must exist which
209

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
are responsible for the formation of the network at c*. It is conceivable that the rods form
adhesive bonds or that they actually form a connected network of fused rods, as has been
proposed by Cates [32]. In such situations two different types of networks have to be distinguished,
namely saturated and interpenetrating networks. In the first case the mean distance
between the knots or entanglement points is equivalent to the meshsize but in the second
case this distance can be much larger than the meshsize between neighbouring rods.
The viscosities above c* increase abruptly following the power law (Eq. 8) with an exponent
x>3.5, while the exponent of the power law for the shear modulus is always about
2.3. This behaviour has been treated in detail by Cates et al. [33]. The structural relaxation
times are affected by both, reptation and bond breaking processes. Cates treats three different
kinetic mechanisms. The first consists of the break of a rod with the formation of two
new end caps. In the recombination step the rods have to collide at the ends in order to fuse
into a new rod. In the second mechanism the end cap of one rod collides with a second rod
and in a three-armed transition state a new rod and a new end cap is formed. In the third
mechanism two rods collide and form two new rods through a four armed transition state. It
is obvious that stress can release all three mechanisms. These mechanisms lead to somewhat
different power laws for the kinetic time constant t according to
t /
c x
: …10†
c
But in all cases x is between 1 and 2. Mechanism 3 is less likely in systems with low
c* values and stiff rods. For this argument it is likely that mechanism 2 or 3 is effective in
the more concentrated region of the pure C x DMAO solutions.
For C 16 DMAO and ODMAO solutions the slope of the log (Z 0 )-log (c) plots suddenly
changes at a characteristic concentration c**. For both regions the same scaling law for the
shear modulus with an exponent of about 2.3 is found while the power exponent for the relaxation
times changes from 1 to zero. From the constant exponent for G 0 it can be concluded
that the structures in both concentration regions are the same. The change in the
slope must therefore be due to a new mechanism which becomes effective above c**. The
independence of t of the concentration makes it likely that in this region the dynamics are
governed by a purely kinetically controlled mechanism and that a reptation process is no
longer possible. This situation has not yet been treated theoretically. Cates mentioned, however,
that there might be situations where the reptation loses its importance. The more effective
mechanism in this range could be the bond interchange mechanism.
For the C 14 DMAO solutions with C 10 OH and the CPySal system at the minimum
viscosity the extremely low power law exponent of 1.3 for the viscosity and an exponent of
–1 for the structural relaxation times are found. A detailed explanation for this behaviour
which has also been described by other authors [30] has not yet been given. The explanation
for this behaviour could be that the cylindrical micelles for systems with such low exponents
are very flexible. In such a situation the persistence length would be much shorter than the
contour length between two neighbouring entanglement points. Furthermore, the persistence
length should be independent of the concentration. The diffusion of the rods can therefore
be described by a constant diffusion coefficient D. For two arms to collide they have to diffuse
a distance x and for two neighbouring rods to undergo a bond exchange process they
have to diffuse at least the average distance x between two arms. The time constant t D for
210

11.4 Viscoelastic solutions with multilamellar vesicles
the diffusion should be proportional to x 2 /D. Since the meshsize x decreases with the square
root of the concentration one obtains for the structural relaxation time t the observed law
t !1/D7c, which is identical with Eq. 9. We therefore conclude that for systems with the
low exponent 1.3 the viscosity is governed by a diffusion-controlled bond interchange mechanism.
The absolute values of Z 0 and t can still vary from system to system because the
persistence length l p of the rods should depend on the particular conditions of the systems.
With increasing chain length l p should decrease and D increase. For such situations we
would expect to find the lowest activation energies for the viscosity.
A similar mechanism could be based on the assumption of connected or fused threadlike
micelles as crosslinks. These crosslinks could be visualized as disc-like micelles from
which the rods extend. This means that the transient intermediate species in the various
bond interchange mechanisms are now assumed to be stable. In this situation all end caps
could be connected. The resulting network could be in the saturated or unsaturated state.
The crosslink points could then slide along the thread-like micelles like a one-dimensional
diffusion process with a concentration-independent diffusion coefficient. A knot can be dissolved
if two network points meet on their random path. If the structural relaxation time is
determined by this random movement a similar equation t !1/c can be derived. Both models
can describe the low exponent of 1.3 for the scaling law for Z 0 and for both models reptation
is no longer necessary for the release of stress. The mechanisms could probably be
distinguished by the concentration dependence of the self-diffusion coefficients D s of the
surfactant molecules. In a solution with a connected network a surfactant molecule should
be in the same situation as in a L 3 phase for which it has been shown that D s is independent
of the surfactant concentration [34]. As Kato et al. [35] showed that the D s values increase
with concentration for a corresponding system with rods a diffusion limited bond interchange
mechanism is more likely for the explanation of the scaling law of the structural relaxation
time than the assumption of connected networks of thread-like micelles.
11.4 Viscoelastic solutions with multilamellar vesicles
11.4.1 The conditions for the existence of vesicles
Thermodynamically stable vesicles have been found recently in many solutions with various
types of surfactants [36–39]. Such vesicles occur especially in ternary systems of zwitterionic
or non-ionic surfactants, aliphatic alcohols as cosurfactants, and water. As explained in
detail in Section 11.4.2, this is due to the spontaneous curvature of the micellar surface
which is continuously lowered with increasing amounts of the cosurfactant because of its
very small headgroup area [3]. The systems try to come as close as possible to the spontaneous
mean curvature without causing much bending energy by adjusting the two main curvatures
on the micellar aggregates. As a consequence the micellar system undergoes several
phase transitions with increasing cosurfactant concentration.
211

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
The vesicle phase occurs within a wide range of the total surfactant and cosurfactant
concentration but only within a small range of the molar ratio of cosurfactant/surfactant
around 1 :1. If the surfactant bilayers are charged by the addition of an ionic surfactant the
phase diagram becomes somewhat simpler because some mesophases are suppressed by the
charge. As can be seen from the Fig. 11.11, the vesicle phase is still found under these conditions
but it is shifted towards higher cosurfactant concentrations. Thermodynamically
stable vesicles have also been found in ternary systems of non-ionic alkylpolyglycol surfactants,
cosurfactants, and water [10] and also in the binary system of the double-chain surfactant
didodecyldimethylammoniumbromide (DDABr) and water [42].
600
500
400
L 1
+L 2
L α
L 1
+L 3
300
L 3
L 3
L 1
+L α
L αh
L 1
*
200
L αl
L * 1
+L 1
L*
1
100
L 1
+L*
1 L 1
L 1
0,0 0,2 0,4 0,6 0,8 1,0 0
X C 14 TMABr
c C6 OH /mM
Figure 11.11: Section of the phase diagram of the quaternary system 100 mM C 14 DMAO/C 14 TMABr/
C 6 OH/H 2 Oat258C. For details see Refs. [9, 13, 40, 41].
11.4.2 Freeze fracture electron microscopy
The vesicles can be made visible by freeze fracture transmission electron microscopy (FF-
TEM). Figure 11.12 shows the vesicles in a system of 90 mM C 14 DMAO, 10 mM tetradecyltrimethylammoniumbromide
(C 14 TMABr), 220 mM n-hexanol (C 6 OH), and water. The
cationic surfactant can also be replaced by the anionic surfactant SDS without causing a
change of the vesicles or of the rheological properties. From this electron micrograph it is
possible to recognize some general features which are of relevance for the properties of the
212

11.4 Viscoelastic solutions with multilamellar vesicles
Figure 11.12: Electron micrograph of vesicles in the system of 90 mM C 14 DMAO, 10 mM
C 14 TMABr, 220 mM C 6 OH, and water (the bar represents 1 mm).
systems. The vesicles have a rather high polydispersity; some seem to be rather small unilamellar
vesicles, while others consist of about 10 bilayers. The interlamellar spacing is fairly
uniform and is in the range of 800 Å. The vesicles are very densely packed and the whole
volume of the system is completely filled with them. They have a spherical shape even
though the outermost shell can have a radius of several 1000 Å. Some of them do not consist
of concentric shells but have defects. The larger vesicles have typical sizes in the range
of 1 mm and the wedges, which result from the dense packing, are completely filled with
smaller vesicles. Thus, each vesicle is sitting in a cage from which they cannot escape by a
simple diffusion process without deforming their shells. Therefore, the system must have
viscoelastic properties.
11.4.3 Rheological properties
In Fig. 11.13 the viscoelastic properties of a vesicle phase of 90 mM C 14 DMAO, 10mM
C 14 TMABr, 220 mM C 6 OH, and water are demonstrated by plots of the storage modulus G',
the loss modulus G@, and the magnitude of the complex viscosity |Z*| as a function of the oscillation
frequency f. G' is much larger than G@ and almost independent of f in the whole frequency
range. The system behaves like a soft solid and must have a distinct yield stress. This
can be seen from the plot (Fig. 11.14) of the shear rate _g as a function of the applied shear
stress s. As can be seen from Fig. 11.15 both, shear modulus G 0 , the frequency-independent
value of G', and yield stress s y, increase with increasing total surfactant concentration. Both
quantities disappear at total concentrations below 1 wt%. This means that the vesicles are no
longer densely packed and they can move around each other easily under shear flow. For
higher concentrated systems s y varies linearly with G 0 and is about one tenth of G 0 . This
means the vesicles must be deformed by about 10% before they can pass each other under
shear. Similar results are obtained for the vesicles in the binary system of DDABr and water.
213

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
10 2 G´
G´´
|η * |
10 2
G´, G´´ / Pa
10 1
10 1
10 0
|η*| / Pas
10 0
10 -1
10 -2 10 -1 f/Hz
10 0 10 1
Figure 11.13: Double logarithmic plot of storage modulus G', loss modulus G@, and the magnitude of
the complex viscosity |Z*| vs. frequency f for a vesicle phase of 90 mM C 14 DMAO, 10 mM
C 14 TMABr, 220 mM C 6 OH, and water at 25 8C.
4
3
σ/Pa
2
1
0
0,0 0,1 0,2 0,3 0,4 0,5
γ
.
/s -1
Figure 11.14: Plot of the shear rate _g vs. applied shear stress s for the same vesicle phase as in
Fig. 11.13, showing a distinct yield stress value s y at 1.1 Pa.
G´ / Pa
50
40
30
20
10
σ y
G´
5
4
3
2
1
σ y /Pa
0
50 100 150 200
c surfactant /mM
0
Figure 11.15: Plot of shear modulus G 0 and yield stress s y vs. surfactant concentration c surfactant for a
vesicle phase of C 14 DMAO and C 14 TMABr with a molar ratio of 9 : 1 and C 6 OH at 25 8C.
Figures 11.16 a and 11.16 b demonstrate the influence of charge density on the bilayers
on G 0 . The modulus increases with increasing amounts of ionic surfactant and saturates at
about 10 mol% of the ionic compound (Fig. 11.16 a). On addition of electrolyte the modulus
decreases again linearly with the square root of the ionic strength (Fig. 11.16 b).
214

11.4 Viscoelastic solutions with multilamellar vesicles
25
100 mM Surfactant (C 14 DMAO + C 14 TMABr), 220 mM C 6 OH
G´ / Pa
20
15
10
5
0
0 5 10 15 20
c C 14 TMABr
Figure 11.16a: Plot of shear modulus G 0 vs. the concentration of the ionic surfactant C 14 TMABr for a
vesicle phase with a total concentration of 100 mM C 14 DMAO + C 14 TMABr, and 220 mM C 6 OH at
25 8C.
G´(0.1 Hz) / Pa
70
60
50
40
30
20
10
0
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0
Figure 11.16b: Plot of shear modulus G 0 vs. the square root of the concentration of added NaCl for a
vesicle phase with 85 mM C 14 DMAO, 15mMC 14 TMABr, and 300 mM C 6 OH at 25 8C.
(c NaCl /mM) 1/2 215
The effect of the chain length of the surfactant compound can be seen from Fig. 11.17. For
systems with the same concentrations the modulus increases with increasing chain length from
C 10 to C 16 and decreases for C 18 . This means that the modulus is not only determined by the electrostatic
repulsion between the bilayers but also depends on the thickness of the bilayers.
The headgroup of the surfactant in the vesicle phase has only a moderate effect on
rheological properties. For example, a vesicle phase of 90 mM C 12 E 6 ,10mMSDS, and
250 mM C 6 OH shows a very similar behaviour as the corresponding phase of 90 mM
C 14 DMAO, 10mMC 14 TMABr, and 220 mM C 6 OH. The absolute values of G', G@, and
|Z*| are also very similar for both systems. The same can be found for systems where the
concentration of the cosurfactant (Fig. 11.18) or the chain length of the cosurfactant is changed
(Fig. 11.19). The figures show that the moduli and the yield stress increase slightly with
the cosurfactant concentration and the chain length of the cosurfactant. But this increase is
small in comparison with the strong dependence of these values on the concentration and
the chain length of the surfactant compounds. The temperature has only a very small effect
on both, G 0 and s y , between 10 8C and 60 8C.
An interesting effect is shown in Fig. 11.20 where the shear viscosity Z as a function of
the shear rate _g and the magnitude of the complex viscosity |Z*| as a function of the angular

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
10 3 x=10 x=12
x=14 x=16
x=18
G´ / Pa
10 2
10 1
10 -2 10 -1 10 0 10 1
f/Hz
Figure 11.17: Double logarithmic plot of storage modulus G' vs. frequency f for vesicle phases of
90 mM CxDMAO, 10mMC 14 TMABr, and 220 mM C 6 OH at 25 8C with various chain lengths x of
the zwitterionic surfactant.
Figure 11.18: Plot of storage modulus G', loss modulus G@, and yield stress s y against the cosurfactant
concentration for a vesicle phase of 90 mM C 14 DMAO, 10mMSDS, and varying amounts of C 6 OH at
25 8C.
Figure 11.19: Double logarithmic plot of storage modulus G' vs. frequency f for a vesicle phase of
90 mM C 14 DMAO, 10mMC 14 TMABr, and 160 mM C n OH for cosurfactants with various chain
lengths at 25 8C.
216

11.4 Viscoelastic solutions with multilamellar vesicles
|η*|, η/Pas
|η*|(ω)
η(γ )
10 2 90 mM C 12 E 6 ,10mMC 14 TMABr,
280 mM C 6 OH
|η*|(ω)
η(γ )
90 mM C 14 DMAO, 10 mM C 14 TMABr,
10 1
220 mM C 6 OH
10 0
10 -1 10 0 10 1 10 2 10 3
Figure 11.20: Double logarithmic plot of shear viscosity Z vs. shear rate _g and of the magnitude of the
complex viscosity |Z*| vs. angular frequency o for two different vesicle phases at 25 8C.
frequency o are compared for two vesicle phases. The diagram shows the important difference
to viscoelastic solutions of thread-like micelles [43]. They do not always fulfill the Cox-
Merz rule, stating that for all values of _g the shear viscosity Z is equal to |Z*| at the corresponding
value of o in the shear thinning region [44]. At low shear rates or frequencies both
viscosities have the same value, while at higher shear rates Z (_g) is larger than |Z*|(o = _g). The
curve for the zwitterionic system shows two breaks at characteristic shear rates. For shear
rates above these characteristic values it is likely that the multilamellar vesicles undergo transformations
to new structures as has recently been proposed by Roux et al. [45].
11.4.4 Model for the shear modulus
γ /s -1 , ω /rads -1 217
In previous publications the shear modulus for the multilamellar phases was considered to
be the result of the interactions of hard sphere particles [46–48]. In this picture each
charged multilamellar vesicle is treated as a hard sphere. The theoretical treatment of the
samples would then be similar to latex systems. The modulus of the systems depends on the
chainlength of the surfactants that are used for the preparation of the systems if all other
parameters like charge density, salinity, and concentration of surfactants and cosurfactants
are kept constant. It can be argued that the differences of the moduli result from a change of
the particle density of the vesicles. But these values are not known exactly. Systems with different
chainlengths have similar conductivities which suggests that the particle density is
also similar and therefore not responsible for the different shear moduli.
Furthermore the birefringence looks the same, too. If the particle density of the vesicles
decreases and if the mean size of the vesicles increases then the birefringence should increase.
But this is not the case. The different moduli must therefore have a different origin. We propose,
consequently, a different model for the explanation of the magnitude of the shear moduli. For
the treatment of multilamellar vesicle phases and L a phases this model was proposed by E. v. d.

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
Linden [49]. To our knowledge the theory has not yet been applied to experimental results. E. v.
d. Linden assumes that multilamellar vesicles (droplets) are deformed in shear flow from a
spherical to an elliptical shape. Turning into the deformed state the energy of closed shells is
shifted because their curvature as well as their interlamellar distance D are changed. Due to the
interaction of the bilayers, expressed by the bulk compression modulus B, the inner shells are
deformed and the total deformation energy E of the lamellar droplet gets minimized. Assuming
that the volume of a droplet is not modified by the deformation, the surface A must increase.
One can define an effective surface tension s eff = E/DA. E. v. d. Linden obtains:
ef f ˆ 1
2 …KB†1 = 2 ; …11†
where K is the bulk rigidity which is correlated to the bilayer’s bending constant k by K = k/D.
We can relate this effective surface tension to the shear modulus G of a vesicle with radius R.
Using the identity G =2s eff /R yields:
1 =2
k
G ˆ … KB†1
= 2 D B
ˆ
R R
: …12†
Both, bulk compression modulus and bending constant, depend on the charge density
of the bilayers and the shielding of the charges with excess salt.
This means the theory of E. v. d. Linden results in a calculation of the geometrical
average of the compression E B and bending energy E K per unit volume.
The expression (Eq. 12) can be squared to
G 2 ˆ
k
D B
R 2 : …13†
With n = R/D which denotes the number of bilayers in a vesicle we obtain:
G 2 ˆ nk
R 3 B ˆ E k E B :
…14†
Now we can try to find adequate expressions and values for the quantities B and K by
other theories.
For the bending constant as a function of the charge density we can use the expression
(Eq. 15) that has been given by H. Lekkerkerker [50],
K el ˆ kBT …q 1†…q ‡ 2†
2pQk …q ‡ 1†q
…15†
p
with q ˆ p 2 ‡ 1
218
and p ˆ 2pQjsj
.
ke

11.4 Viscoelastic solutions with multilamellar vesicles
k B is Boltzmann’s constant, T the absolute temperature, Q the Bjerrum length, k the reciprocal
Debye length, and s the surface charge density.
For the bulk compression modulus we can use the expression that is often used to describe
the interaction between two charged particles
K ˆ f a f s n p k 2 d 2 V…d† ;
…16†
where V(d) is the energy of interaction between a pair of spherical particles
V…d† ˆz2 e 2
exp…ka† 2
exp… kd†
; …17†
4p" …1 ‡ ka† d
d denotes the separation of particles and a their radius. f a and f s are numerical factors [51].
There is a further possibility to get an expression for the compression modulus B.
This quantity may be simply given by the osmotic pressure between the bilayers. According
to a theory of Dubois et al. [52] we have calculated the osmotic pressure using the equation
P ˆ c m kT :
…18†
where c m is the concentration of ion particles at the midplane between the bilayers, calculated
by the Poisson-Boltzmann equation for the current conditions. In a previous paper we
have discussed the possibility to identify the shear modulus with the osmotic pressure. This
seems to be obvious because the osmotic pressure qualitatively increases like the shear modulus
with increasing charge density of the bilayers and decreases on salt addition. But this
attempt failed because the calculated values, which were in the order of several thousand
pascals, were too large. In the current context, however, it looks reasonable to identify the
osmotic pressure with the compression modulus B from Eqs. 11 and 12.
Now there is a concept which is appropriate to reproduce the characteristic features of
our experimental results. These are the influence of the charge density and the influence of
salinity on the shear modulus.
The growth of the shear modulus with the charge density can mainly be attributed to
the increase of the bending constant and the osmotic pressure with the charge density. For
small salinities the decrease with the salinity seems to come from the pair potential V (d)
whereas the linear decrease at higher salinities seems to come from the bending constants.
However, we should keep in mind that for both changes the vesicular structures do not remain
constant. Therefore we cannot give a complete quantitative interpretation of the experimental
results and the data cannot be fitted precisely to an exact theoretical model.
At the end of this paragraph we check whether Eq. 12 gives reasonable results for our
primary data. If we assume d = 80 nm for the interlamellar distance, R = 0.5 mm for the radius
of a vesicle, 10% for the charging degree (i. e. a surface charge density on the bilayers
of e 0 /500 H 2 , where e 0 is the elementary charge), k = 0.56 k B T for the electrical contribution
of the bending modulus (calculated according to Eq. 15) [47], and B = P = 3000 Pa for
the compression modulus (calculated according to Eq. 18) [46] then Eq. 12 yields the shear
modulus G&18 Pa. This value is very close to the experimental one and demonstrates that
a correct approach for a theoretical description of the rheological properties of the solutions
seems to be found.
219

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
11.5 Ringing gels
11.5.1 Introduction
For a large variety of amphiphilic compounds cubic phases are a particular class of surfactant
systems normally observed in the more concentrated region of the phase diagram [53].
Such phases may occur in binary systems (surfactant and water) as well as in ternary or
pseudo ternary systems, where usually a hydrocarbon is the third component. In addition
they may contain a long alcohol chain as a fourth component (cosurfactant). Such a composition
reminds one of that of microemulsions and for that reason they are also called microemulsion
gels [54]. However, the denomination cubic phase is insofar more instructive since
it relates directly to the structure of the corresponding systems.
These phases are thermodynamically stable, optically isotropic, transparent, and highly
viscous which distinguishes them easily from conventional L 1 or L 2 phases, i. e. micellar or
reverse micellar phases. Often they possess a yield stress and exhibit elastic properties for
not too large deformations [55]. In particular these elastic properties are responsible for the
other name – ringing gels – which sometimes is used for these systems because samples of
these phases usually show an acoustical resonance phenomenon (ringing sound) after being
tapped with a soft object [56]. This effect is not necessarily associated with a cubic phase
but quite frequently observed within this class of surfactant systems. The ringing phenomenon
is observed for all the cubic phases studied by us.
It might be mentioned here that such phases have been used already for pharmaceutical
and cosmetical applications without having detailed structural information regarding the
corresponding systems at that time [57]. More recently it has been claimed that cubic phases
may play a key role in the fusion process of biological membranes [58, 59] and they are frequently
formed by lipids, obtained from membrane extracts.
The main perspective of our investigations was first to determine the detailed microstructure
of the corresponding systems and then to relate them to the observed macroscopic
properties. For that purpose two surfactant systems were chosen. Both have a cubic phase
but at different locations in the phase diagram. In the first system – tetradecyldimethylaminoxide
(C 14 DMAO)/hydrocarbon/H 2 O – the cubic phase is located between the isotropic L 1
phase and the hexagonal phase at a constant surfactant/hydrocarbon ratio (Fig. 11.21 a) [55].
In the second system – bis-(2-ethylhexyl)sulfosuccinate (AOT)/1-octanol/H 2 O – it is situated
between the lamellar phase, at lower octanol content, and an isotropic L 2 phase, at higher
octanol and lower water content, or a reverse hexagonal phase at higher AOT content
(Fig. 11.21 b) [60].
These systems are insofar similar as they both are next to an isotropic phase. Moreover
a binary system containing triblock copolymers of the polyethyleneoxide/polypropyleneoxide/polyethyleneoxide
(PEO/PPO/PEO) type has been studied for which we also
have found a formation of a cubic phase with location in the phase diagram similar to the
aminoxide case.
220

11.5 Ringing gels
Decane
20
5
80
T=25°C
50
1
2
3
4
50
2
+
S
80
20
H 1
L α S
G
H L 2
O
1
C 14
DMAO
0 20 N c 40 60 80 100
Figure 11.21a: Phase diagram of the ternary system C 14 DMAO/decane/H 2 Oat258C. Isotropic water
continuous phase (L 1 ), nematic phase (N c ), cubic phase (G), hexagonal phase (H 1 ), lamellar phase (L a ),
crystals (S), other phase regions (1, 2, 3, 4, 5).
Octanol
20
80
50 50
L 2
80
I 2
D
H AOT
2
O
0 20 40 60 80 100
L 1
Figure 11.21b: Phase diagram of the ternary system AOT/1-octanol/H 2 Oat258C (in wt%). Isotropic
water continuous phase (L 1 ), lamellar phase (D), (bicontinuous) cubic phase (I 2 ), isotropic oil continuous
phase (L 2 ), reverse hexagonal phase (F), other phase regions without symbols.
F
20
11.5.2 The aminoxide system
For the aminoxide system investigations were done along a line of constant C 14 DMAO/decane
ratio. Upon going along such a line close to the solubilization capacity of the surfactant one can
cross from the L 1 phase into the cubic phase by increasing the surfactant concentration. Samples
located on such a dilution line contain spherical aggregates of constant size. This was shown by
means of static and dynamic light-scattering experiments as well as by SANS measurements
[55, 61]. The interactions are very well described by a hard sphere model. In Fig. 11.22 ex-
221

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
3000
2000
1
I(q) in a.u.
1000
2
3
6 54
0
0,00 0,05 0,10 0,15 0,20 0,25
qin1/Å
Figure 11.22: SANS intensity curves as a function of the magnitude of the scattering vector q for samples
of constant C 14 DMAO : decane weight ratio of 5.4 : 1 and different total concentrations of
C 14 DMAO plus decane were 8 wt% for 1; 16 wt% for 2; 24 wt% for 3, 28 wt% for 4, 32 wt% for 5,
37.1 wt% for 6. Samples 1–5 are in the L 1 phase whereas sample 6 is located in the cubic phase.
perimental SANS curves for various total concentrations (constant weight ratio of
C 14 DMAO :decane = 5.4 :1) are given which can be fitted curves in good agreement with the
hard sphere model. Furthermore the SANS experiments show that the microemulsion droplets
are fairly monodisperse, i.e. possess a polydispersity index of about 0.1 [62].
For the cubic phase the SANS experiment shows still the same type of aggregates but
more concentrated. However, here the packing exceeds the critical volume fraction of
53 vol%, which is typical for a hard sphere crystallization [63, 64]. In the cubic phase the
spherical aggregates are packed similar to metal atoms in a cubic lattice. From SANS investigations
of samples in the cubic phase it seems that the packing is not primitive cubic but
either face-centred cubic (fcc) or body-centred cubic (bcc). Between these two possibilities
an experimental distinction was not feasible [65].
Samples of the cubic phase show a very distinct SANS behaviour when the scattering
intensity is detected two-dimensionally. Then an isotropic pattern is no longer observed.
One finds more or less pronounced spikes superimposed on a symmetric correlation ring.
These spikes are distributed randomly on this ring. The occurrence of such scattering patterns
is commonly observed with cubic phases as demonstrated by a typical example in
Fig. 11.23 [61]. These relatively sharp peaks indicate a long-range ordering in the cubic
phase. This scattering pattern can be explained by the presence of relatively large crystalline
domains [66].
A similar series of samples as in the SANS experiments was studied in cooperation
with the group of Prof. Wokaun by NMR self-diffusion experiments. The pulsed field gradient
spin echo (PGSE) method [67, 68] allows the determination of the self-diffusion coefficient
of each of the individual constituent components in particular water, surfactant, and
hydrocarbon. Here, in order to obtain simpler NMR spectra the hydrocarbon was cyclohexane.
The molar ratio of C 14 DMAO :cyclohexane was chosen to be 1:1.2, with three samples
in the L 1 phase and three samples in the cubic phase.
The obtained self-diffusion coefficient of water shows a continuous, approximately
linear decrease with increasing volume fraction of the micellar aggregates and with no dis-
222

11.5 Ringing gels
Figure 11.23: Three-dimensional plot of the scattered intensity in a SANS experiment for 32.2 wt%
C 14 DMAO/6.0 wt% decane/61.8 wt% D 2 O.
continuity at the phase transition: L 1 phase ? cubic phase. Such a behaviour is in good
agreement with the model of a continuous aqueous phase where the diffusion of the water
molecules is simply hindered by the steric restrictions imposed by the presence of micellar
aggregates [69].
A much different picture is obtained for the diffusion of the surfactant and the hydrocarbon
(Fig. 11.24). Within the L 1 phase both diffuse with the same coefficient. This is in
good agreement with the diffusion coefficient of the micellar aggregates calculated from
their size, determined by scattering methods. Again some decrease of the diffusion coeffi-
100
10
D s /10 -12 m 2 /s
1
0,1
a)
b)
c)
d)
e)
L 1 -phase
cubic
phase
0,01
0,0 0,1 0,2 0,3 0,4 0,5
Φ
Figure 11.24: Self-diffusion coefficient D s of surfactant and hydrocarbon in logarithmic representation
as a function of the micellar volume fraction F for the system C 14 DMAO/cyclohexane/D 2 O. The molar
ratio of C 14 DMAO : cyclohexane was always 1:1.2. a) D s of (N(CH 3 ) 2 ); b) D s of (CH 2 ); c) D s of (CH 2 )
for samples with deuterated cyclohexane; d) D s of H determined by means of a different instrument
[69]; e) D s of H for samples with deuterated cyclohexane [69].
223

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
cient with increasing micellar volume fraction occurs, as the movements of the aggregates
get hindered increasingly. However, upon crossing into the cubic phase the situation is dramatically
altered. The self-diffusion coefficient of surfactant and hydrocarbon drops suddenly
by more than a factor of 20 for the cyclohexane and a factor of 200 for the surfactant.
This is consistent with the picture from above, i. e. now the aggregates are frozen in and are
no longer able to move in the cubic lattice which is formed. Hence the diffusion coefficient
does not describe any more the motion of aggregates but is due to the diffusion of the individual
molecules. This explains why the much smaller cyclohexane molecule now diffuses
much faster than the larger C 14 DMAO molecule. Furthermore the very low diffusion coefficient
excludes the possibility of a bicontinuous system since for such a structure a much faster
diffusion would be expected [70].
In summary, this experiment clearly demonstrates that in the L 1 phase and in the cubic
phase the same water continuous structure is present. But while in the L 1 phase the micellar
aggregates are freely mobile their translational mobility is blocked in the cubic phase, i. e.
the microemulsion droplets are condensed into a glass-like liquid crystalline state of high
elasticity.
In general, from the investigation of the more dilute L 1 phase one may already conclude
the microstructure of the cubic phase. This concept can be very useful for the explanation
of macroscopic properties which are related to the aggregate size, like the shear modulus
G 0 . By means of oscillatory rheological experiments one may determine G 0 which is typically
in the range of 10 5 –10 6 Pa for cubic phases. Experimentally one finds for aminoxide
systems that G 0 decreases with increasing size of the respective aggregates [71]. These experimental
values may now be compared to theoretical calculations for hard sphere crystals
[72]. In general, this ansatz will predict a proportionality of the elastic moduli to the particle
density N of the aggregates. Furthermore, one can calculate for a given volume fraction the
factor which relates the particle density to the elastic constants, like G 0 . The calculated values
are in reasonable qualitative agreement with the experimental data [65]. This means
that it is possible to deduce mechanical properties from the knowledge of the microstructure.
Furthermore, this explains the strong dependence on the particle size since the particle density
is proportional to 1/R 3 , i. e. the microstructure of the cubic phase directly determines the
macroscopic elastic properties of the system.
11.5.3 The bis-(2-ethylhexyl)sulfosuccinate system
As stated above, the cubic phase of the AOT system is located differently in the phase diagram
than the aminoxide system, which is due to the fact that AOT as a double-chain surfactant
has a tendency to form reverse phases. Again it was of great interest to investigate the
relation between the isotropic L 2 phase and the cubic phase. Furthermore, this cubic phase
is interesting since here the surfactant concentration can be varied over a large range (30–
76 wt%).
The transition from the L 2 phase into the cubic phase has been studied by a variety of
methods [73]. This can be done easily on a line of constant AOT content (in our case mostly
35 wt%). Then increasing the octanol/H 2 O ratio one crosses from the cubic phase into the
224

11.5 Ringing gels
L 2 phase. Interestingly, measurements of the electric conductivity showed no discontinuity
of the equivalent conductivity (specific conductivity/AOT concentration) upon crossing the
phase boundary. Instead one observes a continuous increase of L with increasing H 2 O content.
The value in the cubic phase is about 40% of that of the free Na + ions, which should
mainly be responsible for the ionic conductivity (since the surfactant counterion will be largely
immobilized being fixed in the amphiphilic film), which indicates that the structure
must be water continuous.
NMR PGSE self-diffusion studies on similar samples also showed a continuous increase
of the water self-diffusion coefficient with rising water content. The value in the cubic
phase is again about 40–50% of the bulk water diffusivity, confirming the water continuous
structure. For the alkyl chains of octanol and AOT a separation was not possible, but
the values of (2–4)610 –7 cm 2 /s are relatively large, more than one order of magnitude larger
as in the case of the aminoxide, and show that the structure ought to be bicontinuous
and similar to the neighbouring L 2 phase [73].
SANS spectra of the cubic phase show the typical spikes on the isotropic diffraction
ring (compare Fig. 11.22). From the position of the peaks and from the total scattering intensity,
it can be concluded that the size of the structural units decreases with increasing AOT
(water) content, at constant octanol/water (AOT/octanol) ratio.
Measurements of the shear modulus G 0 on a series of samples with constant ratio of
octanol/water showed that G 0 is proportional to c(AOT) 3.1 , i. e. again, as in the case of the
aminoxide system, the elastic modulus increases with decreasing size of the structural units,
since the SANS experiments show that the size of the structural units decreases with increasing
AOT concentration. Correspondingly the particle density N of these units increases and,
as stated above, G 0 should be proportional to N. This means that the determination of the
microstructure already allows the prediction of the elastic properties, as in the case of aminoxide.
Upon heating the cubic phase will melt yielding a lowly viscous L 2 phase. This phase
transition was studied in some detail by differential scanning calorimetry (DSC). It was
found that the higher the melting temperature the higher the surfactant content is, with melting
temperatures in the range of 50 to 95 8C. The melting enthalpies DH show the same
trend with typical values of 100–300 mJ/g. Similar values were also found for the aminoxide
system. The maximum value, DH = 1034 mJ/g, was found for the cubic phase in the binary
system of 75.8 wt% AOT and 24.2 wt% H 2 O at the melting temperature of 89.5 8C. For
AOT concentrations close to the minimal value (around 28 wt%) necessary for forming the
cubic phase, however, the situation becomes somewhat more interesting. Here, DH can be
extremely small, below 0.5 mJ/g. This means that the macroscopically observable phase
transition from the solid-like cubic phase to the lowly viscous L 2 phase is associated with almost
no enthalpy difference. Furthermore, for samples with c (AOT) < 37 wt% another very
striking phenomenon occurs. Here a transition from the cubic phase to the L 2 phase can be
achieved by heating as well as by cooling! For the transition at lower and at higher temperature
the transition enthalpy is very similar. Both cases are endothermic processes. Such a reverse
melting process is quite unique for this type of cubic phase and has formerly only
been reported for triblock copolymers of PEO/PPO/PEO-type (Section 11.5.4) [74–76] and
PEO/PPO diblock copolymers [77]. However, for the block copolymers the situation is insofar
different as the cubic phase is made up from normal micellar aggregates, which are
transformed into a cubic phase upon heating since here the effective volume fraction of the
225

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
spherical aggregates increases [75]. In contrast to the cubic phase the AOT system is bicontinuous
and transformed into a bicontinuous L 2 phase by a different and still unknown mechanism.
11.5.4 PEO/PPO/PEO block copolymers
PEO/PPO/PEO triblock copolymers exhibit properties similar to typical surfactants, i. e.
they reduce surface and interfacial tension of aqueous solutions and form micellar aggregates
above a critical micellar concentration [74, 78]. For some compounds of this type, like
P104 (EO 18 PO 58 EO 18 ), P123 (EO 20 PO 70 EO 20 ), and F127 (EO 106 PO 70 EO 106 ), a similarly
located cubic phase like the one of the aminoxide systems has been found in the binary aqueous
system [74, 79].
Structural investigations on these systems have shown that the block copolymers form
spherical micellar aggregates in the L 1 phase close to the cubic phase. Here the PPO block
is the hydrophobic moiety. It forms the core of the micellar aggregate and is surrounded by
the more hydrophilic PEO blocks which act as the hydrophilic part of the polymeric surfactant
molecules. The temperature dependence of this system is interesting. At low temperatures
the block copolymers dissolve in water as unimers. Upon increasing the temperature
the PO groups are dehydrated rendering them more hydrophobic. This increased hydrophobicity
is responsible for the aggregation of the monomers into micellar aggregates. As has
been determined by DSC measurements this micellization process occurs over a fairly large
temperature range of typically 15–30 K [79]. For this dehydration process the enthalpy
changes are about 3 kJ/mol per PO group. This uncommon temperature behaviour is also responsible
for the interesting, already mentioned, reverse melting transition, i. e. in a corresponding
concentration range the lowly viscous L 1 phase is transformed into the solid-like
cubic phase upon heating. The simple explanation for this effect is that upon raising the
temperature more and more monomers will aggregate in micelles. If the volume fraction F
of the micellar aggregates exceeds the required one for the formation of a cubic phase
(F = 0.53) then the gelification process occurs and a cubic phase, composed of individual
micelles, is formed [64]. However, this phase transition is associated with an enthalpy about
two orders of magnitude smaller than the heat of micellization – typically between 25 and
100 mJ/g, i. e. similar in extent as for the other cubic phases described above. Its nature is
endothermic, i. e. the transition is associated with an increase of entropy. This transition can
also be nicely monitored by SANS measurements. For the micellar solution an isotropic
scattering ring is found where the typical spikes of the cubic phase can be observed if the
transition temperature is crossed [74].
More recently it has been found that the formation of the cubic phase can be suppressed
by the admixture of a simple surfactant, such as SDS [80]. This effect is due to cooperative
binding of SDS molecules on the block copolymer molecules. By doing so the micelles
are dissolved in favour of monomeric units until the volume fraction of the micelles
gets to small to form a cubic phase, i. e. one can melt the cubic phase by adding surfactant.
For example, for transforming the cubic phase of 25 wt% F127 a SDS concentration of
100 mM is required.
226

11.6 Lyotropic mesophases
Summarizing, cubic phases are a type of surfactant systems that can be observed for a
large variety of different surfactants. Depending on the molecular structure of the surfactant
they may be located in different places of the phase diagram with correspondingly different
microstructures. They can be composed of an array of individual micellar aggregates or be
of bicontinuous structure. However, even such a bicontinuous structure still will be characterized
by a very well-defined typical size of the structural units with long-range ordering.
At this point it is interesting to note that this size can already be obtained by studying isotropic
phases close to the cubic phase. This is insofar advantageous since they are often much
more amenable to experimental studies than the corresponding cubic phase itself. The
knowledge of the size of these units (no matter whether they are individual aggregates or
only structural repeat units of a bicontinuous structure) enables the prediction of the elastic
moduli which will be proportional to the particle density of these units. For a given structural
build-up theoretical relations can be used to calculate the moduli quantitatively. Therefore
a detailed knowledge of the microstructure of the corresponding systems allows a quantitative
understanding of macroscopic properties. Cubic phases are a nice example for a system
where by now such structure property relations are well established.
11.6 Lyotropic mesophases
11.6.1 Introduction
In general, at higher concentrations surfactant aggregates show a nearest neighbour order under
the influence of the intermicellar interaction. Upon further increasing the concentration
(mostly above 30–40 wt%) a long-range order between the micelles takes place and lyotropic
mesophases are formed. Depending on the kind of micelles cubic phases are formed
from globular micelles, rod-like micelles form hexagonal phases, and disc-like micelles
form lamellar phases. At surfactant concentrations significantly above 50 wt% inverse hexagonal
and cubic phases can also be built. Generally speaking, with increasing concentration
the phase sequence: cubic > hexagonal > lamellar > inverse hexagonal > inverse cubic is
found (Fig. 11.25) [81]. Figure 11.25 shows also schematically the structures of the different
lyotropic mesophases.
Figure 11.25 shows that between the hexagonal and the lamellar phases further isotropic
phases, which also are cubic, can exist, although they do not consist of globular particles but
show bicontinuous structures [73, 82]. Of course, not all these phases must be present in each
surfactant solution.In most of the systems the first cubic phase is missing up to high surfactant
concentrations because globular micelles are only present in few cases. Double-chain surfactants
often have a lamellar phase as the first mesophase [83], but all systems show principally
the given sequence of the mesophases. This is also valid for more-component systems, where
according to the constitution of the further components, certain phases may be favoured or
suppressed. Hydrocarbons, for example, favour globular micelles and hence cubic phases [55,
227

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
Figure 11.25: Schematic phase diagram of a binary system surfactant/water. Inverse hexagonal phase
H II , lamellar phase L a , hexagonal phase H I , isotropic (cubic) phases a, b, c, d.
61, 84]. But cosurfactants, like aliphatic alcohols, with an intermediate chain length support
the formation of disc-like micelles and thus of lamellar phases [85].
The aggregates of the phases in Fig. 11.25 show a long-range order with respect to
their orientation and to their centre of masses. Nevertheless, also lyotropic mesophases exist
which are built up by rod-like or disc-like micelles, respectively. They show only a longrange
order of their orientation, while their mass centres are statistically distributed in the
solutions. These phases are called nematic calamitic (N c ) or nematic discotic (N d ) phases
and exist at lower concentrations than the corresponding hexagonal or lamellar phases.
Those nematic phases are very interesting because they can be uniformly oriented by weak
external fields due to the lack of long-range order of their mass centres. This gives rise to
the development of single crystals instead of the polycrystalline mesophases with differently
oriented domains. On addition of chiral components the nematic phases can be transformed
to lyotropic cholesteric phases with a helical twist of the orientation, as has been found for
the corresponding thermotropic phases [86].
11.6.2 Nematic phases and their properties
Lyotropic nematic phases have been discovered a long time after the finding of the lyotropic
mesophases. In 1967 first evidence of such a phase was published [87] but it took more than
10 years to prove unambiguously the existence of these phases [88]. The first nematic
228

11.6 Lyotropic mesophases
phases have been found by fortune in complicated ternary and quaternary systems consisting
of surfactants, cosurfactants, electrolytes, and water. Systematic studies on different surfactant
systems have finally shown that lyotropic nematic phases do not occur so rarely as
could be concluded from their late discovery. At the same time criteria could be established
which allow a fairly precise prediction of the existence of a nematic phase and its position
in the phase diagram [89]. This also allows the direct preparation of different types of the
nematic phases and the study of their properties. Important conditions for the existence of
nematic phases are the voluminous hydrophobic part and the headgroup area of the hydrophilic
group which must be within certain values, for example 60–90 Å 2 for double-chain
ionic surfactants with a N c phase under these conditions. The headgroup area can be lowered
by adding cosurfactants (aliphatic alcohols with an intermediate chain length). This leads to
the formation of a N d phase. Both phases border to the corresponding hexagonal or lamellar
phase from the side of lower concentrations. For smaller headgroup areas a lamellar phase is
formed first while for larger headgroup areas a hexagonal or a cubic phase is the first mesophase
[89]. For example, numerous nematic phases have been found in solutions of anionic
perfluoro surfactants with their voluminous hydrophobic groups which mostly have been
identified as N d phases [90, 91]. Figure 11.26 shows a typical phase diagram of the binary
system perfluoro-surfactant/water. It can be seen that the nematic phase exists only in a
small concentration and temperature range which was probably the reason for the late detection
of these phases. From Figure 11.26 the marked thermotropic behaviour of the nematic
and the lamellar phases can also be seen. Thus it is possible by increasing the temperature
to go through the phase sequence: lamellar > lamellar/nematic > nematic > nematic/
isotropic > isotropic. These thermotropic phase transitions are reversible. In most cases DSC
measurements have shown that the phase transitions are of first order with very small heat
transitions. Further experiments have shown that the nematic phases can be oriented uniformly
in a magnetic field according to the anisotropy of the diamagnetism of the surfactant
molecules. Isolated anisotropic aggregates cannot be oriented in micellar solutions because
the energy of the magnetic field is much smaller than the thermal energy which achieves a
random orientation of the micelles in this case. The lamellar phases, on the other hand, also
cannot be oriented by the magnetic field because this would require the simultaneous orientation
of a sheet of lamellae for which the energy of the magnetic field is not sufficient.
70
60
nematic
50
isotropic
T/°C
40
30
lamellar
20
crystalline
10
0 10 20 30 40 50
wt%
Figure 11.26: Section of the phase diagram of the system C 8 F 17 CO 2 N(CH 3 ) 4 /D 2 O.
229

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
But it is possible to orient the phases uniformly in the nematic region and to freeze
the orientation by cooling the system to the lamellar region. This orientation remains without
the magnetic field [92]. This allows a simple study of such phases and of the type of
particles present in the phases. It is also possible to solubilize anisometric dye molecules in
the aggregates and to orient them together with these micelles in the nematic phase by a
magnetic field, leading to a dichroism of the systems [93]. The oriented nematic and lamellar
phases show also a strong anisotropy of their electric conductivity, i. e. increase of the
conductivity parallel and decrease of the conductivity perpendicular to the large axis of the
aggregates. N c phases have also been found in solutions of the cationic double-chain surfactants
of the series C x C y N(CH 3 ) 2 Br with x =14or16andy = 1–4 which fulfill the established
conditions with respect to the headgroup area and the volume of the hydrophobic alkyl
chain. On the other hand, surfactants with y = 6 do not form mesophases up to concentrations
of 75 wt%, while surfactants with y>8 show first a lamellar mesophase [89].
These mesophases also exist only in a small concentration range near the hexagonal
phase, as can be seen, for example, in Fig. 11.27. Above a characteristic temperature they
show a reversible phase transition of first order into an isotropic phase. Figure 11.27 also
shows that a thermotropic transition hexagonal > nematic cannot be observed in these systems.
Furthermore it could be shown for these phases that on addition of a cosurfactant a
transition of the N c into a N d phase can take place. For one system it was found that both nematic
phases can be present in equilibrium. From orientation experiments of these nematic
phases in a magnetic field it could be concluded that the sign of the anisotropy of their diamagnetism
is positive. If the Br counterion was substituted by benzene sulfonate, the nematic
phases are kept but the sign of the anisotropy of the diamagnetism changes. Hence it
is possible to prepare and to study all four possible nematic phases N + c,N – c,N + d, and N – d with
these surfactants.
Figure 11.27: Section of the phase diagram of the system CTAB/H 2 O.
The identification of the N c and N d phases can be archived by polarization microscopy
based on their different textures which are presented in Fig. 11.28 a and 11.28 b. 2 H-NMR
spectroscopic studies and orientation experiments on the nematic phases in a magnetic field
are suitable for their detection, especially if SANS measurements are done on oriented samples.
For these experiments the phases have to be prepared with D 2 O instead of H 2 O. This
230

11.6 Lyotropic mesophases
Figure 11.28a: Texture of a calamitic nematic (N c ) phase in the binary system of 25 wt% hexadecyltrimethylammoniumbromide
(CTAB)/water at 35 8C.
Figure 11.28b: Texture of a discotic nematic (N d ) phase in the binary system of 47.5 wt% tetradecylpyridiniumheptanesulfonate
(C 14 PyC 7 SO 3 )/water at 26 8C.
does not affect the nematic phases except a small shift of their existence region towards
lower surfactant concentrations. Furthermore, SANS and SAXS measurements show a high
degree of orientation of the nematic phases by the appearance of a third order scattering
peak.
Finally it shall be mentioned that until now no inverse nematic phases could be detected.
Also the existence of nematic phases in non-polar solvents could not be proven for
any studied system. Thus it is not yet possible to replace the thermotropic nematic phases in
displays by lyotropic systems because lyotropic nematics cannot be orientated in an electric
field with low voltages due to the rather high conductivity of the water solvent compared to
non-polar organic compounds.
231

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
11.6.3 Cholesteric phases and their properties
For thermotropic and also for lyotropic nematic phases it is known that the presence of
chiral compounds leads to a transition of the nematic phase to a cholesteric one. This phase
has also only a long-range order with respect to the orientation of the aggregates, but this orientation
shows a twist within a domain, and the pitch depends on the concentration of the
chiral compound. Cholesteric phases can be built up with chiral components (intrinsic cholesteric
phases). The transition nematic > cholesteric can also take place by adding chiral
samples (induced cholesteric phases) [86]. Intrinsic cholesteric phases require the use of
chiral surfactants. For this purpose the group of the non-ionic sugar surfactants has been
chosen. As the sugar head group is mostly very voluminous, it could be expected that only a
few systems could fulfill the conditions for the formation of a nematic or cholesteric phase
with respect to the headgroup area and the volume of the hydrophobic group. Further problems
arose from the poor solubility of sugar surfactants with sufficiently small headgroups
and because many of these surfactants are strongly sensitive to hydrolysis. Nevertheless, the
first intrinsic cholesteric phase in a binary system surfactant/water could be detected for an
alkylpolyglucoside with 12–14 C atoms and 1,1 glucose units on an average [94]. By orientation
in a magnetic field this phase could be identified as a cholesteric discotic phase with
negative anisotropy of its diamagnetism. On the other hand, it is no problem possible to
transform N c and N d phases in a ternary system of the zwitterionic surfactant tetradecyldimethylaminoxide
(C 14 DMAO), a cosurfactant (an aliphatic alcohol with 7–10 C atoms),
and water into the corresponding induced cholesteric phases by adding chiral compounds
[95]. This transition could be observed with a non-polar chiral additive (cholesterol), which
can be solubilized only in the micellar aggregates, and with a polar chiral compound (tartaric
acid), which remains dissolved in the aqueous phase.
The cholesteric phases could be identified by their fingerprint textures in the polarization
microscope, which are characteristic for non-oriented cholesteric phases. Uniformly oriented
cholesteric phases show characteristic stripe textures. Both textures are presented in
Figs. 11.29 a and 11.29 b. From the distance of the stripes the pitch of the phase can be cal-
Figure 11.29a: Texture of a non-oriented cholesteric phase in the binary system of 60 wt% alkylpolyglucoside
(APG 1)/water at 25 8C.
232

11.6 Lyotropic mesophases
Figure 11.29b: Texture of a oriented cholesteric phase (aligned in a magnetic field of 2 T) in the binary
system of 60 wt% alkylpolyglucoside (APG 1)/water at 25 8C.
culated. The experiments showed a linear relation between the reciprocal pitch and the concentration
of the chiral additive. Until now, the reason for the pitch caused by the chiral additive
is not understood. An interesting fact about these cholesteric phases is that above a
certain temperature the pitch disappears and a nematic phase develops reversibly. This transition
temperature is lower than the temperature for the transition nematic > isotropic. It
could not be detected by DSC experiments, very likely due to its small enthalpy change.
11.6.4 Vesicle phases and L 3 phases
From Fig. 11.25 it can be seen that lyotropic lamellar phases normally exist at surfactant
concentrations above 50 wt%. According to the theory of surfactant aggregation [4] surfactants
with small headgroup areas and big hydrophobic groups are known to form lamellar
phases already at concentrations far below 10 wt%. Such systems are perfluorosurfactants
with strongly binding counterions [96] or double-chain surfactants [83]. In mixtures of surfactants
with small headgroup areas like zwitterionic or non-ionic surfactants and cosurfactants
– for example aliphatic n-alcohols with intermediate chain lengths – the geometrical
conditions for the formation of lamellar phases in highly dilute systems are also fulfilled.
Furthermore it is possible in these systems to change the natural curvature and the flexibility
of surfactant lamellae simply by variation of the mixing ratio surfactant : cosurfactant. According
to this, these ternary systems exhibit a typical phase behaviour with lamellar phases
at the water rich corner, which has already been explained in detail in Section 11.4.1.
The most powerful method for the identification of the different phases and for the determination
of their microstructures is the transmission electron microscopy. This method requires
a sample preparation using the freeze fracture and etching technique. With these measurements
the microstructures of the various phases can be visualized directly. This is shown
for three examples in Fig. 11.12 (Section 11.4.2) and in Figs. 11.30 a and 11.30 b [13, 41, 97].
233

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
Figure 11.30a: Electron micrograph of a lamellar phase in the system of 100 mM C 14 DMAO, 190 mM
C 7 OH, and water (the bar represents 2.5 mm).
Figure 11.30b: Electron micrograph of a L 3 phase in the system of 100 mM C 12 DMAO, 220 mM
C 6 OH, and water (the bar represents 140 nm).
With these technique it can also be demonstrated in the two phase regions that both phases
with their different structures are really coexisting. This result can be supported by self-diffusion
measurements with the pulsed NMR field gradient method from which the existence
of bicontinuous phases and their topology can be concluded [73, 98].
The electron micrograph in Fig. 11.12 shows the existence of thermodynamically
stable vesicles in the L al phase which can be formed as small unilamellar vesicles besides
large multilamellar vesicles (liposomes). Depending on the composition of the ternary system
the vesicles can also show structural faults like holes (perforated vesicles) [37]. The vesicle
phases show a significantly reduced electric conductivity because a part of the water
phase together with the ions is included in the interior of the vesicles. With stopped-flow experiments
and optical or conductivity readout it is thus possible to determine the permeability
of the vesicle membranes for dissolved compounds [99]. Vesicle phases show often high
viscosities and viscoelasticity due to the mutual hindrance of the vesicles in sheared solu-
234

11.6 Lyotropic mesophases
tions. Especially if the vesicle membranes are charged by incorporation of certain amounts
of added ionic surfactants, the phases show with increasing charge of the surfactant film an
increasing yield stress [40, 100]. The vesicle phases usually are not or only weakly birefringent
and thus cannot be identified in the polarization microscope.
From Fig. 11.30 a and for the L ah phases the presence of flat surfactant lamellae like in
normal lamellar phases is evident. These phases are birefringent but they often do not develop
their characteristic texture in the polarization microscope due to their low concentration. Until
now it could not be proven whether there is a phase boundary between the L a1 and the L ah
phase or a continuous transition from L al to L ah with increasing cosurfactant concentration.
This problem is very difficult due to the high viscosity and turbidity of the phases.
The bicontinuous structure of the L 3 phase can also be seen directly from Fig. 11.30 b.
This phase shows a low viscosity without elasticity and is optically isotropic but shows a
strong flow birefringence. According to the similarity with normal micellar or reversed micellar
solutions (L 1 and L 2 phases) these phases have been called L 3 phases. The surfactant
lamellae are arranged as branched tubes similar to a sponge. The low viscosity results from
the very high flexibility of the lamellae which break and reform easily.
As mentioned in Section 11.2, the phase sequence in Fig. 11.1 can be understood regarding
the properties of the surfactant films. With low amounts of cosurfactant the natural
curvature of the film is convex and the flexibility is usually low. Hence micellar aggregates
are present in the L 1 phase. With increasing amounts the curvature of the films decreases
and their flexibility increases due to the small headgroup area of the cosurfactant. Thus transition
into a vesicle phase and with more cosurfactant into a normal lamellar phase can take
place. Further increase of the cosurfactant increases the flexibility of the films and permits
the formation of the L 3 phase. Adding even more cosurfactant finally leads to the separation
of the cosurfactant phase. This concept also allows to understand the influence of further additives
on the phase behaviour of the surfactant/cosurfactant/water systems. Adding an ionic
surfactant to the L 3 phase leads to a reduction of the flexibility and to a convex curvature of
the surfactant films due to the electrostatic repulsion between the ionic headgroups. Thus a
transition of the L 3 phase to a vesicle phase is observed [38]. These vesicles have rather stiff
membranes and the phase often shows a yield stress, as demonstrated in Section 11.4.3. But
addion of an electrolyte to these systems shields the charge of the headgroups and gives rise
to an increased flexibility and decreased curvature of the films. Hence the vesicle phase is
transformed again into a L 3 phase. Addition of an ionic surfactant to phospholipid vesicles,
which normally must be prepared by sonification of the aqueous phospholipid dispersions
[101], also leads to a destruction of the vesicles and to a transition to rod-like micelles above
a certain concentration of the ionic surfactant [25].
Finally, it is worth mentioning that thermodynamically stable vesicles can also be
found in the binary systems surfactant/water if the hydrophobic group is sufficiently voluminous.
As already mentioned, this has been observed for the double-chain surfactant didodecyldimethylammoniumbromide
DDABr in water [102]. Recently also a L 3 phase has been
found in the binary system of a polyoxyethylene-polyoxypropylene-polyoxyethylene triblock
copolymer and water at high temperatures. This block copolymer has a composition
EO 5 PO 30 EO 5 and hence a very small hydrophilic headgroup. It forms a lamellar phase at
room temperature which is transformed into the L 3 phase above a certain temperature [103].
The reason for this transition is probably the increased flexibility of the block copolymer
film at the high temperature.
235

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
11.7 Shear induced phenomena
11.7.1 General
In this Section we discuss a remarkable and puzzling phenomenon that, because of its complicated
nature, is not yet completely understood. The phenomenon has, however, considerable potential
for technical applications where it is necessary to control the flow behaviour of aqueous
solutions. In all applications where large amounts of water have to be circulated for cooling or
heating purposes the energy expense for the pumping is a major economic factor. Usually one is
interested in pumping as fast as feasible so that the flow in the water pipes is generally in the turbulent
flow region. Under these conditions it is possible to reduce the friction coefficient by
polymer additives or by drag-reducing surfactants (Fig. 11.31). In recycling operations polymers
have the big disadvantage that they deteriorate under shear because the molecules break
under shear forces. Surfactants do not have this disadvantage because the micellar structures
which produce this effect are self-healing. Pilot operations in Europe have been running for
months without loss of efficiency. The energy costs have been cut to less than a half.
pressure drop [hPa]
360
300
240
180
120
60
12
10
8
6
4
2
0
0 1 2 3 4 5
0
0,0 0,2 0,4 0,6 0,8
flow velocity [m/s]
Figure 11.31: Plot of the pressure drop vs.the flow velocity in a capillary in the laminar and turbulent
flow regions for water (solid line) and for a drag reducing surfactant solution (750 ppm
C 14 TABr + NaSal at 27.5 8C, dashed line).
11.7.2 Under what conditions do we find drag-reducing surfactants?
The phenomenon occurs often in surfactant solutions in which small rod-like micelles are
formed that are charged and the charge is not screened by excess salt [104]. Many surfactant
systems have been found where such conditions exist. Typically the length of the rods is
236

11.7 Shear induced phenomena
smaller than the mean distance between them. From this point of view the micellar solutions
can be considered dilute even though there is repulsive interaction between the rods. Because
of this repulsion the micelles try to be as far away from each other as possible and set
up what is called a nearest neighbour order. The result of this order is a correlation peak in
scattering experiments. In typical conditions the surfactant concentration is a few mM (about
0.1–0.2 wt%), the rods are a few hundred angstroms long and their mean separation is
somewhat larger. Because there is no steric hindrance between the rods they can undergo
Brownian rotations with rotation times of a few microseconds. Such conditions can easily be
set up when charged surfactants are mixed with zwitterionic surfactants. Usually mixtures of
ionic (10–30 mol%) and zwitterionic surfactants are favourable for the effect. These features
of the micelles are the necessary prerequisites for the occurrence of the effect. They are,
however, not the only ones, as will become clear.
In shear measurements one expects the described solutions behave like normal Newtonian
aqueous solutions. This is in fact the case for small shear rates (Fig. 11.32). In Fig. 11.32
the shear viscosity, which was measured in a capillary viscometer, is plotted vs.the shear rate.
One observes a sudden rise of the viscosity at a characteristic shear rate _g c and for _g > _g c the
solutions show some shear thickening behaviour. Obviously something dramatic has happened
to the micelles in the solutions. Some conclusions about what has happened can be drawn
from flow birefringence measurements. Some typical results of flow measurements from a
Couette system are shown in Fig. 11.33. We note a sudden increase of the flow birefringence
at a critical shear rate. For _g < _g c no flow birefringence could be detected.
In flow experiments, besides the birefringence it is also easy to measure the angle of
extinction, which is the angle between the direction of flow and the mean orientation of the
rods. In normal flow orientation this angle decreases smoothly from 458 to zero with increasing
flow rate because the rods become more and more aligned. In the drag-reducing solution
the situation is very different. In the Newtonian region for _g < _g c the solution remains isotropic
and no preferential alignment can be detected. However, if _g > _g c the angle of extinction
is close to zero, i. e. the new structures which are produced by the shear are completely
aligned. Obviously the newly found structures must be much larger than the original small
rods which were not aligned.
η[mPas]
8
6
4
2
60 mM
80 mM
100 mM
120 mM
T=25°C
0
10 1 10 2 10 3
.
γ[s -1 ]
Figure 11.32: The shear viscosity Z vs. the shear rate _g for mixtures of C 14 DMAO : SDS = 6 : 4 with
various total concentrations in a capillary viscometer at 25 8C.
237

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
-∆n/10 -6
10
8
6
4
2
30 mM
60 mM
80 mM
100 mM
0
0 100 200 300 400 500 600 700 800 900
.
γ /s -1
Figure 11.33: The flow birefringence Dn vs. the shear rate _g for the same solutions as in Fig. 11.32.
It is believed now that the small rod-like micelles undergo collisions in the shear flow
and they stick together for some time because of their interfacial properties. In this way long
necklace-type structures are formed under shear and at the same time get aligned in the
shear flow. This situation is schematically sketched in Fig. 11.34. These necklace-type structures
act like high molecular weight polymers and give rise to drag reduction.
These results show that shear is an important variable for micellar structures. We are
aware that generally temperature, ionic strength, concentration, and cosurfactants can change
micellar structures and hence the properties of surfactant solutions. We also should be aware
that shear can change and influence micellar structures and even mesophases. It has been
observed that micellar solutions can be transformed into liquid crystalline phases and single
clear phase solutions become turbid as well as biphasic under shear. On the other hand, biphasic
solutions can turn into a single phase under shear. A very striking example of a
shear-produced transition is the transformation of a dilute L 3 phase into a L a phase with
bright iridescent colours [105]. All these effects are based on the fact that the shape and size
of the micelles depend to some degree on the intermicellar interaction energy, which itself
depends on the mutual orientation of the micelles. When the interaction energy is changed
the system responds with a change of structure and properties. These changes can be unexpected
and large. They can, however, be used to our advantage.
Figure 11.34: Model for the explanation of the shear induced micellar structures. The small rod-like
micelles can form long necklace-type structures under shear.
238

11.8 SANS measurements on micellar systems
11.8 SANS measurements on micellar systems
A large variety of our surfactant systems has been investigated by means of small angle neutron
scattering (SANS) experiments mainly in cooperation with the group of Prof. Kalus
(Universität Bayreuth). SANS is a method particularly suited for the study of self-aggregating
colloids since its spatial resolution is typically in the range of 10–1000 Å, which is the
size range of micellar aggregates. In the course of the investigation a shear apparatus was
constructed in the group of Prof. Kalus which allows SANS experiments under shear. This
has the advantage that one can align anisometric aggregates in the shear field and from the
scattering curves of the aligned particles one can deduce more detailed information regarding
their structure and their dynamic behaviour in the shear field. In addition, systems can
be studied that exhibit shear induced structures, which are very interesting since they show
drag-reducing behaviour. In the following we give just a short, exemplary overview over our
large number of SANS experiments (some of them are discussed in the corresponding chapters,
e. g. the work on cubic phases).
The SANS is a method for the detailed determination of size and shape of the corresponding
surfactant assemblies. For instance, from such experiments we found that tetramethylammoniumperfluoroctanesulfonate
(TMAFOS) forms rod-like structures with a radius
of 22 Å and a length of 200–300 Å [106]. For perfluorated surfactants the SANS
method is only particularly advantageous because the refractive index of water and perfluorated
compounds are very close. Therefore for these systems the powerful light scattering
experiments for studying micellar systems will not work. Furthermore, detailed structural information
might not be accessible for perfluorated surfactants, which are an interesting class
of surfactants since normally they are even more surface-active than their hydrocarbon counterparts.
However, elongated micelles are by no means restricted to perfluoro surfactants but
also commonly found with conventional hydrocarbon alkyl surfactants. An interesting system
is the mixed cationic/anionic surfactant tetradecylpyridinium-heptanesulfonate. Here SANS
measurements have shown that below 160 mM charged rod-like aggregates are present which
get shorter with increasing concentration. For these samples always a correlation peak is observed
in the spectra but for more concentrated samples this peak will disappear. The reason
is that with increasing concentration the degree of dissociation is decreasing. Therefore the
charge density on the micelles decreases thus decreasing the electrostatic repulsion between
the aggregates until they are effectively uncharged at high concentration [107].
Another interesting type of surfactant are double-chain amphiphiles. As an example of
this surfactant-type hexadecyloctyldimethylammoniumbromide (C 16 C 8 DMABr) has been
investigated. Here just above the cmc globular aggregates are formed whereas above a second
transition concentration relatively short (150 Å) rod-like micelles are formed [108].
However, anisometry of the micellar aggregates is a necessary (yet not sufficient) prerequisite
of liquid crystal formation. Indeed, at higher concentrations many perfluoro surfactants
are known to form nematic, lyotropic, liquid, and crystalline phases [109]. One such
system is the tetramethylammoniumperfluorononanoate (TMAPFN) which exhibits a nematic
phase over a relatively large concentration range but only in a small temperature interval
[91]. This system has been studied in detail by SANS [92]. The experiments showed that
239

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
the nematic phase is build up from disc-like micellar aggregates, i. e. the local structure of
the amphiphilic film is planar. This phase can be oriented in an external magnetic field,
thereby allowing for a more detailed analysis of the underlying micellar structures. The
thickness of the disk was found to be 37 Å and from a rocking experiment the order parameter
S & 0.85 was deduced. In addition, it is interesting to note that only the nematic phase
can be oriented in an external magnetic field. But upon cooling the samples are transformed
into the corresponding lamellar phase. This phase could not be oriented but it is highly ordered
and also keeps its orientation without applying a magnetic field.
As just discussed, lyotropic nematic phases can easily be aligned in an external magnetic
field and it is of course of interest to study this alignment process as a function of
time, i. e. to study the dynamics of the nematic phase. This has been done for a system
made up from an aqueous solution of 10.2 wt% hexadecyltrimethylammoniumbromide
(CTAB), 9.9 wt% hexadecyltrimethylammoniumbenzenesufonate, and 2.45 wt% decanol in
D 2 O. This system forms a nematic phase that consists of disc-like micelles. Such a sample
had been prealigned in a magnetic field of 7 T. The highly ordered phase gives a strongly
anisotropic scattering pattern with two sharp peaks (Fig. 11.35). The preoriented sample
which was in the neutron beam was then exposed to a magnetic field (1.4 T) oriented perpendicular
to the axis of the originally employed magnetic field as well as to the neutron
beam. Of course, this new magnetic field wants to reorient the texture of the sample and
this reorientation process has been monitored by SANS. In Fig. 11.35 we see the time evolution
of the scattering pattern: upon turning on the perpendicular field the two original peaks
start to disintegrate into four smaller peaks which move on a ring to form finally again two
narrow peaks which are then oriented perpendicular to the original peaks. The typical time
for reorientation is about 60 min [110].
a) b)
c) d)
Figure 11.35: SANS curves of a prealigned nematic phase of a CTAB system (composition see text).
the original aligned sample (a), the sample in a perpendicular magnetic field B 1 after 20 (b), 50 (c), and
90 min (d). (B 0 =7T,B 1 = 1,4 T).
240

11.8 SANS measurements on micellar systems
Of course, anisometric micelles can also be oriented by shear and this again allows a
more detailed study of the micellar structure as well as the dynamic alignment process. One
such system has been cetylpyridiniumsalicylate in 20 mM NaCl D 2 O solution which at 20 mM
concentration has been shown to contain rod-like micelles with a radius of 21.5 Å and a length
of 500–750 Å [111]. Under shear an anisotropic scattering pattern is observed. It relaxes to
the isotropic pattern after switching off the applied shear field. Time resolved measurements
(time resolution of 250 ms) showed that the scattering curves during this relaxation process
can be described by a single parameter, namely the rotational diffusion coefficient D rot .Itwas
observed that D rot is time-dependent and decreases with time due to the interaction between
the charged rod-like aggregates, i. e. at the beginning the aligned rods have the largest electrostatic
repulsion giving a large driving force for the disorientation because for this aligned arrangement
the electrostatic potential energy has its highest value. The less aligned the system
the smaller becomes this driving force (since the electrostatic interaction becomes weaker)
and correspondingly the rotational diffusion coefficient becomes smaller [112].
The experimental scattering curves can be explained by an orientation distribution function
of the rods depending on the applied shear gradient [113]. Under the experimental conditions
the product of shear gradient _g and structural relaxation time t, which is increasing with
the length of the aggregates, was always much larger than 1, i. e. a high orientational ordering
was achieved because the ordering force becomes stronger than the diffusive force.
Generally all anisometric aggregates can become oriented in the shear field but systems
with shorter relaxation times t require higher shear gradients. This effect has been studied
with the above mentioned C 16 C 8 DMABr at a concentration of 50 mM. Here the unsheared
solution shows a correlation ring which becomes increasingly anisotropic with increasing
shear rate. Figure 11.36 shows the scattering patterns for various shear gradients. In
addition a higher order peak becomes visible. Finally for shear rates above _g = 2000 s –1 the
scattering pattern hardly changes any more. In this system not only the relatively short rodlike
aggregates become weakly aligned but also a second type of micelle is formed in the
shear field, where their relative concentration increases with rising shear rate. Therefore in
this system a shear-induced transformation of micellar aggregates is observed [114].
A similar behaviour has also been observed for the system tetradecyltrimethylammoniumsalicylate
(C 14 TMA-Sal) [115] and seems to be quite common for viscoelastic surfactant
systems. A more detailed analysis of this system indicates that the sharp peaks that occur
upon applying the shear are due to a hexagonal array of cylindrical micelles, i. e. again now
2 types of micelles are present: the originally present short rod-like micelles and the very long
rod-like aggregates that are formed in the flow field [116]. This lattice-type array is formed
within about 2 min as could be seen from time-resolved shear experiments [116]. This result
agrees also well with flow birefringence experiments. At a given shear rate an equilibrium between
the short rod-like micelles, which normally are only weakly aligned, and the long rodlike
aggregates is present. This equilibrium depends on the shear rate. The higher the shear
rate the more the equilibrium is shifted in favour of the long aggregates [117].
The growth process of the large micellar structures, which are strongly aligned, has
been studied in more detail by transient SANS experiments. In these experiments the shear
rate for the samples was raised stepwise from zero to a certain finite value. These experiments
showed that the large micelles grow according to the Avrami law
c…t† ˆc inf …1 exp… kt †† : …19†
241

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
a) b)
c)
d)
Figure 11.36: SANS patterns of C 16 C 8 DMABr in D 2 O (50 mM) for various shear gradients (the momentum
transfer is given in units of nm –1 ): _g =0s –1 (a), _g = 100 s –1 (b), _g = 400 s –1 (c), _g = 2000 s –1 (d).
The neutron beam is perpendicular to the direction of shear.
Originally this equation was used to describe nucleation and growth in metals and alloys.
For the given system (C 14 TMA-Sal) the exponent n was found to be between 2 and 2.5
[118]. The exponent n should be i + 1 for i-dimensional growth, which means that the
growth process observed here is close to a one-dimensional one as should be expected since
the micelles grow in length without change of dimension.
Such shear-induced structures (SIS) are an interesting phenomenon in particular for
self-aggregating systems like micelles where the equilibrium structure often depends very
subtly on small energetic changes. Of course, these structural changes have a profound influence
on the properties of these systems, especially on their flow behaviour. For instance one
may observe a shear-thickening behaviour that is coupled to drag-reducing properties of the
system. Shear-induced transitions have been found in a variety of micellar systems [105]. Of
particular interest in our studies have been systems that in the unsheared state contain small
rod-like micelles and systems which show a strongly anisotropic behaviour beyond a critical
threshold shear rate _g c . Above this shear rate a strong increase of flow birefringence and
viscosity together with a large anisotropy of the electric conductivity are observed. At the
same time the scattering patterns of the SANS experiments exhibit also a strong anisotropy.
This shear-induced effect will already occur at _g t rot P 1, where t rot is the rotational time
constant of the small type of micelles, i. e. in a range where the shear field should not be
able to orient the small rod-like aggregates significantly. This means that the observed anisotropy
is not due to the orientation of these originally present micelles but that larger oriented
micellar aggregates have to be present in the solutions. So far the mechanism for formation
of the SIS is not fully understood and several different mechanisms have been postulated
[119–121].
242

11.9 A new rheometer
The tetradecyldimethylaminoxide/sodiumdodecylsulfate (C 14 DMAO/SDS) system has
been studied in much detail. This system shows a pronounced SIS formation around a molar
mixing ratio of 7 : 3 for C 14 DMAO/SDS [122] – and it might also be noted here that for this
composition the nematic phase, which is found for those systems, extends to the lowest surfactant
concentration [123]. In order to find relations between the macroscopic behaviour of
the system and the structure of the micellar aggregates, SANS study was performed [124].
Changing the contrast condition in the micellar aggregate by using both hydrogenated and
deuterated SDS, yields detailed information regarding the structure of the micelles. The
SANS experiments show that at the mixing ratios where the length of these aggregates reduces
with increasing SDS content and where SIS is observed, elongated micelles are present
which are best described by a three-axes prolate ellipsoid. From a contrast variation experiment
using both, deuterated and hydrogenated SDS, it could be concluded that the buildup
of these micelles is homogeneous and no internal segregation of the surfactant molecules
within the aggregate could be deduced [124]. Such an internal segregation by having an enrichment
of SDS at the end caps (here the relative area per molecule is necessarily larger because
of the larger curvature and one could imagine that the SDS with its larger hydrophilic
head group would preferentially be located at this position) would have been conceivable
and, of course, such a build-up that would contain two more highly charged ends would
have been an important factor to consider for the explanation of the SIS. However, this evidently
is not the case and SIS formation has to be explained starting from originally short,
homogeneously charged, elongated aggregates.
If we summarize we find that SANS is a very powerful tool to investigate self-aggregating
structures. It is perfectly adapted with the size range to be observed and in addition it
allows a detailed observation of anisotropic samples which might be oriented by a magnetic
field, shear field, order effects close to the wall of the cells, etc. Scattering patterns of such
oriented samples contain even more information regarding the intra and interparticle structure.
Furthermore, the contrast conditions in the samples can in principle be changed because
they consist to a large degree of hydrogen and because the scattering properties of H
and D nuclei differ largely. Therefore isotopic substitutions, that normally have only a very
minor effect on the properties of the respective systems, will enable us to get much more
structural information than would be accessible via other methods. This method is called the
contrast variation. Those experiments have shed a lot of light on structural properties of micellar
systems with respect to some of their dynamic properties.
11.9 A new rheometer
From electric birefringence measurements it is known that surfactant solutions, like those of
tetraethylammoniumperfluorooctanesulfonate, show different time constants in the range
from microseconds up to some seconds [24, 94, 125]. The shortest time t 1 is attributed to
the free rotation of rod-like micelles and amounts to 10 –5 to 10 –6 s. It can be found in almost
the whole concentration range. The second time t 2 is the birefringence which has a dif-
243

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
ferent sign and comes to 10 –4 s. It can only be detected in the narrow concentration region
where the rods begin to overlap (7 mM–15 mM). The longest time constant t 4 (about some
seconds) represents the structural relaxation time. It can be found in the electric birefringence
and rheological measurements at sufficiently high concentrations (c >30 mM) when
a network is formed. The remaining time constant t 3 (in the range of milliseconds) could
not yet be detected by rheological measurements because the frequency range of commercial
rheometers ends typically at frequencies of about 10 Hz. Therefore it is not possible to determine
rheological time constants shorter than about 0.1 s.
In order to overcome this problem, it was necessary to improve the frequency range
for the dynamic rheological measurements. For this purpose a dynamic rheometer (HF rheometer)
with a frequency range from 1 Hz up to 1 kHz was built. For samples with a high
modulus (G&1 kPa) the measurements can even be extended to 2 kHz.
With this apparatus measurements on viscoelastic surfactant solutions were carried
out. As expected, these solutions show short rheological time constants t 3 . They correspond
well to those determined with the electric birefringence.
The new HF rheometer is based on a prototype that was acquired from a group of the
Universität Ulm, Abteilung Angewandte Physik, Prof. W. Pechhold. The sensitivity of the
apparatus could be notably improved by numerous technical modifications. A schematic
drawing of the mechanical part of the apparatus is shown in Fig. 11.37.
Figure 11.37: Schematic drawing of the mechanical part of the HF rheometer.
244

11.9 A new rheometer
The rheometer works with a concentric cylinder geometry. The gaps between the cylinders
are 50 mm or 100 mm, respectively. The inner cylinder is driven by an electromechanical
converter (shaker) and performs linear harmonic oscillations. The force is transferred
by the sample to the outer cylinder and is detected by a very sensitive piezoelectric force
transducer producing a signal which is amplified by a charge amplifier. The amplitude of
the inner cylinder is determined by an inductive displacement transducer coupled with a carrier
frequency measuring amplifier. A lock-in amplifier is used for the measurement of the
force signal, the displacement signal, and the phase angle. In the present state measurements
on viscoelastic samples with a modulus of 50 Pa, 100 Pa, and 1 kPa are possible up to
900 Hz (50 mM gap), 1.3 kHz and 2 kHz, respectively.
The reliability of the apparatus was tested and the frequency range was experimentally
determined by measurements on Newtonian liquids (silicon oils, glycerol/water mixtures).
The viscosities of Newtonian liquids can be well detected down to 50 mPas. In this case the
frequency range extends to 1.6 kHz.
Solutions of tetraethylammoniumperfluorooctanesulfonate (C 8 F 17 SO 3 NEt 4 ) were studied
with the HF rheometer. A typical rheogram is shown in Fig. 11.38. The measurement
was carried out on a 90 mM solution with the 50 mM gap in the frequency range above
10 Hz. Below 10 Hz a Bohlin CS 10 rheometer with a cone plate geometry was employed.
10 2 G´
G´, G´´ / Pa, |η*| / Pas
10 1
10 0
10 -1
10 -2
10 -3
G´´
|η*|
10 -4
10 -2 10 -1 10 0 10 1 10 2 10 3
f/Hz
Figure 11.38: Dynamic rheogram of a 90 mM solution of C 8 F 17 SO 3 NEt 4 at T =208C. Below 10 Hz
the measurement was performed with a Bohlin CS 10 rheometer, above 10 Hz with the HF rheometer
and the 50 mM gap (with 1% deformation).
In the lower frequency range the samples show Maxwell behaviour. G' rises with the
slope 2, G@ with slope 1. At frequencies above the crossover of G' and G@, G' levels out and
reaches a plateau value, whereas G@ first decreases and then increases again. At high frequencies
both, G' and G@, increase. Even though a second plateau value of G' at high frequencies
could not be found it was possible to fit the data by a Burger model (four parameter
Maxwell model):
G 0 o 2 t 2 4 o 2 t 2 3
…o† ˆG 1
1 ‡ o 2 t 2 ‡ G 2
4
1 ‡ o 2 t 2 3
and G 0 ot 4 ot 4
…o† ˆG 1
1 ‡ o 2 t 2 ‡ G 2
4
1 ‡ o 2 t 2 3
: …20†
245

11 The Micellar Structures and the Macroscopic Properties of Surfactant Solutions
The structural relaxation time t 4 (the indices have been chosen for correspondence
with previous results) decrease with rising concentration from 25 ms (70 mM) to 0.2 ms
(300 mM) and the short time constant t 3 decrease from 0.35 ms (70 mM) to 0.1 ms
(250 mM). Therefore, they correspond well with the time constants determined by dynamic
electric birefringence measurements.
For each concentration the minimum value of G@ is about a factor of 2.5 lower than
the plateau value of G'. According to the theory of Granek and Cates [126] this yields a value
of 2.5 for the ratio of the mean contour length of the micelles to the entanglement length
(Eq. 7)). A decrease of the mean micellar length with rising concentration can be concluded
on the basis of this value and from the decreasing structural relaxation times.
Furthermore, mixtures of tetraethylammoniumperfluorooctanesulfonate and the pure
perfluorooctanesulfonic acid (C 8 F 17 SO 3 H) were studied with the HF rheometer. In Fig. 11.39
a rheogram is shown of the 150 mM solution with the salt : acid ratio of 7 : 3.
10 3 G´
G´, G´´ / Pa, |η*| / Pas
10 2
10 1
10 0
10 -1
10 -2
G´´
|η*|
10 -3
10 -2 10 -1 10 0 10 1 10 2 10 3
Figure 11.39: Dynamic rheogram of a 150 mM solution of C 8 F 17 SO 3 NEt 4 /C 8 F 17 SO 3 H = 7:3 at
T =218C. The deformation was 1% again. The minimum of G@ is lower than in the case of the pure
C 8 F 17 SO 3 NEt 4 .
f/Hz
From a qualitative point of view the rheogram does not differ too much from the one
presented in Fig. 11.4a. It can be noticed that after substitution of salt by acid the minimum
of G@ is more pronounced. This means a larger ratio of the mean contour length to the entanglement
length (here about 4), calculated by Eq. 7.
Generally one observes that up to a mole fraction of 40% of the acid the structural relaxation
time t 4 increases by a factor 10 (from 7 ms to 70 ms). From this result the growth
of the micellar aggregates can be concluded. The short time constant t 3 does not notably
change and amounts to about 0.1 ms. At a molar ratio of more than 50 % of the acid the
viscosity breaks down due to a decreasing length of the micelles. In this case it is not possible
anymore to measure the samples with the HF rheometer.
The high frequency increase of the moduli can be interpreted on basis of the following
models:
a) In addition to the continuous network there also exist unentangled shorter rods due to
the equilibrium distribution of the micellar lengths. These shorter aggregates do not significantly
influence the rheological behaviour of the samples at lower frequencies. At higher frequencies,
however, they contribute to the moduli.
246

References
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of the entanglement length cannot relax anymore by Rouse diffusion and a glass process
starts causing an increase of the moduli.
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250

12 Photophysics of J Aggregates
Herrmann Pschierer, Hauke Wendt, and Josef Friedrich
12.1 Introduction
Dye molecules of the pseudoisocyanine (PIC)-type form, under certain conditions, linear
long chain aggregates [1, 2]. The conditions concern the solvent, temperature, and freezing
procedure and depend, in addition, on the dye molecule and its counterion [3]. At sufficiently
low concentration and quick freezing aggregate formation is hampered and one obtains
the monomer. The spectral properties of the monomer show the usual features of dye
molecules in frozen solution, namely a rather broad structureless inhomogeneously broadened
absorption band. It is centred around 18850 cm –1 (Fig. 12.1). On the other hand, if
aggregates are formed, the spectral properties change dramatically. The wavenumber is
shifted from the monomer band by about 1450 cm –1 to the red and one or several extremely
sharp bands appear (Fig.12.1), whose intensities depend on the formation procedure. These
sharp bands are the so-called J bands [2]. They have exciton-like character with large coherence
lengths. It is the large coherence length which determines – with respect to localized
states – the unusual optical properties, namely the narrowing phenomena in the inhomogeneous
band, the molecular superradiative properties [4–7], the characteristic temperature de-
Figure 12.1: Low temperature (4.2 K) absorption spectra of the pseudoisocyaninechloride (PIC-Cl) aggregate
and monomer.
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
251

12 Photophysics of J Aggregates
pendence of the homogeneous line width [7, 8], and non-linear optical features [9–12]. In
this paper we focus on the narrowing phenomena.
A physically transparent interpretation of this narrowing phenomenon is based on fast
moving excitons which average the inhomogeneities of the host glass to a large degree. The
magnitude of the band narrowing depends on the coherence length.
p
If the number of coherently
coupled monomers is N c the narrowing is approximately 1/ N c . From the ratio of the
inhomogeneous band widths of monomer and aggregate, N c is estimated to be of the order
of 100. The problem with this estimation is that the inhomogeneous broadening cannot be
determined accurately enough since phonons and molecular vibrations may be hidden below
the inhomogeneous envelope. Another problem may arise from correlation effects in the disorder,
see below. Even the nature of the inhomogeneity in the J band is not known exactly.
There are various suggestions in the literature: chain length distribution [13], conformational
inhomogeneity [14], and coherence length distribution [15, 16].
In this paper we present a comparative pressure and electric-field tuning hole burning
study between PIC monomers and aggregates. It is our goal to gain information on
the coherence length of the excitons on the aggregate chains from these comparative experiments.
12.2 Basic aspects of pressure and electric-field phenomena in hole
burning spectroscopy of J aggregates
Pressure has a twofold influence on spectral holes: it shifts the hole and, in addition, broadens
it. The pressure shift of spectral holes in aggregates is quite strong. This strong pressure
shift is also related to the coherence length [17], because it scales with the polarizability of
the exciton chain. In this paper, however, we focus on the pressure broadening which has a
very direct relation to the coherence length. In small molecules, like the PIC monomer,
pressure broadening of spectral holes arises because the local configuration of host molecules
surrounding the probe is changed a little bit when the lattice is compressed. These
configurational changes signal a lack of spatial correlation among the glass-forming molecules.
The respective broadening is inhomogeneous in nature. Hence, it will be motionally
narrowed by a fast moving exciton, very much in the same way as the inhomogeneous band
is narrowed. As a consequence, by comparing the pressure broadening s M of the monomer
with the respective s A of the aggregate, we directly get the number N c of molecules within
the coherence length [4–6, 11],
r
s M
ˆ 2…N c ‡ 1†
: …1†
s A 3
The advantage of a hole burning experiment is that the change in line width under
pressure can be measured much more accurately than the inhomogeneous bands. No correc-
252

12.3 Experimental
tion due to underlying phonons, or vibrations, or bandshape asymmetries is necessary. Hole
burning in J aggregates was demonstrated first by de Boer et al. [18].
As to the influence of an electric field the situation is somewhat different: the monomer
has a dipole moment which gives rise to a first order Stark effect. The result is a splitting
and a broadening of the hole. In addition, the local environment determined by the host
glass may induce a dipole moment which can make a significant contribution to the field
broadening as well [19–21].
In the aggregate, a variety of things can happen: the structure of the chain may have
an inversion center. As a consequence, the broadening will be reduced. In addition, similar
to the pressure phenomena, the field broadening due to the environmentally induced dipole
moments may be motionally narrowed. In this case motional narrowing can be interpreted in
the sense that the influence of the matrix fields averages to zero within a scale given by the
coherence length.
12.3 Experimental
So far, we performed comparative pressure experiments for two different PIC systems,
namely PIC-Cl and PIC-I in an ethyleneglycol-water glass with a mixing volume ratio of
1:1. In the following, PIC stands for 1,1'-diethyl-2,2'-cyanine. Hole burning was performed
with a ring dye laser pumped by an argon ion laser. The laser band width was about 1 MHz.
The respective scan range covered 30 GHz. The holes were detected in transmission. In the
J band, typical burning times were of the order of one minute. The respective power levels
varied between 0.01 and 0.1 mW. The hole burning efficiency is very low in the monomers.
Hence, burning times of the order of 20 minutes were used at power levels of about 100 mW.
The aggregate spectra were measured at 4.2 K, whereas the monomer spectra were
measured at 1.5 K. The reasons are the following: in the monomer, the holes are subject to
thermal line broadening, hence are much broader at 4.2 K than at 1.5 K. This lowers the accuracy.
As mentioned above, the photoreactive quantum yield in the monomer is extremely
low. Hence, it is much harder to burn a hole at 4 K than at 1.5 K. However, at 1.5 K the
pressure range is rather limited because He solidifies at a level of 2.4 MPa. The narrow
pressure range at 1.5 K, on the other hand, limits the accuracy for the aggregates. In this
case pressure broadening is so low that one needs a larger range to get significant results.
Consequently, the respective experiments were performed at 4.2 K. We note that thermal
line broadening between 1.5 K and 4.2 K is of no concern for the aggregate. Due to the molecular
superradiance, the line width is not affected by phonons. Hence, there is no significant
change with temperature in this range [8].
For the Stark experiments on J aggregates holes were burned with the ring dye laser
but scanned with a pulsed dye laser system because of its larger scan range. The aggregate
solution was dispersed on an indium tin oxide (ITO) coated glass substrate. Maximum field
strengths were about 300 kV/cm.
253

12 Photophysics of J Aggregates
The monomers were investigated in a glass cuvette placed between two electrodes. In
this case, maximum field strengths were about 10 kV/cm and both, burning and scanning of
the holes, were performed with a pulsed dye laser system.
12.4 Results
In Fig. 12.1, we compare the inhomogeneous long wavelength absorption bands of PIC-Cl
monomer and aggregate. Figure 12.2 shows for both cases as holes deform under isotropic
pressure conditions.
Figure 12.2: Behaviour of spectral holes in the aggregate (left) and monomer (right) under different
pressures. Burn-frequencies: 17334 cm –1 for the aggregate, 18727 cm –1 for the monomer. Temperature
4.2 K (aggregate) and 1.5 K (monomer). Lorentzian fit curves are indicated.
Figures 12.3 and 12.4 show the influence of pressure on the line width of monomer
and aggregate. In the insets, it is demonstrated for the monomers on an expanded scale that,
despite the small pressure range, line broadening is linear with pressure and that the respective
slope can be determined with significant accuracy. For the monomer of PIC-Cl we measured
(0.18 +0.02) cm –1 /MPa. For PIC-I a value of (0.21 +0.01) cm –1 /MPa was found. We
stress that values of the same order of magnitude have been found for a series of similar
sized molecules in organic glasses [22–24].
Going from the monomer to the aggregate a dramatic decrease in pressure broadening
occurs. We measured (0.020 +0.001) cm –1 /MPa and (0.026+0.001) cm –1 /MPa for PIC-Cl
and PIC-I aggregates, respectively. The respective ratios in the broadening per pressure are
(9+1) and (8 +1).
Figure 12.5 shows the broadening of a spectral hole for the PIC-I monomer (a) and
aggregate (b) in an electric field. For the monomer the field-induced broadening is quite
254

12.4 Results
Figure 12.3: Pressure broadening of spectral holes for PIC-Cl monomer and aggregate. The inset shows
the data for the monomer on a smaller scale.
Figure 12.4: Pressure broadening of spectral holes for PIC-I monomer and aggregate. The inset shows
the data for the monomer on a smaller scale.
strong (about 1 GHz cm/kV) and linear with the applied Stark field E St . In the aggregate the
field-induced broadening is dramatically reduced. The fitted curve in Fig. 12.5 has a strong
quadratic component but there is a linear contribution as well. Comparing the linear regime
of the aggregate with the respective one of the monomer, it is obvious that the field-induced
broadening is reduced by a factor of about 150.
255

12 Photophysics of J Aggregates
Figure 12.5: Electric field-induced broadening (Stark effect) of spectral holes for pseudoisocyanineiodide
monomer (a) and aggregate (b). Note the different scales of the applied Stark field for monomer
and aggregate.
12.5 Discussion
12.5.1 Pressure phenomena
First, let us address broadening of the holes in the monomer band. The reason for pressure
broadening is based on the fact that a large variety of microscopic environments in amorphous
host materials can correspond with the same absorption energy of the probe molecule.
This degeneracy is partly lifted under pressure and is reflected in pressure broadening. The
magnitude of this broadening is determined through the magnitude of the probe solvent interaction
and through the similarity of the microscopic probe-lattice interaction configurations
with their respective change under pressure [25]. This similarity is expressed in a specific
degree of correlation. In a rather perfectly ordered crystal, for instance, this degree of
correlation is close to 1 and, correspondingly, the pressure broadening is close to zero as has
been observed [26]. In amorphous solids, on the other hand, the respective correlation is
low and broadening is rather large. Typical values are of the order of 0.1 to 0.2 cm –1 /MPa
[22–24]. As is obvious from the discussion above, pressure broadening in solids is inhomogeneous
in nature.
In the aggregates the wavefunction is delocalized over N c monomer units. Then, we
can consider two cases:
a) the microenvironments of the individual monomers in the aggregate are statistically independent.
In this case the site energy of molecule n is completely independent from molecule
n +1;
b) there is a finite correlation length l 0 , i. e. within l 0 the site energies of the molecules
forming the aggregate are correlated to some degree.
256

12.5 Discussion
For case a), with statistically independent microenvironments, it was shown by several
authors that the inhomogeneous line width of the monomers q forming the aggregate is narrowed
according to Eq. 1, i. e. roughly by a factor N c
1 . This narrowing is known as exchange
narrowing and it can be interpreted in a way that a fast moving exciton on an aggregate
chain averages over the local inhomogeneities. The amount of averaging is determined
by the coherence length of the exciton. Hence, the coherence length can be determined from
the ratio of the inhomogeneous band widths of aggregate and monomer. The exact determination
of these band widths is a problem, because of hidden states and band asymmetries.
However, we stressed above that pressure broadening of spectral holes is inhomogeneous in
nature. It reflects in a specific way the disorder in the local environments and, hence, should
be narrowed by a fast moving exciton in quite the same way as the inhomogeneous band. If
so, we can determine the number N c of monomers within the coherence length from Eq. 1,
by comparing the pressure broadening in the monomer with the respective one in the aggregate.
We found (120+32) for PIC-Cl and (101+24) for PIC-I. These numbers may
slightly depend on the solvent and freezing procedure. Generally speaking they fit to what is
known of J aggregates.
For case b), with correlated site energies of the monomers in the chain, the situation
becomes quite different. Equation 1 has to be substituted by
s
s M
ˆ 1 b
s A 1 ‡ b 2…N c ‡ 1†
: …2†
3
Equation 2 is approximative and holds only for sufficiently small b values.
Again, b is a degree of correlation between the site energies of the monomers in the
chain. It can be written as [27]
b ˆ exp
1
l 0
…3†
with l 0 being the correlation length, i. e. beyond l 0 the site energies of the two monomers are
statistically independent. As b goes to zero, we have case a) discussed above. As b goes to
unity, s A approaches s M and there is no motional narrowing at all. This latter case cannot be
directly derived from Eq. 2, since b = 1 is beyond the respective validity range.
The main point is that, if there were correlation in the site energies, it would not be
possible any more to determine N c from a linear optical experiment alone because Eq. 2
would contain two unknown quantities. There are indications from non-linear experiments
that l 0 is finite [28]. Another way to estimate b is via the ratio of the radiative rates of
monomer and aggregate. This ratio is also determined by the number N c of coherently
coupled molecules [7, 29] but is not influenced by correlation effects in the site energies,
g A ˆ N c g M ;
…4†
where g A is the radiative rate of the aggregate and g M the radiative rate of the monomer. In
an earlier paper we measured the homogenous line width down to 0.300 K [8]. The mea-
257

12 Photophysics of J Aggregates
sured value corresponded with a lifetime of about 30 ps. Since the fluorescence quantum
yield in the J band is supposed to be close to unity, the aggregate lifetime is governed by radiative
processes. With g M –1 = 3.7 ns [30], we get from Eq. 4 an N c value of about 123 which
is close to what we found from our pressure tuning experiments. Hence, according to this estimation,
b should be close to zero.
We stress however, that the results for N c in the literature, as obtained with various
techniques, are subject to large variations. Clearly, more work must be done to achieve unambiguous
results.
12.5.2 Electric field-induced phenomena
The electric field-induced phenomena in the J band are much more complex than the pressure
phenomena and not yet understood. Firstly we note that the reduction of the linear component
in the Stark effect is in agreement with what we expected from motional narrowing
effects. What was not expected at all, however, is the order of magnitude which is more than
a factor of 10 larger as compared to the pressure experiments. We conclude that additional
mechanisms must play a role. One such mechanism is the loss of the permanent dipole moment
upon aggregate formation. The absence of a dipole moment has severe structural implications
which are not clear at the present time. Our result is also at variance with experiments
by Misawa et al. [31], who reported huge changes in the static dipole moment of J aggregates,
which were attributed to their enhanced size. However, since these experiments
were performed with spin-coated J aggregates, it may possibly be that geometric effects play
an important role.
Acknowledgements
The authors thank the Deutsche Forschungsgemeinschaft (Sonderforschungsbereich 213,
B15) and the Fonds der Chemischen Industrie for financial support.
258

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26. P. Schellenberg, J. Friedrich, J. Kikas: J. Chem. Phys., 100, 5501 (1994)
27. J. Knoester: J. Chem. Phys., 99, 8466 (1993)
J. Knoester: J. Lumin., 58, 107 (1994)
28. J. R. Durrant, J. Knoester, D. A. Wiersma: Chem. Phys. Lett., 222, 450 (1994)
29. J. Grad, G. Hernandez, S. Mukamel: Phys. Rev. A, 37, 3835 (1988)
30. H.-P. Dorn, A. Müller: Chem. Phys. Lett., 130, 426 (1986)
31. K. Misawa, K. Minoshima, H. Ono, T. Kobayashi: Chem. Phys. Lett., 220, 251 (1994)
259

13 Convection Instabilities in Nematic Liquid Crystals
Lorenz Kramer and Werner Pesch
13.1 Introduction
Pattern formation in hydrodynamic instabilities has been studied intensely over the last decades
[1, 2]. Although Rayleigh-Bénard convection (RBC) in simple fluids has been the
prime example [3], the rich variety of scenarios found in nematic liquid crystals (LCs) has
attracted increased attention.
LCs are materials made up of highly anisotropic organic molecules in a phase that reflects
this anisotropy. The class of nematic LCs (nematics) are fully liquid without longrange
translational but with long-range uniaxial orientational ordering of the molecules. As
a result of the coupling of the molecular alignment axis (described by the director ^n) with
the (mass) flow electric or thermal current the hydrodynamic equations involve numerous
non-linearities (Section 13.2), which easily lead to instabilities when a state of non-equilibrium
is maintained [4]. Convective flow can be driven electrically through space charges
that naturally arise in an anisotropic conductor in the presence of spatial variations, the electrohydrodynamic
convection (EHC), or thermally through buoyancy forces, the Rayleigh-Bénard
convection (RBC). EHC has attracted more attention and will play a major role in this
review.
The study of EHC by Williams [5], Kapustin and Larinova [6] in 1963 initiated extensive
experimental work in the typical thin-layer geometry shown in Fig. 13.1a. The nematic
is sandwiched between glass plates (separation d * 10–100 µm) with transparent electrodes.
The surfaces are treated to provide (ideally) uniform anchoring of the director, in most
cases along the x direction (planar or homogeneous alignment) but sometimes also in the z
direction (homeotropic alignment) 1 . Above an applied voltage V c > 10 V (typically low-frequency
ac) convection rolls appear with associated director distortions, which are easily detected
optically. The spacing of the rolls is of order d except in the higher frequency dielectric
range. Figure 13.1b shows a typical pattern with the rolls in the y direction normal to
the undistorted director.
1 We will concentrate here on the planar case. Some remarks about other alignments are made in
Section 13.5.
260 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

13.1 Introduction
E || z^
^ y
z^
^ z
x^ v
Director n
Figure 13.1: a) Cell geometry with section of a roll pattern for EHC (planar configuration). E = electric
field, v=velocity; b) Normal roll pattern for EHC with a dislocation.
The mechanism for instability in EHC based mainly on space charges generated by
preferential conduction along ^n (charge focusing, see Section 13.2) was suggested by Carr
[7] and then incorporated into a first one-dimensional model by Helfrich [8]. Subsequently
the linear theory, giving the onset of the instability and in principle the pattern up to degeneracy,
was generalized to include the common case of ac driving [9, 10], a rigorous two-dimensional
analysis [11], and finally a full three-dimensional treatment, signalizing the beginning
of a renewed interest in the subject. For references see e. g. [12], for a comparison between
the experimental and theoretical threshold see Fig. 13.2 a. Unfortunately already the linear
theory is a numerical problem, but useful analytic approximations are possible [12–15].
The need for the full three-dimensional theory became particularly evident from the
experimental work of Ribotta and co-workers [16, 17], where three-dimensional structures
(oblique rolls) were observed at threshold, and from systematic measurements by Kai and
co-workers [18, 19] under well-defined conditions. Figure 13.2b shows the oblique rolls,
100
Threshold Voltage [V]
80
60
40
20
0
0 10 20 30 40
Driving Frequency f [Hz]
Figure 13.2: a) Threshold curve for EHC as function of frequency. Experimental points from [20];
b) Zigzag pattern after increasing the voltage in Fig. 13.1b.
261

13 Convection Instabilities in Nematic Liquid Crystals
which in this case bifurcated from normal rolls after an increase of V. Note the appearance
of grain boundaries separating domains with the two symmetry degenerate directions (zig
and zag). At threshold the boundary would become less sharp.
As the theory progressed expanding into a weakly non-linear analysis providing the
Ginzburg-Landau amplitude equation description [12, 14] and eventually also including
mean flow effects that allow to capture the transition from ordered periodic to weakly turbulent
patterns [21–24] there was also a revival of experimental activity, For a phenomenological
treatment, see Refs. [25, 26]. Some milestones were the identification of the Eckhaus instability
(Section 13.3) that gives limits of the stable wave number band [27–29], the characterization
of the instability that may lead to pattern turbulence [36, 37] (Fig. 13.3b), and
the structure and dynamics of single dislocations and their interaction [30–33]. For a comparison
of the experiments [32, 34] with the results obtained from the Ginzburg-Landau theory
[35] see Fig. 13.3a.
8
5
7
6
exp.
exp. ξ 1 , ξ 2 , τ
theo. ξ 1 , ξ 2 , τ
4
3
2
1
0
-1
-0.01 0.01 0.02 0.03 0.04
0.00
∆q [µm -1 ]
Figure 13.3: a) Climb velocity of a single defect vs. wave number mismatch. Parameters of the GLE
are either from experiments [32] (solid curve) or from hydrodynamical calculations [12] (broken line);
b) Snapshot of a defect turbulent pattern.
Further highlights were the identification of thermal noise slightly below threshold by
Rehberg et al. [38–40] and finally the clear identification of a Hopf bifurcation leading to
travelling rolls or waves in sufficiently thin layers, below about 50 µm and clean material
(low conductivity) by Refs. [18, 41–43]. It is not possible to explain the Hopf bifurcation
within the conventional theoretical framework the standard model (SM), see Section 13.2,
where the LC is treated as an anisotropic ohmic conductor. Indeed some of the theoretical
effort, in particular the inclusion of the rather complicated flexoelectric terms [14, 44–46],
was aimed primarily at resolving this problem. The situation is further complicated by the
fact that the bifurcation is often observed to be slightly subcritical, i. e. with a very small
hysteresis [38–40, 47], whereas the theory predicts a supercritical bifurcation.
Very recently an extension of the standard theory, the weak-electrolyte model (WEM),
has been worked out where electric transport in the nematic is described in terms of two mobile
ion species of opposite charge which are coupled via a slow dissociation-recombination
reaction and whose densities are treated as dynamic variables [48]. Some results of the
262

13.1 Introduction
WEM will be discussed in Section 13.5 together with experiments in the material I52 [49].
Whether the WEM can also capture the subcritical bifurcation remains unclear.
In almost all the measurements the standard reference material 4-methoxybenzylidene-4'-n-butyl-aniline
(MBBA) or a mixture, Merck Phase V, have been used, sometimes
doped with an ionic substance. MBBA is the only room-temperature nematic with dielectric
anisotropy e a < 0 where all the material parameters have been measured. For tabulated values
see e. g. Ref. [12]. Unfortunately, it is a Schiff base and rather unstable when exposed
to moisture. Therefore, it is difficult to control the long-time conductivity in situ. Thus the
recent successful introduction of the very stable material 4-ethyl-2-fluoro-4'-[2-(trans-4-npentylcyclohexyl)-ethyl]-biphenyl
(I52) doped with iodine is very promising [50, 51]. This
material exhibits at low external frequencies strongly oblique travelling rolls which bifurcate
supercritically, leading to a particularly interesting scenario. 2
In RBC the traditional instability mechanism by buoyancy forces is enhanced considerably
by preferential conduction of heat parallel to the director leading to a heat focusing
effect, as was first shown by Dubois-Violette in 1971 [52]. After some amount of experimental
and theoretical work, for a summary see Ref. [53], Feng et al. [54] recently presented a
fully three-dimensional treatment including the weakly non-linear analysis. Subsequent experiments
in thick layers (some millimetres) with an additional stabilizing magnetic field H x
in the x direction (Fig. 13.1a) have substantiated several predictions [55]. In Fig.13.3 a a zigzag
pattern is shown which arises with increasing magnetic field after normal rolls (n is horizontal).
Further increase of H x leads to a larger angle of obliqueness with superposition of
the two roll systems (Fig. 13.3b) and eventually to rolls which are parallel to n. The very
common nematic 4-n-pentyl-4'-cyanobiphenyl (5CB) was used, whose relevant material
parameters have all been measured. 3
Figure 13.4: a) Oblique roll pattern for RBC in the planar configuration [55]; b) Superposition of zig
and zag for RBC [55].
2 In the note added at the end of this Chapter some very recent developments in EHC have been summarized.
3 Some very recent progress is summerized at the end of Section 13.4.
263

13 Convection Instabilities in Nematic Liquid Crystals
In this review we will address mostly the developments of the past twelve years, characterized
by substantial progress in qualitative as well as quantitative understanding of the
various instabilities. After introducing and explaining the basic equations (Section 13.2) the
theoretical concepts of the linear and weakly non-linear analysis will be presented
(Section 13.3). In Section 13.4 and Section 13.5 the results for the two systems, introduced
above, are discussed in particular in the light of recent experiments. Finally, we will list topics
which are omitted due to space limitations and we comment on some perspectives for
future work (Section 13.6).
For a classical review of convective instabilities in LCs see Ref. [56] and for EHC one
may consult the books of Blinov [57], Pikin [58], and recent review articles [14, 15, 59–63].
RBC has been reviewed by Barrat [53] and very recently by Ahlers [64].
13.2 Basic equations and instability mechanisms
13.2.1 The director equation
The macroscopic nematodynamic equations describe the dynamics of the slowly relaxing
variables, which usually are either connected with conservation laws or with the Goldstone
modes of the spontaneously broken symmetries. To formulate them we will follow the traditional
approach [65–67] rather than the one based more directly on the principles of hydrodynamics
and irreversible thermodynamics [68]. In the nematic state isotropy is spontaneously
broken and the averaged molecular alignment singles out an axis whose orientation
defines the director ^n, i. e. an object that has the properties of a unit vector with ^n =–^n.
The static properties are conveniently expressed in terms of a free energy density whose orientational
elastic part is given by [69]
f elast ˆ 1
2 k 11…r ^n† 2 ‡ 1 2 k 22…^n …r^n†† 2 ‡ 1 2 k 33…^n …r^n†† 2 : …1†
The elastic constants k 11 , k 22 , and k 33 pertain to the three basic deformations splay,
twist, and bend, respectively. For typical nematics with prolate molecules one has
k 33 > k 11 > k 22 and k ii * 10 –11 N.
Electric and magnetic fields E and H exert torques on ^n that can be derived from an
additional contribution to the free energy
f em ˆ 1
2 m 0w a …^n H† 2 1
2 e 0 e a …^n E† 2 E P f lexo : …2†
Here w a = w k – w k and e a = e k – e k are the relative diamagnetic and dielectric anisotropies,
so that the uniaxial susceptibility and dielectric tensors can be written in Cartesian coordinates
in the form
264

13.2 Basic equations and instability mechanisms
w ij ˆ w ? d ij ‡ w a n i n j ; e ij ˆ e ? ij ‡ e a n i n j …3†
and the flexoelectric polarization is [70]
P f lexo ˆ e 1 ^n…r ^n†‡e 3 …^n r†^n :
…4†
Static director configurations are obtained by minimizing the total free energy
F = R dV ( f elast + f em ) under the condition n j n j = 1 with suitable boundary conditions, which
results in equating to zero the torque density G on the director. The generalization to dynamic
situations is usually done by defining a vector
S ˆ S rate ‡ S cons ‡ S diss
…5†
and requiring the balance of torques [65, 71, 72]
ˆ ^n S ˆ 0 :
…6†
S has the form
S rate ˆ g 1 N;
1
N ˆ d^n=dt …r v†^n ;
2
…7†
S cons ˆ dF=d^n ; …8†
S dissji
ˆ g 2 A ij n j ; A ij ˆ 1
uv i
‡ uv
j
:
2 ux j ux i
…9†
Here d/dt = u t + v 7r denotes the usual material derivative and in N the rigid rotation
part x = 1 2 …r v† of the fluid has been subtracted out. A notation like ux j(n i )=u j n i = n i,j is
used freely. g 1 and g 2 are called rotational viscosities, which couple the flow field v to the director.
To confirm that Eq. 6 is indeed only a rate equation for the director one may take the
vector product of Eq. 6 with ^n leading to
g 1 ‰ d^n=dt 1=2…r v†^n Š ˆ …1 ^n^n T †…S cons ‡ S diss † : …10†
The typical relaxation time of the director in the thin-layer geometry of EHC is easily
seen to be
t d ˆ g 1 d 2 =…p 2 k 11 † ;
…11†
where we chose the splay elastic constant as a representative; obviously g 1 > 0 has to hold.
t d is typically in the order of 1 s.
It can be seen that Eq. 6 and Eq. 10 have only two independent components which
can be made explicit by transforming into one of the two local coordinate systems
^n; ^x ^n; ^n …^x ^n† ;
…12†
265

13 Convection Instabilities in Nematic Liquid Crystals
^n; ^z ^n; ^n …^z ^n† :
…13†
The first one becomes singular when ^n is parallel to ^x and is thus not suitable for planar
alignment. Similarly, the other one is not practical for homeotropic alignment.
13.2.2 The velocity field
The generalized Navier-Stokes equation for the velocity field v follows from momentum
balance
r m
dv
dt ˆ f ‡rT ;
…14†
where r m is the mass density, f the bulk force to be discussed below, and T the stress tensor
with components
T ij ˆ pd ij
uF
un k;i
Eˆ0
n k;j ‡ t ij ;
…15†
and p is pressure. The viscous stress tensor for an incompressible nematic contains the six
Leslie shear viscosity coefficients a i [72],
t ij ˆ a 1 n k n m A km n i n j ‡ a 2 n i N j ‡ a 3 n j N i ‡ a 4 A ij ‡ a 5 n i n k A kj ‡ a 6 n j n k A ki :
…16†
It is instructive to consider the three simple geometries for plane parallel shear flow,
the Miesovicz geometries [73], corresponding to the orientations of the director relative to
the flow axis and the shear gradient. Choosing v = u (z) ^x one has the effective shear viscosities
4
1) for ^n = ^z along the shear gradient ?Z 1 =(a 4 + a 5 – a 2 )/2, G y = a 2 u z ;
2) for ^n = ^x along the flow axis ?Z 2 =(a 3 + a 4 – a 6 )/2, G y = a 3 u z ;
3) for ^n = ^y along the shear gradient ?Z 3 = a 4 /2, G = 0.
The value of the non-vanishing component of the torque G on the director is also given.
Another positive effective viscosity is Z 0 = a 1 + a 4 + a 5 + a 6 . All shear viscosities are
typical of order 10 –1 kg m –1 s –1 .
From the Onsager reciprocity relations one finds [74]
a 6 a 5 ˆ a 2 ‡ a 3 …17†
4 Unfortunately the definitions of Z 1 and Z 2 are not universally accepted. Our definition is adopted by
Blinov [57] whereas the definitions in the review articles [67] and [73] are in the reverse order.
266

13.2 Basic equations and instability mechanisms
and also
g 1 ˆ a 3 a 2 ; g 2 ˆ a 3 ‡ a 2 : …18†
Note that the flow-alignment parameter l =–g 2 /g 1 is a reversible quantity. For l >1
the director can align in the flow plane at a fixed angle b with respect to the velocity (xaxis),
where tan 2 b =(l–1)/(l +1)=a 3 /a 2 .Forl < 1 there is tumbling [73].
The pressure is to be determined from the incompressibility condition
rv ˆ 0 :
…19†
An efficient way to implement this relation is to represent the velocity field in terms
of a toroidal and a poloidal potential g and f, respectively [75]
v…x; y; z† ˆr^zg ‡r…r^zf †ˆeg ‡ df
…20†
with
e T ˆ…@ y ; @ x ; 0† ; d T ˆ…@ 2 xy ;@2 yz ; @2 xx @ 2 yy † ; …21†
where z corresponds to the coordinate perpendicular to the plane of the layer. Applying e
and d on Eq. 14 gives two equations for g and f eliminating the pressure.
The typical relaxation time for a velocity field in a thin-layer geometry is
t visc = r m d 2 /(p 2 Z 1 ). This time is usually of the order of 10 –5 s, much shorter than the others.
Thus the velocity field can usually be treated adiabatically (neglect of inertial terms).
13.2.3 Electroconvection
13.2.3.1 The standard model
Now we come to the additional equations that are specific to the processes driving the instability.
In EHC the bulk force in the Navier-Stokes Equation 14 is derived from the Maxwell
stress tensor, which here reduces to
f ˆ r el E ‡…P r†E; P ˆ D E : …22†
The equation determining the charge density r el is obtained from charge conservation
and the Poisson law
dr el
dt
‡rj el ˆ 0; j el ˆ s E ;
r el ˆrD; D ˆ e E ‡ P f lexo :
(23)
267

13 Convection Instabilities in Nematic Liquid Crystals
In the SM the usual assumption of an anisotropic but fixed ohmic conductivity is
made. The conductivity tensor r has the same form as the other material tensors with
s a = s k – s k , see Eq. 3.
Equations 23 are easily seen to lead to charge relaxation with the time scale
t q ˆ e 0 e ? =s ?
…24†
which is typically of the order of 10 –3 s. Sometimes a dopant is added to the LC to obtain
sufficient or well controlled conductivity (not much is needed).
Now we are in a position to discuss the basic driving mechanism for EHC. The important
point is that in almost all nematics s a is substantially positive, typically s a /s k & 0.3–1.
Choosing materials with negative or only slightly positive dielectric anisotropy e a , here the
materials show great diversity, one easily sees that in the presence of an applied field E 0 and
with a small spatial variation (fluctuation) of the director n a space charge r el results. For r el =0
there is no solution of Eqs. 23. Roughly speaking, the charges are focused at locations where
the director bends. The bulk force in the Navier-Stokes equation may then overcome viscous
stresses and drive a velocity field v. Via the viscous coupling this may enhance the spatial variation
of the director and thus generate an instability. The threshold voltage is for low frequency
and for materials with not too large dielectric anisotropy of order V c 2 &p 2 k 11 /(s a t q )
and the introduction of the reduced control parameter R = V 2 (s a t q )/(p 2 k 11 ) is often useful.
13.2.3.2 The weak electrolyte model
The SM fails in particular to describe the Hopf bifurcation leading to travelling rolls, which
is observed quite frequently. For a Hopf bifurcation two processes that compete on a comparable
time scale are necessary. In our case, however, the director relaxation is much slower
than the other processes and thus determines the dynamics. Charge relaxation can usually
also be treated adiabatically. In addition, director relaxation and charge relaxation do not
compete but rather support each other usually, which also excludes a Hopf bifurcation.
Thus another slow process appears to be operating. A recently proposed model assumes
that this process is a relaxation of the mobile ion densities n + and n – on the time scale
t rec , which may result from a dissociation-recombination reaction. One then gets if singlecharged
ions are assumed
where
r el ˆ e…n ‡ n †; s ij ˆ ss 0 ij ; …25†
s ˆ e…m ‡ ? n‡ ‡ m ? n †;
s 0 ij ˆ d ij ‡ s a
s ?
n i n j ;
…26†
Here m + k, m + k are the ionic mobilities perpendicular and parallel to the director, respectively.
For simplicity the anisotropies were assumed to be the same for both types of
ions so that s a /s k = m + k /m + k.Sonows is an additional variable. From the balance equations
for n + and n – one easily recovers Eqs. 23 which now read
268

13.3 Theoretical analysis
ds
dt ‡r
m‡ ? ‡ m
? s ‡ m
‡
? m ? r el
" #
ˆ seq s ‡ m
1
? r el s m? r el
1
m ‡
s s eq ‡
? m ?
2t rec t rec 2
s 2 eq
r el
; …27†
where s eq = e (m + k + m – k)n eq contains the equilibrium ion density n eq . The last expression is
obtained by linearization in the quantities n + – n eq and n – – n eq .
Thus, in this model ion accumulation effects are included whereas ionic diffusion is
neglected as in the SM. The charge accumulation counteracts the standard (Helfrich) mechanism
of generation of space charges. If t rec is sufficiently slow one can find an oscillatory
behaviour of the system at threshold, i. e. a Hopf bifurcation (Section 13.5).
13.2.4 Rayleigh-Bénard convection
In RBC the bulk force in the Navier-Stokes equation is f = rg. In the spirit of the Boussinesq
approximation one has for the mass density r = r m [1 – a (T– T 0 )], where g is the gravitational
acceleration and a the thermal expansion coefficient. One needs in addition the heat
conduction equation
dT
dt ‡rj T ˆ 0; j T ˆ krT ; …28†
with k ij = k k d ij + k a n i n j , k a = k || – k k (k *10 –7 m 2 s –1 ). As first pointed out by Dubois-
Violette [52, 76] the conventional instability mechanism operative in isotropic fluids is here
enhanced considerably by preferential conduction of heat parallel to the director (k a >0)
leading to a focusing effect in the presence of out-of-plane fluctuations of the director. For
anisotropies of order one the reduction of the threshold is of order F = t d /t therm &10 3 ,
where t therm = d 2 /(p 2 k k ) is the vertical thermal diffusion time. Other important dimensionless
quantities besides F are the Prandtl number Pr = t therm /t visc &10 3 and the Rayleigh
number R = ag t visc /t therm (T/d), which is the traditional control parameter.
13.3 Theoretical analysis
We here present the general methods used to extract the relevant information from the basic
equations. It is convenient to introduce the notation V=(n,v, …) for the collection of all
field variables involved in the specific problem. We choose the fields in such a way that
269

13 Convection Instabilities in Nematic Liquid Crystals
V = 0 corresponds to the non-convecting basic and primary state. Then the set of macroscopic
equations, as presented in the previous Section, can be written in the following symbolic
form:
LV ‡ N 2 …VjV†‡N 3 …VjVjV†‡ ˆ…B 0 ‡B 1 …V†‡B 2 …VjV†† uV
ut :
…29†
The vector operators N 2 , N 3 … denote quadratic, cubic … operators in V and its spatial
derivatives, whereas the operators L and B i represent matrix differential operators of the
indicated order on V. Computer algebra can be used to perform the expansion.
A direct simulation of the coupled system of the partial differential equations, Eq. 29,
with appropriate boundary conditions (v = 0, n prescribed at the confining plates, etc.) is at
the limits of the supercomputers of today. It will turn out that in the liquid crystal systems a
rich scenario of patterns, including spatio temporal chaos, develops already near threshold
so that perturbational calculations are useful.
The onset of the instability is obtained from a standard linear stability analysis of the
basic (primary) state. The problem can be diagonalized with respect to horizontal coordinates
x=(x, y) associated with the directions of idealized infinite extent by a Fourier transform,
V…x;z;t† ˆ R d 2 qe iq
x U q …†e z
l… q†t
;
with q=(q x , q y ). From Eq. 29 with boundary conditions at z=± d/2 one arrives at an eigenvalue
problem,
lB 0 …iq; u z ; R†U q …z† ˆL…iq; u z ; R†U q …z† ;
…30†
where R=R, S, … are the control parameters of the system. 5
In the typical scenario the real part s of one of the eigenvalues l (q, R) =
s (q, R) ± io (q, R) crosses zero upon increase of the main control parameter R at fixed q
beyond a value R 0 (q) while the real parts of all other eigenvalues remain negative. Thus the
neutral surface R=R 0 (q), which separates the unstable (R > R 0 (q)) from the stable
(R < R 0 (q)) modes, is given by the condition of vanishing growth rate s(q, R) = 0. Minimizing
R 0 (q) with respect to q gives the threshold R c = R 0 (q c ) with the critical wave-vector
q c =(q c , p c ) and the critical frequency o c = o (q c ), which is zero for a stationary bifurcation,
which is the more common case, while non-zero for a Hopf (oscillatory) bifurcation.
We will encounter situations where different minima of R 0 (q) coincide. This multicritical
behaviour is either accidentally or caused by some symmetry and calls for a special treatment.
In isotropic systems q c is even continuously degenerate on a circle.
In planarly aligned nematics, i. e. in an axially anisotropic system, one can distinguish
two fundamental cases.
5 We will treat R as the main control parameter whose increase carries the system across the instability,
i. e. the squared voltage in the case of EHC and DT in the case of RBC (both non-dimensionalized).
270

13.3 Theoretical analysis
a) If q c is parallel to one of the symmetry directions, p c =0orq c = 0, one speaks of normal
(Fig. 13.1 b) or parallel rolls, respectively.
b) If q c is at an oblique angle, one speaks of oblique rolls (Figs. 13.2 b and 13.4a).
Clearly, we get the two symmetry-degenerate directions zig and zag, which may superpose
to give rectangles (Fig. 13.4 b). In the case of a Hopf bifurcation one has degeneracy
between travelling waves in opposite directions, which may also superpose to give standing
waves. Oblique rolls arising via a Hopf bifurcation give four degenerate modes.
In EHC, for the usual case of driving with a pure ac field of frequency o =2/pf, the
eigenvector U q of Eq. 30 inherits the additional periodic time dependence and the eigenvalue
l becomes a Floquet coefficient. Then there is an additional discrete symmetry
(z, t) ? (–z, t + 1/(2f )) and each component of U q has a definite parity. Generally the conductive
mode (for even parity the out-of-plane component of director and v z ; for odd parity
the in-plane components of velocity) destabilizes first at low frequencies f. For materials
with negative dielectric anisotropy, e a < 0, there exists a cut-off frequency f c so that the dielectric
mode with the other parity destabilizes first for f > f c , where f c ?? for e a ?0. The
existence of these two regimes was first pointed out by Orsay’s group [9, 10]. For further details
see Refs. [14, 61].
The linear problem (Eq. 30) with realistic boundary conditions has to be solved numerically,
often by treating the z-dependence by a Galerkin expansion with a suitable cutoff.
In particular for stationary bifurcations a cut-off at lowest non-trivial order, i. e. one trial
function for each component of U q gives approximate analytic expressions for the neutral
surface and the growth rate, from which the interplay of the many material parameters becomes
transparent [12–15, 48, 54, 77].
The basic idea of the weakly non-linear analysis [1, 78–80] in its rather general form
[54, 81, 82] is to reduce the phase-space dimension of the system by choosing an appropriate
basis of states, characterized as the dynamically active ones [80]. We demonstrate the
method only for stationary bifurcations, where the relevant time scale becomes slow near
threshold but a generalization to Hopf bifurcations is straightforward. One expands V in
Eq. 29 at lowest order as a wave packet of the eigenmodes U q of Eq. 30 at the neutral surface
R 0 (q) and with the growth rate s =0,
V V 1 ˆ R
dq A…q†U q …z†e iqx ‡ c:c: ;
…31†
D ‡
where A (q) denotes the amplitude order parameter, which vanishes at threshold and c.c. is
the complex conjugate of the preceeding expression. The integration domain consists of
small areas D ± centred at ±q c which need not be specified precisely at this point. We will arrive
at an order parameter equation for A (q) by expanding Eq. 29 up to order A 3 . In an intermediate
step, one needs the contributions V 2 * A 2 , generated by quadratic interaction from
V 1 . They are determined from the relation LV 2+ N 2 (V 1 |V 1 ) = 0 derived from Eq. 29. Here
terms from the right-hand side can be neglected, since they contain in addition a (slow) time
derivative.
The solution V 2 contains separate contributions with wave-vectors near ±2q c and 0. In
the sector near q=0the so called mean flow or mean drift can be isolated systematically
[24, 83]. The mean flow arises from long wavelength variations – on a scale much larger
271

13 Convection Instabilities in Nematic Liquid Crystals
than the spacing of the rolls – along the roll axis (undulations) leading to a lateral pressure
gradient, whose spatial average across the cell is non-zero [1, 2]. A second amplitude B is
introduced to describe the resulting Hagen-Poisseulle-like shear flow, also characterized by
the non-zero vertical vorticity (curl v) z . The equations can be closed at order A 3 by inserting
V 2 in Eq. 29 and projecting onto the subspace spanned by the linear modes V 1 . One arrives
at two coupled integral equations for the amplitudes A (q) and B (s):
a 1
dA…q;t†
dt
Z Z
ˆ a 2 A…q†‡ dq 0 dq 00 a 3 A…q 0 †A…q 00 †A…q q 0 q 00 †
‡ R dsb 1 B…s†A…q s† ; …32†
c 1 s 2 x ‡ c 2 s 2 y B…s† ˆR dqb 2 A…q†A…s q† : …33†
The q-integrations are confined to the regions D = D + & D – and s is near zero. The
coefficient functions a i (q), b i (q, s) and the constants c i depend on the material parameters.
They involve z-integrations and have to be calculated numerically.
The above procedure guaranties that all coefficients are smooth functions of the wavevectors.
Since the field B satisfies an anisotropic Poisson equation, transformed to Fourier
space, its long-range character is evident. In nematics its effect turns out to enhance transverse
modulations in distinct contrast to isotropic fluids in most cases. As will become clear
below, the mean flow contributions can be neglected in the immediate vicinity of threshold.
Stationary periodic roll solutions with wave-vector q 0 can be calculated by the ansatz
A r (q) = c r d (q – q 0 )+c r * d (q + q 0 ). Then the double integral on the right hand side of
(Eq. 32) becomes trivial and one finds that B : 0. One also easily sees that |c r | 2 is proportional
to (R – R 0 (q 0 ))/R 0 (q 0 ), the reduced distance from the neutral surface. A subcritical bifurcation
is signalized by a negative proportionality factor. Then higher powers in A would
have to be included.
The stability analysis of the periodic roll solution [54] is performed by introducing a
small perturbation dA (q, t) of the amplitude A r with a modulation wave-vector s in Eqs. 32
and 33,
dA…q;t†ˆ…c 1 …q; s†d…q q 0 s†‡c 2 …q; s†d…q ‡ q 0 s††e Lt ; …34†
corresponding to a perturbation dV 1 in Eq. 31.
Roll solutions with wave-vector q 0 are unstable if the maximum of Re (L (q 0 , s)) with
respect to s is positive. Well-known long wavelength destabilization mechanisms involve:
a) local dilation and compression of the roll pattern, the Eckhaus process (E) [84], i. e.
s || q 0 ,
b) undulations along the roll axis, the zigzag process (ZZ) [86], i. e. skq 0 ,
c) combinations of both processes, the skewed varicose process (SV) [86].
In the last case mean flow is decisive. In addition one has short wavelength instabilities
where |s| is of the order of q c . They are well-known in systems that are isotropic in the
plane, where they can lead from rolls to squares, hexagons or bimodal structures. A different
272

13.3 Theoretical analysis
version is found in our anisotropic systems (planar director alignment) when oblique rolls
become unstable with respect to a superposition of zig and zag (Fig. 13.4 b).
Many features become more transparent when formulated in real (position) space in
terms of amplitude (envelope) or Ginzburg-Landau equations (GLE). Then one sees that the
important information is really condensed in a few parameters and the universal aspects of
the systems become apparent. By model calculations, which can often be performed analytically,
stability boundaries and secondary bifurcation scenarios are traced out. The real space
formulation is essential when it comes to the description of more complex spatio-temporal
patterns with disorder and defects, which have been studied extensively in EHC slightly
above threshold (Figs. 13.1b, 13.3b). One introduces a modulation amplitude A(x) defined
as
A…x† ˆ R
dqA…q†e i…q qc†x : …35†
D ‡
Near threshold one expects that only a small region D + (i. e. |q-q c | P q c ) is relevant in
Eq. 35 and that correspondingly the amplitude A(x) varies on a slow scale. The various coefficients
of the order parameter equations (Eqs. 32 and 33) can now be expanded into Taylor
series around q c with respect to q and around zero with respect to s. Since all non-analyticities
have been absorbed in the amplitude B the expansion is smooth and powers of the components
of (q-q c ) and of s can obviously be identified with spatial derivatives of A(x) and
B(x), which is constructed in analogy to Eq. 35.
If the q-dependence is taken into account only in the linear coefficient a 2 of Eq. 32 –
all other coefficients taken at q=q c and s=0, where b 1 = b 2 = 0 – one ends up with the famous,
slightly generalized Ginzburg Landau equation [87]
u t A ˆ l…q c ir; e†A gjAj 2 A; …36†
where l is the linear growth rate (Eq. 30). Clearly l is zero at threshold e =(R – R c )/R c =0
and r = 0 and should be expanded in both arguments. For our purposes at lowest order it is
sufficient to keep the following terms
l…q c ir; e† e ‡ x 2 1 u2 x ‡ 2ax 1x 2 u x u y ‡ x 2 2 Wu2 y iZx 1 x 2 2 u xu 2 y x 4 2 u4 y : …37†
The reason for keeping higher powers in u y than in u x will become clear shortly. On
this level the expansion can be cast into an overall expansion scheme in terms of e 1/2 ,or
equivalently A. In the anisotropic case, where there is no continuous degeneracy of the critical
mode(s), one may in general assume e *A 2 *u t *u 2 x *u 2 y, so that the higher order terms
*u x u 2 y and u 4 y drop out. Then Eq. 36 becomes uniformly of order e 3/2 .
The remaining mixed derivative term in Eq. 37, that vanishes anyway for normal as
well as for parallel rolls, can always be transformed away by rotating the coordinate system.
Moreover, by rescaling x and y the differential operator becomes proportional to the Laplacian,
so that Eq. 36 finally reduces to
tu t A ˆ eA ‡ x 2 DA gjAj 2 A: …38†
273

13 Convection Instabilities in Nematic Liquid Crystals
This constitutes the simplest GLE [88, 89]. Until now we have not specified explicitly
the new spatial scaling. By appropriate scaling the time, length, and A all parameters can be
scaled away. Note that the conventional way to derive the GLE starts from a multi-scale analysis
in space and time, expanding systematically in powers of e 1/2 .
Going back to Eqs. 36 and 37 with a = 0 it is easy to see that changing W from positive
to negative – by changing some secondary control parameter like the frequency in EHC
– describes a normal pitch fork bifurcation from normal to oblique rolls. 6 For W < 0 the
maximum growth rate of plane wave solutions of Eq. 36 occurs at wave-vectors with nonzero
y component. Details of this transition, which is the analogue of an LP in the theory of
equilibrium phase transitions, have been discussed elsewhere [12, 21, 89, 90]. The corresponding
uniform scaling *e 3/2 as in Eq. 38 is recovered with W*e 1/2 *u x and now
u y *e 1/4 . This corresponds to the scaling adopted in isotropic media and in fact the wellknown
Newell-Whitehead-Segel amplitude equation for isotropic systems [91, 92] can now
be obtained as the special case W=0 and Z=±2 in Eq. 37.
The general form of the amplitude equations is known a priori from symmetry, translation
in space and time as well as rotation and reflexion symmetry within the plane of the
layer, which manifests itself in the linear growth rate function. The coefficients for a specific
problem have to be determined only for quantitative comparison with experiments.
So far we concentrated on the simplest version of a stationary bifurcation where only
a single mode becomes critical. One may have (near) degeneracy due to symmetry or accidentally
by tuning of a secondary control parameter. If there is a n-fold degeneracy of marginal
modes one is led to a coupled system of n amplitudes. Here degeneracy due to symmetry
occurs in the oblique roll case (n =2). From the ratio of the two non-linear coefficients
for self and cross coupling one then deduces if rolls zig or zag with the possibility of domain
boundaries (Fig. 13.2b) or if their superposition (Fig. 13.4b) is stable.
In the case of a Hopf bifurcation, as observed in EHC, one has degeneracy due to reflection
symmetry with two modes, corresponding to left and right travelling waves (rolls).
Also, the coefficients of Eq. 38 become complex and from the linear dispersion relation one
gains a term ± itv g 7rA (the signs ± pertain to left and right travelling waves), which describes
a group velocity v g . From a phenomenological point of view changes of complex
coefficients and group velocity stem from the absence of reflection symmetry in the solutions.
Destroying reflection symmetry by an external perturbation has a similar effect. Depending
on the ratio of the real parts of the non-linear coefficients there are again two possibilities:
either travelling waves with the possibility of domain boundaries (sources and sinks)
or a superposition of the two wave systems leading to standing (oscillating) rolls. In the first
case one is essentially left with the celebrated complex Ginzburg-Landau equation (CGLE),
which exhibits transitions, e. g. at the Benjamin-Feir instability, to various forms of spatiotemporal
chaos and is presently studied intensely. For general reviews see Refs. [2, 80]. The
CGLE is applicable also to systems that show a Hopf bifurcation leading to a spatially homogeneous
state (q c = 0), which is mainly found in oscillatory chemical reactions. The point defects
of CGLE correspond to the famous spirals. In our system there is also the possibility of
oblique travelling waves, which have in fact been observed in Merck Phase V [47] and in
I52 [49, 51] and are known to lead to an interesting type of four-wave interaction [93].
6 Now the higher powers in u y become essential; if we wanted to describe the transition between parallel
and oblique rolls, higher powers in u x would have to be retained.
274

13.4 Rayleigh-Bénard convection
For the description of modulated roll patterns away from threshold (possibly only
slightly) the amplitude B must be included, now also transformed to real space. A uniform
scaling in e is then no more possible and coupling terms like Au y B appear in Eq. 36 as well
as derivative terms in the cubic non-linearities. The gradient expansion starting from Eqs. 32
and 33 is systematically truncated in such a way that all O(s 2 ) contributions to the growth
rate (Eq. 34) are included [24, 83]. The additional equation for B is of the form
…c 1 u 2 x ‡ c 2u 2 y †B ˆ q 1u x u y jAj 2 ‡ q 4 u y …iA u 2 y A ‡ c:c:†‡...
…39†
The terms occurring are those allowed by symmetry. Due to the anisotropy more terms
appear than in isotropic systems [94]. The clue for the characteristic appearance of the zigzag
instability as a secondary bifurcation is that q 4 is typically negative in nematics [23, 24]
leading, in contrast to isotropic fluids, to amplification of transverse modulations of roll patterns.
Model calculations that include this feature [25, 26] were quite successful in describing
qualitatively the secondary instability and the behaviour beyond.
13.4 Rayleigh-Bénard convection
In the following two Sections we will discuss and compare theoretical and experimental studies
on RBC and EHC. We often use non-dimensionalized units. Thus we write wave numbers
as q i = q i 'p/d, where the prime is sometimes omitted, and magnetic fields as H i = h x H f
with the splay Freédericksz transition field H f =(p/d)[k 11 /(m 0 w a )] 1/2 . Other quantities have
been introduced before. Note that for the Cartesian components of the wave-vector we use
two symbols, q=(q x , q y )=(q, p).
In the case of RBC we will mainly present an analysis of recent experimental investigations
[55] in comparison with theoretical results [54]. In Fig. 13.5a the critical Rayleigh
number R c (continuous curve) normalized to the value R c0 = 1707.37 for isotropic fluids is
shown as a function of a stabilizing magnetic field h x = H x /H f (R c /R c0 ? 1 for h x 2 /F p1).
The symbols are experimental results for the nematic 5CB. The linear theory predicts for increasing
field a lower Lifshitz point (LP), where the roll orientation changes from normal
( p c = 0) to oblique and at higher field an upper LP with a transition from oblique to parallel
rolls (q c = 0). In Fig. 13.5b the experimental results for the squared wave numbers (q c , p c )
together with |q c | 2 are plotted and compared with the prediction of the theory (solid curves).
The agreement is quite remarkable considering that there is no adjustable parameter.
Weakly non-linear theory predicts two tricritical points (TP) with a subcritical bifurcation
in the field range 5 < h x < 26. Note that the upper TP at h x = 26 is slightly above the
lower LP. One may expect rather complex non-linear behaviour in that range, which has not
been worked out in detail for 5CB so far. The situation should be simpler for MBBA where
the TPs (h x = 4.15 and 31) are below and well-separated from the Lifshitz points (h x =36
and 62).
275

13 Convection Instabilities in Nematic Liquid Crystals
Figure 13.5: a) Rayleigh number as a function of the stabilizing field h x in the planar configuration.
b) The squared components of q c and |q c | 2 as a function of h x .
In Figs. 13.6a, b stability diagrams of MBBA in the e-q plane (normal rolls, p =0)
and in the e-p plane (oblique rolls, q=q c ) are shown for a magnetic field h x = 34 between
the upper TP and the lower LP (e =(R – R c )/R c ). Rolls exist inside the neutral curves (NC)
and are stable in a region bounded by the E, SV, and ZZ lines. The lowest-order theory
(GLE) would only give the Eckhaus instability. Including the higher order terms with mean
flow produces the skewed varicose instability SV, which however is hard to distinguish from
the Eckhaus instability E. More important is that the stability regime for normal rolls is now
limited from above by a ZZ line. When the ZZ line for normal rolls is extended on the left
it joins the neutral curve at a point L which one may call a LP on the neutral curve. To the
left of that point the growth rate for oblique rolls becomes larger than for normal rolls, or
equivalently, the curvature of the neutral surface in the p-direction becomes negative. With
0.02
0.02
ε=(R-R c
)/R c
0.01
E
L
ZZ
SV
NC
0.00
–0.2 –0.1 0.0 0.1 0.2 0.3
q-q c
P
ε=(R-R c
)/R c
0.01
ZZ
SV
NC
E
0.00
0.0 0.5 1.0 1.5
p
Figure 13.6: a) Stability diagram for normal rolls slightly below the lower LP; b) Stability diagram in
the transverse (oblique) direction for q=q c .
276

13.4 Rayleigh-Bénard convection
increasing h x the point L moves down along the neutral curve until it reaches q c , e =0at
h x = 36. This is the lower LP beyond which oblique rolls appear at threshold. In the vicinity
of the LP the scenario can be described by the extended GLE as discussed in the last Section.
The parameter W in Eq. 37 changes sign when the LP is crossed. From this equation
without higher order terms one obtains a ZZ line that goes vertically up [89]. Coupling to
the mean flow enhances the ZZ instability (Section 13.6) and consequently tilts the curve to
the right. By contrast, in simple fluids, where the ZZ line always emanates from q c , mean
flow effects, which are important for small to medium Prandtl numbers, turn the line to the
left. Thus anisotropy is responsible for moving the LP away from q c along the neutral curve
and mean flow tilts the line to the right. This seems to be a general scenario which also applies
to EHC.
Let us turn attention to the point P on the neutral curve NC in Fig. 13.6a, where several
stability limits merge. P corresponds to a tricritical point on the neutral curve at q tri .
For q6q tri the solution bifurcates subcritically from the neutral curve, i. e. a small amplitude
solution exists only outside the neutral curve. With decreasing magnetic field P moves
down along the neutral curve and at h x = 31 it reaches e = 0 and q = q c , signalizing the
change to a subcritical bifurcation with further decrease of h x . Interestingly, at small magnetic
fields, where the bifurcation becomes supercritical again, the tricritical point moves upwards
on the left branch of the neutral curve.
Figure 13.6 b shows that normal rolls can escape the ZZ instability by undergoing a
secondary transition to oblique rolls. Their stability range is bounded by two roughly parabolic
curves. This transition has indeed been observed [55] and is quite well-known in EHC.
There exists in addition a tertiary bifurcation (dotted lines), where the oblique rolls become
unstable against a short wavelength mode which appears to lead to oblique rolls with p
roughly reversed. At present it cannot be predicted into which state the system evolves. It
could be a superposition of zig and zag rolls, as observed in the experiments, or a complex
dynamic state where the system oscillates between the two states separated by grain boundaries
(and maybe other defects) generated persistently. Also, in the light of the results for
EHC (see below), one may expect that under some conditions in large aspect-ratio systems
spatio-temporal chaos develops when the stability limit of normal rolls is exceeded.
The non-linear scenarios have not been systematically studied in experiments so far.
Actually investigations at small fields (h x ^ lower TP) would be very interesting. According
to the theory the upper limitation of the stability regime is now of the SV-type, beyond
which the system cannot evade into stable oblique rolls and complex behaviour seems unavoidable.
Very recently a generalized weakly non-linear analysis including homogeneous twist
of the director has been worked out [171]. Then the above mentioned SV instability changes
into a ZZ instability in agreement with experiments [167]. In the non-linear regime one has
a transition to abnormal rolls and bimodal convection as is also found in EHC, see note
added at the end.
277

13 Convection Instabilities in Nematic Liquid Crystals
13.5 Electrohydrodynamic convection
The theoretical results are obtained with the SM (Section 13.2) unless otherwise stated.
13.5.1 Linear theory and type of bifurcation
As pointed out before, most experiments were done with the material MBBA, sometimes
doped with an ionizing substance. Typical material parameters can be found in the literature
[12] which look fairly reliable except the flexoelectric coefficients. Unfortunately in most experiments
the conductivity, which may vary strongly, has not been measured. One therefore
has to determine the charge relaxation time t q , e. g. by fitting the cut-off frequency f c where
the crossover between the low-frequency p conductive and the higher frequency dielectric regime
occurs. The formula 2 pf c t q = C applies, where C is a function of s|| /s k , s || /s k , and
the viscosity ratio [12]. For MBBA C is about 6.3. In Fig. 13.2 a we show the threshold curve
for a rather thick (d = 100 µm) and clean cell showing quite good agreement between theory
and experiment. In the calculations the flexoelectric coefficients were taken considerably
smaller than those estimated from some measurements. Otherwise one would find in contrast
to the observations oblique rolls at a threshold in the dielectric regime. The conductivity was
chosen as s k = 0.28610 –8 O –1 m –1 and s || /s k = 1.65. For more details see Ref. [14]. The theory
correctly describes many qualitative features like the occurrence of oblique rolls at threshold
at sufficiently low frequencies for materials with not too negative e a . MBBA (e a &–0.5)
is a border-line case [14], Merck Phase V (e a & –0.3) has well developed oblique rolls and
the newly introduced material I52 (e a & 0) exhibits strongly oblique rolls [50, 51].
For materials with e a < 0 the above behaviour with the conductive threshold apparently
diverging at f c is a reasonable approximation if the charge relaxation time t q is much smaller
than the director relaxation time t d . Actually the conductive threshold curve always becomes
vertical at a frequency f m ^ f c and bends back at higher voltages giving rise to restabilization
of the conductive mode at high voltages. The effect becomes apparent in thinner and cleaner
specimens. The fact that the conductive mode becomes ineffective for high voltages as well as
for large ac frequencies is a consequence of the competition of the (destabilizing) electrohydrodynamic
torques and the dielectric torques, which are stabilizing when e a 0
the homogeneous (q c = 0) Freédericksz transition competes with EHC [12] and in fact always
preempts the latter at sufficiently high frequencies. For a recent investigation see [95].
Actually the quantitative agreement between theory and experiment over a large frequency
range including the dielectric regime, suggested from Fig. 13.2a, is somewhat deceptive.
For the more common thinner specimens we are not always able to fit experimental
curves satisfactorily over the whole frequency range by adjusting the conductivity, the flexocoefficients,
and (to a lesser extent) the ratio s || /s k . When adjusting the conductivity so that
f c (and thereby essentially the whole conductive range) is described correctly, the dielectric
threshold tends to come out too low. One could increase the conductivity to fit the dielectric
threshold, which is roughly proportional to s k d 2 , but then f c becomes too large.
278

13.5 Electrohydrodynamic convection
This discrepancy is in line with other observed quantitative discrepancies such as in
the threshold behaviour when a stochastic component is present in the applied voltage [96].
As already mentioned, the most drastic non-standard behaviour, which is not understood
from the SM, is the observed Hopf bifurcation in sufficiently thin and clean specimens [18,
41–43, 49–51] and the very small hysteresis sometimes observed at threshold [38–40, 47].
In fact a little further above threshold (but still near it) the amplitude of the pattern does appear
to coincide reasonably with the results of the weakly non-linear theory [59].
Since the measurements on RBC exhibit such beautiful quantitative agreement with
theory one has concluded that the electrodynamic part of the basic equations needs improvement.
This was the motivation for introducing the WEM (Section 13.2). The linear stability
calculations for the conductive mode have been carried out within this model [48, 49] using
the same approximations which led to the analytic threshold formulas within the SM [12–
15]. It is found that there is an upward shift in the threshold, which may be quite small, and,
more importantly, a Hopf bifurcation with critical frequency
s
o H t d ˆ 2pf H t d ˆ Rc ~aC
1
1 ‡ o 02
…1 ‡ o 02 †l s t 2
d
R c ~aC
; …40†
if the expression under the square root is positive. Here R = V 2 s a t q
p 2 k 11
is the reduced control parameter,
~a 2 = m km + kg – 1 p 2 /(s a d 2 ) is proportional to the geometric means of the mobilities, and
o' = ot q b, with b = …s jj=s k † q 02 ‡p 02 ‡1
…e jj =e k
, is a reduced frequency.
† q 02 ‡p 02 ‡1
Moreover, l s =–[t –1 rec + t –1 d R~a 2 b/(1 + o' 2 )] < 0 is the damping rate of the (new) WEM
mode. Its dominant contribution is usually just determined by the ion recombination rate
1/t rec . Thus for the Hopf bifurcation to occur the quantity t d /(~at rec ) must not be too large.
This requires that the recombination of ions is sufficiently slow and that the layer is sufficiently
thin and clean. Note, the Hopf condition is always satisfied near the cut-off for materials
with negative dielectric anisotropy, where R c diverges at the cut-off frequency (in the
approximation used). But the Hopf frequency, which is then just given by the prefactor of
the square root from Eq. 40, becomes large there. This appears to be consistent with the experiments
[18, 41, 42]. Moreover, the prediction of the theory that for materials with vanishing
dielectric anisotropy the Hopf condition and Hopf frequency become essentially independent
of the external frequency has been verified experimentally using the material I52
[49]. I52 has the property that e a changes from negative to positive when the temperature
passes through T&60 8C.
13.5.2 Results of Ginzburg-Landau equation
The large aspect-ratios, which can be achieved comparatively easily in EHC, make this system
particularly well suited to test predictions of GLEs. Experimental results for the existence
and stability of normal roll patterns in the q-e plane (Busse balloon) are shown in
Fig. 13.7 [28, 36]. Similar experiments were reported by Ref. [29]. The symbols give the experimental
points and the curves represent parabolic fits for the neutral curve (N) and the
279

13 Convection Instabilities in Nematic Liquid Crystals
Defect Turbulence
ε
0.2
E
II
ZZ Pattern
0.1
0.0
–0.2 –0.1 0.0 0.1 0.2
q-q c
Figure 13.7: Experimental stability diagram (see text).
N
Normal Rolls
I
Eckhaus stability limit (E). The secondary and tertiary instabilities that limit the regions
from above will be discussed further below.
The wave number q can be controlled within certain limits on the small wave number
side by the so-called frequency-jump technique [32]. Since q c increases with external frequency
one can prepare first a state for the desired wave number by choosing the appropriate
frequency and then by jumping to the frequency and e values. In the course of the experiment
e can still be varied. Outside of the neutral curve the rolls decay whereas inside they
grow up to their non-linear saturation. The neutral point is then determined by extrapolation.
In this way also the GL relaxation time t was determined.
The coherence length x 1 in Eq. 37 can be obtained from the curvatures of the neutral
surface at the minimum. x 1 pertains to the direction parallel to the wave-vector of the pattern
and is therefore often denoted as x || . Both, x 1 and x 2 (x 2 = x k for normal rolls) have been measured
by imaging the core of defects (see below) [32]. In the same paper comprehensive measurements
on MBBA are presented for all parameters of the GLE as a function of external frequency
in the conductive range. The comparison with theory shows quite good agreement.
Another elegant method for measuring the linear parameters of the GLE makes use of thermal
fluctuations very slightly below threshold [38, 39, 97, 98]. Then the dynamic structure function
is exploited. The agreement with theoretical results [12] is generally, maybe except some
cases of thin layers, satisfactory including the predicted variations with ac frequency.
Slightly above onset the pattern should be dominated by the critical mode at q = q c
and according to the
p
weakly non-linear analysis (Section 13.3) the pattern amplitude should
grow proportional to e . This behaviour has been confirmed in experiments, where the optical
contrast of the roll pattern was monitored as function of e [59] using the shadowgraph
method [34, 38, 99]. The proportionality factor, which is determined by the non-linear (cubic)
coefficient g in the GLE, agreed satisfactorily with theoretical results [12].
In a next step the Eckhaus instability boundaries (curves E in Fig. 13.7), which probe
non-linear aspects of the system, could be identified by observing the destabilization of the
pattern via longitudinal modulations after a frequency jump [29, 36, 37] 7 . Subsequently the
7 The forcing of a pattern into a wave number state with q(q c can also be done by the use of digitized electrodes
and led to the first detailed investigation of the Eckhaus process in a remarkable experiment [27].
280

13.5 Electrohydrodynamic convection
pattern evolves into a state with wave number in the stable range by spontaneous creation of
defects either at the surface or in pairs. p The prediction of the GLE that the Eckhaus curve is
narrower by the universal factor 1/ 3 than the neutral curve, both in parabolic approximation,
has been confirmed. Actually also the full evolution process from an unstable to a
stable wave number can be analyzed with the GLE [100].
A particularly beautiful example for the usefulness of the GLE is the description of
the structure and dynamics of point defects (dislocations), see Fig. 13.1 b. A defect is characterized
by a zero of the amplitude A (x) where it behaves as |x| exp (±if), where f is the polar
angle and the topological charge is ±1 in the isotropic scaling. A stationary, isolated defect
solution exists only for a pattern with the background wave-vector q = q c , i. e. with
modulation wave-vector Q = q – q c = 0 in the GL description. Otherwise the defect moves
with constant velocity V perpendicularly to Q. Each defect crossing the system carries away
one periodic unit and brings the system nearer to its globally stable state Q=0(or q = q c )
by an amount 2 p/L, where L is the system length. Since the GLE has a minimizing potential
one can speak in such terms and the force on the defect is in fact the analogue of the Peach-
Kohler force in solids [101, 102]. Due to the two-dimensional nature of the problem V (Q) is
non-analytic at V=Q= 0. One has V log (V t/(3.29 x)) = 2 (x 2 /t)Q for small Q [88, 103].
The full universal curve V (Q) can be obtained numerically [14] and is compared with detailed
experimental results in EHC for motion of defects along the rolls [32] (Fig. 13.3a).
Also the interaction of defects, which is repulsive for equal topological charge and attractive
otherwise, was investigated and compared with experiments [33, 104].
The theoretical results discussed above pertain to the normal-roll regime with wave
number changes confined to the longitudinal component, i. e. the rolls remain normal. Some
rather qualitative experiments with transverse wave number changes, showed that defects
moved perpendicular to the rolls, as theoretically expected [30]. One also gets a neutral
curve and a (generalized) Eckhaus stability limit roughly as expected [19]. In the latter experiments
a magnetic field, applied in the plane of the layer at an oblique angle, was used to
influence q c .
13.5.3 Beyond the Ginzburg-Landau equation
From the foregoing discussion we know that the weakly non-linear behaviour of EHC in the
range of validity of the GLE is understood fairly well. The situation is not quite as satisfactory
when it comes to secondary bifurcations, which confine the q- range and which are captured
in the theory only if corrections to GLE are included (Section 13.3) or a full evaluation
is done to the hydrodynamic equations.
13.5.3.1 Experimental results
From the most recent experiments it is quite clear that, beginning with normal rolls at
threshold and increasing e, the ZZ instability, where long wavelength modulations with
wave-vector s = (0, s y ) perpendicular to q =(q, 0) start growing, provides the most impor-
281

13 Convection Instabilities in Nematic Liquid Crystals
tant destabilization mechanism, which sets in at quite low e (^0.2). This has been observed
for MBBA (Fig. 13.7) [29, 36, 37, 105], where there are usually normal rolls at threshold
for all frequencies. Furthermore, in Merck Phase V, where one has an LP at moderately low
frequencies [105], and in I52 too, where oblique rolls at threshold extend to higher frequencies
[50, 51]. Actually in the first clear observation of oblique rolls [106] they appeared after
a secondary bifurcation.
It also seems fairly clear now that under sufficiently ideal conditions – very homogeneous
large aspect-ratio system, slow increase of e – beyond the ZZ instability the system
can settle down in an (ideally) stationary oblique roll state. Usually zig and zag domains
seem to persist. For MBBA the angle of obliqueness remains small, below about 108. The
necessity to change sufficiently slowly has been stressed in particular by Ribotta [167].
Otherwise the system tends to remain in a dynamic defect turbulent state, which was called
fluctuating Williams domains by Kai and co-workers [19]. It is characterized by continuous
generation and annihilation of dislocation pairs modulating the (normal) roll pattern, see
e. g. the snapshot shown in Fig. 13.3b. Thus one seems to have two coexisting attractors.
The transition to normal roll defect turbulence takes place at e above e zz (diamonds in
Fig.13.7) [36, 37]. At even higher values of e a transition to a rectangular or grid pattern is
typically observed before the onset of the strong turbulence types with creation and annihilation
of disclinations in the director field [17, 62].
According to some measurements of (doped) MBBA the SV instability also plays a
significant role, particularly in the higher frequency part of the conductive range, where it
appeared to replace the ZZ instability as the first roll destabilization mechanism [29, 37].
When increasing e slowly beyond the SV instability the system settled down at first in an interesting
regular defect-lattice structure [37, 107, 108]. At higher values of e a transition to
defect turbulence again sets in. For an ac driving o around the crossover frequency o cr between
the two types of behaviour a (fairly) direct transition from normal rolls to defect turbulence
is observed. Below o cr
the SV instability manifested itself as transients in experiments
where e was increased suddenly, starting out from below the ZZ line. Whereas immediately
above the ZZ line the first destabilization took place via the ZZ instability as expected.
It changed over to a SV destabilization at higher values of e thus defining a SV line
in the q-e plane. With increasing o the SV line moved closer to the ZZ line and apparently
joined it at o cr . Although defect-lattice structures of the above type have apparently been
observed by other researchers [168], they did not characterize quantitatively the existence
range. Possibly their cells with undoped MBBA behaved somewhat differently.
We only mention in passing that various undulated structures have been reported by
Ribotta and co-workers [17, 105] but could not always be reproduced by others [169]. Actually
such structures can be obtained near the LP even within the GLE approximation, but
they then turn out to be metastable since their (generalized) potential is higher than that of
straight (usually oblique) rolls [14, 89, 90, 109, 110]. This may explain why they are difficult
to observe.
13.5.3.2 Theoretical results and discussion
The experimental scenario can be partly understood with the SM [24, 111, 112]. One finds
that the general shape of the measured ZZ instability limit in the q-e plane, shown in
282

13.5 Electrohydrodynamic convection
Fig. 13.7 (triangles), is reproduced by the theory in a rather robust fashion, whereas the actual
position depends crucially on the material parameters and is very sensitive to approximations.
Thus using standard MBBA parameters the ZZ instability comes out too high
within the order-parameter approach, which involves an expansion up to cubic order in the
pattern amplitude (Section 13.3) [24, 111]. One needs the full Galerkin computations
(Section 13.3), which lead to rather good quantitative agreement (triangles and broken line
in Fig. 13.8 a) [112]. We conclude that higher order terms in the amplitude become important
already at quite low values of e.
In any case corrections to the GLE approximation in particular the coupling of the
pattern amplitude to the mean flow [24] (Eq. 39), which is made explicit in the order parameter
approach, has the effect of reinforcing the ZZ instability with increasing amplitude, in
distinct contrast to the effect in isotropic fluids 8 . This results in a ZZ line roughly horizontal
in the q-e plane in the range where it determines the stability boundaries. A convenient characterization
of the ZZ line is its position at q = q c , which is called e zz . On the low-q side one
expects the ZZ line to extend to the neutral curve and join it at a point L, which is sometimes
called an LP on the neutral curve, because below normal and above oblique rolls become
critical on the neutral curve. This point can be calculated from the linear theory and is
also shown in Fig. 13.8a. Following the ZZ line through the Eckhaus unstable range on the
low q side there is a delicate numerical problem and the curve may exhibit rather unexpected
variations (Fig.13.8 a).
Figure 13.8: a) Stability diagram for normal rolls for a rather thick cell with MBBA at ot q = 0.2. Full
Galerkin computation. For details see text; b) Stability diagram for rolls escaping in the oblique direction
at fixed q=q c . The unstable bubble was tested by using Galerkin expansion with more terms (Z).
For details see text.
In any case, if decreasing the external frequency the LP on the neutral curve typically
moves down and may eventually cross the minimum at q c leading to oblique rolls at threshold.
This is the case for the materials Merck Phase V and I52. Then, simultaneously e zz
moves down to zero. Of course shifts of e zz and the LP on the neutral curve can also be effected
by changing the material parameters. Figures 13.9a,b give an impression of the de-
8 In technical terms it is the opposite sign of the coefficient q 4 of Eq. 39 which is responsible for the
different behaviour
283

13 Convection Instabilities in Nematic Liquid Crystals
0.08
0.06
ωτ 0
= 0.5
ε a
= –0.53
d = 55µm
0.070
ε ZZ
0.065
0.060
ε ZZ
–1.0 –0.8 –0.6 –0.4 –0.2 0.0
σ a
= 0.5
d = 55µm
0.04
0.02
0.055
0.050
0.045
0.00
0.1 0.3 0.5
σ a
0.7 0.9
Figure 13.9: Onset of the ZZ instability at band centre as a function of (a) the anisotropy of the conductivity
s a and (b) the anisotropy of the dielectric constant e a .
0.040
ε a
pendence of e zz on the anisotropies of the conductivity and the dielectric tensor. The trend
shown in Fig. 13.9b is in qualitative agreement with experiments when comparing MBBA
to Merck Phase V and I52, which have e a & –0.5, –0.3, 0, respectively. 9
The mean flow also has the effect of transforming the Eckhaus instability on the large
q side into a SV instability. This effect becomes noticeable only in the upper part as the ZZ
line is approached because at smaller e the ratio s y /s x is very small, see the broken line in
Fig. 13.8a. In fact the SV instability then turns around and joins the ZZ line smoothly. This
effect indicates that the SV instability may become relevant and could be a clue to the observations
quoted above [29, 37, 108].
In order to show that the system can escape at e zz into an oblique roll state, as is actually
observed, the stability of rolls with q, p&0 was investigated. An example of the stability
diagram for MBBA is shown in Fig. 13.8b, where q is fixed at q c . The point ZZ on the
axis corresponds to e zz . It can be seen that the smallest roll angle arctan ( p/q c ), where rolls
are stable, first increases with increasing e but saturates at about 88. This is in agreement
with experiments [36, 37, 113]. With increasing e the minimal angle decreases again and,
surprisingly, beyond the point denoted by SV normal rolls could restabilize. It is interesting
to note that the order parameter approach gives the ZZ destabilization qualitatively correct
but fails to reproduce the restabilization of normal rolls [24]. We conclude that at e zz a forward-type
bifurcation to oblique rolls can occur as has been observed.
From the curve CR in Fig. 13.8b we see that at larger e a short wavelength instability
comes into play, i. e. a roll system with a different wave-vector, particular in different orientation,
starts growing. This may saturate the often-observed rectangular patterns or, for a
non-symmetric superposition, lead to the sometimes-observed oblique modulated structures
[17, 105]. In order to examine this possibility one would have to test the stability of such
patterns by a suitable Galerkin procedure, which was done for normal rolls. However, since
there are other possibilities, in particular turbulent states, this approach is not exhaustive
and has to be complemented by simulations of the dynamics.
9 Recently most material parameters of Merck Phase V have been measured, see note added at the
end. The material parameters of I52 have either been measured directly or fitted to EHC measurements
[49].
284

13.5 Electrohydrodynamic convection
Unfortunately, solving the full hydrodynamic equations is impossible. Therefore the
order parameter approach, applied here, becomes useful. As mentioned before, this approach
[24] (Section 13.3) captures all features of the secondary bifurcation scenarios found in rigorous
calculations, though the ZZ destabilization line comes out too high for realistic material
parameters. This may be corrected by using a larger value of s a /s k . When formulated
in real space one is led to coupled amplitude equations which have been simulated numerically
[21, 24] 10 . Below the ZZ line the system approaches rapidly the stable normal roll attractor
when starting from random initial conditions. Beyond the ZZ line the situation becomes
complicated. Starting slightly above the ZZ line, from small random initial conditions,
one could either end up in a stationary zigzag pattern (Fig. 13.10 a) for s a /s k , e = 0.1,
and ot q = 0.5) or in states with shorter and more sinusoidal undulations (Fig. 10 in
Ref. [24]). Then a few slowly moving defects appeared to persist. There is no clear-cut separation
between undulations and zigzag, analogue to the situation near the LP without the
mean flow mentioned above.
For larger e (e = 0.2), on the other hand, the pattern evolves into a state similar to the
one shown in Fig. 13.10 b. This is a time-dependent state with alternating zigs and zags
where defects are continuously generated in pairs and is reminiscent of some cases of the
observed defect turbulence, although in the experiments the rolls often tend to be more
aligned in the normal direction. This discrepancy could be an artefact of the order parameter
approximation, where normal rolls do not restabilize at higher e (see above), so the roll orientation
remains more oblique. The generation and annihilation of defects can be understood
from an advection of the roll pattern by the mean flow, which amplifies small undulations.
Because the anisotropy counteracts the bending of rolls the stress is released by
straightening the rolls and dislocations are left behind.
The situation is reminiscent of Rayleigh-Bénard convection in isotropic fluids where
stable roll attractors apparently compete with complex patterns, spiral defect turbulence
[115–117]. It has been shown very recently that if anisotropy is introduced into this system
by inclining the convection cell, a normal roll pattern with dislocation defect turbulence occurs,
which looks quite similar to patterns observed in EHC [118].
Figure 13.10: (a) Stationary zigzag pattern for e = 0.1. (b) Snapshot of a defect turbulent zigzag pattern
at e = 0.2.
10 Similar results have been obtained from quite simple model equations if certain parameters are adjusted
ad hoc [25, 114].
285

13 Convection Instabilities in Nematic Liquid Crystals
13.6 Concluding remarks
Clearly this review could not be comprehensive, so we first list here some of the omitted
material. Recently some experiments in planarly aligned samples with an additional destabilizing
magnetic field in the z-direction (parallel to the electric field) were performed [119,
120]. Some earlier work is discussed in [60]. For not too high magnetic fields the EHC
threshold is merely reduced in accordance with the standard theory. For fields that are larger
than a critical value, however, the Freédericksz transition comes first and then EHC sets in
as a secondary instability, similarly to case F in homeotropic alignment, see below. The voltage
threshold should then increase with increasing field, because destabilization of the
Freédericksz distorted state becomes increasingly difficult, which is in qualitative agreement
with experiments. Also the change to a subcritical bifurcation appears to be born out by theory
[121].
In the experiments with I52 oblique travelling rolls were observed. In this four-wave
scenario (left-right and zigzag) a number of superpositions are possible, depending on the
non-linear coefficients [93]. The left and right travelling roll systems always appear to separate,
whereas in some parameter range the zigs and zags make a transition from separation
to coexistence [51]. However, the superposed state is disordered (weakly turbulent). One can
show theoretically that just beyond this codim-2 point a Benjamin-Feir-type modulational instability
(Section 10.3) becomes generic, depending only on the sign of coefficients and not
on their magnitude [122]. Presumably the above experiments were done in that parameter
range.
EHC is particularly suited to apply space or time-modulated forcing. The effect of
spatially periodic forcing of stationary patterns in a quasi one-dimensional situation was studied
by using structured electrodes [27, 123]. In particular, commensurate-to-incommensurate
transitions were observed as predicted by theory based on phenomenological amplitude
equations [124, 125]. Additional two-dimensional effects in spatially forced isotropic systems
were studied theoretically [126]. Experiments appear to be lacking at this time. The effect
of resonant time-periodic forcing of travelling (normal) rolls was first discussed theoretically
using normal-form equations [127, 128] and was then confirmed experimentally [43].
Interesting scenarios including a transition to standing, oscillating rolls can be induced in
this way. Similar effects for travelling oblique rolls, where one has four-wave mixing, have
also been considered [47, 129]. The effect of stochastic driving was investigated theoretically
[96, 130] and experimentally [131, 132].
Some statistical properties of defect turbulent states in EHC were studied experimentally
[42, 133, 134] and, as pointed out before, the role of thermal fluctuations slightly below
threshold was measured and analyzed. Of the numerous investigations of far-off threshold
effects we only mention the study of phase waves in the oscillatory bimodal state [135, 136]
and the transition between strongly turbulent states [60]. 11
Besides the case of planar surface alignment of the director, considered here, one can
have homeotropic alignment where by appropriate treatment of the confining plates the di-
11 Very recently a multifractal analysis of electroconvective turbulence has been performed [170].
286

13.6 Concluding remarks
rector is oriented perpendicular to the plane. Then the system is isotropic in the plane of the
layer and therefore from a symmetry point of view more related to convection in simple
fluids. Both, RBC and EHC, have recently been investigated theoretically using the methods
described before [81, 137, 138]. In RBC, when heated from below, a subcritical Hopf bifurcation
is predicted, which at high stabilizing magnetic fields first transforms into a stationary
bifurcation and subsequently becomes supercritical. The results are consistent with experiments,
mostly without visualization of the patterns, by Guyon et al. [139]. One also obtains
convection when heating from above leading to stationary squares. Application of a
weak planar magnetic field renders the system anisotropic and then one has rolls in a small
range above threshold. Experiments were performed in [140].
In homeotropic EHC one has to distinguish two cases. For materials with positive or
only very slightly negative dielectric anisotropy, like e. g. the newly introduced material I52
in an appropriate temperature range [49–51], theory predicts a direct transition to convection
(case C) in the form of stationary squares, except at large e a where one finds rolls. The
linear results can be described in good approximation by an analytic threshold formula
[138]. For materials with more negative e a , like MBBA, one has first a bend Freédericksz
transition (case F) leading to a planar component of the director, which (ideally) singles out
spontaneously a direction in the plane. Subsequently one has a transition to EHC, which on
the linear level exhibits similar scenarios as in the planar case. Some experimental results
have been reported in [107]. The tendency towards oblique rolls is more pronounced in the
homeotropic case, and in fact MBBA should show oblique rolls at low frequency. Experiments
without [141] and with a stabilizing magnetic field [142] have verified these predictions.
On a deeper (non-linear) level there is, however, a very essential difference that makes
this system rather unique. The direction singled out by the Freédericksz transition in the
plane of the layer is the result of a spontaneously broken symmetry in the absence of an applied
planar magnetic field. This makes itself known in experiments, of course, by the fact
that the planar component is not really uniform over the cell but varies slowly in space being
pinned by inhomogeneities. For a description (under ideal conditions) one has to couple the
EHC mode from the beginning on with the Freédericksz Goldstone mode. For normal rolls
this situation may often not be very important, although it could lead to a destabilization of
the pattern from the very beginning. But for oblique rolls there is certainly a drastic consequence.
One easily understands that oblique rolls exert a mean torque on the director via the
velocity field. This torque cannot be balanced. So the director has to react by a rotation
which in turn acts back on the rolls reducing their amplitude. Clearly there cannot be a static
state even under ideal conditions and indeed in the experiments a pattern with slow dynamics
shows up [141]. The weakly non-linear theory is being worked out now. Hopefully
there will be more experiments to study the transition between order and chaos as the magnetic
field is decreased and a better characterization of the chaotic state.
Besides the planar and homeotropic alignment there is also the possibility of hybrid
anchoring, i. e. planar on one side, homeotropic on the other. Then, in the basic state the director
bends by p/2 when going from one plane to the other and there are in fact two symmetry
equivalent directions for the bend to occur. Usually the preparation procedure of the
probe, i. e. the filling with the LC, will single out one direction. Thus the basic state is not
reflection symmetric and the pattern is expected to drift. Recently, this has been shown theoretically
[143]. Under dc driving drift can also be imposed by non-ideal planar alignment,
287

13 Convection Instabilities in Nematic Liquid Crystals
i. e. by a pretilt [144, 145]. Indeed a small pretilt is hard to avoid. There is some similarity
to systems with an externally imposed drift that has been studied recently in Taylor vortex
flow in simple fluids [146, 147].
Finally we point out that the uniaxial symmetry of planarly oriented cells can be perturbed
in at least two ways: by applying a planar magnetic field at an oblique angle to the
alignment direction, which has been utilized to some extent in [19], or by imposing a twist
on the director by having different planar alignment directions on the two plates. In this case
one also has to break reflection symmetry around the midplane of the layer, which may be
done by applying a dc voltage (the flexoelectric effect is linear in the field). As a consequence
one loses the sharp separation between normal and oblique rolls and in fact the roll
direction should turn smoothly by changing any parameter, e. g. the voltage. This has indeed
been demonstrated experimentally [148]. Actually, twisted cells have interesting properties
also under ac excitation [149]. We point out that twisted cells are employed in the most
common type of LC displays. But the materials used there have strongly positive dielectric
anisotropy that leads to a Freédericksz transition instead of EHC.
Although during the last 12 years much progress could be achieved, there still remain
many open problems. For planar alignment there exist some quantitative discrepancies with
the SM which are possibly not explained by the WEM. It is also not yet clear if that model
will explain the (very weakly) subcritical bifurcation, observed under some conditions. But
this should be resolved in the near future.
A different source of space charges is operative in the very beautiful electroconvection
experiments in thin freestanding smectic films [150, 151]. Here the free surface charges and
the inhomogeneity of the applied field certainly play a crucial role. Very recently a linear
and weakly non-linear theory has been worked out [152].
Acknowledgements
We have benefited from discussions with G. Ahlers, A. Buka, W. Decker, M. Dennin,
A. Hertrich, S. Rasenat, I. Rehberg, A. Rossberg, H. Richter, M. Treiber, and W. Zimmermann.
A. Buka, I. Rehberg, H. Richter, and A. Rossberg have also kindly provided graphs
of their results. We are grateful to W. Decker, A. Hertrich, and F. Schmögner for help in preparing
the manuscript. Financial support by Deutsche Forschungsgemeinschaft (Sonderforschungsbereich
213) is gratefully acknowledged. L. Kramer wishes to thank the Center
Emile Borel, Institut Henry Poincare, and the ECM, Universidad de Barcelona, and W. Pesch
the LASSP, Cornell University, where part of this work was performed, for their hospitalities.
288

Note added
Note added
We here list some recent developments in EHC. Firstly, on the level of the SM, the weakly
non-linear theory for homeotropic surface alignment in the usual case F, where one first encounters
a bend Freédericksz transition, has been worked out [153, 154]. One obtains in the
normal roll regime one GLE for the patterning mode (two equations in the oblique roll
range) coupled to a dynamic equation for the Goldstone mode that signifies the broken continuous
symmetry. In the normal roll case the equations essentially scale uniformly in e. It
turns out that in the absence of any external symmetry breaking (no stabilizing magnetic
field) no stable periodic roll solutions exist and one seems to have a disordered state in all
cases. Theory suggests that the state is always dynamic, and the type of spatio-temporal defect
chaos has been termed soft-mode turbulence [155, 156], some experiments appear to
show in the normal roll range for very small e static (frozen) disorder [141, 142]. The destabilization
of normal rolls is brought about by the important fact that, although the growth
rate is maximal for the rolls oriented perpendicular to the (undistorted) director, they exert
an abnormal torque around the z-axis on the director that amplifies fluctuation-induced misalignments
away from the normal orientation. The abnormal torque is a signature of the
non-variational nature of the system, even at onset. In the presence of a (weak) stabilizing
magnetic field H there are stable rolls up to e c *H 2 . For normal rolls the destabilization occurs
in a frequency range o L < o < o AR at e ZZ by a ZZ instability, and for higher frequencies
by a spatially homogeneous transition (in the x-y plane) at e AR to abnormal rolls, where
the director is rotated away from its normal position, either to the left or to the right. The
term abnormal rolls has been introduced earlier in the context of experiments without magnetic
field [166]. The rotation is again a manifestation of the abnormal torque. The abnormal
rolls destabilize at 1.5 e c (in the weak-field limit) and then one has defect turbulence.
Interestingly, at even higher values of e, the coupled GLEs describe in a substantial parameter
range a new spatial superstructure where defects of equal topological charge tend to
assemble along chains aligned parallel to the magnetic field. The chains form a periodic
stripe pattern and the topological charge alternates from chain to chain. Between the chains
the rolls (and the director) are rotated alternatively to the right and to the left. The roll angle
is related to the defect density along a chain by a simple topological constraint. These structures
are reminiscent of the chevrons, observed for more than 25 years, in the dielectric range
of EHC in planarly aligned cells – for photographs see any of the standard textbooks [57, 65,
66] – and it appears that a similar mechanism is operative there. The formation of chevrons
can be modelled as a Turing-like instability arising in the defect turbulent state [157].
Secondly, it has been realized recently that also in the conductive range of planarly
aligned cells the transition to abnormal rolls occurs generically at e = e AR * 0.1 [158]. As in
the homeotropic case the transition occurs at high frequencies in the stable range and can be
preceded for lower frequencies (o L < o < o AR ) by a ZZ instability. In contrast to the (nearly)
isotropic homeotropic case one finds for o < o AR restabilization of abnormal rolls above a
certain e ARstab . Thus, in the oblique roll range and in the normal roll range below o AR normally
oriented rolls reappear as abnormal rolls. The restabilization had been found before in
the full Galerkin computations, described in Section 13.5.3 (Fig. 13.8b), without understanding
the physical basis. Although in planarly aligned systems abnormal rolls cannot be distin-
289

13 Convection Instabilities in Nematic Liquid Crystals
guished from normal rolls by the usual optical techniques, there exists indirect evidence for
their occurrence [158]. Increasing e further, the abnormal rolls are destabilized by a short
wavelength bimodal varicose instability that transforms at large frequencies into a long wavelength
instability of the SV-type.
Finally, we mention some recent progress in the application and analysis of the WEM
model. The linear analysis was shown to describe quantitatively recent systematic measurements
on Merck Phase V of V c , q c , and the Hopf frequency o c as a function of o for cells
of different thickness [159]. This agreement is particularly noteworthy because it is based on
recent measurements of the viscosities a 2 – a 6 [160] as well as on independent measurements
of the elastic constants. The only remaining SM parameter a 1 was fitted to give the correct
Lifshitz frequency. So it appears that the WEM can describe at least the three systematically
studied materials MBBA [161], Merck Phase V, and I52 [49]. Meanwhile it has also become
clear that the WEM model can explain the weakly subcritical nature of the bifurcation
observed under some conditions in MBBA and Merck Phase V [38–40, 47] and, more recently,
also in I52 [162–164]. Whereas the results for I52 show that the bifurcation is subcritical
only in the stationary regime near the crossover to the Hopf bifurcation, in agreement
with the predictions [165], one observes hysteretic behaviour in the other materials
also in the Hopf range. This can be understood in terms of the subcritical stationary branch
persisting in the non-linear range even in the region where the Hopf bifurcation precedes
slightly the stationary one.
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294

14 Preparation and Properties of Ionic and Surface
Modified Micronetworks
Michael Mirke, Ralf Grottenmüller, and Manfred Schmidt
14.1 Introduction
In recent years micronetworks have gained increasing interest in academic and industrial research.
Like dendrimers microgels could principally serve as molecular containers in order
to transport guest molecules (drug delivery) or as microscopic reaction sites for the formation
of nanosized colloidal particles.
In the present work we have
a) investigated the preparation and surface modification of microgels via crosslinking reactions
in microemulsion;
b) utilized microgels as the core in core-shell and star-like structures;
c) prepared and characterized ionic micronetworks.
14.2 Polymerization in normal microemulsion
14.2.1 Mechanism and size control
Antonietti et al. [1] and Wu [2] have reported on the size control of the polymerized microemulsion
in the system: styrene, di-isopropenylbenzene (cross-linker), cetyltrimethylammoniumchloride
(CTMACl) or a similar surfactant, and water. According to a simple model
developed by Wu [2] the resulting latex size is a function of the fleet ratio s = m s /m M with
m s and m M , the masses of surfactant and monomer respectively, given by
S 1 ˆ NA r
3M s
a 0 R ‡ C …1†
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
295

14 Preparation and Properties of Ionic and Surface Modified Micronetworks
with N A the Avogadro number, M s the molar mass of the surfactant, a 0 the surface area per
surfactant molecule, r the mean particle density, and the constant [3]
C ˆ NA r
3M s
a 0 2d 1 …2†
with d the surfactant microgel interpenetration depth.
Whereas Eq. 1 is almost perfectly confirmed by the experimental data of Antonietti et
al. and Wu, the present data do not follow Eq. 1 and show much more scatter than literature
results (Fig. 14.1). Variation of the ionic strength of the continuous phase, of the reaction
temperature, of the total monomer content, and of the concentration of crosslinking agent
has essentially no effect on the resulting microemulsion.
6
5
4
S -1
3
2
1
0
-1
0 5 10 15 20 25 30 35 40 45 50 55 60
R h /nm
Figure 14.1: Inverse fleet ratio S –1 versus size R h for AIBN initiated, polymerized latices: data from
Ref. [1] (_), Ref. [2] (M), and present results (y).
As shown in Fig. 14.2 differences, however, were observed for different initiators, e. g.
AIBN, redox system (K 2 S 2 O 8 /K 2 S 2 O 5 ), and dibenzylketone. Although much effort was taken
to improve the reproducibility and size control we could never reproduce the literature
data for AIBN initiation particularly at small S. We presently cannot offer a satisfactory explanation
for this discrepancy.
Concerning the mechanism of particle formation the polymerization starts in a two
phase or in a monophasic region of the phase diagram depending on the total monomer concentration
and on the fleet ratio [4]. The biphasic regime is easily detected by a bimodal decay
of the time correlation function g 1 (t) as recorded by a dynamic light scattering instrument
whereas for the monophase an extremely fast, monomodal decay curve is observed.
The hydrodynamic radius R h , deducted from the measured diffusion coefficient, is less than
1 nm which is – within experimental error – identical to the radius of empty micelles formed
by the pure surfactant in water at the respective concentration. It is tempting to interpret
these results in terms of a bicontinuous monophase rather than in terms of monomer-filled
microdroplets. Since similar investigations are currently performed in Hoffmann’s group [5],
296

14.2 Polymerization in normal microemulsion
6
5
4
S -1
3
2
1
0
-1
0 5 10 15 20 25 30 35 40 45 50 55 60
R h /nm
Figure 14.2: Inverse fleet ratio S –1 versus size R h for differently initiated polymerized lattices: AIBN
(y), redox (o), and photo initiation by dibenzylketone ($).
we have not further persued the phase diagram of the unpolymerized emulsion but concentrated
on the changes during polymerization.
As shown in Fig. 14.3, the bimodal decay of the correlation function has merged into a
single, almost monoexponential curve after 90 min reaction time, which does not change with
monomer conversion any more. The final latex size is already reached after 90 minutes or at
about 50 % monomer conversion! Again, this observation is not easily understood, because it
means that polymerized particles do not grow beyond a certain size and new particles are
formed even during the later course of reaction. Surface tension measurements as function of
the reaction time and as function of the fleet ratio only show the well-known surface activity
of styrene as compared to polystyrene, which is still significant at 50 % monomer conversion.
At this point also the rate of conversion exhibits a maximum as function of time which is
1.0
0.5
g 1 (t)
0.1
0 100 200 300 400 500 600 700 800 900
t/msec
Figure 14.3: Time correlation function g 1 (t) for AIBN initiated latices at different polymerization
times: 0 min (T), 30 min (y), 90 min (o), 150 min (M), and 2 days (6).
297

14 Preparation and Properties of Ionic and Surface Modified Micronetworks
usually interpreted in terms of a two stage mechanism. At the beginning of the polymerization
more and more growing particles are formed until a certain number of particles is reached.
Then the reaction rate decreases because the monomer concentration in the system decreases
or the number of active polymerization sites is diminished. In view of the discussion above,
the hypothesis of the constant number of growing particles appears questionable.
Qualitatively similar results are obtained for emulsions which were initiated by light
or redox initiators. Despite the extreme lack of understanding and reproducibility in most
cases the resulting microemulsions were monodisperse and spherical. Therefore they are
principally well suited as starting material for surface-functionalized spherical particles.
14.2.2 Surface functionalization of microgels
A suitable functional group on the surface of the microgel could be an azo-moiety which
can be utilized to start a grafting from reaction in order to synthesize core-shell or star-like
structures.
Here, we have taken advantage of the cosurfactant properties of the methacrylic acid
ester (Fig. 14.4) which is simply added with DTMACl to the reaction mixture. During the
polymerization the methacrylic group is bound into the microgel thus forming the pendent
azo group. Since we wish to preserve the azo group during microgel formation the microemulsion
was redox initiated. Redox initiation with the system K 2 S 2 O 8 /K 2 S 2 O 5 introduces ionic
SO – 3 groups to each primary chain, thus creating an ionic micronetwork. In fact, increasing
the initiator to monomer ratio results in an increased conductivity of the microgels dissolved
in DMF.
O
O
N N CN
CH CN
3
Figure 14.4: Cosurfactant containing the azo-moieties methacrylic acid [4-(1,1-dicyanoethyl)azo]benzylic
ester.
The presence of ionic groups makes the particle characterization somewhat ambigious,
since now interparticle interactions might seriously influence the interpretation of the
light scattering data. One example is shown in Tab. 14.1, where the measured apparent radius
of gyration R g,app of the latex in water changes from negative values to about 17 nm with increasing
salt concentration, whereas the hydrodynamic radius R h remains approximately constant
at about 8.5 nm. This large ratio of R g /R h = 2 is not compatible with the value of
0.775, expected for a spherical structure.
Also, most of the redox-initiated structures do not dissolve in toluene after careful removal
of the surfactant and the solution characterization was performed in DMF. Again, the
experimental R g /R h values were larger than 1.5 indicating that no spherical structures were
298

14.3 Polymerization in inverse microemulsion
Table 14.1: Influence of added salt on the light scattering results of redox polymerized latices.
c (NaCl)/mol/l R g,app /nm R h,app /nm R g /R h
0 –10.5 8.1 –
4 610 –3

14 Preparation and Properties of Ionic and Surface Modified Micronetworks
Br
O
CH 3 CH 3
O
N
+
O
O
CH 3
N
CH 3
O
Monomer I
CH 3 CH 3
CH 3 Br
O
N Br -
+ CH 3
Monomer II
O
O
O - Na + + O S
O
O
O SO 3 - Na +
Monomer III
O
O
N
H
OH
N
H
O
Crosslinker I
Figure 14.5a: Structure of monomers and cross-linking agents employed in this study: methacrylic
acid-[ethyl-dimethyl-benzyl-ammonium bromide]ester (Monomer I),: methacrylic acid-[trimethylammonium
bromide]ester (Monomer II), methacrylic acid-[propyl-3-sodium sulfonate]ester (Monomer III),
bis-(acrylamido)acetic acid (Crosslinker I), and 1,2-dihydroxyethylene-bis-acrylamide (Crosslinker II).
O
N
H
OH
OH
H
N
Crosslinker II
O
O
HO
HO
HO
O
O
O
O
y
O
z
O
x
O
O
Tween 80
O
HO
O
HO
O
Span 20
HO
Figure 14.5b: Structures of surfactants employed in this study: polyoxyethylene(20)-sorbitane-monooleate
(Tween 80), sorbitane monolaurate (Span 20).
In Fig. 14.6 the measured apparent particle radii are shown as function of the ionic
strength. The ionic strength dependence does not clearly emerge from the plot because above
a concentration of 0.01 m salt the particles start to aggregate and eventually precipitate, and
below 0.001 m salt electrostatic effects prohibit a proper size evaluation, as will be discussed
in detail below. It appears, however, that due to the high cross-linking density the particles
300

14.3 Polymerization in inverse microemulsion
Table 14.2: Characterization of the ionic microgels in aqueous 0.01 m KBr solution.
S –1 M w 710 6 g mol –1 R g /nm R h /nm R g /R h Monomer 1)
0.3 3.97 2) 16.8 18.2 0.92 I
0.3 1.48 12.9 16.4 0.78 II
0.3 1.32 18.7 21.8 0.85 III
0.35 3.97 2) 13.5 18.9 0.71 I
0.35 1.44 16.0 18.3 0.87 II
0.35 1.84 23.2 23.8 0.97 III
0.4 5.99 2) 14.9 21.1 0.71 I
0.4 1.45 13.5 18.5 0.73 II
0.4 1.83 20.3 22.7 0.89 III
0.45 6.26 2) 15.4 21.9 0.70 I
0.45 2.22 15.4 21.2 0.73 II
0.45 2.27 20.3 25.6 0.79 III
0.50 7.42 2) 16.0 23.2 0.69 I
0.50 2.76 15.8 19.8 0.80 II
0.50 3.69 22.9 27.8 0.82 III
0.55 10.2 2) 17.8 24.7 0.72 I
0.55 5.45 25.4 25.6 0.99 II
0.60 15.3 2) 20.0 25.4 0.79 I
0.60 5.33 22.8 25.0 0.91 II
1) Monomers as shown in Fig. 14.5: I: 15 mol % crosslinker I
II: 10 mol % crosslinker II
III: 10 mol % crosslinker I
2) Approximate molar masses measured in methylformamide solution with an estimated dn/dc = 0.1 cm 3 /g
radius [nm]
70
60
50
40
30
20
10
0
0.0001 0.001 0.01 0.1
log ( c (KBr)
[mol/l ] )
Figure 14.6: Ionic strength dependency of the microgel dimension for monomer II with 10 wt% of
crosslinker I: R h (y), R g (T).
do not swell much. This is a bit unexpected since conductivity measurements have indicated
the salt concentration within the microgel to be in the order of 10 –1 m. Although this largely
reduces the electrostatic expansion of the chains between the crosslinks, it induces an enormous
osmotic pressure of water towards the inside of the microgel which should cause a
301

14 Preparation and Properties of Ionic and Surface Modified Micronetworks
0.05
0.04
η sp /c [l/g]
0.03
0.02
0.01
0.00
0 1 2 3 4 5
c[g/l]
Figure 14.7: Reduced viscosity of the anionic microgel (monomer III, 10 wt% crosslinker I) with molar
mass M w = 4.97610 6 g/mol. Salt-free water (y), 0.01 m potassium bromide solution (T).
strong swelling. This point is still under investigation along with peculiar interparticle, electrostatic
interaction effects affecting the solution properties of the micronetworks, as already
discussed above.
Here, we have investigated the interparticle effects in more detail. In Fig. 14.7 the reduced
viscosity Z sp /c of the sulfonated microgel is shown as function of the microgel concentration
c in pure water and in 0.01 m salt solution. For the salt-free solution a maximum
in the reduced viscosity is observed, as already reported for linear polyelectrolytes and for
charged spheres [7]. In Fig. 14.8 the reduced scattering intensities are plotted versus q 2 ,
where q is the scattering vector, in a concentration regime similar to the viscosity measurements.
The filled squares represent the measurements in 0.02 m salt solution and yield the
true radius of gyration R g = 24.5 nm. This value is never observed in salt-free solution, even
if the microgel concentration is chosen as low as 10 mg/ml, where R g = 21.0 nm is evaluated.
It is to be noted that the size of the microgels is expected to increase with decreasing
salt in contrast to the measured apparent values. Increasing the concentration to larger than
0.2 mg/ml R g,app vanishes below the measurable value of 10 nm. This behaviour can be only
interpreted in terms of the interparticle structure factor. The observed linearity of the scattering
curves is somewhat surprising and suggests that the structure factor S (q) is able to precisely
compensate the angular dependence of the intraparticle formfactor P (q), since for
spheres
K c
ˆ 1 1
R q M w P…q†S …q† :
…3†
In the limit of small q, P (q) = 1 – 1/3 R g 2 q 2 which leads to
S (q) =S (0)/(1 – x 2 q 2 ) (4)
if the scattering curve remains linear in q 2 as observed in Fig. 14.8. For large concentrations,
obviously x 2 = 1/3R g 2 , which yields the angular-independent scattering intensity.
302

14.4 Conclusion and relevance to future work
K·c/R Θ
[mol/g]
3.6·10 -7
3.4·10 -7
3.2·10 -7
3.0·10 -7
2.8·10 -7
2.6·10 -7
2.4·10 -7
2.2·10 -7
2.0·10 -7
0 2.0·10 10 4.0·10 10 6.0·10 10
q 2 [cm -2 ]
Figure 14.8: Light scattering results of the microgel described in Fig. 14.7 in 0.02 m potassium bromide
solution (solid squares, c = 0.02g/l). Further in salt-free water at different concentrations for the
microgel in g/l: 0.017 (_), 0.026 (E), 0.1 (l), 0.20 (:), 0.47 (o), 0.55 (M), 1.02 (y).
These experiments support the recent claims [7] that the conformational change of the
polyions is not responsible for the anomalous viscosity behaviour. The observed maximum
in the reduced viscosity is most likely caused by a maximum in the electrostatic interaction
energy. Light scattering experiments at concentrations larger than the concentration of the
highest viscosity are currently performed and will provide a more detailed insight into such
complex electrostatic phenomena.
14.4 Conclusion and relevance to future work
Despite a big lack of understanding polymerization in microemulsion leads to model micronetworks
which are easily surface-functionalized to radical macroinitiators or to ionic microgels.
The macroinitiators have been utilized to obtain core-shell or star-like structures.
The ultimate goal of the investigations on ionic microgels is the preparation of a coreshell
structure via electrostatic interaction: a cationic microgel is mixed with a telechelic
oligo or polystyrene bearing one sulfonic acid group at one chain end. Since complex formation
between the oppositely charged species will occur the ionic microgel will be sur-
303

14 Preparation and Properties of Ionic and Surface Modified Micronetworks
rounded by an unpolar polystyrene shell which is dense enough in order to provide perfect
solubility in unpolar solvents such as toluene. Here, the microgel as a molecular container is
locked because water (or any other polar species) is prevented to diffuse out of the microgel
by the polystyrene layer. Again, in preliminary experiments the principle is shown to work
perfectly. Any quantitative work, however, has to be subject of more extensive work in the
near future.
References
1. M. Antonietti, W. Bremser, D. Müschenborn, Ch. Rosenauer, B. Schupp, M. Schmidt: Macromolecules,
24, 6636 (1991)
2. C. Wu: Macromolecules, 27, 298 (1994)
3. C. Wu: Macromolecules, 27, 7099 (1994)
4. V. H. Pérez-Luna, J.E. Puig, V.M. Castano, B.E. Rodriguesz, A.K. Murthy, E.W. Kaler: Langmuir, 6,
1040 (1990)
5. H. Hoffmann, G. Sühler: Vortrag anläßlich des Abschlußsymposiums zum BMBF-Projekt „Mesoskopische
Polymer-Systeme“, Frankfurt (1995)
6. F. Candau, J.M. Copart: Colloid Polym. Sci., 271, 1055 (1993)
7. S. Förster, M. Schmidt: Adv. Polym. Sci., 120, 51 (1995)
304

15 Ferrocene-Containing Polymers
Oskar Nuyken,Volker Burkhardt, Thomas Pöhlmann, Max Herberhold,
Fred Jochen Litterst, and Christian Hübsch
15.1 Introduction
Classical organic polymers consist of carbon and hydrogen. Some of them contain beside C
and H additional atoms such as N, O, Si, S, P, or halogens. A relatively small number of publications
deal with metal-containing polymers despite the fact that more than 40 metals are
available [1]. However, metal-containing polymers have emerged as an important class of
polymers in the last twenty years. One reason for this development is that many of them became
available via conventional synthetic routes [2]. Further motivation for developing these
materials is based on the premise that polymers are expected to posses properties significantly
different from those of conventional organic polymers and that they show electrical conductivity,
magnetic behaviour, thermal stability, non-linear optical properties etc. [3–5].
This field of activity can be seen as crossing the conventional boundaries of organic,
inorganic, and polymer chemistry as well as physics and material science.
The aim of our work was the synthesis of tailor-made polymers containing ferrocene
units at different positions of the polymer chain.
From Mößbauer spectroscopy of ferrocene-labelled polymers one could expect valuable
information about the polymer structure, diffusion, and fluctuation processes. In earlier
experiments we have been able to show remarkable differences in Mößbauer spectra taken
from solvent-free and swollen polymers [6].
Basically there are two strategies for the synthesis of metal-containing polymers. One
approach is to synthesize derivatives of organic polymers. The second pathway involves the
conversion of monomeric ferrocenes into polymers. We have concentrated on the latter approach.
A. Polymers with pendent ferrocene units
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
305

15 Ferrocene-Containing Polymers
B. Polymers with ferrocene units in the main chain
C. Polymers with ferrocenyl end groups
15.2 Addition polymers
Vinylferrocene (VFc) was synthesized in 1955 and its polymerization behaviour has been
studied under radical [7, 8], cationic [9], and Ziegler-Natta conditions [9]. It was claimed
that VFc is inert to anionic initiation [10].
15.2.1 Radical polymerization
VFc undergoes oxidation with peroxide initiators for which azoinitiators have been used almost
exclusively. Unlike most vinyl monomers the molecular weight of poly(vinylferrocene)
(PVFc) does not increase with the decrease of the initiator concentration. This is the consequence
of an anomalously high chain-transfer constant [11].
The experimental results can be described under the assumption that termination is an
intramolecular reaction,
P
! k t
P ;
and that transfer processes play an important part in the radical polymerization of VFc,
P ‡ M k tr;M
!
P ‡ Sol k tr;Sol
!
P ‡ M ;
P ‡ Sol :
306

15.2 Addition polymers
In this case the degree of polymerization (DP n ) is given by the equation
DP n ˆ
k p c…VFc†
k t ‡ k tr;M c…VFc†‡k tr;Sol c…Sol† :
…1†
Since the transfer to the solvents applied in these investigations is known to be small,
one can simplify the last equation and write
1 k t
ˆ
DP n k p c…VFc† ‡ C tr;M
…2†
C tr;M ˆ ktr;M
k p
; …3†
where k t is the rate constant of termination, k tr,M the transfer constant to monomer, k tr,Sol the
transfer constant to solvent, and k p the rate constant of propagation.
Transfer to polymer was also considered unimportant because monomer conversion
was kept below 10% in all experiments.
From the following graph (Fig. 15.1) one can determine k t /k p = 0.056 mol l –1 and
the transfer constant C tr,M = 3.5610 –2 , which is about three times higher than found in literature
[11].
1/DP) 10 –2
1/DP) * 10 -2
*
10
8
6
4
0 0.4 0.8 1.2
1/[VFc] /l*mol -1 –1
1 / / l *
Figure 15.1: DP –1 n =f(c (VFc) –1 for the radical polymerisation of VFc in toluene at 70 8C, c (AIBN)
= 1.5610 –2 mol l –1 .
At present one can only speculate about the nature of the intramolecular termination.
However, macroradicals can be considered as strong electrophiles, comparable to o-chloroanil
or tetracyano ethene [12]. It is also known that ferrocene is oxidized by o-chloroanil or
tetracyano ethene [13, 14]. Therefore it seems reasonable to assume termination via an intramolecular
electron transfer step according to the following scheme:
307

15 Ferrocene-Containing Polymers
Scheme 15.1: Termination via intramolecular electron transfer.
This view is supported by the detection of Fe 3+ species via Mößbauer spectroscopy
[15–18]. Moreover, in order to get high conversion one had to add initiator portion by portion.
The following scheme may explain the effect of the initiator on the basis of intermolecular
electron transfer:
Scheme 15.2: Intermolecular electron transfer between an initiator fragment and a deactivated chain end.
The glass transition temperature of PVFc was determined by DSC and was found in the
area of about 220 8C. From TGA studies we learned that this polymer is stable up to 400 8C.
15.2.2 Radical copolymerization
VFc has been copolymerized with common monomers such as styrene [19], methyl methacrylate
[19], N-vinylpyrollidone [20], and acrylo nitrile [19]. The electron richness of VFc
has been demonstrated in its copolymerization with maleic anhydride, in which an alternating
composition of the copolymer was observed over a wide range of feed ratios [16, 19].
The Q and e values of VFc were determined. The value of e = –2.1 again emphasizes the
electron rich nature of the vinyl group of VFc [21]. Therefore it looked rather hopeless to
apply anionic initiators for the polymerization of VFc.
15.2.3 Anionic polymerization of VFc
Many attempts were necessary in order to find conditions which allowed anionic polymerization.
Finally, we found n-BuLi, s-BuLi, and distyryl dianion in THF most suitable for the
308

15.2 Addition polymers
polymerization of VFc. However, polymerization was not possible in toluene, methylene
chloride, or dioxane. The reason for these differences are due to the complexation of Li + by
THF and the formation of highly reactive naked carbanions. This view is strongly supported
by the fact that polymerization in dioxane became possible by addition of 12-crown-4, which
functioned as complexing agent [16].
15.2.3.1 Living polymerization
The living polymerization is characterized by the following properties
a) spontaneous initiation
b) control of molar masses by the ratio c (monomer) : c (initiator)
c) linear molar mass conversion relationship
d) continuous growth of the molar mass when monomer is added portion by portion
e) narrow molar mass distribution.
They are fulfilled in the case of the polymerization of VFc in THF initiated by n-BuLi.
Typical results are shown in Figs. 15.2 and 15.3.
The accuracy of the control of molar masses is shown in Tab. 15.1.
Molar masses of the samples were determined by field-desorption mass spectrometry
(FD-MS), vapour pressure osmometry (VPO), and GPC. GPC was calibrated with PVFc.
Both, calibration via an oligomer sample in which oligomers from n = 3 through n =11
could be identified and via polymers produced by living anionic polymerization (characterized
by VPO or FD-MS), gave similar results and excellent agreement with theoretical values.
The FD-MS did not show only the oligomers from n = 3 through n = 11 but satellite
peaks of M + 79 for each molar mass.
Mn/g * mol -1
8000
7000
6000
5000
4000
3000
2000
addition of monomer
1000
0
0 5 10 15 20
converted VFc/mmol
Figure 15.2: Monomer addition experiment, M n = f (converted monomer), in THF, T = –25 8C, c (n-
BuLi) = 0.015 mol l –1 (addition of monomer: after complete consumption of the first portion of monomer
a second portion was added).
309

15 Ferrocene-Containing Polymers
IV
III
II
I
I: M n = 2450 g/mol –1
M w =M n = 1.14
II: M n = 4200 g/mol –1
M w =M n = 1.15
III: M n = 5650 g/mol –1
M w =M n = 1.14
IV: M n = 6450 g/mol –1
M w =M n = 1.17
M n is the number average of the molar
mass and M w the weight average of the
molar mass.
21 24 27 30 33
Elution volume /ml
Figure 15.3: Polymerization of VFc with n-BuLi in THF at –25 8C, GPC of samples taken at different
reaction times.
Table 15.1: Polymerisation of VFc, initiated with n-BuLi. Monomer addition experiment.
Time/min converted VFc M n (calc.) M n M w M w =M n
/mmol /g mol –1 /g mol –1 /g mol –1
7 0.8253 295 375 430 1.15
30 2.6207 936 1070 1240 1.16
80 5.5384 1979 2070 2360 1.14
127 6.3357 2265 2340 2640 1.13
178 6.5010 2324 2400 2740 1.14
206 6.7242 2403 2400 2730 1.13
248 6.8705 2455 2450 2810 1.14
Addition of Monomer
269 9.2446 3304 3110 3540 1.14
292 12.2361 4373 4180 4800 1.15
364 13.8921 4969 5030 5690 1.13
446 15.1741 5423 5650 6440 1.14
1122 18.4713 6601 6450 7546 1.17
For these findings we suggest the following explanation:
Scheme 15.3: Possible explanation for satellite peaks in the FD-MS with M + 79 (79: THF + Li).
310

15.2 Addition polymers
n=10 n= 9 n=11
n= = 8
n= = 7
n= = 6
n= = 5
26 28 30
Elution volume / ml
Figure 15.4: Identification of oligomeric (VFc) n with n = 3 through n = 11 by GPC; polymerisation
conditions: T =–408C, solvent: THF, M n = 2400 g mol –1 , M w =M n = 1.11.
n= = 4
n= = 3
This again would strongly support the view of complexation of the gegenion by THF.
All attempts to start an anionic polymerization of VFc initiated with naphthyl – K +
failed. However, it was possible to apply distyryl dianion as initiator. The conversion vs.
time is shown in Fig. 15.5.
The molar masses are again controlled by the ratio of monomer :initiator and the molar
mass distribution is narrow.
80
conversion/%100
60
40
20
0
0 50 100 150 200 250 300
Time/min
Figure 15.5: Polymerisation of VFc initiated with naphthyl-K/styrene at –30 8C in THF with
c (VFc) = 1 mol l –1 , c (naphthyl-K) = 0.1 mol l –1 , c (styrene) = 0.1 mol l –1 .
311

15 Ferrocene-Containing Polymers
IV
III
II
I
I: M n = 2400 g/mol –1
M w =M n = 1.11
II: M n = 3500 g/mol –1
M w =M n = 1.12
III: M n = 4200 g/mol –1
M w =M n = 1.11
IV: M n = 7500 g/mol –1
M w =M n = 1.14
24 26 28 30 32 34
Elution volume / ml
Figure 15.6: Polymerisation of VFc in THF, initiated by naphthyl-K/styrene at –30 8C, GPC for samples
taken at different reaction times.
15.2.3.2 Block copolymers
One of the main reasons for the interest in living polymerization is its potential for the
synthesis of block copolymers.
ABA-type block copolymers are available if VFc is added after the polymerization of
styrene (PSt) has reached 100%:
Scheme 15.4: Synthesis of ABA-type block copolymers; A = poly(vinylferrocene) segment; B = poly(-
styrene) segment.
312

15.2 Addition polymers
IV
III
II
I
I: M n = 7500 g/mol –1
M w =M n = 1.20
II: M n = 10500 g/mol –1
M w =M n = 1.19
III: M n = 18000 g/mol –1
M w =M n = 1.22
IV: M n = 25000 g/mol –1
M w =M n = 1.21
24 26 28 30 32 34
Elution volume / ml
Figure 15.7: GPC of ABA-type block copolymers; A = c (PVFc), B = c (PSt), solvent THF, T = –30 8C.
Typical results are shown in Fig. 15.7.
AB-type block copolymers can be synthesized by adding a second monomer (B) to
preformed polyvinylferrocene anion (A), which was synthesized as described above. Styrene,
methyl methacrylate, and propylenesulfide have been applied as monomer B. Typical results
are given in Tab. 15.2 and Fig. 15.8.
Table 15.2: Anionic block copolymerization in THF initiated by n-BuLi; T = –25 8C, c(VFc) = 0.173 mol/l,
c(n-BuLi) = 0.015 mol/l, c (Monomer B) = 2 mol/l.
Monomer A Monomer B Poly. temp./8C a Poly. time/min a M n /g mol –1 b M w /g mol –1 b MWD c
VFc Styrene –70 30 30000 33900 1.13
VFc MMA –70 30 25500 29800 1.17
VFc Propylene- 0 15 11900 13100 1.11
sulfide
a) after addition of the second monomer; b) determined by GPC with internal standard PSt; c)M w /M n
molecular weight distribution.
It is interesting to note that the block copolymers show only one instead of two glass
transition temperature (Tab. 15.3). This might be due to the fact that the molar mass of segment
A is still too small for being able to establish its own phase.
313

15 Ferrocene-Containing Polymers
IV
III
II
I
I: M n = 2300 g/mol –1
M w =M n = 1.10
II: M n = 10000 g/mol –1
M w =M n = 1.09
III: M n = 20000 g/mol –1
M w =M n = 1.08
IV: M n = 30000 g/mol –1
M w =M n = 1.07
24 26 28 30 32 34 36
Elution volume / ml
Figure 15.8: Anionic block copolymerization of VFc with styrene in THF at –70 8C.
Table 15.3: Glass transition temperature of homopolymer A, homopolymer B, and block copolymers AB.
Glass transition temperature a (DSC) of
Monomer A Monomer B Homopolymer A/8C Homopolymer B/8C Block copolymers AB/8C
VFc Styrene 219 100 112
VFc MMA b 219 105 127
VFc Propylene sulfid 219 –40 –29
a) constant heating rate with 10 K/min; b) methylmethacrylat.
15.2.4 Polymeranalogeous reactions
Polymeranalogeous reactions which do not need more than one modification step are an interesting
alternative for the preparation of polymers with pendent ferrocene units. Therefore
it is not surprising that a wide range of conventional organic polymers has been used for this
purpose [22]. These reactions have the advantage of simplicity and variability concerning
molar masses and composition, which we could demonstrate for the esterification of poly
(MMA-co-MACl) [23] (Tab. 15.4).
314

15.3 Polymers with ferrocene units in the main chain
Scheme 15.5: Esterification of poly(MMA-co-MACI) with hydroxymethylferrocene.
Table 15.4: Typical results of the esterification of poly(MMA-co-MACl) with hydroxymethylferrocene.
MMA : MACl a ferrocene b M n g/mol c M w g/mol c
mol ratio
mol%
90 : 10 6 70000 200000
70 : 30 6.4 45000 61000
60 : 40 10.7 20000 42000
0 : 100 85 40000 66000
a) from IR; b) from 1 H NMR; c) from GPC calibrated with poly(MMA) standards
15.3 Polymers with ferrocene units in the main chain
15.3.1 Polycondensation
Numerous bifunctional ferrocene derivatives have been synthesized and used for polycondensation
reactions [24]:
Scheme 15.6: Candidates for polycondensation reactions.
X = COOH, COCl, CH 2 OH, CH 2 CH 2 OH, CH 2 CH 2 NH 2
315

15 Ferrocene-Containing Polymers
Reaction of 1,1'-dichlorocarbonyl ferrocene with diamines such as 1,6-diamino-hexane
and 1,8-diamino-octane yielded polyamides which were soluble in DMSO, DMF, and m-cresol
[16].
Reaction of 1,1'-dimercapto-ferrocene with succinc acid chloride and sebacinic acid
chloride yielded polymers with molar masses of M n = 12000 g mol –1 which were determined
by end group analysis assuming OCH 3 end groups and by GPC calibrated with a series
of homologous oligomers identified in the GPC graph [25].
15.3.2 Polymers by addition of dithiols to diolefins
15.3.2.1 Radical reaction
The addition of dithiols to diolefins is a well-established method for the synthesis of polymers
[26–28]. Reaction of 1,1'-dimercapto-ferrocene with norbornadiene in toluene in the
presence of AIBN as initiator and heating to 70 8C yielded a yellow powder after precipitation
in methanol. The GPC shows a homologous series of oligomers in which the first
members were identified by FD-MS with m/e = 682, 930, 1179. Instead of the expected alternating
copolymers from ferrocene and norbornene, bridged by sulphur, only oligomers
of ferrocenyl disulphides with norbornene or nortricyclyl end groups were detected
(Scheme 15.7).
Scheme 15.7: Reaction of norbornene and 1,1'-mercaptoferrocene.
15.3.2.2 Base catalyzed reactions
It has been known for a long time that thiols can react with activated olefins such as unsaturated
ketones in the presence of a basic catalyst [29]. This reaction has also been used for
the synthesis of polymers [30] (Scheme 15.8).
316

15.3 Polymers with ferrocene units in the main chain
22 24 26 28 30 32 34 36 38
Elution volume / ml
Figure 15.9: GPC diagram of oligo(ferrocene disulphide)s detected by UV absorption.
Table 15.5: Molecular weight of the oligomers, calculated M n;theor and found M n;exp .
Number of ferrocene units M n;theor g/mol M n;exp g/mol
2 682 682
3 930 930
4 1178 1179
5 1426 1420
6 1674 1690
7 1922 1930
8 2170 2170
9 2418 2420
10 2666 2670
11 2914 2920
12 3162 3140
Scheme 15.8: Base-catalyzed addition of thiols onto activated olefins.
1,1'-dimercapto-ferrocene was used as dithiol and 1,4-butanol-dimethacrylate was applied
as diolefin. From both, GPC and end group analysis with the assumption of olefinic
end groups, M n = 4000 g mol –1 was determined. The structure of the polymer was confirmed
by elemental analysis, IR, 1 H, and 13 C NMR. Similar results were found for divinylsulfone
(M n = 2700 g mol –1 ) and dibenzylidene acetone (M n = 6850 g mol –1 ) [25].
317

15 Ferrocene-Containing Polymers
Scheme 15.9: Reaction of 1,1'-bis(mercaptopropylthio)ferrocene with divinyl sulfone.
1,1'-bis(2-mercapto-propylthio)-ferrocene reacts with divinylsulfone according to the
following scheme [31] (Scheme 15.9).
Molar masses were determined by GPC (M n = 2900 g mol –1 ) and end group analysis
(M n = 2400 g mol –1 ).
15.3.2.3 Acid catalyzed reactions
Markovnikov addition of thiols to olefins is observed, if acids are applied as catalysts according
to the following mechanism:
Scheme 15.10: Acid-catalyzed reaction of a thiol with an olefin.
Olefins suitable for this reaction are activated by substituents such as OR, SR, NR 2 .
Polymer formation is exemplified by the reaction of 1,1'-dimercapto ferrocene with a divinyl
ether [25] (Scheme 15.11).
Molar masses of M n = 8900 g mol –1 were determined by GPC and end group analysis,
assuming OCH 3 end groups. These end groups are a clear indication that H + does not only
function as catalyst for polymer forming but also for polymer degradation.
318

15.3 Polymers with ferrocene units in the main chain
Scheme 15.11: Reaction of 1,1'-dimercapto ferrocene with divinyl ether.
15.3.3 1,1'-dimercapto-ferrocene as initiator
Ring opening polymerization of propylene sulphide by 1,1'-dimercapto ferrocene yields
polymers which contain one ferrocene unit in the center of the macromolecule (Scheme
15.12).
Until now there is no good explanation for the discrepancy between the GPC value
(M n = 28,000 g mol –1 ) and the 1 H NMR analysis value (M n = 6000 g mol –1 ), which is very
close to the theoretical value (M n = 6300 g mol –1 ). Maybe the calibration of GPC with polystyrene
standards is not suitable in this particular case.
Scheme 15.12: Ring opening of propylene sulfide initiated with ferrocene-1,1'-dithiolat.
319

15 Ferrocene-Containing Polymers
15.3.4 Reductive coupling
Reductive polycondensation of 1,1'-diacetyl ferrocene with low-valent titanium, synthesized
by a reaction of TiCl 4 with Zn powder [32, 33], yield polymers with ferrocene-vinylene
repeating units [34]:
Scheme 15.13: Reductive polycondensation of 1,1'-diacetyl ferrocene.
Molar masses up to M n = 6800 g mol –1 were obtained in the low molar mass area of
the GPC, which was calibrated with oligomers given above.
15.4 Mößbauer studies of polymers containing ferrocene
The study of motional dynamics close to the transition from viscous liquid to a glassy state
is a highly important issue for understanding its physical origin and its applicational consequences.
Recently the glass transitions in synthetic polymers and other glass forming systems
have been extensively investigated via neutron scattering [35, 36]. We have employed
57 Fe Mößbauer spectroscopy for the study of polymers containing ferrocenyl groups (Fc) [6,
37, 38]. Apart from the vibrational behaviour as derived from the temperature dependent
Mößbauer-Lamb factor f giving a measure of the mean square atomic displacement, these
studies also allow to detect relaxation processes in a time window of 10 –9 –10 –6 s, i. e. close
but complementary to neutron spectroscopy (10 –12 –10 –9 s).
Ferrocene units have been used as side group, as part of the backbone, as terminating
group within various homo and copolymers, crosslinked and swollen polymers, block polymers,
and telecheles. For comparison also monomeric compounds and crystallized polymers
were studied. Only a few examples may suffice to be mentioned:
poly(vinyl Fc), poly(dimercapto Fc) poly(Fc dioxymethylene terephthalate), poly(itaconic
anhydrid-co-vinyl Fc), poly(methylmethacrylate-co-vinyl Fc), polystyrene terminated
by vinyl Fc, poly(vinyl Fc-block-propylene sulphide) (PROPS), poly(ferrocenelene-dimethylvinylene),
telechel of norbonadiene, and 1,3-dimercapto-benzole with Fc end groups.
The analysis of f below the glass transition T g reveals very similar behaviour for all the
polymers in the temperature range between 50–170 K where nearly harmonic behaviour is
found with a temperature y –2 & 40–50 K, corresponding to the second moment of the
320

15.4 Mößbauer studies of polymers containing ferrocene
vibrational frequency distribution. The motion of Fc is the same if included in the backbone or
as side group thus reflecting the motion of the backbone. Fc at the chain end, however, reveals
a reduction of y –2 to about 30 K. At lower temperatures the amorphous samples show anharmonicities,
which are interpreted to be due to soft quasi-localized modes, which is also typical
for other amorphous materials. The width of the local potentials is up to about 0.08 nm.
Deviations from harmonic behaviour are also found above about 200 K, however, only
for the amorphous samples. These high temperature anharmonicities occur often far below
T g , which is typically around 300–350 K. They are supposed to be caused by residual solvents
in the polymer matrix. We have also studied f (T) for some
p
polymers with a relatively
low T g of 250–300 K. ln f (T) decreases rapidly following a T C T dependence as predicted
by mode coupling theory (MCT). This is interpreted as the onset of local processes.
T c represents the transition from non-ergodic to ergodic behaviour, which occurs typically
30–150 K above the macroscopic glass transition temperature T g . In Fig. 15.10 we show
f (T) for PROPS. The MCT fit is indicated by the broken line yielding T c = 306+12 K
whereas T g &240 K. Simultaneously with the onset of anharmonic behaviour of f the Mößbauer
resonance lines broaden and quasi-elastic lines appear close to T c .
Figure 15.10: Temperature dependence of –ln f. Intramolecular contributions of Fc have been subtracted.
We have performed an analysis of the entire Mößbauer line shape taking into account
a density fluctuation correlation function of the Kohlrausch type y(t) =a exp (–|t |/t) b with
b = 1/2 and the relaxation time t. We could thus also follow the onset of the structural relaxation
a process related to the dynamic glass transition. The relaxation times decrease
from about 10 –6 satT g to 5610 –8 s at 300 K. Details of these studies especially concerning
the behaviour of swollen polymers will be given elsewhere.
During the course of these studies, which concentrated on the motional dynamics, it
was recognized that many polymers did not only reveal the well-known hyperfine interaction
321

15 Ferrocene-Containing Polymers
of Fc (a quadrupole doublet with a splitting of about 2.35 mm/s) but also have an additional
doublet with a splitting of about 0.8 mm/s which is still a typical isomer shift for a Fe III species
[39]. The relative intensity of the Fe III resonance increases at high temperatures, at
lower temperatures it is usually hardly visible in the spectra (Fig. 15.11). Actually these two
species have already been detected earlier in similar polymers and were attributed to Fe II
(Fc) and Fe III with different f factors. Belov [40] related the Fe III to a Fc with both rings included
as crosslink in the macromolecule. George and Hayes [41] explained it by Fe III
formed in a special termination step during radical polymerization. Our investigations
showed that the Fe III resonance occurs irrespective of the way of polymerization, e. g. also in
polycondensation. In addition, the bonding of both rings to the backbone turned out to be
neither a necessary nor sufficient condition for the formation of Fe III . A careful analysis reveals
that at low temperatures Fe II is dominantly present whereas it is partly transformed to
Figure 15.11: Mößbauer absorption spectra of poly(ferrocenelene-dimethyl-vinylene) at different temperatures.
322

References
Fe III at higher temperatures. This transformation is completely reversible. A quantitative description
of this valence transformation is achieved within a model assuming a thermally activated
excitation of the highest populated molecular orbital of Fc to an unoccupied orbital
of ligand character, or a narrow band, which may be supposed from the semiconducting
properties of many polymers containing Fc. The relative population of the Fe III can then be
expressed as
p III ˆ
2
exp…=kT†‡1 :
…4†
For all the studied compounds p we obtain the excitation energy D = 1.50+4 meV.
Notably y –2 is about 2 times that for Fe II . This indicates a duplication of the vibrating
mass. We therefore propose that the Fe III species is related to crosslinks formed by two Fc
units. Whether these are bound via a bridging oxygen or something else is still unclear. We
note that the Fe III cannot be identified with ferrocenium. No relaxational averaging or broadening
is observed. This means that the mixed valent state is fluctuating slowly (610 –6 s) unlike
usual mixed valence systems with much faster fluctuations.
The details of the electronic structure responsible for the mixed valence behaviour and
the relation to the electron delocalization found in several of these polymers is still unclear
and need further investigation. In particular the excitation energy of about 1.5 meV has to
be verified spectroscopically.
Work was supported by Deutsche Forschungsgemeinschaft grant Li 244/8-1,2.
References
1. C.E. Carraher, C.U. Pittman, in: J.E. Sheats, C.E. Carraher, C.U.Pittman (eds.): Metal-containing
Polymeric Systems, Plenum Press, New York, p. 1 (1985)
2. E.W. Neuse: Adv. Macromol. Chem., 1, 1 (1968)
3. K.E. Gonsalves, X. Chen, in: A. Togni, T. Hayashi (eds.): Ferrocenes, Verlag Chemie, Weinheim,
p. 497 (1995)
4. P.D. Hale, T. Inagaki, H.I. Karan, Y. Okamoto, T.A. Skotheim: J. Am. Chem. Soc., 111, 3482 (1989)
5. M.E. Wright, E.G. Toplikar: Macromolecules, 25, 6050 (1992)
6. F.J. Litterst, A. Lerf, O. Nuyken, H.A lcala: Hyperfine Interactions, 12, 317 (1982)
7. F.S. Arimoto, A.C. Haven: J. Am. Chem. Soc., 77, 6295 (1955)
8. J.C. Lai, T. Rounsefell, C.U. Pittman: J. Polym. Sci. A-1, 9, 651 (1971)
9. C. Aso, T. Kunitake, T. Nakashima: Makromol. Chem., 124, 232 (1969)
10. C.U. Pittman, C.C. Lin: J. Polym. Sci. Chem. Ed., 17, 271 (1979)
11. Y. Sasaki, L.L. Walker, E.L. Hurst, C.U. Pittman: J. Polym. Sci. Polym. Chem. Ed., 11, 1213
(1973)
12. G. Henrici-Olivé, S. Olivé: Makromol. Chem., 68, 219 (1963)
13. R.I. Collins, R. Pettit: J. Inorg. Nucl. Chem., 29, 5ß3 (1967)
14. C.U. Pittman, P.L. Grube: J. Appl. Polym. Sci., 18, 2269 (1974)
15. H.J. George, G.F. Hayes: J. Polym. Sci. Polym. Lett. Ed., 11, 471 (1973)
323

15 Ferrocene-Containing Polymers
16. V. Burkhardt: PhD thesis, Univerität Bayreuth (1992)
17. R. Feyerherm, F.J. Litterst, V. Burkhardt, O. Nuyken: Solid State Commun., 82, 141 (1992)
18. W. Wagener, M. Hilberg, R. Feyerherm, W. Stieler, F.J. Litterst, T. Pöhlmann, O. Nuyken: J. Phys.
Condens. Matter, 6, L391 (1994)
19. J.C. Lai, T.D. Rounsefell, C.U. Pittman: J. Polym. Sci., A1, 9 (1971)
20. C. Simionescu, T. Lixandru, T. Mazilu: Makromol. Chem., 163, 59 (1973)
21. T. Alfrey, T.T. Price: J. Polym. Sci., 2, 101 (1947)
22. I.R. Butler, W.R. Cullen, N.F. Han, F.G. Herring, N.R. Jagannathan, J. Li: Appl. Organomet.
Chem., 2 (1988)
23. O. Nuyken, V. Burkhardt, T. Pöhlmann, M. Herberhold: Makromol. Chem. Macromol. Symp., 44,
195 (1991)
24. K.E. Gonsalves, R.W. Lenz, M.D. Rausch: Appl. Organomet. Chem., 81 (1987)
25. T. Pöhlmann. PhD thesis, Universität Bayreuth (1993)
26. C. S.Marvel, A. Kotch: J. Am. Chem. Soc., 73, 481 (1951)
27. O. Nuyken, T. Völkel, T. Pöhlmann: Makromol. Chem., 192, 1959 (1991)
28. T. Völkel: PhD thesis, Universität Bayreuth (1990)
29. T. Posner: Ber. dtsch. Chem. Ges., 34, 1395 (1901)
30. O. Nuyken, T. Völkel: Makromol. Chem., 191, 2469 (1990)
31. O. Nuyken, T. Pöhlmann, M. Herberhold: Macromol. Reports, A29, 211 (1992)
32. T. Mukaiyama, T. Sato, J. Hanna: Chem. Lett., 1041 (1973)
33. D. Lenoir: Synthesis, 883 (1989)
34. R. Bayer, T. Pöhlmann, O. Nuyken: Makromol. Chem., Rapid Commun., 14, 359 (1993)
35. W. Petry, E. Bartsch, F. Fujara, M. Kiebel, H. Sillescu, B. Farago: Z. Phys. B, 83, 175 (1991)
36. D. Richter, R. Zorn, B. Farago, B. Frick, L. J. Fetters: Phys. Rev., 68, 71 (1992)
37. R. Feyerherm, F.J. Litterst, V. Burkhardt, O. Nuyken: Solid State Commen., 82, 141 (1992)
38. M. Hillberg, W. Stieler, F.J. Litterst, V. Burkhardt, O. Nuyken: Hyp. Int., 88, 137 (1994)
39. W. Wagener, M. Hillberg, R. Feyerherm, W. Stieler, F.J. Litterst, Th. Pöhlmann, O. Nuyken: J. Phys.
Cond. Matt., 6, L 391 (1994)
40. V.F. Belov et al.: Dokl. Akad. Nauk SSSR, 159, 831 (1964)
41. M.H. George, G.F. Hayes: J. Polym. Sci. Lett., 11, 471 (1973)
324

16 Transfer of Vibrational Energy in Dye-Doped Polymers
Johannes Baier, Thomas Dahinten, and Alois Seilmeier
16.1 Introduction
Photoexcitation or radiationless relaxation processes supply a lot of thermal excess energy
to polymer systems. In particular, dye molecules dispersed in polymers reach transient excess
temperatures of several 100 K by absorbing visible light [1]. The equilibration of the
excess energy is of interest in the discussion of the photostability and of the kinetics of polymer
dye interaction [2] of such systems. In the condensed phases the energy relaxation is determined
by microscopic transfer processes proceeding on a picosecond time scale.
In recent years many investigations have been performed on intra and intermolecular
energy transfer in the gas phase and in liquids. Whereas in low-pressure gases time constants
extending to the microsecond range have been reported, transfer times from a fraction of a
picosecond to several picoseconds are observed in the liquid state [3]. There the considerably
higher density results in larger collision frequencies and shorter relaxation times.
In this paper, the transfer of vibrational energy in polymers doped with dye molecules
is investigated. Similar experiments have been already discussed after exciting C-H stretching
modes around ~n = 3000 cm –1 [4]. Here we present data for the first time where the
more important skeletal modes between ~n = 1500 cm –1 and ~n = 1800 cm –1 are pumped.
Two types of experiments are discussed:
a) The dye molecules are vibrationally excited by an ultrashort infrared pulse. Intramolecular
energy redistribution and intermolecular energy transfer to the surrounding polymer are
studied.
b) In experiments, where the infrared frequency is tuned to vibrational bands of the polymer,
equilibration of the excess energy in the polymer is investigated. In this case, the dye
molecules serve as sensors for the transient temperature increase of the system.
The experimental data are taken on polycarbonate doped with the dye coumarin 6.
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
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16 Transfer of Vibrational Energy in Dye-Doped Polymers
16.2 Experimental
Excess energy is supplied to the system by an infrared light pulse either via vibrational excitation
of the dye molecules (Fig. 16.1a) or via vibrational excitation of the polymer matrix
(Fig. 16.1b). In the first case a first rapid energy redistribution occurs within the dye molecule
with a time constant in the order of one picosecond, which is characteristic for the excited
state. The energy is subsequently transferred to the polymer matrix via intermolecular
processes. The instantaneous state of excitation is monitored by a second delayed light pulse
which promotes the excited dye molecules into the fluorescing S 1 state. The frequency of
the probing pulses is chosen in the long wavelength edge of the electronic transition. In this
way only vibrationally excited molecules populated either by photoexcitation or by thermal
excitation contribute to the signal. In the following we discuss the transiently enhanced
fluorescence signal due to photoexcitation of the vibrational manifold. Generally, a biexponential
decay curve is expected representing the two relaxation processes.
In the second type of experiments the polymer is vibrationally excited by the infrared
pulse (Fig. 16.1b). The excitation energy is redistributed giving rise to an increased temperature
of the pumped volume [5]. The energy equilibration in the polymer system involves
intermolecular transfer of energy to the dye molecules which serve as sensors for the equilibration
process. The instantaneous temperature is monitored via electronic transitions and
the subsequently emitted fluorescence signal of the dispersed dye molecules as discussed
above. Information is obtained on the time for complete energy equilibration but not on the
individual relaxation steps which can be investigated e. g. by an IR double resonance technique.
The experiments are performed with a modelocked Nd :YLF laser system of 10 Hz repetition
rate. Tunable infrared pulses are generated by difference frequency mixing between
Figure 16.1: Energy transfer processes involved in the energy equalization after vibrational excitation
a) of the dispersed dye molecules and b) of the polymer matrix by an infrared pulse of frequency n pump .
326

16.3 Results and discussion
pulses at the fundamental frequency of the laser and the idler pulses of an optical parametric
oscillator driven by the second harmonic of the Nd laser [6]. In this way infrared pulses tunable
between 5 mm and 10 mm with energy of about 1 mJ are produced [7]. The probe pulses
are generated by frequency doubling of a weak part of the laser pulses. The fluorescence signal
is measured in a spectral region from 19,800 cm –1 to 20,100 cm –1 with the help of a
spectrometer and a photomultiplier. The duration of both pulses amounts to approximately
3 ps. The zero point of the time scale and the time resolution of the system is determined by
simultaneously measured cross correlation curves of the two pulses.
The experiments are performed on polycarbonate films with high optical quality that
are prepared as follows. Commercially available 2,2-bis(4-hydroxy phenyl)propane-polycarbonate
granulate (Bayer, m = 1.5 g) and coumarin 6 (m = 120 mg) are dissolved in trichloromethane
(V = 10 ml). The solution is carefully spread over a cellulose acetate sheet
with a thickness of 200 mm. After drying a film of about 3 mm results with a dye concentration
of 0.3 M. The thin film is detached from the cellulose acetate substrate and used as a
free standing film.
16.3 Results and discussion
Before time-resolved data are presented the infrared properties of our samples should be discussed.
In Fig. 16.2a the infrared absorption of the investigated polycarbonate film is presented.
The two strong lines are assigned to an aromatic C-C ring mode (~n = 1506 cm –1 )
and to the C=O stretching mode (~n = 1774 cm –1 ) of the polycarbonate matrix. The weaker
infrared absorption bands can be identified as vibrational bands of the dispersed coumarin
molecules by comparing the spectrum with that of coumarin 6 in a C 2 Cl 4 solution
(c = 2.5610 –3 M) in Fig. 16.2 b. The strongest vibrational lines of coumarin 6 in the spectral
range presented in Fig. 16.2 are due to aromatic C-C ring modes (~n = 1516 cm –1 ,
~n = 1590 cm –1 , and ~n = 1616 cm –1 ) and due to the C=O stretching mode (~n = 1717 cm –1 ).
The observed absorbance of A *0.2 of the coumarin absorption lines in Fig. 16.2a nicely
compares with estimates on the basis of the absorbance shown in Fig. 16.2b, taking into account
the dye concentration and the sample thicknesses in the film and the C 2 Cl 4 solution,
respectively.
In time-resolved experiments of the first type (Fig. 16.1 a) vibrational bands of the
coumarin 6 molecules are resonantly excited. Experimental data for two pump frequencies
~n pump = 1590 cm –1 and ~n pump = 1717 cm –1 are shown in Figs. 16.3 and 16.4, respectively. The
curves are measured with probe pulses polarized parallel (I) and perpendicular (y) to the
pump pulses. The signal plotted here is the enhancement of the fluorescence signal (F–F 0 )/F 0
due to photoexcitation relative to the thermally induced fluorescence signal F 0 . The observed
relaxation behaviour is very similar to that found in liquid solutions [8]. Figure 16.3 shows a
first rapidly decaying signal with a time constant e1 ps followed by a longer lived component
with a relaxation time of (7+1) ps. The C–C stretching mode of molecules is directly excited
327

16 Transfer of Vibrational Energy in Dye-Doped Polymers
Figure 16.2: a) Infrared absorbance of a polycarbonate film with 0.3 M coumarin 6. The strong lines
are due to the polycarbonate matrix. b) Infrared absorption of coumarin 6 in a C 2 Cl 4 solution which
clearly shows the absorption bands of the dye molecules.
Figure 16.3: Temporal evolution of the excess fluorescence signal versus delay time. A C–C stretching
mode of coumarin 6 is excited. Results for a probe pulse polarized parallel and perpendicular are
shown. The broken curve represents the cross correlation curve of both pulses.
328

16.3 Results and discussion
Figure 16.4: Change of the fluorescence signal as a function of the delay time. The C=O stretching
mode of coumarin 6 is pumped. Data are shown for a parallel and perpendicularly polarized probe
pulse. The broken line gives the time resolution of the experimental system.
with the probe frequency ~n probe = 19,100 cm –1 , giving rise to a rapidly decaying signal in the
first few picoseconds. Intramolecular processes lead to a new energy distribution within the
molecule which can be described by an increased internal temperature of approximately
350 K. The relaxation of this increased temperature with a time constant of about 7 ps via intermolecular
energy transfer is monitored at longer delay times.
At short delay times the signal amplitude is determined by the population of the C–C
mode, by its Franck-Condon factor, and by the lifetime of the vibrational state. At longer delay
times the decrease of the internal temperature of the vibrational manifold is observed.
The probe pulse measures the absorption in the long wavelength edge of the S 1 absorption
band which is in good approximation proportional to the Boltzmann factor exp(hn/kT )in
coumarin 6. Consequently the temperature decrease is measured via the fluorescence signal
which is proportional to exp[h (n probe – n 00 )/kT (t)] and which does not depend on the primarily
excited vibrational state [9].
Figure 16.4 shows the situation after direct excitation of the C=O stretching mode.
The observed signal at delay 0 is one order of magnitude smaller. This fact is due to an infrared
absorption reduced by a factor of two and due to a considerably smaller Franck-Condon
factor of the C=O mode [8]. The reduced signal amplitude of the fast component does
not allow a clear separation of two relaxation components. However, a better fit of the experimental
data is obtained with two time constants of about 2 ps and about 7 ps as compared
to a monoexponential curve. Similar time constants have been identified in liquid solutions
of coumarin 6 [8]. Whereas the signals at delay time 0 are strongly influenced by the
Franck-Condon factors of the excited vibrations, the long-lived component is only dependent
on the absorbed infrared energy which is roughly the same for ~n pump = 1590 cm –1 and
~n pump = 1717 cm –1 . Consequently the slowly relaxing component is of comparable magnitude
in Figs. 16.3 and 16.4.
329

16 Transfer of Vibrational Energy in Dye-Doped Polymers
Of special interest is the polarization dependence of the transient signals. In Fig. 16.3a
signal increased by approximately a factor of two is found for parallel polarization of the
pump and the probe pulse as compared to perpendicular polarization. Such a behaviour is
expected for a situation where the infrared transition moment of the C–C mode is oriented
nearly parallel to the electronic dipol moment. In contrast in Fig. 16.4, where the C=O
stretching mode is pumped, the signal is smaller for parallel polarization of the two pulses
indicating an angle between the C=O dipole moment and the electronic dipole moment larger
than the magic angle of 54.78 [10]. The polarization dependence will be discussed in
more detail elsewhere.
Next, we want to discuss experiments where the polymer matrix is vibrationally excited
by resonant infrared absorption. Figure 16.5 shows time-resolved data for an infrared
excitation frequency of ~n pump = 1774 cm –1 , where only the C=O stretching vibration of the
carbonyl group of polycarbonate is populated. The fluorescence signal in Fig. 16.5 represents
the momentary excess energy transferred to the probe molecules coumarin 6. We find
a signal rising with a time constant of approximately 40 ps. After 150 ps the signal is nearly
constant and a quasiequilibrium with a temperature increased by about 5 K is established in
the excited volume. Obviously, the energy equalization process in the excited volume via intermolecular
energy transfer processes is completed within this time constant. Relaxation of
the increased temperature is expected to occur with a time constant in the millisecond range
via macroscopic heat conduction.
Of special interest is the origin of the time constant of 40 ps observed in Fig. 16.5.
Several processes are involved in the energy equalization process:
a) vibrational relaxation of the primarily excited C=O mode,
b) energy redistribution within the polymer matrix, and
c) energy transfer from the excited matrix to the probed dye molecules.
Figure 16.5: Enhancement of the fluorescence signal as a function of the delay time. The C=O stretching
mode of polycarbonate is vibrationally excited. The rise of the signal monitors the energy transferred
to the dye molecules. The solid curve is calculated for a transfer time of 40 ps.
330

16.3 Results and discussion
The latter step is believed to be much faster because energy transfer between dye molecules
and the polymer was found to occur on a time scale shorter than 10 ps (Figs. 16.3
and 16.4). Energy redistribution in and between the polymer chains is expected to be also relatively
efficient due to the high densities of vibrational states. Consequently, there is indication
that the observed time constant is governed by the depopulation of the primarily excited
C=O stretching mode which obviously does not couple very efficiently to the vibrational
manifold of the polymer. This fact nicely compares with the result on coumarin 6 (Fig. 16.4
and Ref. [8]) where also a relatively long lifetime is found for the C=O stretching mode.
The transient fluorescence signal due to vibrational excitation at ~n pump = 1516 cm –1 is
depicted in Fig. 16.6. According to Fig. 16.2 both, the polymer matrix and the probe molecules,
are excited at this frequency.
Figure 16.6: Time dependence of the excess fluorescence signal after excitation at ~n = 1516 cm –1 .
Both, the polycarbonate matrix and the coumarin molecules, are directly excited. Deconvolution of the
two relaxation processes gives the dashed curves.
Consequently, a relatively complex relaxation behaviour is observed. At short delay
times the directly excited coumarin molecules are detected whereas at longer delay times the
equalization of the excess energy in the matrix is monitored. We observe a first time constant
in the order of a picosecond representing the intramolecular energy redistribution in
coumarin 6 and a second time constant of about 7 ps which reflects the intermolecular transfer
to the matrix. In contrast to Fig. 16.3, a residual signal remains at longer delay times.
Here we observe the increased temperature of the excited volume due to the infrared energy
supplied to the polymer matrix. From the fit curves a time constant shorter than about 20 ps
is deduced for the equalization process which is clearly shorter than the time constant observed
after excitation of the C=O mode. The corresponding dashed line in Fig. 16.6 is calculated
for a time constant of 15 ps. Obviously, the energy transfer from the initially excited
C-C stretching mode of polycarbonate to the vibrational manifold is more efficient.
331

16 Transfer of Vibrational Energy in Dye-Doped Polymers
16.4 Summary
In this paper vibrational energy transfer in polymers with dispersed dye molecules is investigated.
Energy is selectively supplied to the system via resonant excitation of skeletal modes
around ~n = 1500 cm –1 . For the energy transfer, between the dye molecules and the polymer
matrix, time constants of several picoseconds are measured. The relaxation of excess energy
is governed by the depopulation of the initially pumped vibrational state or by the intermolecular
energy transfer depending on the excited vibrational state. It is interesting to see that
the energy can be stored for a few tens of picoseconds, e. g. in the C=O stretching mode, before
the energy is redistributed. This fact is of relevance for structural relaxation processes
in polymers.
References
1. W. Wild, A. Seilmeier, N. H. Gottfried, W. Kaiser: Chem. Phys. Lett., 119, 259 (1985)
2. C. D. Eisenbach, K. Fischer: Amer. Chem. Soc. Polym. Prepr., Polym. Chem. Div., 29/1, 501
(1988)
3. A. Seilmeier, W. Kaiser: Ultrashort Intramolecular and Intermolecular Vibrational Energy Transfer
of Polyatomic Molecules in Liquids, in: W. Kaiser (ed.): Ultrashort Laser Pulses and Applications,
Springer, Berlin, p. 279 (1988)
4. A. Seilmeier, J. P. Maier, F. Wondrazek, W. Kaiser: J. Phys. Chem., 90, 104 (1986)
5. P. O. J. Scherer, A. Seilmeier, W. Kaiser: J. Chem. Phys., 83, 3948 (1985)
6. R. Laenen, K. Wolfrum, A. Seilmeier, A. Laubereau: J. Opt. Soc. Am. B, 10, 2151 (1993)
7. T. Dahinten, U. Plödereder, A. Seilmeier, K. L. Vodopyanov, K. R. Allakhverdiev., Z. A: Ibragimov:
IEEE J. Quant. Electron., QE-29, 2245 (1993)
8. H.-J. Hübner, M. Wörner, W. Kaiser, A. Seilmeier: Chem. Phys. Lett., 182, 315 (1991)
9. F. Wondrazek, A. Seilmeier, W. Kaiser: Chem. Phys. Lett., 104, 121 (1984)
10. H. Graener, G. Seifert, A. Laubereau: Chem. Phys., 175, 193 (1993)
332

17 Picosecond Laser Induced Photophysical Processes of
Thiophene Oligomers
Dieter Grebner, Matthias Helbig, and Sabine Rentsch
17.1 Introduction
Within the conjugated polymers polythiophene is an important material for future technical
applications, e. g. for LEDs, electronic devices, or optical switches [1]. In this context studies
of light induced processes are of interest, especially details of the degradation of electronic
energy within the ultrashort time scale of picoseconds and sub-picoseconds.
These processes mostly proceed within the molecules and are presteps to transfer processes
of energy or electrons between different molecules, which dominate in technical devices.
A polymer molecule with about 100 monomer units is a complex system which corresponds
to an inhomogeneous distribution of subsystems with slightly different optical properties,
called effective conjugated segments within a polymer chain.
In this study we had the opportunity to investigate systematically the spectroscopic
properties of a series of linear thiophene oligomers [2–8]. The idea was to observe the fast
light induced processes of these small constituents of a polymer as starting point for deeper
understanding of processes in the polymer itself.
We studied oligothiophenes from 2 to 6 monomers in dilute solution because in this
case the interaction between the oligomer molecules is negligible. Therefore the dominance
of intramolecular processes was to be expected.
To carry out the intended time resolved studies on oligothiophene a picosecond laser
spectrometer, based on a Nd :YLF oscillator with amplifier system was, build up. The spectrometer
is described in the experimental part of the report. Moreover, the report contains a
brief description of the steady state spectroscopic properties of thiophene oligomers and the
results of the picosecond time resolved spectroscopic studies.
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
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17 Picosecond Laser Induced Photophysical Processes of Thiophene Oligomers
17.2 Experimental
Our picosecond absorption spectrometer operates with optical single pulses generated by a
passively mode locked Nd : YLF laser with a negative feedback loop for stabilisation of
single pulse energy. Figure 17.1 shows the experimental setup of the Nd :YLF laser spectrometer.
The resonator is formed by two flat mirrors, M1 (R = 99%) and M2 (R = 52%), and
by an AR-coated lens L1 ( f = 1250 mm). The total resonator length amounts to 1500 mm.
A single flashlamp-pumped 76.2 mm long Nd : YLF rod with a diameter of 6.35 mm is used
as the active element (AE). The aperture (A) ensures the TEM oo mode structure. The dye
cell (DC) with non-linear absorber is placed in contact with the mirror M1 to achieve passive
mode locking. To get higher pulse stability the non-linear absorber circulates slowly in
a special circuit. We used the absorber dye No.3274y in methanol (relaxation time 6 ps).
The resonator was designed for optimum beam intensity in the non-linear absorber and in
the laser rod by variation of the lens position. All elements of the cavity are wedged to avoid
additional resonators.
Figure 17.1: Experimental setup of the Nd:YLF laser spectrometer.
The negative feedback loop consists of a single crystal Pockels cell (PC), a polarizer
(P1), a fast photodiode (PD1), and the feedback electronic (FE). A constant voltage (around
l/8 voltage) is applied on the electrodes of the PC. The laser radiation has a circular polarisation
after a double pass through the PC and therefore several tens of percent are decoupled
by the polarizer and detected by PD1. The feedback electronic drives an additional voltage
on the electrodes of the PC depending on the signal level. So the round trip losses are increased
if the signal level rises and therefore the pulse intensity is nearly constant for many
round trips. Figure 17.2 a shows a typical pulse train of the passive feedback controlled
mode locked Nd : YLF laser. The pulse train has a duration of approximately 20 µs with
single pulse energies smaller than 2 µJ.
334

17.2 Experimental
a) b)
c)
Figure 17.2: Pulse trains of a passive feedback controlled mode locked Nd : YLF laser: a) passive feedback
controlled mode locked pulse train without Q factor control (5 µs/div); b) passive feedback controlled
mode locked pulse train with Q factor control after 5 µs (5µs/div); c) nanosecond pulse train
with decoupled single pulse (50 ns/div).
A single energy pulse of several microjoules is too small for application in our picosecond-spectrometer.
To increase the pulse energy we can actively control the Q factor of the
resonator as follows:
The electrical delay line (DL) is triggered by the second photodiode (PD2). The constant
and dynamical voltages on the PC are switched off in several tens of nanoseconds after
an adjustable delay time. Thereby the Q factor of the resonator abruptly increases and the
pulse saturates the gain in a few round trips (Fig. 17.2b). A high intensity pulse train with
nanosecond duration (Fig. 17.2 c) is the result. In our case the time delay for optimum pulse
shortening is about 5 µs (500 pulse round-trips). We achieved a 80 ns pulse train (FWHM)
with a total energy of 2.5 mJ.
One of the first pulses in the train is extracted by an electro-optical pulse selector
(PS). The single pulse has a duration of 7 ps (measured by autocorrelation) and a total energy
of 130 µJ with a pulse to pulse standard deviation of 2%. Amplification of the selected
pulse to an energy of 20 mJ is achieved in three single pass amplifiers. The laser system
works with a repetition rate of 10 Hz.
335

17 Picosecond Laser Induced Photophysical Processes of Thiophene Oligomers
The second harmonic (SH) (l = 523 nm) and the third harmonic (TH) (l = 349 nm)
of the fundamental wave (l = 1047 nm) with type I phase matching for SHG and type II
phase matching for THG are generated in two non-linear KDP crystals. We accomplished
conversion coefficients of 29% for SHG (E 2o = 5.8 mJ) and 11% for THG (E 2o = 2.2 mJ).
The second or third harmonic radiation, used as the excitation pulse, is filtered by the
filter 1 (F1) to remove other wavelengths and to adjust the intensity. Then the beam passes
an optical DL and is focused by the lens 4 (L4) into the sample cell (SC).
The unconverted part of the fundamental wave, separated from the harmonics by
mirror 3 (M3) (HT 523,349 nm ,HR 1047 nm ), is focused by lens 2 (L2) into a 10 cm long D 2 O
filled cell. The generated white light pulse (400–970 nm) passes the IR filter 2 (F2) and is
recollimated by lens 3 (L3). It is necessary, that the probe beam is polarized under the magic
angle (54.78) to the pump beam to neglect orientation relaxation effects in the probe. Therefore,
the white light beam passes the polarizer 2 (P2) followed by the beam splitting device
(BD). One of the two probe beams passes through the sample cell to monitor the absorbance
change upon excitation. The other beam is used as reference. The pump pulse reaches the
sample cell after passing the optical delay with a maximum time difference to the white
light pulse of 2 ns. Test and reference beams are focused onto the entrance slit of a low dispersion
polychromator. Both spectra are detected and recorded by an OMA system (OMA-
Vision, EG & G) with a resolution of 4 nm. The used CCD array with 5126512 pixels
(9.769.7 mm 2 ) allows to detect spectra in the range from 200 nm to 1100 nm with a selected
wavelength interval of 200 nm.
To obtain a high accuracy the detector channel matrix is cooled by a Peltier element
to get a signal-to-noise ratio more than 100. Moreover, we average 40 test and reference
spectra to determine the difference spectrum at each delay setting. From the transmission
measurements with excitation (we) and with no excitation (ne) the difference of the optical
density DD(l,t) is calculated by:
D…l;t†ˆ
2
log6
4
P 40
I t
iˆ1
P 40
I r
iˆ1
3
7
5
2
‡log6
4
P 40
I t
jˆ1
P 40
I r
we jˆ1
3
7
5
ne
; …1†
where I t , is the intensity of the test beam and I r of the reference beam. The lowest variation
of the pulse energy from the fundamental pulse and average procedure gives an accuracy of
DD(l,t) better than 0.01.
The picosecond pump probe technique was build up during the running time of the
project. Therefore, in parallel to the build up of the spectrometer, measurements were carried
out with a version of a Nd : YLF laser without negative feedback. In this case we
achieved a time resolution of 25 ps.
336

17.4 Results
17.3 Spectroscopic properties of oligothiophenes
Oligomer molecules exhibit a benzoid structure with a single bond between the rings. This
structure allows various conformers in solution at room temperature which show deviations
from planarity but only small energy differences [9].
These properties result in the observed wide structureless absorption spectra. In contrast
to the absorption the fluorescence band is well structured. This is explained by the planar
quinoid structure of the fluorescence state. In quantum chemical calculations [6] the
bonds between thiophene rings exhibit electron densities like double bonds.
Both, absorption and fluorescence bands, shift to longer wavelengths with increasing
oligomer size as shown in Fig. 17.3.
a) a)
Figure 17.3: a) Steady-state absorption and b) fluorescence spectra of thiophene oligomers nT with
n = 2–6 monomers in dioxane.
It can be seen from the absorption spectra that all oligothiophenes (nT) except 2T can
be excited well with the third harmonic of a Nd : YLF laser (349 nm). 2T was excited with
the second harmonic of a sub-picosecond dye laser system at 308 nm [7].
17.4 Results
17.4.1 Picosecond-transient spectra of oligothiophenes in solution
The transient spectra DD(l, t) ofnT in dioxane are shown in the Figs. 17.4 and 17.5. At first
for 3T the transient absorption (A1) at 600 nm arises during the excitation. Moreover, the in-
337

17 Picosecond Laser Induced Photophysical Processes of Thiophene Oligomers
a) b)
Figure 17.4: Transient spectra DD(l,t) of 3T (a) and 5T (b) in dioxane after excitation at 349 nm.
Figure 17.5: A2 bands of 3T, 4T, 5T, and 6T1 (substituted with alcyl groups) at 2 ns delay to excitation.
duced fluorescence (F) is to be seen in the wavelength region between 430 and 480 nm.
After excitation a new absorption (A2) arises at that spectral region. After about 400 ps the
band A2 dominates, whereas the A1 band is vanished.
For oligomers nT with n = 4, 5, 6, also fast arising A1 bands occur at longer wavelengths.
Thereby induced fluorescence was also registered. The induced fluorescence of 5T
is very well seen in Fig. 17.4b.
We compared the band shape of the measured induced fluorescence band with the
well-known stationary fluorescence band, shown in Fig. 17.3b. Thereby we found a good
agreement of the maximum position but deviation in band form. This difference is mainly
caused by the different Einstein coefficients for spontaneous and stimulated emission.
The bands A2 also appear at the larger nT, but more delayed as at 3T. They shift more
to longer wavelengths than the corresponding F spectra, so that the superposition of F and
A2 becomes weaker. On the other hand, the A1 bands become wider and thereby they overlap
the A2 bands. At long delay the A2 bands solely exist, whereas A1 and F decrease.
These bands are identical with triplet absorption bands reported from nanosecond to
microsecond laser flash experiments [10].
338

17.4 Results
17.4.2 Time behaviour of transient spectra
The time behaviour of transient spectra is very essential for the assignment of these spectra
to energy levels or species. If possible, independent experimental methods should be used to
gain data of the time behaviour. Therefore, we performed time resolved fluorescence measurements
on the oligothiophenes and obtained the fluorescence lifetimes of nT [3]. Today
other references are also available [11].
The kinetic curves DD(t) at selected wavelengths for 3T (Fig. 17.6) could be fitted
very well with a single exponential function for wavelengths within the A1 absorption band.
Figure 17.7 shows semilogarithmic plots of DD of A1 versus time for 2T–6T.
Figure 17.6: Time behaviour of the absorption A1, A2, and the fluorescence F at selected wavelengths
of 3T in dioxane.
Figure 17.7: Semilogarithmic plot of DD of A1 versus time t for nT, n =2–6.
The obtained lifetimes for the A1 band agree with the fluorescence lifetimes [3, 11],
as demonstrated in Tab. 17.1.
From these facts we conclude for all oligothiophenes that A1 is an absorption from
the excited fluorescing state S 1 .
The development of A2 could be observed at all the longer oligothiophenes (n = 3–6).
The A2 spectra are detectable after the decay of A1 and F in times of about 100 ps (for 3T)
339

17 Picosecond Laser Induced Photophysical Processes of Thiophene Oligomers
Table 17.1: Decay time s A1 of the transient absorption A1 and fluorescence decay time s S for nT in dioxane.
s A1 nT s S
(51 ± 5) ps 2T 40/100 ps [11]
(135 ± 13) ps 3T 150 ps [11]
(531 ± 53) ps 4T (530 ± 53) ps
(880 ± 88) ps 5T (870 ± 87) ps
(1018 ± 102) ps 6T1 (1100 ± 110) ps
and 500 ps (for 4T–6T) and rise to their terminal value during the used delay time. A2 is a
long living absorption. The A2 spectra of 3T to 6T at a delay to excitation of 2 ns are shown
in Fig. 17.5.
Figure 17.8 shows the scheme of processes which proceed after occupation of S 1 by a
one photon absorption process (EPA). The excited state absorption (ESA) indicate a transition
from S 1 to S N and the triplet absorption (TTA) from T 1 to T R , respectively. We assume
that the internal conversion (IC) within the triplet ladder is fast. The rate of S 1 depletion is
k = k R + k ISC (internal conversion is neglectible at nT).
Figure 17.8: Scheme of S 1 depletion and triplet formation after excitation.
The formation of triplets is demonstrated for 3T. According to Fig. 17.8 we consider
the decay processes after d-like excitation with simple rate equations (Eqs. 2 and 3) for the
number density of the excited singlet S1 and triplet T1. The solutions (Eqs. 4 and 5) yield
the time behaviour of the excited states.
dc… S1†
ˆ
dt
k S c… S1† …2†
dc… T1†
ˆ‡k ISC c… S1† …3†
dt
c…
S1† ˆ c… S1† tˆ0 exp… k S t† …4†
c…
T1
k T
† ˆ c…
S1† tˆ0 …1 exp… k S t†† ˆ c… S1† k tˆ0 F T …1 exp… k S t†† …5†
S
340

17.4 Results
2,303 D
ˆ
z
2,303 D
z
ˆ
s F c…
S1†‡s TTA c… T1† …6†
s F c… S1† tˆ0 exp… k S t†‡s TTA c… S1† tˆ0 F T …1 exp… k S t†† …7†
Both, number densities and their emission s F or absorption cross section s TTA , influence
the DD spectra, as can be seen from Eqs. 6 and 7. If triplets originate directly from the
S 1 state, only the S 1 depletion rate determines the DD(t) curve of superposed fluorescence
and triplet absorption. This was proved in a fitting procedure as shown in Fig. 17.9.
Figure 17.9: Experimental results (x) and fitting curve (- -) which describes fluorescence decay and triplet
formation with one common rate constant (k = 1/135 ps).
17.4.3 Size dependence of spectroscopic properties of oligothiophenes
Size dependent steady state spectroscopic properties of thiophene oligomers have been reported
by a series of authors. The goal of our studies was the investigation of the behaviour
of excited levels or species in dependence of the oligomer size.
In the literature mostly you can find a graphic representation of spectral energies in
dependence of the inverse of the monomer or double bond number. This graphic yields a
fast convergence of values (with 1/n) to polymer chains of infinite length. More subtle is
the question at which finite number of monomers the optical properties do not show further
alteration. This gives hints for practical preparation of optical devices and it allows to prove
theoretical models. At our studies of the series of small oligomers, n = 2 … 6 we utilized
various models for the representation of size dependence, e. g. the Lewis-Calvin formula for
p
the absorption wavelengths l = n , the energy gap dependence DE & 1/n, and the extended
FEMO model of Kuhn [13] which yields for the energy gap,
DE ˆ
h2
8mL 2 …2DB ‡ 1†‡V 0 1
1
; …8†
2DB
341

17 Picosecond Laser Induced Photophysical Processes of Thiophene Oligomers
where h is Planck’s constant, m the electron mass, L the box length, DB the double bond
number, and V 0 a constant value.
This formula was estimated for a linear polyene chain. We tried to apply it for the oligothiophenes
and found that this formula excellently fits our experimental results.
Figure 17.10 shows our fitting results for the different models.
Figure 17.10: Description of the HOMO-LUMO transition (fluorescence FL) by different models.
Not only the stationary band maxima of absorption and fluorescence match this formula
but also the radical absorption and even the triplet and singlet excited state absorption.
Thereby the fitting parameters have been altered.
The extension to larger DB values of the curve DE (DB) allows a comparison with the
well-known absorption and fluorescence bands of polythiophene. These values match the
curve at a number of 50 double bonds which corresponds to 25 thiophene rings. This means
larger oligomers show no relevance to optical spectra.
17.4.4 Size dependence of the kinetic behaviour of oligothiophenes
We found excited state absorption and fluorescence bands, both originating from the excited
singlet level S 1 for all nT. During its decay a triplet absorption band was observed. Both processes
slow down with increasing oligomer size.
In other words, the fluorescence quantum yield increases because the triplet quantum
yield decreases. Other degradation processes except emission and intersystem crossing were
not detectable in our experiments.
342

References
17.5 Discussion
The laser pulse initiated processes as observed from sub-picosecond to nanaosecond time
range on a series of oligothiophenes in solution started with the increase of the A1 band during
the absorption of the pulse energy. The A1 bands are S n /S 1 absorptions, identified by
their decay times which are identical to the fluorescence decay times measured by time resolved
fluorescence measurements (Tab. 17.1). The observed induced fluorescence F roughly
decays with the same rate constant as A1 if it is not superposed by absorption bands of different
time function.
The nature of the delayed appearance of the band A2 was enlightened. Thereby it was
helpful to find recent works [10–12] on nanosecond laser flash experiments in which triplet
spectra with microsecond to millisecond lifetimes have been identified. A comparison of
these spectra with our A2 spectra showed that both types of spectra are identical in spectral
shape and position.
This means, in our picosecond experiments we have observed for the first time the
formation process of triplets in the nT series.
It seems to proceed an effective singlet-triplet intersystem crossing process in nT. Reasons
for such a fast intersystem crossing may be the influence of the sulfur atom on the spinorbit
coupling elements on one side. Moreover, we assume that there is a higher lying triplet
level neighbouring the excited singlet state S 1 . This is likely accompanied by thermal initiated
transitions from higher occupied vibronic levels of S 1 to the corresponding triplet states.
References
1. International Conference on Science and Technology of Synthetic Metals, Göteborg, 1992, Synthetic
Metals, 55–57, (1993)
2. H. Chosrovian, D. Grebner, S. Rentsch, H. Naarmann: Synthetic Metals, 52, 213 (1992)
3. H. Chosrovian, S. Rentsch, D. Grebner, D. U. Dahm, E. Birckner: Synthetic Metals, 60, 23 (1993)
4. D. V. Lap, D. Grebner, S. Rentsch, H. Naarmann: Chemical Physics Letters, 211, 135 (1993)
5. D. Grebner, D. V. Lap, S. Rentsch, H. Naarmann: Chemical Physics Letters, 228, 651 (1994)
6. R. Colditz, D. Grebner, M. Helbig, S. Rentsch: Chemical Physics, submitted
7. D. Grebner, D. V. Lap, S. Rentsch: International Conference on Molecular Spectroscopy, Essen,
1994, Journal of Molecular Structure, 348, 433 (1995)
8. D. Grebner, M. Helbig, S. Rentsch: Journal of Physical Chemistry, 99, 16 991 (1995)
9. G. Zerbi, L. Chierichetti, G. Ingänas: Journal of Chemical Physics, 94, 4637 (1991)
10. V. Wintgens, P. Valat, F. Garnier: Journal of Physical Chemistry, 98, 228 (1994)
11. P. Emele, D. U. Meyer, N. Holl, H. Port, H. C. Wolf, F. Würthner, P. Bäuerle, F. Effenberger: Chemical
Physics, 181, 417 (1994)
12. R. Rossi, M. Ciofalo, A. Carpita, G. Ponterini: Journal of Photochemistry and Photobiology, A70,
59 (1993)
13. H. Kuhn: Journal of Chemical Physics, 17, 1198 (1949)
343

18 Topospecific Chemistry at Surfaces
Hans Ludwig Krauss
18.1 Introduction
18.1.1 The problem
Surfaces of solids exhibit different sites – a triviality. Even the simplest model, the ideal
NaCl crystal, shows faces, edges, and corners. Different coordinative situations result in a
different behaviour in all reactions. Of course, systems that are more complex contain a
greater variety of different surface species. Finally, we should expect that in amorphous solids
the topological situation of the surface is almost chaotic – apart from general restrictions
given by the stoichiometry, bond lengths, and bond angles of the specific compound.
Very frequently the surface is chemically different from the bulk’s composition: highly
unsaturated situations are more or less moderated by accommodating molecules (as a whole
or in parts) from the surrounding media like H-OH from moist atmosphere. In case of substances
with high specific surface area, these additional surface groups may form a substantial
part of the whole solid. Silica, made from aqueous solutions of silicates for instance, is
completely saturated at its surface by SiOH groups that react in the usual way, as silanol
groups, with respect to their immobility and to their specific topology in the surface.
Amorphous solids with large specific surface area play an important role in many chemical
processes, especially as supports for heterogeneous catalysts. Therefore, the general
problem is the behaviour of surface compounds as a function of their individual topological situation
that provides information of great practical value, e. g. for the modelling of catalysts.
18.1.2 Preparative and analytical methods
The use of selected surfaces as educts for chemical reactions is very common, e. g. in semiconductor
modelling. As another example the intercalation may be mentioned as a typical
344 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

18.1 Introduction
reaction of layer compounds. The preparation of catalysts however usually does not include
crystalline substrates. Here it is the main objective to provide a large specific surface, as offered
e. g. by an amorphous substrate; that includes the inevitable presence of a variety of
many topologically different surface sites or surface groups. When these groups react as anchoring
sites for the specific catalytic agent, e. g. a metal compound, the topology of the
support at a given site controls the physical and chemical behaviour of this individual surface
compound. Vice versa the metal may act as a probe to allow a characterisation of the
site in question.
We must keep in mind that real surfaces are part of a usually disordered solid and in
addition act as an inseparable ensemble of sites. Therefore, the analytical approach is restricted
to methods that do not require free mobility of the target group or a long distance
order of the solid. So the most powerful tools of the modern analytical chemistry, NMR (except
MASNMR), and X-ray structure analysis, are ruled out. Others may be used with considerable
limitations only.
18.1.3 Industrial applications
Surface chemistry can follow two different objectives: the production of a constructional
matter with special properties on its border to the environment, e. g. a coated polymer, or
the preparation of a material with special functions to the environment, e. g. a catalyst. While
the former aspect has attracted the attention of chemists increasingly in the last years, the research
in catalysis at surfaces is an old field. The advantages of heterogeneous catalysts
compared with their homogeneous analogues are well-known and have caused an extensive
search for such systems. On the other hand, in many cases the mechanisms of the reactions
at the active surfaces are not well understood – even if the process is used on a large-scale
production worldwide. The Haber-Bosch process e. g. finally could be explained after almost
a century of research. The mechanism of the Ziegler-Natta process needs still further studies
and the understanding of the Phillips process [1, 2] is insufficient yet.
18.1.4 The standard procedures of the Phillips process
The enormous variety of parameters in surface chemistry requires a restriction to typical
systems. On the background of our earlier work on chromium and its redox chemistry [3],
the Phillips catalyst for olefin polymerization seemed to be a good example for the general
problem of reactions with topologically defined educts. Here we find an inert oxidic support
with a large specific surface (usually silica), the chemical fixation of a metal compound
(usually chromium) at this surface, a thermal or chemical treatment to convert some of the
Cr surface centres to catalytically active species, and finally several uncleared catalytical
processes at these species. In short, we find all the elements of a challenging chemical problem
under the aspect of a topological regulation. Of course we varied the standard prepara-
345

18 Topospecific Chemistry at Surfaces
tion of the catalyst, i. e. impregnation of silica with CrO 3 in aqueous solution and thermal
activation in air, as given in the first patents [4] almost 40 years ago – not to achieve better
yields of the catalytic reactions but to understand the influence of the topologic parameters.
18.1.5 Earlier work
Since its discovery in 1956, the Phillips process [5] was an object of research in many parts
of the world. Our own activity started in 1965 with redox studies [6]. These lead us to the
discovery of coordinatively unsaturated chromium(II) as a substantial constituent of the surface
chromium in active catalysts [7, 8].
The catalytic activity was of course one of the important properties to be determined
as a function of the preparation conditions (and hence the topological situation) of the metal
surface compounds. The variation of the catalyst as well as the variation of the reaction to
be catalysed was within the scope of the continuation of earlier work. Our interest was and
still is focussed here on the mechanism of the catalytic reactions. With respect to the restricted
analytical situation, the spectrum of the products was an important but often ambiguous
source of information.
In the next paragraphs our results of the last decade will be reported, arranged by following
the steps of the catalyst’s preparation: starting from a selected support, preparing a
variety of surface species, exploring the different reactions of these species. However, the
chronological progress of the work did not follow this one-dimensional pattern. Actually, the
variation of single parameters causes usually changes in a multidimensional network of responses.
For drawing a coherent picture, old and new results should be put together – like
pieces of a jigsaw puzzle just one as we hope.
18.2 The support
18.2.1 Unmodified silica
Silica with its constituents Si, O, and H may be regarded as one of the simplest supports.
The huge number of varieties however shows that the architecture of the material is manifold
and sensitively dependent on the preparation. Attempts to construct silica or just an amorphous
SiO 2 by a computer program gave unsatisfactory results [9]. Since the basic work of
Peri and Hensley [10], a simplified model of cristobalite planes is used to explain the properties
of the silica surface, neglecting the aspect that the fractal dimension of the latter may
reach almost the value three. The thermal behaviour of geminal, vicinal, or isolated OH
346

18.2 The support
groups can be interpreted with this model. Experimentally it was shown earlier that the release
of water from vicinal pairs around 680 8C (DTG experiment) is the most important dehydration
process [11]. New work using MASNMR for 29 Si confirmed the existence of geminal
pairs and predominantly single (vicinal, isolated?) OH groups in the untreated gels [12].
After thermal treatment, unfortunately, no well-resolved spectra were obtained.
By electron microscopy we found a general three-stage organisation of the gels: raspberries
with 100–200 nm diameters were composed of small balls with ca 40 nm diameters,
these again consisting of smaller irregular units with diameters of about 10 nm [13]. The
compact particles of some of the gels exhibit surfaces looking like a cobble stone pavement
of narrow-packed 40 nm spheres.
Usually we worked with the supports Merck 7733, Grace 952, Grace XWP types, and
Degussa Aerosil 100 or 200. Different gels had to be used for different experiments. For instance
rigid gels, like Merck 7733, give very good UV-VIS-NIR spectra in reflectance. However,
they are not suitable for ethylene polymerization catalysts, which should easily undergo
a structural degradation (popping) during the formation of the solid polymer. On the other
hand they work well when the polymer is soluble: The polytest with 1-octene, see below,
showed an almost linear dependency of the catalyst’s activity on the specific surface but no
correlation with pore volume or pore diameter [14].
18.2.2 Modified silica
Although industrial applications recommend the use of silica supports with built-in ions of
Ti(IV), Zr(IV), Al(III), Mg(II), and fluoride to increase the activity of polymerization catalysts,
our polytests with 1-octene showed no influence in the cases of Ti, Zr, and Mg. Probably
their effect is restricted to the polyethylene case, here disturbing and destabilising the
structure of the support (easier popping!). The use of Al doped silicas, e. g. Grace 13–110,
increased the activity of the catalyst under otherwise equal conditions by a factor of 2.5 to 3
[14]. This effect may be explained by an electrostatic influence of the Lewis acid Al(III) enhancing
the electronegativity of the gel. By 27 Al MASNMR experiments we could show that
in these gels Al(III) occurs primarily in a tetrahedral configuration. Thus, Al-OH groups are
not detected [12]. Hence, a later fixation of the transition metal as Al-O-M surface compound
may be neglected. The effect of fluoride doping again might be due to an electrostatic
effect. Unfortunately MASNMR measurements were not possible because of interference
with Teflon parts.
Because these modified systems did not promise new evidence in the basic question
of topology but even increased the number of parameters, further work was restricted to the
use of pure silicas as support.
347

18 Topospecific Chemistry at Surfaces
18.2.3 Others
The use of other supports like gels of Al 2 O 3 , AlPO 4 (isoelectronic to SiO 2 ), and activated
carbon was shown to be possible. In the latter case, the reduction of the added Cr(VI) compound
to the active species is carried out by the support itself at higher temperatures [15].
At present no significant advantage over the conventional systems was found.
18.3 Transition metal surface compounds
18.3.1 The metals
The formation of surface compounds from genuine surface groups, e. g. Si-OH, and a metal
compound usually is a condensation releasing water, HCl, NH 3 , and similar products. After
impregnation and drying, this preparation step is performed mostly at higher temperatures
(activation). The thermal treatment does not only allow a fixation of the metal compound at
topologically and energetically different sites but simultaneously includes the condensation
of Si-OH groups among each other, as stated above. Table 18.1 lists the metal applications
used in this work.
Table 18.1: Metal applications used in this work.
Metal Compd. Medium T act 8C Atm. act Ox.Nr. Ref.
Ti TiCl 3 water 500 oxygen &4 16, 17
V (NH 4 ) 3 VO 3 water 800 oxygen &5 16, 18–21,
VOCl 3 gas phase 800 oxygen &5 20
Cr CrO 3 water ^800 oxygen &5.9 13, 22
Cr 2 ac 4 72H 2 O NH 3 /H 2 O 800 oxygen &5.7 23
Fe FeCl 2 , FeCl 3 water 800 vacuum &2.2 24
Cu Cuac 2 7H 2 O water 800 oxygen &2 25
Mo Mo-diglycolate water 800 oxygen &6 26
(NH 4 ) 6 Mo 7 O 24 water 800 oxygen &6 27, 28
Mo 2 ac 4 NH 3 /H 2 O 800 oxygen &6 23
Mo/Cr CrMoac 4 NH 3 /H 2 O ^800 oxygen &6 23
Ru “RuCl 3 7H 2 O” acetone ^600 oxygen ? 29
(For Mn, Co and Ni see Ref. [30, 31])
In the following, the systems with the metals Cr, V, and Mo will be discussed in more
detail.
348

18.3 Transition metal surface compounds
18.3.2 Impregnation and activation
As reported in Table 18.1, the impregnation was mostly carried out from aqueous suspensions
where the metal compound was dissolved in a concentration suitable for the experiments.
Usually for chemical work, the applied concentrations are smaller than 1 wt% of
metal in the final product. Since the support offers just a limited number of sites for the
accommodation of the metal a further increase of the metal compound concentration is not
incorporated into the surface reaction. These free metal compounds may lead to a decomposition
if heated during activation. In the case of CrO 3 an oxidation state of 6 in the activated
product is reached only if c(Cr) ?0. In the case of 1% Cr and activation in oxygen
the product has an oxidation number near 5.9. Free CrO 3 is decomposed to Cr 2 O 3 , as
shown by REM, EDX [6, 58, 59]. (Only the surface bond Cr(VI) is a candidate for the later
reduction to the coordinatively unsaturated sites (c.u.s.) Cr(II), see below.) The average
oxidation number approaches finally the value 3 for very high loads; there are no intermediate
steps.
Another information on the oxidised form of the surface compounds can be derived
from the photoluminescence, which is observed in the cases of vanadium, chromium, and
molybdenum. Considering the best investigated case, surface chromium(VI), we find a drastic
increase of luminescence intensity at an activation temperature around 600 8C (Fig. 18.1).
The emission shows now a well-resolved fine structure while an unstructured background
decreases in intensity [50].
Figure 18.1: The influence of T act on different activities of surface Cr(VI) and the later c.u.s. system:
T and solid curve: photoluminescence [50], & chemoluminescence [50], y polymerization activity (formation
of poly-1-octene, 1/t [14]), _ Cr s /Cr(II) under standard conditions [72].
349

18 Topospecific Chemistry at Surfaces
Thermal treatment of products that were impregnated with metals in lower valences,
e. g. Fe(II), results as well in a fixation of the metal – releasing now the protonated anion
and water, see below. The impregnation with Cr(II)/Mo(II) [23] requires a pretreatment of
the support with aqueous ammonia to increase the nucleophilic character of the SiO
groups. The question of fixation of aggregated metal units, e. g. dichromate(VI), has not
been generally settled yet. For the vanadium case there is evidence from UV, VIS, NIR,
and Ra spectra that polymeric surface vanadates(V) are formed already at very low concentrations.
Thus chain and band structured vanadates can be found [21, 32]. This tendency
continues up to the formation of separate V 2 O 5 phases at higher loading. It may not
be excluded that the reduced species exhibit a certain mobility in the surface and perform
a redistribution at higher temperatures to reach the thermodynamically most favourable
situation.
18.4 Coordinatively unsaturated sites
18.4.1 Reduction of saturated surface compounds
By the pretreatment with oxygen at high temperatures the surface compounds contain the
metal in a high oxidation state and are coordinatively saturated. By reduction the situation is
changed. Oxidation and coordination numbers drop drastically, the metal forms a c.u.s. For
instance the surface chromate(VI) – anchored via two oxygens to the support and saturated
by two other oxygens – is reduced to a surface chromium(II) with only two links to the support
and four vacant coordination sites:
surface(Si-O) 2 CrO 2 +2CO ! surface(SiO) 2 Cr W 4 +2CO 2
This idealised formulation neglects the fact that even in the rather immobile situation
of the metal in the surface there will be at least the chance of a partial saturation by interaction
with neighbouring groups in this surface, e. g. SiOH or eventually SiOSi [33, 34]. The
extent of this saturation will depend on the existence and the topological accessibility of
such groups. The important parameter is therefore, besides the concentration of the metal in
the surface, the relative concentration of SiOH groups next to the metal. Decisive are the activation
and reduction temperatures which influence the amount of SiOH and the mobility of
the Cr(II) ions respectively. At a given Cr concentration high activation temperatures of
800 8C and low reduction temperatures of 350 8C (800 8C/350 8C) favour the formation of
highly unsaturated metal sites. Low activation temperatures and high reduction temperatures
(e. g. 500 8C/500 8C) on the other hand foster a coordination of surface groups to the metal,
yielding a certain saturation. Catalysts prepared with these two different thermal combinations
will be used in the following as standards.
350

18.4 Coordinatively unsaturated sites
Considering the amorphous structure of the support, a large multitude of different individual
positions of the metal on the surface was to be expected. By spectroscopic studies [35–
37] we found surprisingly that surface chromium(II) occurs in only two discrete topological
situations, which we have called A and B. Our standard catalysts (800 8C/350 8C and 500 8C/
500 8C) are dominated by the species A or B respectively. But there is no way to prepare samples
that contain only one of these species. The determination of the population profile of A
and B species in different Cr catalysts by TPD with innocent ligands confirmed this result
[14, 38]. Since there is a certain (energetic) bandwidth of both species they better should be
seen as ensembles. Analogue observations were made with surface compounds of Cu, Fe, Mo
[16]. Vanadium is a special case because of its agglomeration tendencies [32].
The generally accepted interpretation at present is the assignment of the coordination
numbers 2 for the Cr(II)A and 3 for the Cr(II)B, where the third site coordinates oxygen of
neighbouring SiOH or SiOSi. A third species (ensemble) of surface chromium was reported
by the Torino group [39] in 1987. If a reduced catalyst is heated in vacuo or in inert atmosphere
to higher temperatures, say 600 8C, a substantial part of the Cr(II)A is converted into
an almost inert Cr(II)C with a proposed coordination number of four. It is assumed by the
authors that highly exposed Cr(II) ions may migrate to a thermodynamically more favourable
position in the surface-near part of the bulk, reaching there a four-fold coordination.
Consequently the new species behaves semisaturated and may be distinguished from A by
its lack of reactivity vs. certain ligands. Cr(II)B is not affected by the thermal treatment.
In the light of new results we believe that this concept of Cr(II)(A,B,C) has to be reconsidered,
as will be discussed below. Here we will just emphasise the surprising fact that
the surface of amorphous silica offers only two or three topologically different positions for
a two-valent metal ion. The origin from a covalent precursor – in the chromium case the
mixed anhydride of poly-silicic acid and chromium(VI) acid – is not essential, since systems
impregnated with Cr(II), Cu(II), and Fe(II) show the same phenomenon. We can argue that
the different topology is inherently coming up in the activation step already (where we see
the support attacked by a twotined fork of the metal compound). However, it needs the unsaturated
character of the low valence ion for its chemical expression.
How about the other metals? In the case of molybdenum a simultaneous irradiation
with UV light is necessary to get the Mo(VI) attacked by CO [26, 27]. Even then, the average
oxidation number reached was not lower than 2.7 (oxygen titration [27]). Spectroscopic
evidence shows that the product contains Mo(IV) and Mo(II), formed by the reduction of octahedral
and tetrahedral Mo(VI) [28]. Vanadium(V) requires a very long time and higher
temperatures (600 8C) for the reduction by CO [19, 40] (slow de-aggregation?). For iron and
copper see Section 18.4.2.
So far, we have formulated the reductions with CO as the reducing agent. Of course
other means are as well applicable. Each has its specialities: CO for instance must be removed
from the catalyst at the reduction temperature by vacuum or argon to avoid complex
formation. Hydrogen does not form complexes but produces water which can hydrolyse the
bond between metal and support of the not yet reduced units. The reductive effect does not
surpass the one of CO. Other reducing agents like SO 2 do not offer any advantage either.
The use of Li(AlH 4 ) or of activated magnesium was not successful.
To prepare the catalytic centres for a later reaction of olefins with the c.u.s., hydrocarbons
(alkanes, alkenes) may be used as reducing agents. Here the extent of the reduction is
usually lower. Cr reaches oxidation numbers near three [14]. The great variety of organic re-
351

18 Topospecific Chemistry at Surfaces
action products makes an interpretation difficult. Since for some polymerization processes
the addition of a cocatalyst (AlR 3 , AlR n Cl 3–n , SnR 4 ) is useful or even necessary, the reductive
potentials of these metal alkyls were tested as well. Here we observed a well-defined
two step reaction from Cr(VI) via Cr(IV) to Cr(II) [41].
18.4.2 Elimination of ligands
The removal of the oxygens of surface Cr(VI) may be classified as a reductive elimination
of ligands. If from the beginning the metal is applied in a low valence state the formation of
the c.u.s. requires just the (thermal) removal of the outer ligands from the anchored surface
compound:
surface(Si-O) 2 ML n
! surface(Si-O) 2 M_ n +nL.
Fixation and ligand desorption usually are not separated processes, but are characterized
by an overlap that can be studied by analysing the exhaust of a TPD experiment. In
Fig. 18.2 this is shown for FeCl 2 . The reaction can be carried out in an inert, i. e. oxygenfree,
gas stream (Ar) or under high vacuum conditions. Cr and Mo as well may be also applied
to the support in low valence state as Cr(III) [42–47] or Cr(II)/Mo(II) [23].
Figure 18.2: Desorption of ligands from the system FeCl 2 7aq/silica [24]. Release of H 2 O and HCl vs. T.
352

18.5 Physical properties of the coordinatively unsaturated sites
18.5 Physical properties of the coordinatively unsaturated sites
18.5.1 Topologically different sites
As stated earlier, the preparation of the c.u.s. on amorphous substrates does not yield a uniform
surface compound. Depending on the individual position of the metal on the surface,
usually two discrete, topologically different ensembles of species are formed simultaneously.
Their relative amounts depend on a variety of parameters. The temperatures of activation
and reduction have the most serious influence. Of practical importance are further the nature
of the support, its thermal history, the concentration of the metal, the metal compound used
for impregnation, and (if applied) the reducing agent.
18.5.2 Optical and magnetic properties
For measuring the relative amount of the different species, physical methods play the most
important role since they do not influence their ratio. The impressive colours of the catalysts
recommend the use of optical (reflectance) spectroscopy. In the chromium case, for example,
the characteristic absorptions (Fig. 18.3) allow a deconvolution and – with certain assump-
Figure 18.3: UV-VIS-NIR reflectance spectra of surface chromium on silica. — T act = 800 8C and T red
350 8C, green product; - - - T act = 500 8C and T red = 500 8C, blue product.
353

18 Topospecific Chemistry at Surfaces
tions with respect to the relative e values – an estimation of the relative amounts of the A
and B species [13, 37, 38]. If we vary the activation temperature at otherwise constant preparation
conditions the intensities of the characteristic bands for Cr(II)A and B indicate that
the A type is increasing up to 600 8C with a corresponding decrease of the B type
(Fig. 18.4). This coincides roughly with the release of water from vicinal SiOH groups
(680 8C, DTG). Heating the reduced sample (vacuum, 600 8C) diminishes the intensity of
the Cr(II)A bands but does not cause the appearance of new absorptions to be attributed to a
Cr(II)C – provoking some doubts about this species, see below.
Figure 18.4: Areas F rel of the main absorption bands of Cr(II)A at 6.8610 3 cm –1 and 11.9610 3 cm –1
(black), Cr(II)B at 8.6610 3 cm –1 and 13.5610 3 cm –1 (open) in deconvoluted UV-VIS-NIR reflectance
spectra vs. T act .
The results with other metals are less advanced. The study of the iron system suggested
the use of Mößbauer spectroscopy. Unfortunately, the spectra showed a very modest
resolution because of the experimental conditions (c (Fe) < 8%, airtight cuvette, no enrichment
of Fe-57 isotope). The spectra cannot be interpreted conclusively [24].
354

18.6 Chemical properties of the coordinatively unsaturated sites
18.6 Chemical properties of the coordinatively unsaturated sites
The different grade of saturation defining the two species A and B is of course reflected by
their chemical reactivity.
The most striking observation offers the reaction with oxygen: while Cr(II)B dominated
samples show here a dark red glowing chemoluminescence, Cr(II)A dominated samples
react under emission of a bright blazing red light [48, 49]. Both phenomena parallel the
features shown by the photoluminescence of the activated samples, mentioned above, especially
the critical dependence on the activation temperature (Fig. 18.1). The reaction pathway
is well understood, including the intermediate formation and decomposition of peroxo species
[50]. Surface Cr(III) does not react with oxygen. As to other metals, the reduced surface
species of vanadium and molybdenum show a green or a blue chemoluminescence respectively.
These reactions are not yet explored in detail. An intermediate peroxide formation is
observed in the vanadium case [30, 40].
Addition of donor ligands results in the formation of well-defined complexes. Varying
the preparation conditions of the catalysts and p, T of the ligand addition, the resulting UV-
VIS-NIR spectra allow a discrimination of the different band systems together with the determination
of the correct parameters for the deconvolution [37, 38]. This requires some instinct.
In the chromium case, together with the 2 bands for Cr(III) and 263 bands for the
two Cr(II) species there appear the new bands for Cr(II)A.L n with n = 1, 2, 3 and Cr(II)B.L m
with m = 1, 2. As shown by the stoichiometric range, the A species offers one more vacant
coordination site. The complex formation occurs with practically every n-donor (including
even R 3 SiOSiR 3 ) and p-donor (including C 6 H 6 ). The extent of the reaction is a function of
the A/B ratio, the donor properties of L, the concentration (partial pressure) of L, and the
temperature [33, 37, 38]. The formation of complexes with CO and NO was studied thoroughly
in the IR elsewhere [39, 51–53]. There are indications for the occurrence of neighbouring
Cr centres that accommodate CO as a bridge [54].
It should be mentioned that organic p-systems, like the double bond in olefins, benzene,
or the cyclopentadienyl rings are added as well. The formation of 1:1 or 1 : 2 complexes
is obviously the first step of all interactions of olefins with the catalytic centre [55].
Not all of the additions are completely reversible. The temperatures necessary to volatilize
the ligand are often too high to keep the system redox inert. Some of the ligands, e. g.
NO, are irreversibly added, others as N 2 O are decomposed at moderate temperatures under
vacuum conditions (here to N 2 and surface Cr(IV)=O) [38]. Surface Cr(II)(H 2 O) 2 undergoes
a redox reaction at 600 to 800 8C to form surface Cr(III) and hydrogen [38, 56].
18.6.1 Survey of catalytic reactions
The early work with the c.u.s. already included the study of catalytic reactions [57]. On the
one hand, these were redox cycles like
CO + 1/2 O 2 ! CO 2 or SO 2 + 1/2 O 2 ! SO 3
355

18 Topospecific Chemistry at Surfaces
that include both, the oxidised and the reduced form of the metal. On the other hand, there
are reactions that concern the c.u.s. only, like the polymerization / oligomerisation, isomerization
and metathesis of olefins as well as the cyclisation of acetylenes and the Fischer-
Tropsch-like processes.
18.6.2 Olefin polymerization
These reactions include the Phillips process, i. e. the polymerization of ethylene at
T&105 8C and pressures around 30 bar [4, 5]. Of course, this procedure attracted our special
attention. In spite of its worldwide use, which makes more than 1/4 of the world production
of HDPE, and after almost four decades of research the mechanism of the reaction is
still a matter of controversial discussions [1, 2]. The varying results are partially due to different
experimental starting points (e. g. different supports), occasionally also to the use of
inadequate measurements, and the neglect of involved parameters.
If we consider the case of ethylene polymerization, the main experimental obstacles are
the necessary application of pressure and the insolubility of the high molecular product. A
REM picture (Fig. 18.5) shows the growing polyethylene worms [58, 59]. We tried to avoid
these drawbacks studying the polymerization of higher (liquid) 1-olefins, preferentially 1-noctene
[60, 61]. With n C 6 4 the polymers are easily soluble in alkane solvents. This allows
to apply high concentration of the monomer at normal pressure, to analyse the reaction mixture
at any wanted time, and to use classical procedures for the study of the products [61].
According to these experiments the polymerization of olefins is caused by a member
(all members?) of the A species, while the B type centres cause some isomerization of the
monomer. With catalysts of the same preparative history the specific polymerization activity
is not dependent on the Cr concentration up to 1% [61, 62]. The reaction rate (as measured
by the decrease of the monomer) is decreasing with n C from ethylene to 1-hexene [22].
Figure 18.5: REM picture of growing polyethylene on surface Cr(II) [58, 59] with a rock of inert
Cr 2 O 3 . The bar corresponds to 1 mm.
356

18.6 Chemical properties of the coordinatively unsaturated sites
Higher olefins polymerize practically with the same rate as C 6 . The comparison was carried
out at deep temperatures to allow the gas phase olefins to be present in solution in the same
starting concentration as the liquid members. This offers the chance of copolymerizing different
olefins with n 6 6 without discrimination [60]. As to the products, the 1-olefins form
comb-like polymer molecules with a practically unbranched backbone. The broad distribution
of the molecular weight (MW) around 10 5 shows at low polymerization temperatures a
three-modal profile that collapses at higher reaction temperatures to a mono-modal distribution
with a lower MW value for its centre [61]. The chains contain one double bond per molecule
[61].
The use of the new system at ambient pressure and at moderate temperatures allows
to follow the reaction kinetics by analysing the concentration of educts and products as a
function of time. Batch experiments showed that there is a marked induction period of slow
reaction progress (Fig. 18.6) [22, 60, 63]. The systems exhibit the characteristics of an autocatalytic
reaction. Addition of new monomer up to the original starting concentration during
the conversion of the monomer does not result in a new induction period but shortly after
completion of the reaction the system answers to another addition of monomer with a new
induction period [14, 64]. In steady state experiments the polymerization was proceeding
over days with constant reaction rate, i. e. without decrease of the catalyst’s activity. The re-
Figure 18.6: Reaction of surface Cr(II), T act = 800 8C and T red = 350 8C, with 1-octene at room temperature
[22]. Decrease of monomer and increase of polymer show a marked induction period; the formation
of isomers (triangles) is marginal.
357

18 Topospecific Chemistry at Surfaces
action was shown to be of first order in monomer, solving an old controversy in the literature
[63]. The ethylene system does not allow clear answers since the solid reaction product
causes difficulties in the data interpretation, not to speak of problems with collecting the
samples at high pressure and high temperature conditions.
A standard reaction with 1-octene (polytest) was used for a quick characterisation of
the single catalysts. The time t for 50% consumption of the monomer measures the general
activity in 1/t. The time for 10% consumption gives an information on the induction period.
The GC analysis after the completion of the reaction shows an additional formation of low
MW products, for example isomers [14].
Of course it was tempting to connect the observed catalytic properties with the variation
of certain parameters. Primarily the old question of oxidation number and coordination
number of the active centres deserved new studies. By variation of the reducing agent – now
including methane, ethene, ethylene, and isobutene – we learned that an oxidation number
of 2.0 is not favourable. Surprisingly catalysts with oxidation numbers of 3.0 +0.5 were by
far superior to the fully reduced samples [14]. This result agrees to our earlier observation
that surface Cr(III) species, prepared from surface Cr(II)(H 2 O) 2 by heating in argon, are
good catalysts for ethylene polymerization [56]. Even sheer heating of a Cr(II) catalyst to
800 8C in vacuo increases the activity considerably while the oxidation number rises simultaneously
[23]. Lunsford [43] published results, which show a clear coincidence of the activities
of a special surface Cr(III) and the heated Cr(II) catalysts. According to Garrone et al.
[34] the latter should contain the species Cr(II)C besides residual Cr(II)A and the original
share of Cr(II)B, which is not involved in the polymerization reaction. We are inclined to
postulate that Cr(II)C does not exist at all but rather is a Cr(III) species. This is supported
by the fact that there are no Cr(II)C bands in the UV-VIS-NIR spectra. The absorptions may
be hidden in the Cr(III) bands. If so, the transition Cr(II) to Cr(III) must produce a reduction
product, too. Hydrogen from neighbouring SiOH is the only candidate, because of the lack
of H 2 O molecules that are not present anymore:
(surface SiO) 2 Cr(II) + surface SiOH ! (surface SiO) 3 Cr(III) + 1/2 H 2
Heating of Cr(II) impregnated catalysts [23] to 800 8C in Ar produces contacts with
the high oxidation number 2.80! that polymerize olefins without induction period. If the
same starting product is activated in oxygen and reduced in CO (O.N. 2.14) it reacts with induction
period, as usual. So we might conclude that the appearance of an induction period is
caused by a slow transition of a precursor surface Cr(II) to surface Cr(III). However, the active
centres in the first mentioned sample are obviously different from active species that
are produced from Cr(II) during the induction period. Only the latter are transformed back
to their precursors by lack of monomers [14].
This behaviour demands a reconsidering of the Cr(II)A concept. Up to now the A species
was regarded to be well off from reactive surface OH groups. The marked and steady increase
of the A/B ratio up to an activation temperature of 600 8C, roughly correlated with the
disappearance of vicinal SiOH at 680 8C [11], seems to support this assumption. Now this
model may still be applied for some subspecies of Cr(II)A, but obviously not for that substantial
part of the A centres which may undergo a redox reaction with neighbouring SiOH.
There is another experimental access to this problem. If the support was treated in the
beginning with D 2 O (H/D exchange 50 %), the final catalyst forms partially deuterated poly-
358

18.6 Chemical properties of the coordinatively unsaturated sites
1-octene and polyethylene [65]. Thermal deactivation (see below) yields hydrocarbons that
contain one or two deuterium atoms [66]. Therefore the active centre had to have some
SiOD surface groups in reach. Therefore the addition of ligands should be rather considered
as a kind of ligand replacement.
This new model is indeed a very old one (Hogan, 1956) [5] and includes a simple formulation
of the starting step of the polymerization cycle yielding an active species with
Cr(IV),
(surface SiO) 2 Cr(II) + surface SiOH + C 2 H 4 ! (surface SiO) 3 Cr(C 2 H 5 ).
The consumption of the hydrogen by formation of an ethyl group is, however, in competition
with the use of this hydrogen for the formation of an active surface Cr(III) from a
Cr(II) precursor as formulated above. Further arguments against the ethylated Cr species
come from the IR spectroscopy of centres in polymerization. In careful studies the Torino
group [34] showed that there is no n CH of a CH 3 (end)group in the IR spectra. Correspondingly,
the authors support a chromacycle model for the growing chain. If both, C 2 H 4 and
CO, are present at the virginal Cr(II) centre a surface chroma-cyclohexanone-1 is formed, as
is proved by IR [34]. Finally, according to the NMR spectra of polymers, formed with deuterated
catalysts, the incorporation of the D atoms does not result in a formation of –CH 2 D
groups [65].
To make the picture even more complicated the steady increase of A-type centres with
T act going from 300 8C to 600 8C is not paralleled by some other data which are ascribed to
the Cr(II)A species or its Cr(VI) precursor. As shown in Fig. 18.1, the structured chemoluminescence,
photoluminescence, polymerization activity, and content of CrC s-bonds in the
working catalyst 1 rise steeply in the range of 450 8C to 700 8C with a turning point near
600 8C. At this temperature the Cr(II)A bands in the VIS spectra are already fully developed
(Fig. 18.4).
What is Cr(II)A? Originally defined by its reactivity vs. CO (in contrast to the more inert
Cr(II)B [35]), the highly unsaturated character was quantified by measuring DH of the catalyst’s
reaction with O 2 [55, 67–69]. The temperature-programmed desorption of innocent ligands
follows the same basic concept (population profile) [13, 38]. Another approach is the
deconvolution of the spectra of real catalysts, first used by Blümel [37]. However, the rough
coincidence of changes towards A properties by certain treatments is too inexact a measure. A
comparison of Fig. 18.1 with Fig. 18.4 shows that further structuring is needed. Especially we
have to become acquainted to the idea that at least part of the Cr(II)A is positioned in the
neighbourhood of OH groups. This part is converted to Cr(II)C/Cr(III) by heating.
Under topological aspects an impregnation of the support with dimeric Cr(II)acetate
and its Mo and Cr/Mo analogues were of interest. There are clues that a pairwise fixation is
obtained and pertained during the following preparation steps [23]. However, the differences
in catalytic behaviour vs. standard catalysts are not very pronounced.
1 Such bonds were shown to be present and determined quantitatively by quenching the polymerization
by oxygen and analysing the aldehydes formed [70]. They represent up to 10% of the Cr(II) [71, 72].
The details are not yet fully understood. With 1-octene as monomer the s-content is rising linearly
with the progress of the reaction [73]. A well-expressed maximum of the s-bond formation is shown
at a Cr concentration of 1%, maybe due to a topologically special situation [72].
359

18 Topospecific Chemistry at Surfaces
As expected all metals have their individual catalytic profile. The chromium system is
by far the most suitable for the use as polymerization catalyst. Vanadium(III) without cocatalyst
requires UV irradiation to yield at least some polyethylene besides oligomers [19].
With addition of Al(R,Cl) 3 polyethylene is formed after a long induction period in reasonable
yields but with very high MW of about 6610 6 [21]. Higher 1-olefins form dimers in
good yields even if the double bond is in an internal position [19]. Alkanes (solvent!) may
block the free coordination sites at surface V(III) and cause very long induction periods [19,
20]. The reaction of reduced surface molybdenum with 1-olefins produces metathesis products
besides oligomers [27]. With copper the catalytic reaction (with ethylene) was shown
to be restricted to the Cu(II)A species. After a prior complex formation the polymerization
produces oligomers with even and odd C numbers. Additionally benzene, toluene, and ethylbenzene
are found [16, 74, 75].
Since there is no obvious connection to topological facts, the effect of Lewis acids
(Al(R,Cl) 3 , SnCl 4 ) as cocatalysts is mentioned here only marginally. The induction period is
diminished and the reaction is accelerated while the spectrum of products is shifted towards
isomerization [22, 23]. Furthermore the Lewis acids may attack the SiOSi bonds of the support
and so – by loosening its rigidity – make the popping of the catalyst grains easier (polyethylene
process!). At high concentrations of the cocatalyst, a splitting of SiOM bonds may
occur as well [21]. Finally Al(R,X) 3 acts as reducing agent with surface Cr(VI) [41].
18.6.3 Other catalytic reactions
As already mentioned the catalysts show a certain isomerization activity that may become
the main reaction under certain experimental conditions. This is the only activity of the
Cr(II)B species vs. olefins. The mechanism looks like an acid catalysis, using the protons
from to chromium coordinated OH groups. Addition of Lewis acids supports this reaction.
As mentioned above, Mo containing catalysts can react with olefins under metathesis
[23, 27]. This shows that the carbene mechanism is at least not excluded for this type of surface
compounds.
Using mixtures of olefins and hydrogen as educts the olefins undergo hydrogenation
under mild conditions and with quantitative yields [61]. With Ru catalysts this is the typical
reaction [29].
Finally it should be mentioned that Cr(II) surface compounds may act as catalysts in a
Fischer-Tropsch system. From H 2 and CO or CO 2 they produce methane with some ethane
and propane. The reaction pathway may follow the mechanism given 1976 by Henrici-Olivé
and Olivé [76]. Important steps are here the intermediate formation of a formaldehyde complex
follow C 1 carbene complex [77].
360

18.7 Deactivation
18.7 Deactivation
If the reaction of Cr(II) catalysts with olefins is carried out at higher temperatures, oligomer/polymer
chains with odd carbon numbers appear, often with the same MS intensity as
their even numbered neighbours [70]. Furthermore CH 2 units are transferred [66, 78], H 2 ,
CH 4 ,C 2 H 6 ,C 3 H 8 , and n-C 4 H 10 are formed [66, 79, 80], and finally the catalyst is deactivated.
All the Cr(II)A is converted into a new deep green species that was believed for a
long time to be a bis-Z 3 -allyl-Cr(IV) complex [15, 59, 80]. These processes start slowly at
120 8C [78], while at 300 8C the deactivation reaction is completed within seconds. The reactions
require intermediates that may undergo a C–C cleavage, as it is commonly observed in
carbene complex reactions. Such intermediates, recently proposed for the Ziegler-Natta
polymerization [81, 82], can indeed also be formulated for the Phillips process [83]. Since
the carbene complex moiety can also be formulated as a hydrido metallacycle (no CH 3 !),
this model seems to us to be the most promising hypothesis, although some typical carbene
reactions have not been observed yet [84].
The new Cr surface compounds can be split from the support by Brønsted acids, e. g.
HCl, and washed out with organic solvents (methanol, acetone, etc.). The active centres of
the heterogeneous catalyst – or better their deactivation products – form a deep blue solution
this way [59, 70, 83, 85].
As expected, the material is not uniform but consists of an ensemble of homologue individuals,
separated by CH 2 . The minimal C number is 8. The higher homologues are resulting
from the different extent of the polymerization prior to deactivation. Recently we succeeded
in separating the single fractions and in isolating crystals from the C 8 fraction. X-ray
analysis showed surprisingly that the product is actually a cyclopentadienyl-Cr(III) complex
of the formula (CpmCrCl 2 ) 2 , where Cpm is a 1,2,3-trimethyl-cyclopentadienyl ligand C 8 H 11
(Fig. 18.7).
In solution of donor solvents the dimer is split into monomers,
(CpmCrCl 2 ) 2 +2D ! 2 CpmCrCl 2 .D .
The surprising formation of a substituted Cp ring system may proceed via a ring slippage
reaction from a corresponding metallacycle [86–89].
The product belongs to a well-known group of complexes found by E.O. Fischer in
1963 [90, 91]. The corresponding Cp derivative was also prepared by the reaction of surface
Cr(II) with cyclopentadiene [55]. Unfortunately the conversion of ethylene into a Cp product
is not a straightforward process: The hope coming to a new understanding of the structure
of the active centres and for the appearance of odd C numbers in products from even olefins
was disappointed. In our context we have to notify another example for the difference between
Cr(II)A and B; though all Cr(II)A sites are able to form at least C 8 from C 2 , the B
species are not converted. The dehydrogenation of four ethylene molecules to a C 8 H 11 moiety
is obviously the source of the hydrogen needed to form the H 2 and the alkanes mentioned
above.
361

18 Topospecific Chemistry at Surfaces
Figure 18.7: Structure of [CpmCrCl 2 ] 2 , determined by X-ray analysis [84].
18.8 Summary and outlook
As reported in the previous paragraphs, there is still no commonly accepted conclusive and
coherent picture of the surface chemistry of metal impregnated silica and related systems.
Nevertheless, there are some unequivocal results. It is obvious, that there are different surface
compounds, well differentiated by their oxidation state and by their topological environment.
The latter does not offer a chaotic multitude. It leads to the occurrence of just a few
(usually two) different types for a given metal unit. They can be characterised physically by
their spectra and distinguished chemically by their different degree of unsaturation. The
question of the real chemical environment of the species/subspecies, however, cannot be answered
definitively yet. For instance the preparation procedure suggests to exclude SiOH in
the vicinity of Cr(II)A, but the experiments with deuterated silica show the incorporation of
neighbouring SiOD in the polymerization of 1-olefins at just these Cr(II)A centres. Our standard
polymerization system (surface Cr(II) + 1-octene) provides an easy access to screening
experiments. However, the inevitable large variety of reacting centres at the surface did not
allow to relate unambiguously any response to any parameter. Although we were able to isolate
the deactivation product of an active centre and to get its structure via X-ray analysis,
the step from active to inactive species is still a matter of speculation.
Prospects? Obviously, the multidimensional network of interactions needs further investigations.
Whenever it is possible, additional methods should be incorporated. For example,
the advanced mathematical concepts of building-up structures, like silica from given angles,
coordination numbers, and bond lengths, may allow to explain the A and B-type topology.
On the other hand, the advanced microscopic methods (STM, AFM, and LFM) could
provide literally new insights in the fine structure of hydrated SiO 2 surfaces. A first step in
this direction was successful [92]. The oxygen-covered 111 face of a quasi-cristobalite surface
layer (Fig. 18.8) [91], may also be used for modelling silica surfaces, correspondingly.
If we succeed to incorporate metal ions to these surfaces we might be able to simulate active
centres and to determine their behaviour.
362

References
Figure 18.8: Lateral force microscopy (LFM) of a 111 face of quasi-cristobalite on monocrystalline silicon
[91]. The bar corresponds to 10 Å.
„Woran arbeiten Sie?“ wurde Herr K. gefragt. Herr K. antwortete: „Ich habe viel
Mühe, ich bereite meinen nächsten Irrtum vor.“ Bertolt Brecht [93]
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33. H. L. Krauss, D. Naumann: Z. Anorg. Allg. Chem., 446, 23 (1978)
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35. B. Rebenstorf: PhD thesis, Freie Universität Berlin (1975)
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37. B. Blümel: PhD thesis, Freie Universität Berlin (1979)
38. R. Höpfl: PhD thesis, Universität Bayreuth (1981)
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40. J. Geyer: PhD thesis, Freie Universität Berlin (1981)
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41. H. L. Krauss, B. Hanke: Z. Anorg. Allg. Chem., 521, 111 (1985)
42. V. A. Zakharov, Yu. I. Yermakov: Catal. Rev. Sci. Eng., 19, 67 (1979)
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45. D. L. Myers, J. H. Lunsford: J. Cat., 99, 140 (1986)
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47. B. Rebenstorf: J. Cat., 84, 240 (1983)
48. H. L. Krauss, H. Stach: Z. Anorg. Allg. Chem., 366, 34 (1969)
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50. P. Morys, U. Goerges, H. L. Krauss: Z. Naturforsch., 39 b, 458 (1984)
51. G. Ghiotti, E. Garrone, A. Zecchina: J. Mol. Cat., 65, 73 (1991)
52. G. Spoto, S. Bordiga, E. Garrone, G. Ghiotti, A. Zecchina, G. Petrini, G, Leofanti: J. Mol. Cat.,
74, 175 (1992)
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54. B. Rebenstorf, R. Larsson: Z. Anorg. Allg. Chem., 478, 119 (1981); J. Mol. Cat., 11,
247 (1981)
55. H. L. Krauss, U. Westphal: Z. Naturforsch., 33 b, 1278 (1978)
56. D. Naumann: PhD thesis, Freie Universität Berlin (1979)
57. H. Stach: PhD thesis, Technische Universität München (1968)
58. W. Riederer: PhD thesis, Universität Bayreuth (1989)
59. H. A. Schmidt: PhD thesis, Universität Bayreuth (1992)
60. K. Weiss, H. L. Krauss: J. Cat., 88, 424 (1984)
61. G. Langstein: PhD thesis, Universität Bayreuth (1986)
62. H. L. Krauss, B. Bojer: unpublished
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64. H. L. Krauss, B. Frank, B. Hanke, E. Hums, G. Langstein. D. Naumann, B. Siebenhaar, K. Weiss:
Catalytic Reactions with Reduced Phillips Catalysts, in: Y. Imamoglu (ed.): Olefin Metathesis and
Polymerization Catalysts, Kluwer Academic Publishers, Dordrecht, 375 (1990)
65. Q. Xing: PhD thesis, Universität Bayreuth (1993)
66. E. Hums: PhD thesis, Universität Bayreuth (1981)
67. H. L. Krauss, B. Rebenstorf, U. Westphal: Z. Anorg. Allg. Chem., 414, 97 (1975)
68. H. L. Krauss, U. Westphal: Z. Anorg. Allg. Chem., 430, 218 (1977)
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71. G. Ghiotti, E. Garrone, A. Zecchina: J. Mol. Cat., 46, 61 (1988)
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75. R. Gerritzen: PhD thesis, Freie Universität Berlin (1981)
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77. E. Hums, H. L. Krauss: Z. Anorg. Allg. Chem., 527, 154 (1986)
78. H. L. Krauss, G. Zeitler-Zürner: unpublished
79. H. L. Krauss, E. Hums: Z. Naturforsch., 38 b, 1412 (1983)
80. H. L. Krauss, K. Hagen, E. Hums: J. Mol. Cat., 28, 233 (1985)
81. M. L. H. Green: Pure Appl. Chem., 50, 27 (1978)
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82. K. J. Irvin et al., J. C. S. Chem. Comm., 1978, 604
83. H. L. Krauss, E. Hums: Z. Naturforsch., 34 b, 1628 (1979)
H. L. Krauss, E. Hums: ibid., 35 b, 848 (1980)
84. K. Weiss, K. Hoffmann: J. Mol. Cat., 28, 99 (1985)
85. Q. Xing, H. L. Krauss, W. Milius: J. Mol. Cat., 90, 75 (1994)
86. J. R. Bleeke et al.: Organometallics, 6, 1576 (1987)
J. R. Bleeke et al.: J. A. C. S., 111, 4118 (1989)
87. R. Ferede, T. Allison: Organometallics, 2, 463 (1983)
88. W. R. Roper et al.: J. C. S. Chem. Comm., 1982, 811
89. R. P. Hughes et al.: J. C. S. Chem. Comm., 1986, 1694
90. E. O. Fischer, K. Ulm, P. Kuzel: Z. Anorg. Allg. Chem., 319, 253 (1963)
91. F. H. Köhler, R. de Cao, K. Ackermann, J. Sedlmair: Z. Naturforsch., 38 b, 1406 (1983)
92. Th. Schimmel, V. Popp, J. Küppers, H. L. Krauss: unpublished
93. B. Brecht: in: Geschichten, Suhrkamp Verlag, Frankfurt/M. (1962)
365

III
Biopolymers
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

19 Site-Directed Spectroscopy and Site-Directed
Chemistry of Biopolymers
Stefan Limmer, Günther Ott, and Mathias Sprinzl
19.1 Introduction
In our laboratory several aspects of the protein biosynthesis are analyzed by different biochemical,
molecular biological, and biophysical methods. In particular, research interest focused
on tRNAs and the bacterial elongation factor Tu (EF-Tu). Both molecules specifically
interact with each other in the course of the protein biosynthesis cycle.
As indicated by its name, tRNA transports an amino acid to the ribosome where it is
attached to the growing polypeptide chain. In this way, the tRNA forms the link between the
genetic information being coded in the DNA and its RNA transcript (messenger RNA).
For the exact binding of the aminoacylated tRNA at the A-site of the ribosome a complex
between aminoacyl-tRNA and the active form of EF-Tu is required. Activation of EF-Tu
is achieved by binding one molecule of GTP. By means of GTPase activity, GTP is cleaved
into GDP and inorganic phosphate. This cleavage is accompanied by a dramatic conformational
change of the protein structure [1, 2] whereby EF-Tu turns into its inactive GDP state.
This transition can also be monitored via 1 H NMR studies [3]. In the GDP state, the capability
of binding aminoacyl-tRNA is reduced by several orders of magnitude.
By NMR studies information on structure and structure-function relationships of both,
EF-Tu and tRNA, can be gained. In the case of tRNA, not only the whole molecule was analyzed,
but also truncated structures that represented well-defined parts of the intact molecule
and which still fulfilled certain biological functions, e. g. the acceptor stem of alanine-specific
tRNA from Escherichia coli (E. coli) is correctly recognized by its cognate aminoacyltRNA
synthetase and subsequently charged with alanine.
NMR spectroscopy has proved to be a powerful method for the elucidation of the
structure of relatively small macromolecules (molecular weights of up to ca. 10,000 Da). In
particular, this applies to RNA molecules for which X-ray crystallographic analyses are relatively
rare, mainly due to difficulties with their crystallization. Under certain conditions a total
structure elucidation of these (small) molecules in their native environment (aqueous solution)
at atomic resolution is feasible by means of multidimensional NMR methods [4–10].
Moreover, even distinctly larger molecules, like whole tRNAs, can be at least partially characterized
with reference to their structure, in particular secondary structure [11–14]. As detailed
below, in certain cases NMR spectroscopy can also provide useful information for the
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
369

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
characterization of the interaction of relatively large molecular complexes, if e. g. certain
isotopically labelled molecules act as NMR active probes. By such probes, the sites of a specific
intermolecular interaction can be localized.
In this article we present NMR studies with isotopically labelled aminoacyl-tRNA and
its complexes with EF-Tu 7GTP. Furthermore, we studied by combination of chemical RNAsynthesis
and NMR spectroscopy, the structure of an RNA helix containing a single-stranded
dangling end and a specifically located non-Watson-Crick G-U base pair. By specific fluorescence-labelling
of tRNA the dissociation constants for aminoacyl-tRNA, EF-Tu, and GTP
were determined. Finally, the structure and function of EF-Tu was studied by overexpression
of thermostable EF-Tu originating from thermophilic bacterium Thermus thermophilus in
E. coli, crystallisation, X-ray structure determination and site-directed mutagenesis.
19.2 Site-specific NMR spectroscopy of chemically synthesized
RNA duplexes
One of the four arms of the tRNA secondary structure is the acceptor arm. It consists of seven
base pairs and a single-stranded 3'-terminus. At the 3'-terminal nucleotide (invariably
A76), the amino acid is specifically attached. The aminoacylation of the tRNA occurs by esterification
of the amino acid to the 2' or 3'-OH group of the ribose of adenosine A76 [15].
This reaction is catalyzed by aminoacyl-tRNA synthetases.
Earlier biochemical studies have demonstrated that in many cases the main identification
features or recognition elements necessary for the correct recognition of tRNAs by their
cognate aminoacyl-tRNA synthetases are localized in the acceptor stem [16–19]. Frequently,
these recognition sites are referred to as identity elements since they make this particular
tRNA unique and distinguishable from all other tRNAs for the corresponding aminoacyltRNA
synthetase. For the tRNA Ala from E. coli it could be shown by the groups of Schimmel
[20, 21] and McClain [19, 22–24] that a wobble base pair G3-U70 at position 3 of the
acceptor arm represents the major identity element.
Subsequently, it was demonstrated that even a truncated tRNA, consisting only of the
acceptor arm with the single-stranded 3'-end, is recognized as a substrate by the alanyltRNA
synthetase (ARS) and correctly aminoacylated with alanine [21, 25], provided that the
G3-U70 base pair is present at the correct location.
This leads to the question whether this G-U pair causes local irregularities of the helical
geometry, which are recognized by the ARS [24], or if certain functional groups
(2-amino group of G3) are important in this recognition process [26, 27], or whether possibly
both factors are affecting the ARS interaction with tRNA Ala . In addition, the influence
of the ubiquitous terminal CCA sequence on the stability of the acceptor arm was studied.
Earlier studies [28–30] indicated an ordered arrangement of the single-strand nucleotides.
By varying both the number and the sequence of the single-strand nucleotides, their impact
upon structure and stability of the duplex were analyzed. The duplex variants with altered
370

19.2 Site-specific NMR spectroscopy of chemically synthesized RNA duplexes
stem sequence were also investigated. NMR studies were complemented by recording the
UV melting profiles [31] which permitted the determination of thermodynamic parameters.
Sufficient amounts of cations are necessary for the formation of correct secondary
and tertiary structures as well as for the stabilization of nucleic acid helices. In particular,
divalent cations, especially Mg 2+ , play an important role in folding RNA. Consequently, the
question arises whether short helical duplexes are also capable of creating specific binding
sites for divalent cations. We addressed this problem by using paramagnetic manganese ions
which give rise to line broadening of the proton resonances which are located in the vicinity
of the bound ion [32, 33]. The dependence of the specific ion binding capability on the sequence
context was also analyzed [33].
All ribooligonucleotides described in the following were chemically synthesized using
the H phosphonate method [34, 35]. It was demonstrated that this method allows the preparation
of the relatively large amounts (> 10 mg) of RNA being necessary for 2D NMR studies
at reasonable costs and with the required purity. In addition, as compared to phosphoramidite
chemistry, the H phosphonate chemistry offers the advantage of recovery and reusing
of synthons. This is particularly important for expensive modified or isotopically labelled
nucleotides.
19.2.1 Stability of tRNA-derived acceptor stem duplexes
By checking the imino resonance region of 1 H NMR spectra of nucleic acids information
about the secondary structure can be obtained quickly. Sufficiently narrow resonance lines
will be observed only if stable hydrogen bonds between the paired bases are formed. In that
case the exchange of the protons of the endocyclic base nitrogens with the surrounding
water is drastically slowed down as compared to the non-paired state. Fast exchange of the
imino protons with the water protons gives rise to line broadening which increases with
growing exchange rate. Above a certain limit (several hundreds of Hz, corresponding to exchange
rates above 10 3 s –1 ) these signals become too broad to be detected any more at the
typical signal-to-noise ratios [36, 37].
Usually, a separate imino signal is observed for each regular, stable Watson-Crick
base pair. This resonance may be, however, strongly broadened due to the enhanced base
pair opening rate for terminal base pairs ( fraying), and is often not detectable at moderately
high temperatures. The spectral region of the imino resonances at 10–15 ppm is well separated
from that of the other proton signals as, e. g., aromatic and ribose protons. Thus, in certain
cases, an assignment of all imino protons to the individual base pairs of intact tRNAs
was feasible [11–14].
The unusual G-U pair, in contrast to classical Watson-Crick base pairs, gives rise to
two well-separated imino lines since in this case two imino protons are involved in base pair
hydrogen bonding. The G imino resonance is easily detectable since it is shifted strongly upfield
to about 10–11 ppm. Moreover, the two G-U imino protons display an extraordinarily
strong nuclear Overhauser effect (NOE) due to their close spatial proximity.
371

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
Figure 19.1: Sequence of the acceptor stem duplex of E. coli tRNA Ala with the numbering of base pairs
used in the text.
In our studies the influence of the single-stranded terminus (ACCA) on the stability of
the duplex was analyzed. The starting sequence used in these investigations was that of the
acceptor stem of tRNA Ala from E. coli (Fig. 19.1).
Both, 1 H NMR studies and thermodynamic analyses, have been performed with the
original (wild-type) sequence and with sequence variants having gradually truncated singlestranded
regions, up to the duplex without unpaired nucleotides [32]. In the following the
corresponding sequences are referred to as 18mer, 17mer, etc.
A comparison of the imino region of the 1 H NMR spectra of the five different duplex
variants reveals that after attachment of the first unpaired nucleotide (A73) onto the C72 of
the first base pair (G1-C72) the imino resonance of G1 is substantially shifted as compared
to the 14mer. This is also valid for the imino lines of base pairs 2 (G2-C71), and – though
clearly less pronounced – for the G imino resonance of base pair 3 [32]. This upfield shift is
increased upon attachment of further nucleotides of the single-strand terminus with even the
fourth nucleotide (A76) causing a measurable upfield shift of the imino resonances of base
pairs 1 and 2 (Fig. 19.2).
Figure 19.2: Upfield shift of the imino resonances for base pairs 1 (y), 2 (o), 3G (_), and 3U (V)
upon stepwise attachment of 3' nucleotides onto the base pair 1 of the 14mer at 4 8C. For comparison,
the upfield shifts are given for the variants with an IU base pair at position 3 (solid symbols).
372

19.2 Site-specific NMR spectroscopy of chemically synthesized RNA duplexes
The upfield shifts are reflected also in a quite similar dependence of melting temperature
T m and Gibbs free energy DG of duplex formation (Tab. 19.1) on the number of singlestrand
nucleotides.
Both, melting temperature and Gibbs free energy, systematically increase with rising
number of single-strand nucleotides at the 3'-terminus. The largest stabilization increment is
thereby contributed already by the attachment of the first 3'-nucleotide (A73) though further
single-strand nucleotides, too, affect the total duplex stability to a measurable degree. The
remarkable imino line narrowing of the first base pair upon attachment of the 3'-A73 can be
explained by the stabilizing effect of this nucleotide for both, the individual base pair and
the duplex, as a whole. This nucleotide is by no means as mobile and disordered as it is suggested
by the frequently used term dangling nucleotide. Rather it is stacked almost regularly,
i. e. well ordered above the base pair that terminates the duplex stem.
Table 19.1: Thermodynamic parameters for duplex formation and activation enthalpies for imino proton
exchange of base pair 2. T m is the melting temperature, H a (2) the activation enthalpy for the exchange of
the G-N1H of base pair 2 with water, and G 0 the Gibbs free energy for duplex formation at 37 8C. Typical
experimental errors are +0.5 8C for T m , +2% for DG 0 , and +5% for H a (2) . Single-strand concentrations
in the melting curve analyses were always 3 mM.
Duplex
(2)
DH a T m
DG8
[kJ/mol] [8C] [kJ/mol]
14mer 223 35.9 –33.4
without dangling end
15mer 266 43.3 –41.0
( 3' A)
16mer 256 44.6 –41.5
( 3' CA)
17mer 295 45.6 –43.3
18mer 301 45.9 –43.9
( 3' ACCA)
18mer/5mM 338 50.8 –48.9
MgCl 2
By overlap of the geometrically more extended purine base over the preceding pyrimidine
base of the same strand (C72) with the purine base (G1) of the opposite strand, both
nucleotides of the terminal base pair of the stem are locked together more strongly than
without this non-paired nucleotide. In this way the overall duplex stability is enhanced, and
at the same time the opening rate of the terminal base pair and consequently the imino proton
exchange rate of the G1 are considerably reduced.
The upfield shift, too, is directly related to the ordered stacking of the single-strand
nucleotide upon the terminal base pair. The ring current effects in the conjugated ring systems
of the bases produce strong shielding of the protons in the case of well ordered stacked
nucleotides [38].
373

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
A possible explanation of the incremental upfield shift is based on the assumption that
with rising numbers of single-strand nucleotides the stacking order of the preceding
(5'-neighbouring) nucleotides are successively enhanced. This enhanced stacking order, which
can be visualized by an increase of the residence probability in the ideally stacked state, also
affects the terminal base pairs of the stem and the stability of the duplex as a whole.
Another parameter that reflects the stability of individual base pairs and, to a certain
degree, of the whole duplex is the activation enthalpy of the imino proton exchange. It can
be derived from the temperature dependence of the imino signal line widths, under the assumption
of an Arrhenius-like thermal activation of this exchange process. Such an analysis
has been carried out for several imino resonances that were separable and which could be
detected over a sufficiently broad temperature range. Of particular interest in this context
was the imino proton exchange of base pair 2 which is, on the one hand, not too far from
the single-strand terminus to be totally unaffected by it. On the other hand, however, it is
even without single-strand nucleotides stable enough. Moreover, it is directly adjacent to the
G3-U70 wobble pair.
The determined activation enthalpies (Tab. 19.1) display the same dependence on the
number of single-strand nucleotides as melting temperature, Gibbs free energies, and chemical
shift changes. It has to be concluded that the activation enthalpies do not reflect the base
pair opening behaviour and correspondingly the stability of one individual base pair, but
rather of several (3–4) neighbouring base pairs at a time. Hence it indicates the cooperative
character of this process for which the sequence context plays an important role.
In all duplexes with G-U and I-U pairs (I = inosine) the activation enthalpy for the exchange
of the G and I imino proton were distinctly greater than that of the U imino proton.
This could be attributed to an asymmetric opening mechanism of this base pair or different
accessibilities of the concerned imino protons for the phosphate ions in the buffer that catalyze
the proton exchange [37].
Beyond the duplexes with single-stranded ends of differing length other variants with
altered sequences in the single-strand and in the stem have been studied [32]. Most of these
variations resulted in reductions of duplex stability as compared to the original sequence. As
expected, replacement of the G3-U70 with a G3-C70 base pair yielded an increased duplex
stability. Surprisingly, a duplex with an I3-U70 is substantially less stable than the duplex containing
G3-U70, because I differs from G only by a lacking 2-amino group. This seems to indicate
an involvement of the NH 2 group in the stabilization of the duplex structure, possibly
via the formation of specific hydrogen bonds with an immobilized water molecule that has
been found in an X-ray structural analysis of an RNA duplex containing a G-U pair [39].
19.2.2 Manganese ion binding sites at RNA duplexes
In an X-ray analysis of the yeast tRNA Phe , a specific magnesium-binding site in the vicinity
of a G4-U69 wobble base pair was found. Thus it seemed attractive to study if a G-U pair is
a specific binding site for magnesium ions. Addition of the diamagnetic Mg 2+ cation has
only little, though measurable, effects on the chemical shifts of the imino resonances
(< 0.05 ppm). Since it is known from a number of biological and biochemical studies that
374

19.2 Site-specific NMR spectroscopy of chemically synthesized RNA duplexes
Mg 2+ ions can be replaced by manganese ions, Mn 2+ has been employed as a probe for identification
of metal binding sites in RNA duplexes. The paramagnetic manganese causes line
broadening of resonances originating from protons in the vicinity of the bound Mn 2+ . The
observed line width increase varies, under certain conditions, approximately as r –6 with r
being the distance between paramagnetic ion and the concerned proton [40].
Indeed, specific imino line broadenings were found upon addition of manganese ions
for the tRNA Ala acceptor stem duplexes with and without ACCA terminus (Fig. 19.3).
Figure 19.3: Effect of the addition of Mn 2+ on the imino proton resonance signals at 4 8C of the duplex
without a single-stranded end (14mer) (a) and the duplex with an ACCA 3'-terminus at base pair 1
(18mer) (b). In each case, the lower trace belongs to the solution without Mn 2+ and the upper trace
gives the spectrum after addition of either 7.5 mM (a) or5mM (b) MnCl 2 . The solutions contain 100
mM NaCl and either 7.5 mM (a) or 5 mM (b) MgCl 2 .
It should be noted that the molar ratio of manganese ions to RNA was 1:150 to
1:200. From the appearance of the observed spectra it must be concluded that a fast exchange
on the NMR time scale occurs with rates for the exchange of ions among different
RNA molecules of more than 10 3 s –1 . Thus, the binding is not a strong one. Nevertheless,
the particularly distinct broadenings of the imino resonance of base pair 2 and the U imino
line of base pair 3 (G3-U70) indicate that the manganese ion is preferably found between
base pairs 2 and 3. In conjunction with the information obtained from two-dimensional nuclear
Overhauser spectroscopy (NOESY) experiments (Section 19.2.3.) the most probable
binding site of the manganese ion was located in the major groove between the bases G2
and G3 of the short strand.
These manganese ion-binding studies have been extended by analyzing several other
RNA duplexes that were derived from tRNA acceptor stems [33]. These included sequence
variants of the E. coli tRNA Ala (with G3-C70 and I3-U70 base pairs), E. coli tRNA Gly ,
tRNA His acceptor duplexes, as well as yeast tRNA Phe , and tRNA Asp acceptor stems.
375

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
As mentioned before, tRNA Phe from yeast also has a G4-U69 wobble pair in its acceptor
arm in the vicinity of which a magnesium binding site was determined in the native
tRNA. With the chemically synthesized acceptor duplex line broadening effects upon manganese
ion addition were observed (Fig. 19.4) which are most strongly pronounced for the
imino proton of the third and the U imino proton of the fourth (G4-U69) base pair. The
ranking of the specific broadenings corresponds fully to the pattern that also has been observed
with the tRNA Ala duplex if the G-U pair is chosen as a reference point, i. e. largest
line width increase for the G imino proton of the 5'-neighbour of the guanosine of the G-U
pair, followed by the U imino proton of the G-U pair and so on [33].
1
5
6
2
4(U)
4(G)
A
C
C
A
1 G-C
2 C-G
3 G-C
4 G-U
5 A-U
6 U-A
7 U-A
14
13 12 11
ppm
Figure 19.4: Imino region of the H NMR spectra of the acceptor arm duplex derived from yeast
tRNA Phe before (lower trace) and after (upper trace) addition of 7.5 mM MnCl 2 (RNA concentration
1.65 mM, 7.5 mM MgCl 2 ) at 277 K.
It could be concluded that G-U pairs indeed have a potential for specific binding of
divalent cations. However, similar broadening effects for the tRNA Ala acceptor stem variant
with a G3-C70 instead of the G3-U70 base pair were also observed. There the largest line
width increase is likewise observed for the G imino proton of base pair 2, followed by a
slightly smaller broadening for the G3 imino resonance [33]. Analysis of several further duplexes
led to the conclusion that stretches of four consecutive purines also create sites of
preferred binding of manganese ions.
376

19.2 Site-specific NMR spectroscopy of chemically synthesized RNA duplexes
19.2.3 Structural determination of short RNA duplexes by 2D NMR spectroscopy
Though one-dimensional 1 H NMR studies of the tRNA Ala acceptor arm gave hints to the special
role of the G-U base pair, detailed information on the possible modifications of the local
helical geometry in the vicinity of the G-U pair cannot be obtained by these experiments.
Elucidation of the fine structure of biological macromolecules at atomic resolution is
possible by multi-dimensional NMR [4, 10, 41–43].
Two-dimensional NOESY allows detection of spatial proximity between protons that
are separated by less than 4.5 Å. By the correlated spectroscopy method (COSY) scalar couplings
between protons, which are separated by at most three (in some exceptions four) chemical
bonds, are used. In the case of ribooligonucleotides such couplings consequently can
only be observed between the protons of the ribose rings and the C5 and C6 protons of the
pyrimidine bases of a given nucleotide. The NOESY method, however, permits the detection
of contacts between the protons of different nucleotides.
For molecules with a regular secondary structure, like RNA duplexes, there are numerous
characteristic interproton distances below 5 Å between adjacent nucleotides [41, 42]
which give rise to typical cross peak patterns in the NOESY spectra. For a regular A-RNA,
the distance between the aromatic proton (C6H or C8H) of a given nucleotide and the 2'-ribose
proton of the 3'-neighboured nucleotide, as well as the one between the aromatic proton
and the 3'-ribose proton of the same residue is particularly small (ca. 2.0–2.5 Å). The cross
peaks in the NOESY spectrum caused by the interaction between the corresponding protons
are therefore especially strong.
For mixing times larger than 200 ms contacts between protons which are more distant
than 4.5 Å are detectable. This is due to a process that is referred to as spin diffusion in
which NOE contacts are mediated by an additional proton which acts as a kind of relay station.
In this way, NOE contacts between the 1'-ribose protons and the aromatic protons
(C6H/C8H) of both, the same and the 3'-neighboured nucleotide, can be observed. Thus,
complete assignment paths between 1' and aromatic protons of all the residues within one
strand can be drawn as long as the helical geometry is not perturbed. There are also contacts
between protons of nucleotides in the two different strands of the duplex, e. g. between C2H
of an adenosine and the 1'-ribose proton of the 3'-neighboured nucleotide of the opposite
strand [4].
Making use of these techniques as well as of 13 C and 1 H hetero-COSY experiments an
assignment of the aromatic protons of C5H, C6H, C8H, C2H, and all 1', 2', 3', and part of
4'-ribose protons of the tRNA Ala acceptor stem was achieved (Fig. 19.5) [44].
In the spectra at low mixing time (80 ms) as well as in the frame of the signal-to-noise
ratio, which was attainable with the existing equipment and the available sample concentrations,
only those cross peaks were found that are particularly strong for A-RNA helices, as
2'-ribose-aromatic interresidue cross peaks, beyond the contacts which always give rise to intense
cross peaks independent of helix type (1'-2'-ribose NOEs, 5–6 pyrimidine contacts).
From these results, a regular helical structure was deduced.
At longer mixing times many further contacts become detectable due to spin diffusion
that cannot be used for a quantitative evaluation in terms of proton distances. However, these
contacts allow a qualitative comparison of equivalent interactions – as e. g. 5–5, 6–6, and
5–6 contacts – between different adjacent nucleotides.
377

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
Figure 19.5: Schematic representation of the observed internucleotide NOESY contacts in the E. coli
tRNA Ala derived acceptor arm duplex (18mer/GU) for mixing time of 300 ms at 303 K.
In Fig. 19.5 all the NOE contacts between the protons of adjacent nucleotides detected
for 300 ms mixing time NOESY spectra are compiled.
The number of the internucleotide contacts, visualized in Fig. 19.5 by connecting
lines, gives an approximate impression of the stacking between the individual bases. There
is a particularly efficient stacking between nucleotides G2 through U6 in the purine-rich
(short) strand as well as between G68 and C69, C71 and C72, and, remarkably, between
A73, C74 and C75 of the longer strand. Obviously there is poorer stacking between U70 and
C71 than between most of the other adjacent nucleotides.
The small number of the observed internucleotide NOEs between C72 and A73 suggests
a weak stacking. However, there are cross peaks of the C2 proton of A73 with both,
the aromatic C8H and the 1'-ribose proton of G1 of the opposite strand. The former cross
peaks indicate an overlap of the purine base plane of A73 with that of G1, which corroborates
the before-mentioned interlocking effect of the purine base in position 73 and provides
a structural explanation for the stabilizing effect of 3'-dangling purine nucleotides. At the
same time, the C8 proton of A73 is moved away from the aromatic protons of C72 making
the lack of the aromatic-aromatic cross peaks conceivable. Some NOE contacts have been
378

19.2 Site-specific NMR spectroscopy of chemically synthesized RNA duplexes
observed between C75 and the terminal A76, suggesting at least a partial stacking of A76
upon C75. Thus it can be concluded that the stacking order within the single-strand terminus
is comparable to the one in the interior of the duplex.
19.2.4 NMR derived model of the tRNA Ala acceptor arm
As mentioned above, the cross peak intensities from NOESY spectra taken at long mixing
times cannot be related in a simple and direct way to distances between two protons due to spin
diffusion effects that mask the actual proton distances. A possibility to extract such information
is provided by relaxation matrix analysis that accounts for all dipolar interactions of a given
proton and hence takes spin diffusion effects explicitly into consideration. Several computational
procedures have been developed which iteratively back-calculate an experimental
NOESY spectrum, starting from a certain molecular model that is altered in many cycles of the
iteration process to fit best the experimental NOESY data. In each cycle, the calculated structures
are refined by restrained molecular dynamics and free energy minimization [42, 43].
We have performed an approximate calculation of the duplex structure of the tRNA Ala acceptor
arm on the basis of the 300 ms NOESY spectrum using the iterative relaxation matrix
analysis procedure IRMA [45] of the FELIX software package (Biosym Technologies, San
Diego). The structures were refined employing the DISCOVER module of the INSIGHT II molecular
graphics software package (Biosym Technologies). As a starting structure, a canonical
A-RNA duplex was used. If uracil (U70) is substituted for the C70 of the regular Watson-Crick
duplex with a G3-C70 base pair, there is a distinct base plane overlap not only between the purines
(G4 through G2) but also between the pyrimidines (C69 to C71) (Fig. 19.6 a).
The arrangement calculated for a helix with G3-U70 base pair is given in Fig. 19.6 b.
The most striking difference between the Structures 19.6a and 19.6b is that in the latter a
distinct destacking is found between the bases U70 and C71. Moreover, a regular G-U hy-
Figure 19.6: Arrangement of the three consecutive base pairs G2-C71, G3-U70, and G4-C69; a) in the
starting structure (as created with the BIOPOLYMER module of INSIGHT II) obtained from a regular
Watson-Crick A-RNA duplex with a G3-C70 base pair after subsequent replacement of the C70 residue
by a uridine. No further adjustment of the U70 base position has been made and hence no correct hydrogen
bonding pattern is found; b) in the final (average) structure derived from relaxation matrix analysis
of the 300 ms NOESY spectrum using the IRMA procedure [45] and restrained molecular dynamics
calculation. Note that G3 and U70 now are forming a regular base pair. The bases G3 and
U70 are indicated by thick lines.
379

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
Figure 19.7: Model of the tRNA Ala acceptor arm structure as determined on the basis of the 300 ms
NOESY spectrum using the IRMA procedure [45] and restrained molecular dynamics. Three structures
resulting from different calculations are superimposed. The continuation of the helix geometry into the
single-stranded terminus is clearly visible. The black sphere marks the most probable location of the
bound manganese ion in the vicinity of G3-U70 base pair.
drogen bond geometry for the paired bases is found though this has not been implemented
as a constraint into the calculation. Figure 19.7 displays a superposition of three duplex
structures, obtained for different independent calculations each starting from the same regular
A-type helix. The rmsd value between each two structures amounts to ca. 1.35 Å.
19.2.5 Chemical shifts and scalar coupling as an indicator of RNA structure in the
vicinity of a G-U pair
If the chemical shift differences of the equivalent 1'-ribose and aromatic protons of the wildtype
G3-U70 sequence and the G3-C70 variant sequence are plotted versus the nucleotide
position, only small values (< 0.05 ppm) are found except at the wobble pair site (3–70) or
in its immediate vicinity (Fig. 19.8). The difference of the chemical shift values of the C5
380

19.2 Site-specific NMR spectroscopy of chemically synthesized RNA duplexes
Figure 19.8: Differences Dd in ppm between the chemical shifts in the 18mer/GU and the 18mer/GC
duplex for aromatic and H1' proton resonances at T = 303 K, given separately for both strands.
protons of U70 and C70 amounts to 0.22 ppm. The one for the difference of the C5 protons
of C71 is similarly large. If the intrinsic chemical shift differences between cytidine and uridine
are taken into account, the correct shift difference of position 70 is increased to ca.
0.4 ppm.
Since the upfield shifts of the resonance signals (in particular of the pyrimidine C5
protons) of nucleotides upon transition from the totally structurally disordered coil to the
well ordered duplex state are caused by the ring current effects of the stacked bases the
downfield shift of U70 and C71 proton signals in the G3-U70 duplex can be interpreted in
terms of a reduction of stacking interaction between U70 and C71, as compared to the regular
G3-C70 duplex.
The results presented above demonstrate that there is a deviation from the regular A-
helical geometry in the wild-type tRNA Ala acceptor arm that is mainly characterized by a
displacement of the U70 base. Thereby the base plane overlap between C71 and U70 is dis-
381

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
tinctly diminished in comparison to a regular Watson-Crick duplex with a G3-C70 base pair
in place of the G3-U70 wobble pair. Probably, the discontinuity in the helix geometry introduced
by the wobble pair is recognized by the ARS. Exactly this has been suggested by
McClain et al. [24] from in vivo studies using suppressor tRNAs in E. coli.
19.2.6 Structure of aminoacyl-tRNA and transacylation of the aminoacyl residue
Esterification of the amino acid with the tRNA occurs either at the 2' or 3'-hydroxyl group
of the terminal adenosine residue A76, according to the specificity of the corresponding
aminoacyl-tRNA synthetase [15]. After dissociation of the aminoacyl-tRNA from the
synthetase, the amino acid residue can transacylate between the 2' and 3' positions. The rate
of this process, which correlates with the deacylation rate of the amino acid from the tRNA,
depends on temperature and pH value as well as on the specific type of the attached amino
acid [46]. Methionine that is found at the N-terminus of polypeptides often is formylated in
procaryotes. This formylation takes place after the esterification of the corresponding
tRNA fmet with methionine.
A way to study the effect of formylation on Met-tRNA structure was opened by specifically
labelling the amino acid with 13 C. In the 13 C NMR spectrum of the yeast
[e- 13 CH 3 ]Met-tRNA i Met a single 13 C resonance is observed which corresponds to the methyl
group. This indicates a rapid (on the NMR time scale) exchange of the amino acid residue
between the 2' and 3' acylation positions (Fig. 19.9 a). A quite similar behaviour is found if
E. coli tRNA fMet is aminoacylated with [e- 13 CH 3 ]Met (Fig. 19.9 b).
By contrast, formylation of the methionine gives rise to two separate resonances in the
spectrum of [e- 13 CH 3 ]fMet-tRNA fMet with roughly equal intensity that can be assigned to
the 2' and 3' isomers of the aminoacyl-tRNA, respectively (Fig. 19.9c). Thus, formylation
leads to a drastic reduction of the transacylation rate permitting the detection of signals for
2' and 3' acylation isomers. From the frequency splitting of the two lines (0.35 ppm) it can
be concluded that the transacylation rate in this case has to be less than ca. 150 s –1 .
By analyzing the fMet-adenosine monophosphate, additional information on the transacylation
process was obtained. This molecule corresponds to the aminoacylated 3'-terminal
A76 of the fMet-tRNA fMet . Owing to its low molecular weight and the small number of protons
the resolution of the proton NMR spectrum is naturally much better than for the whole
tRNA. Indeed, the chemical shifts for all protons of the 2' and 3' isomerss can be detected.
From the smallest measured chemical shift difference between the lines corresponding to 2'
and 3' isomers (0.0073 ppm for the formyl proton) an upper limit for the transacylation rate
in fMet-AMP of ca. 11.5 s –1 can be deduced. In contrast to fMet-tRNA fMet , the ratio of the
3' and 2' isomers populations is here about 1.9 : 1. However, in native tRNA the isomer ratio
and the transacylation rate could be different.
A similar experiment as with Met-tRNA was performed with 13 C Val-tRNA Val . In this
case, the amino acid was 13 C labelled at the carbonyl position of the valine. Here, likewise,
two separate carbonyl resonances have been observed that corresponded to the 2' and 3' acylation
positions [47]. The intensity ratio of 3' and 2' isomers amounts to ca. 7 : 3. The slow
transacylation correlates well with the very slow deacylation rate of Val-tRNA Val .
382

19.3 Structure of elongation factor Tu
Figure 19.9: 125.7 MHz proton-decoupled 13 C NMR spectra a) of yeast [e-CH 3 - 13 C]Met-tRNA Met
i ,
b) E. coli [e-CH 3 - 13 C]Met-tRNA fMet , and c) E. coli [e-CH 3 - 13 C]fMet-tRNA fMet .2' and 3' isomers can
be distinguished only after formylation of the a-amino group due to the decrease in the transacylation
rate.
19.3 Structure of elongation factor Tu
Bacterial elongation factor Tu (EF-Tu) is a GTPase which promotes the binding of aminoacyl-tRNA
to the codon-programmed ribosomes. During its functional cycle it consecutively
binds several ligands: GDP, elongation factor Ts, GTP, aminoacyl-tRNA, and ribosomes
[48]. It functions as a timing device, determining the relation between the rate of protein
biosynthesis and its error frequency [49].
The cycle of EF-Tu during protein biosynthesis, emerging from the studies on
T. thermophilus system [2] conducted in our laboratory, is shown in Fig. 19.10. EF-Tu 7 GDP
binds EF-Ts and a tetrameric complex (EF-Tu 7 EF-Ts) 2 is formed. The dissociation constant
for this interaction is about 10 –8 M, in the case of T. thermophilus elongation factors. The
functional reason for the formation of an (EF-Tu 7 EF-Ts) 2 tetramer, also observed with other
elongation factors, is not yet understood. EF-Ts functions as nucleotide exchange factor for
EF-Tu since it promotes dissociation of GDP. Aminoacyl-tRNA in the presence of GTP is required
for dissociation of T. thermophilus (EF-Tu 7EF-Ts) 2 and aminoacyl-tRNA7EF-
Tu 7 GTP ternary complex is formed. The high concentration of GTP (10 –3 M) and aminoacyl-tRNA
(10 –4 M) in the bacterial cell is the main factor which drives the cycle of EF-Tu
383

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
Figure 19.10: Functional cycle of EF-Tu. The cycle is driven by GDP to GTP exchange which is catalyzed
by EF-Ts and GTPase, stimulated by programmed ribosome with A-site bound aminoacyl-tRNA. EF-
Tu 7EF-Ts complex occurs in vitro as (EF-Tu7EF-Ts) 2 tetramer. This fact is not indicated in the figure.
toward the formation of aminoacyl-tRNA7EF-Tu 7 GTP ternary complex. The aminoacyltRNA
is transported to ribosomal A-site where its codon-dependent binding to mRNA-programmed
ribosomes occurs. Correct reading of the codon by the aminoacyl-tRNA induces a
GTPase which is required for the dissociation of EF-Tu 7GDP from ribosomes [50].
19.3.1 Sequence of Thermus thermophilus EF-Tu
The sequence of T. thermophilus EF-Tu was determined by gene analysis and DNA sequencing
[51]. The regulatory GTPases, like other proteins, are composed of structural modules, which we
will call domains. GTP is bound to a nucleotide binding domain containing conserved sequences
which are present in all regulatory GTPases (Fig. 19.11). These sequences GlyXXXXGly-
LysThr 25 , (Arg)XXThr 62 ,AspXXGly 84 ,AsnLysXAsp 139 , and SerAla 175 fulfill a specific function
in nucleotide binding, GTP hydrolysis, and regulation of the conformational switch of EF-Tu.
The L 2 region of the nucleotide binding domain (Fig. 19.11), connecting the C-terminus
of the helix A with the N-terminus of the antiparallel b sheet b, was named the effector
loop since it changes its structure between GDP and GTP forms of the G protein [52]. Most
mutations inhibiting the interaction with the GTPase activating protein (GAP) are located
within this L 2 region [53].
In most GTP/GDP binding elongation factors, the C-terminal part of L 2 contains an
ArgXXThr 62 consensus sequence. The arginine59 of this consensus sequence in T. thermophilus
EF-Tu is a trypsin hypersensitive site. The threonine62 is one of the residues coordinating
the magnesium ion and the g-phosphate of the EF-Tu bound GTP. There is less sequence
homology in the N-terminus of L 2 than in the C-terminus of this loop (Fig. 19.11).
The N-terminal part of L 2 in EF-Tu probably interacts with aminoacyl-tRNA and ribosome
[2]. A remarkably high homology between different bacterial EF-Tus exists in the region
around helix B (amino acid residues 83–106), where the catalytic site for GTPase is located.
The GTPase-dependent conformational changes of EF-Tu are propagated from this highly
conserved region [1].
384

19.3 Structure of elongation factor Tu
Figure 19.11: Sequence homology in eubacterial EF-Tu. The bold capitals above the line show the
homologous sequences occurring in all regulatory GTPases including G-proteins and proteins of ras family.
The capitals above the line give the amino acid residues conserved in all bacterial EF-Tus, derived
from 36 known sequences. The numbers indicate the position of these residues in Thermus thermophilus
EF-Tu. The secondary structure elements of T. thermophilus EF-Tu are shown below the line. Open boxes
(a–r) are b structures and the hatched areas are a-helices. The alternative black area of the helix B occurs
in the EF-Tu7GDP structure and indicates the change in the position of this element upon regulatory
switch from GDP to GTP bound conformation [1].
385

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
There are many conserved aminoacyl residues in domains II and III. The function of
most of these residues is not yet known. Arginine300 [54] of domain II apparently plays an
important role in stabilization of interdomain interactions and threonine394 was suggested
as a phosphorylation site regulating aminoacyl-tRNA binding to EF-Tu 7GTP [55]. The
amino acid residues on the interface between the domains I and II are often conserved and
are important targets for binding antibiotics.
19.3.2 Crystallization, X-ray analysis, and the tertiary structure
The crystallization of native elongation factor Tu isolated from bacterial cells, i. e. in complex
with GDP, proved to be difficult. Useful crystals for X-ray structure analyses could be
obtained only from a partially trypsinized E. coli EF-Tu 7 GDP missing a part of the L 2 region
(amino acid residues 45–58 E. coli numbering). Through laborious efforts of several
crystallographic groups, the structure of the partially proteolyzed E. coli EF-Tu 7 GDP has
been determined up to 2.5 Å resolution [56].
The structure of EF-Tu in the active GTP form, namely that of EF-Tu7GppNHp from
T. thermophilus, was determined up to 1.7 Å resolution [1]. The EF-Tu structures in GDP
and GTP bound forms can now be compared (Fig. 19.12). However, we still have to keep in
Figure 19.12: Structures of E. coli EF-Tu7GDP [56] and T. thermophilus EF-Tu7GppNHp [1]. The nucleotide
binding domains were placed at approximately the same location in order to demonstrate the
movements of the helix B and the domains II/III (dark lines). The L 2 region connecting the helix A with
the antiparallel b sheet b, c is missing in the EF-Tu7GDP structure (arrows).
386

19.3 Structure of elongation factor Tu
mind that the structure of EF-Tu7GDP is not derived from the native protein but from its
proteolytic derivative.
EF-Tu is composed of three domains (Fig. 19.12). Domain I, which harbours the nucleotide
binding site, contains five b structures and six a helices (compare with Fig. 19.11).
Domains II and III are composed exclusively of antiparallel b sheets forming two b barrels.
The most striking difference between EF-Tu 7 GDP and EF-Tu 7 GppNHp is the relative location
of the three domains. In EF-Tu 7 GDP, domain II is separated from domain I, whereas
the EF-Tu7GppNHp structure is compact with an interface formed between all three domains.
Large intramolecular movement must occur during transition from the GDP to the
GTP bound form. Since the interface between domains II and III does not change during
this dramatic conformational change, a movement of domain I takes place involving distances
which are more than one third of the molecular diameter. The GDP form of EF-Tu is
stabilized only by interactions between domains I and III. These interactions must be disrupted
by going from GDP to the GTP form, a process which does not take place spontaneously
since EF-Ts, aminoacyl-tRNA, and GTP are required to promote it (Fig. 19.10).
19.3.3 Nucleotide binding and GTPase reaction
The nucleotide interacts with consensus sequences located in domain I (Fig. 19.11). An important
interaction of guanine nucleotides with magnesium ion is coordinated by the b and
g-phosphates of GTP and four other ligands located in the phosphate loop (Thr25), effector
loop (Asp51, Thr62), and GTPase loop (Asp81).
An intriguing mechanistic question concerns the mode by which the signal induced by
interaction of a GTPase with GAP is transmitted to the GTPase centre to trigger the hydrolysis
of GTP to GDP. In the case of EF-Tu, the rate of intrinsic GTPase activity is stimulated
100,000-fold by the binding of aminoacyl-tRNA and ribosomes [50]. Thus cooperative interaction
of programmed ribosomes and the aminoacyl-tRNA7EF-Tu 7GTP complex is essential
for the EF-Tu-GTPase stimulation. Evidence is available which suggests that the L 2 region
of EF-Tu participates in this process. Cleavage of the L 2 region at arginine59 with trypsin
does not affect the binding of GTP, GDP, or the interaction with aminoacyl-tRNA [57]
but abolishes the ability of ribosomes to stimulate the GTPase [58]. Thus it seems likely that
a cut in the L 2 region intercepts the transfer of information from the GTPase site of the protein
to the ribosomal interaction site.
The central feature of the GTPase mechanism in EF-Tu is the catalytic triad composed
of aspartic acid87, histidine85, and a catalytic water molecule placed in the vicinity of the g-
phosphate of GTP [1] (Fig. 19.13). This isolated water molecule was found in all structures of
GTPases crystallized with slowly hydrolyzable GTP analogues. The intrinsic GTPase of EF-
Tu is (in the absence of ribosomes and aminoacyl-tRNA) probably catalyzed by different mechanisms
than the 100,000-fold stimulated GTPase of the same protein taking place during
the decoding process. Thus the GTPase in EF-Tu is regulated. The action of His85, which acts
as a nucleophile for water activation, is blocked by a lipophilic gate formed by Ile61 and
Val20 as well as by the involvement of Asp87 in a salt bridge with Arg59. Binding of aminoacyl-tRNA
and the ribosomal A-site provides for movement of the effector loop L 2 and the
387

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
Figure 19.13: Schematic presentation of the model for GTPase mechanism in EF-Tu.
loosening of these interactions. This leads to the activation of the catalytic triad. The GTPase
is facilitated in addition by interactions involving Mg 2+ and several other amino acid residues
(Lys24, Thr62, Gly84) with the pentacoordinated phosphate transition state. This mechanism
is supported by experiments with EF-Tu mutated at His85 and Thr62. Their mutations to Ile85
and Ala82, respectively, completely abolish the GTPase [54]. It is remarkable that the two residues
of the L 2 region, arginine59 and isoleucine61, which seem to prevent the optimal formation
of this catalytic triad, are conserved in all elongation factors Tu. This provides supporting
evidence for their involvement in the GTPase trigger mechanism.
19.3.4 Mechanism of GTP induced conformational change of EF-Tu
The EF-Tu 7 GDP and EF-Tu7GTP structures in Fig. 19.12 provide a picture of the dramatic
change caused by the presence of g-phosphate group of GTP in complex with EF-Tu. This
conformational change is propagated from domain I by a flip of the glycine84 peptide bond
and results in a large movement of domains II/III. In the GDP structure the NH group of
this glycine residue is located on the N-terminus of helix B of domain I and stabilizes this
helix by interaction with the side-chain of aspartate87. The peptide bond between proline83
and glycine84 moves by approximately 1508 in the transition between the inactive GDP and
active GTP forms of EF-Tu. A new interaction between the NH group of glycine84 and the
g-phosphate of GTP is formed. Consequently, the first turn of the helix B melts. In addition
helix B changes its location in the structure:
a) it is translated along its axis by one turn,
b) its axis is tilted by about 308, and finally
c) a slight turn around its axis takes place [1].
These movements alter the intramolecular interactions of several amino acids in helix
B. For example, asparagine91 interacts, in the GDP bound form of the protein, with cysteine82
of domain I, whereas in the GTP bound form it interacts with arginine300 of domain
II – a residue playing a pivotal role in stabilization of interdomain interaction [59] –
and isoleucine63 of the GTPase regulating region of domain I. The centrally located helix B
388

19.3 Structure of elongation factor Tu
(Fig. 19.12) thus functions like a cylinder of a lock and is hidden in the open state EF-
Tu 7 GDP. In the locked state EF-Tu 7 GTP it enters the complementary site of the neighbouring
domain II. The moveable parts of this system are the glycines84 and 95. Mutation of
glycine83 of E. coli EF-Tu (equivalent to glycine84 in T. thermophilus EF-Tu) interferes with
the conformational change of EF-Tu and prevents the interaction of EF-Tu 7GTP with aminoacyl-tRNA
[60]. Helix B with its GTP/GDP-dependent open/closed forms is well suited to
interact with an effector protein. Since in the EF-Tu 7 GTP the domains II/III interact with
the helix B, it is most likely that these domains represent here an integrated effector.
19.3.5 Aminoacyl-tRNA in complex with EF-Tu7GTP
The dissociation constants of aminoacyl-tRNA EF-Tu 7 GTP ternary complexes were measured
at equilibrium by several independent methods [61–63]. Site-specific fluorescence labelling
of tRNA on the cytidin74 with AEDANS allowed an equilibrium measurement of
the complex formation between aminoacyl-tRNA and EF-Tu 7 GTP [62]. This method was
used to determine the equilibrium dissociation constants K D for different tRNA isoacceptors
[64], modified tRNAs [47, 65–68], modified EF-Tu, and EF-Tu mutants [54, 69, 70]. These
are summarized in Tab. 19.2. The K D values for different aminoacyl-tRNAs and EF-Tu species
are in the range of 5610 –10 –10 –8 M [64]. The missing aminoacyl residue (tRNA7EF-
Tu 7 GTP interaction) lowers the affinity by four orders of magnitude [63]. About 10 base
pairs of the aminoacyl arm are recognized by EF-Tu 7GTP, as shown by investigation of aminoacyl-RNA
minihelices [68]. Whereas the positional isomers of aminoacyl-tRNA (2' or 3')
do not have a strong effect on complex formation, the attachment of the aminoacyl residue
by an amide bond to the 3'-terminal ribose strongly inhibits the ternary complex formation
(Tab. 19.2a). An aminoacylated-tRNA-like structure containing a pseudoknot in the acceptor
Tables 19.2: Apparent equilibrium dissociation constants K D for aminoacyl-tRNA EF-Tu7GTP ternary
complex as determined by a fluorescence-spectrometric method according to [62].
Table 19.2a: Variation of aminoacyl-tRNA using T. thermophilus EF-Tu7GTP. Asp-tRNA Asp originates
from yeast and is fully modified. A minihelix consists of two RNA oligonucleotides derived from aminoacyl
arm of tRNA. The number of base pair counted from aminoacyl end is indicated in parentheses. His-
TYMV RNA is an aminoacylated tRNA-like structure originating from turnip yellow mosaic virus. AE-
DANS indicates that the fluorescence reporter group was used as an indicator for the measurements of the
interaction with EF-Tu7GTP whereas 2' or 3' dA, 3'NH, and oxi/red denote different non-isomerisable
aminoacyl-tRNAs [28].
Aminoacyl-tRNA K D [10 –10 M] Fold decrease Reference
Asp-tRNA Asp (modified) 16.1 [68]
Asp-tRNA Asp (unmodified) 26.5 1.65 [68]
Asp-Minihelix Asp (12 bp) 62.0 3.85 [68]
Asp-Minihelix Asp (13 bp) 92.1 5.72 [68]
Asp-Minihelix Asp (8 bp) 1900 118 [68]
389

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
Table 19.2a (continued)
Aminoacyl-tRNA K D [10 –10 M] Fold decrease Reference
Asp-Minihelix Asp (7 bp) > 2680 166 [68]
Minihelix Asp > 51200 3180 [68]
His-tRNA His (Yeast) 45.1 2.80 unpublished
His-TYMV RNA (3'-end) 44.7 2.78 unpublished
Ala-tRNA Ala (E. coli) 18.9 unpublished
Ala-Minihelix Ala (12 bp) 110 5.82 unpublished
Ala-Microhelix Ala (7 bp) > 4040 214 unpublished
(AEDANS)Tyr-tRNA Tyr 17.0 [72]
Tyr-tRNA Tyr 17.4 1.02 unpublished
Tyr-tRNA Tyr 2'dA 42.0 2.47 unpublished
Tyr-tRNA Tyr 3'dA 120 7.06 unpublished
Phe-tRNA Phe 22.8 1.34 unpublished
Phe-tRNA Phe 3'dA 189 11.1 unpublished
Phe-tRNA Phe 3'NH 1230 72.4 unpublished
Phe-tRNA Phe oxi/red 1830 108 unpublished
Table 19.2b: Variations of aminoacyl-tRNA using E. coli EF-Tu7GTP. When not indicated tRNAs originate
from E. coli.
Aminoacyl-tRNA K D [10 –10 M] Fold decrease Reference
Tyr-tRNA Tyr (AEDANS) 2.4 [62]
Gln-tRNA Gln 1.9 0.79 [64]
Tyr-tRNA Tyr 5.6 2.33 [64]
Phe-tRNA Phe 6.0 2.50 [62]
AcPhe-tRNA Phe 3800 1580 [63]
tRNA Phe 26100 10875 [63]
tRNA (unfractionated) 28200 11750 [63]
Ser-tRNA Ser (GCU) 7.0 2.92 [64]
Ser-tRNA Ser (UGA) 7.2 3.0 [64]
Ser-tRNA Sec 500 208 [65]
Met-tRNA Met 7.3 3.04 [64]
Met
Met-tRNA f 173 72.1 [63]
Met
fMet-tRNA f 1360 567 [63]
Met-tRNA Met i (Yeast) 6000 2500 [66]
Met-tRNA Met i (Yeast, ox) 110 45.8 [66]
Thr-tRNA Thr 7.7 3.21 [64]
Trp-tRNA Trp 13.5 5.63 [64]
Glu-tRNA Glu 17.0 7.08 [64]
Leu-tRNA Leu (UAA) 22.3 9.29 [64]
Leu-tRNA Leu (GAG) 29.7 12.4 [64]
Asp-tRNA Asp 31.2 13.0 [64]
Leu-tRNA Leu (CAA, E. coli) 33.8 14.1 [64]
Val-tRNA Val (GAC) 36.2 15.1 [64]
Val-tRNA Val (UAC) 47.1 19.6 [64]
Leu-tRNA Leu (CAG) 64.1 26.7 [64]
390

19.3 Structure of elongation factor Tu
arm does not disturb the interaction with EF-Tu [71]. Different aminoacyl-tRNA species interact
with EF-Tu with different affinities. These sequence and aminoacyl-dependent variations
are up to 64-fold (Tab. 19.2b). Peptidyl-tRNAs and initiator Met-tRNAs interact only
weakly with EF-Tu 7 GTP (Tab. 19.2b).
19.3.6 1 H NMR of yeast Phe-tRNA Phe EF-Tu 7GTP complex
For codon-specific binding of aminoacyl-tRNA to the ribosome the tRNA must form a complex
with the elongation factor Tu. The complex formation manifests itself also in the imino
resonance region of the 1 H NMR spectrum of the tRNA. Since the molecular mass of tRNA
(25,000 Da) is only about one third of the molecular mass of the aminoacyl-tRNA-EF-
Tu 7 GTP complex (70,000 Da) the complex formation should become apparent by an increase
of the line widths of the imino resonances. This was indeed observed with a complex
of yeast Phe-tRNA Phe and EF-Tu 7 GTP from T. thermophilus (Fig. 19.14).
Figure 19.14: The imino proton resonances in the NMR spectrum of free Phe-tRNA Phe and in ternary
complex with EF-Tu 7GTP and EF-Tu7GDP [47]. a) 270 µM tRNA Phe , b)270 µM Phe-tRNA Phe EF-
Tu 7GTP ternary complex, c) the same complex as in (b) after GTP hydrolysis and deacylation of PhetRNA
Phe . The dashed lines in (b) and (c) correspond to the 1 H NMR imino proton spectrum of free
tRNA Phe , as shown in (a). The numbers denote the imino proton resonances of the base pairs in the
tRNA Phe acceptor stem as assigned by [74]. The numbers in circles indicate the signals of the first three
imino protons in the acceptor stem, which disappear from the NMR spectrum after ternary complex formation.
391

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
The spectra recorded immediately after complex formation, display resonance signals
which are broadened by a factor of about 2.6 in comparison to free tRNA. In addition, it
should be kept in mind that during the acquisition of the spectrum (about 1 hour), part of
the GTP has already been cleaved to GDP [75]. In the GDP bound form, however, the affinity
of EF-Tu for aminoacyl-tRNA is considerably lower than in the GTP bound form
(Tab. 19.2c).
Table 19.2c: Variation of EF-Tu using Tyr-tRNA Tyr (AEDANS).
EF-Tu (from T. thermophilus) K D [10 –10 M] Fold decrease Reference
EF-Tu . GTP 17.0 [72]
EF-Tu . GDP 98000 5760 [72]
EF-Tu . GTP (Thr62Ala) 618 36.4 unpublished
EF-Tu . GTP (Thr62Ser) 31.7 1.86 unpublished
EF-Tu . GTP (His67Ala) 60.9 3.58 unpublished
EF-Tu . GTP (His85Gln) 260 15.3 [54]
EF-Tu . GTP (His85Leu) 54.6 3.21 [54]
EF-Tu . GTP (Glu55Leu) 16.3 0.96 unpublished
EF-Tu . GTP (Glu56Ala) 49.2 2.89 unpublished
EF-Tu . GTP (Arg59Thr) 36.4 2.14 [47]
EF-Tu7GTP 181–190 24.0 1.41 unpublished
wild type (6His-tag) 82.3 4.84 unpublished
EF-Tu-Domain I >2120 125 unpublished
EF-Tu f 35 2.06 [72]
EF-Tu . GMPPCP 82.0 4.82 unpublished
EF-Tu . GMPPNP 98.9 5.82 unpublished
EF-Tu . f EF-Ts 50.6 2.98 unpublished
EF-Tu (from E. coli) K D [10 –10 M] Fold decrease Reference
EF-Tu . GTP 2.4 [62]
EF-TuA R 37 15.4 [73]
EF-TuB o 56 23.3 [73]
Trypsin cleaved EF-Tu 54 22.5 [73]
TPCK-treated EF-Tu 2400 1000 [73]
The imino spectrum of the Phe-tRNA Phe -EF-Tu 7 GTP complex also reveals several
distinct line shifts in comparison to the spectrum of the free tRNA. Moreover, some of the
imino resonances, namely the ones of base pairs 1 through 3 of the acceptor stem, seem to
be missing. However, it cannot be excluded that these lines overlap with other lines, due to
larger shift changes. The assumption of vanishing imino resonances is also supported by the
work of Heerschap et al. [76], who observed a disappearance of the imino resonances of the
first acceptor stem base pairs upon complex formation of tRNA Phe with E. coli EF-Tu.
As the ternary complex dissociates in the course of the GTP hydrolysis and the PhetRNA
Phe deacylates, the line width of the imino resonances decreases again and the chemical
shifts adopt the positions they had before in the free tRNA. However, even after com-
392

19.3 Structure of elongation factor Tu
plete GTP hydrolysis and total deacylation the line widths do not fully recover the initial values
of the spectra of free tRNA Phe , indicating a complex between tRNA Phe and EF-Tu7GDP
(Fig. 19.14). Since the above-described NMR experiments have been undertaken at complex
concentrations of about 0.3 mM, the still noticeable association of deacylated tRNA and EF-
Tu 7 GDP represents no contradiction to the numerous biochemical studies which typically
use concentrations below ca. 1 µM.
19.3.7 13 C NMR studies of the Val-tRNA Val EF-Tu7GTP ternary complex
E. coli Val-tRNA Val , 13 C labelled at the carbonyl group, was used for 13 C NMR analysis of
the ternary complex formed by Val-tRNA Val and EF-Tu 7 GTP [47]. Labelling the carbonyl
group appeared particularly favourable since it represents the carbon atom which functions
as a linker between the ribose of the tRNA terminal adenosine76 and the amino acid residue.
This linkage is the most important prerequisite for ternary complex formation (Tab. 19.2b).
Free Val-tRNA provides two peaks in the 13 C NMR spectrum around 170 ppm corresponding
to 2' and 3' isomers, with a slight preference for 3' derivative (Fig. 19.15 a). Complex formation
of Val-tRNA Val with EF-Tu 7 GTP leads to a drastic upfield shift to a region around
64 ppm (Fig. 19.15 b). Such an extremely large upfield shift cannot be explained by conformational
changes of the protein or isomerizations of Val-tRNA. Rather it requires the assumption
of substantial alterations in the electronic environment of the carbonyl group. A
change of the hybridization state from sp 2 (carbonyl) to sp 3 (orthoester) upon complex formation
of Val-tRNA Val with EF-Tu 7 GTP is a conceivable explanation for the observed resonance
shift.
Figure 19.15: Part of the proton-decoupled 125.7 MHz 13 C NMR spectrum of a) E. coli [1-C- 13 C] ValtRNA
Val . The resonances at 169.8 ppm and 169.5 ppm represent the 3' and 2' isomers of Val-tRNA Val ,
respectively; b) [1-C- 13 C] Val-tRNA Val EF-Tu7GDP/GTP. The resonances at 63.7 ppm and 63.5 ppm
were suggested to belong to the GDP and GTP forms of the protein, respectively [75].
393

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
Such an orthoester can be formed either with the oxygen atoms of both, the 2' and the
3'-hydroxyl groups of the A76 ribose (Fig. 19.16 a) or with a nucleophilic functional group
of the protein (Fig. 19.16 b). The structure in Fig. 19.16 a seems to be reasonable as during
the (relatively slow) transacylation reaction of the amino acid residue this intermediate state
has to be passed [77]. The structure shown in Fig. 19.16 b would explain the observed chemical
shift as well. In this case a nucleophile leading to an easily cleavable transient bond
should be provided by the protein. A deprotonated carboxylate located in the interface between
domains I and II of EF-Tu [1] is the most obvious candidate for such a function.
A
B
Figure 19.16: Orthoester acid intermediate structure of the aminoacyl residue in the Val-tRNA Val EF-
Tu 7GTP ternary complex. The function of nucleophile is fulfilled a) either by the 2'-OH group or b) by
a functional group of the EF-Tu.
Evidently, EF-Tu stabilizes the orthoester intermediate possibly via electrostatic or hydrogen
bond interactions. This could also be the reason for the strong upfield shift of this resonance,
which is extraordinary as compared to the available model molecules. Moreover, a
strong conformational constraint in the vicinity of the sp 3 hybridized carbonyl group could
contribute to this strong upfield shift.
The relative intensities of the two lines at 63.7 ppm and 63.5 ppm are varying with time
after complex formation. The 63.7 ppm line increases at the expense of the 63.5 ppm signal
[47]. The disappearance of the 63.5 ppm resonance proceeds with the time constant of the
GTP hydrolysis [3]. Thus, the two signals could be associated with two slightly differing conformations
of the orthoester in the GTP and GDP forms of EF-Tu. This assumption is corroborated
by the observation of only one single resonance at 63.7 ppm for a complex of [1- 13 C]ValtRNA
Val with EF-Tu which has bound the slowly hydrolyzable GTP analogue GMPPNP [47].
Positional isomers (2' or 3') of aminoacyl-tRNA do not have a dramatic effect on the
affinity of EF-Tu 7 GTP. However, the replacement of the ester bond (–O–CO–) for the
amide bond (–NH–CO–) attachment of aminoacyl residue to tRNA strongly influences the
affinity (Tab. 19.2 a).
394

19.3 Structure of elongation factor Tu
19.3.8 Role of EF-Tu in complex with aminoacyl-tRNA
Domains II/III function as an effector promoting the binding of aminoacyl-tRNA. In the inactive
GDP conformation these domains are tilted away from the domain I, forming a large
cavity. This conformational change has considerable functional consequences, since EF-
Tu 7 GDP interacts with aminoacyl-tRNA 5000 times less efficiently than EF-Tu7GTP. Thus
it is conceivable to suggest that the binding site of the main determinant of the aminoacyltRNA
binding, namely the aminoacyl-residue, is placed in the binding site which is formed
by the EF-Tu 7GDP to EF-Tu7GTP conformational change. Indeed it was demonstrated that
the reactive e-bromoacetyl-lysyl-tRNA Lys labels specifically a histidine67 residue located
near the interface of domain I and II in EF-Tu 7 GTP [78]. The X-ray structure determination
of aminoacyl-tRNA7EF-Tu 7 GppNHp [79] and aminoacyl-AMP EF-Tu 7 GppNHp ternary
complexes confirm the affinity labelling results and, in addition to histidine67, locate glutamate271
of domain II in the vicinity of aminoacyl-tRNA binding site. Aminoacyl residues
50–60 in the effector loop of EF-Tu Cys82 in domain I [64], arginine300 [54], and threonine393
[55] of domain III are other aminoacyl residues from the domain interfaces which
are involved in aminoacyl-tRNA binding. Mutations in EF-Tu which affect GTPase activity
(Thr62, His85) do not influence aminoacyl-tRNA binding (Tab. 19.2 c).
The aminoacyl domain of tRNA composed of the CCA end, seven base pairs of the
aminoacyl stem and five base pairs of the T stem, is sufficient to promote efficient binding
to the EF-Tu7GTP [68]. About 10 base pairs of the aminoacyl domain are bound to domain
III of EF-Tu [79]. This binding is probably governed by ionic interactions of the RNA-phosphate
backbone. The sequence of nucleotides, however, also plays some role in this process.
EF-Tu 7GTP serves for aminoacyl-tRNA not only as a vehicle transporting it to the ribosome
but in addition as a matrix setting the correct conformation of the L shaped molecule [80].
The precise distance between the anticodon loop and aminoacyl-residue is evidently required
for aminoacyl-tRNA functioning during translation [50].
19.3.9 EF-Tu interaction with EF-Ts
Elongation factor Ts forms stable complexes with EF-Tu and fulfills the role of the nucleotide
exchange protein (NEP). It is not known how this protein binds to EF-Tu and how the
acceleration of the GDP dissociation is achieved. However, it is clear that EF-Tu 7 EF-Ts interaction
changes the structure of nucleotide binding domain. Such effects were identified
especially in the L 2 region of EF-Tu, which alters its protease accessibility upon EF-Tu 7 EF-
Ts complex formation [81]. Replacement of the lysine in the NKXD-guanine binding region
of E. coli EF-Tu for glutamate causes a dissociation of nucleotide and formation of stable
EF-Tu 7EF-Ts complexes. This effect can be reversed by a second mutation in EF-Tu which
prevents EF-Tu 7 EF-Ts interaction. Such mutants were found by replacement of some amino
acids in the C-terminal region of domain I [82] (residues 154–199). The binding of EF-Ts
may dislocate the guanine binding SA consensus element (Fig. 19.2) leading to dissociation
of the bound nucleotide. The interaction of transducin-b with rhodopsin takes place presum-
395

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
ably also in the vicinity of the region of the GTP-binding domains [83]. In addition to domain
I of EF-Tu, EF-Ts apparently interacts also with domains II or III. This is evident from
investigation of EF-Tu species of domains I/II [84] and II/III [72].
19.3.10 Site-directed mutagenesis of EF-Tu
An elegant way to achieve a site-directed replacement of one or more amino acids in the
protein is provided by genetically induced mutations into particular genes followed by expression
of the mutant protein. This method was extensively used in the investigations of
T. thermophilus EF-Tu. A compilation of mutants is presented in Tab. 19.3. In many cases
the replacement of invariant and functionally important amino acids leads to the loss of the
Table 19.3: Derivatives of EF-Tu, prepared by site-directed mutagenesis of the wild type T. thermophilus
tufA gene variants, were overexpressed in E. coli, purified, and fully characterized.
EF-Tu Mutant Properties References
A) Single Mutants
Glu55Leu Ribosome-induced GTPase affected unpublished
Glu56Ala Ribosome-induced GTPase affected unpublished
Arg59Thr Poly(Phe) synth. affected, binding of aa-tRNA affected [47]
Thr62Ser Nucleotide binding is affected unpublished
Thr62Ala
Nucleotide binding is affected,
aa-tRNA binding is affected,
Ribosome binding is defective
unpublished
His67Ala aa-tRNA binding is affected unpublished
His85Gln Ribosome and aa-tRNA binding is affected [54]
His85Gly Protein degradation; cell growth affected [54]
His85Leu Slower GTPase; ribosome binding is affected [54]
Asp81Ala Protein degradation [54]
Cys82Ala As wild type unpublished
Arg300Ile Protein degradation [54]
EF-Tu(NHis6) aa-tRNA binding affected unpublished
EF-Tu (CHis6) Ribosome induced GTPase affected unpublished
B) Double Mutants
His85Gly/Gly233Asp Protein degradation; normal cell growth unpublished
Cys82Ala/Thr394Cys aa-tRNA binding defective unpublished
C) Deletion Mutants
181–190 Thermostability decreases [85]
212–405 (Dom. I) Thermostability decreases [85]
313–405 (Dom. I/II) Higher intrinsic GTPase; thermostable [85]
1–316 (Dom. III) No functional [85]
1–208 (Dom. II/III) Thermostable; EF-Ts interaction [72]
212–405/His85Gly Thermostability decreases; no ribosome interaction [85]
396

19.4 Summary and conclusions
specific function of the protein. Thus the GTPase was affected by replacement of His85 or
Thr62, the binding of aminoacyl-tRNA was impaired when Arg59 or Thr294 were mutated,
and the stability of protein was decreased by mutations at Asp81 or Arg300. Remarkably,
the deletion of domain III led to the stimulation of intrinsic GTPase. Some mutations were
introduced in order to increase the stability of EF-Tu (Arg59) against proteolysis to improve
the quality of crystals (His67) or to allow crystallisation of EF-Tu with affinity reagents and
transition state analogues (Cys82). Many of these investigations are still in progress.
19.4 Summary and conclusions
The size or properties of the molecule often do not allow the use of direct physical or biochemical
methods to study the structure and function of biological macromolecules. Even in
the case of the tertiary structure determination by NMR or X-ray crystallography the structure
dynamics, mechanism of catalytic reactions, and their regulation remain to be studied
by site-specific methods. In the present work we concentrated our effort on the study of
RNA structure, structure and mechanism of a regulated GTPase (represented by an elongation
factor Tu) and on a protein-RNA complex (formed by aminoacyl-tRNA, EF-Tu, and
GTP). Fluorescence spectroscopy (after specific labelling of RNA), electron microscopy
(after attachment of a gold cluster to tRNA), NMR spectroscopy (using specifically introduced
stable isotopes), ESR spectroscopy (with nitroxyl radicals bound to substrates of enzymatic
reactions), affinity labelling of proteins (with reactive substrates), and site-directed
mutagenesis in combination with the X-ray structure analysis of proteins were the utilized
methods. The results of this investigation provided insight into the regulation and function
of GTPases and allowed the discovery of some rules governing the RNA-protein interaction.
Despite the large amount of new information which was gained from this research, most rewarding
is the fact that we are now better prepared to ask new relevant questions for our future
work.
Acknowledgement
The research laboratory in Bayreuth was supported by the Deutsche Forschungsgemeinschaft,
Sonderforschungsbereich 213, D4, and D5. We thank R. Hilgenfeld and J. Nyborg for
cooperation on the study of EF-Tu 7 GppNHp structure, M. Daniel for help with preparation
of the manuscript, and all co-workers who participated in the work. Their names can be
found as co-authors in the referred work from our laboratory.
397

19 Site-Directed Spectroscopy and Site-Directed Chemistry of Biopolymers
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400

20 Spectroscopic Probes of Surfactant Systems
and Biopolymers
Alexander Wokaun
20.1 Introduction
Characterization of surfactant systems is a challenging task because these systems, which often
represent stable and homogeneous phases on the macroscopic scale, are typically consisting
of microscopic or mesoscopic aggregates, such as micelles. Tremendous advances in the
understanding of these systems has been made through direct visualization by freeze-fracture
electron microscopy and by the application of scattering techniques rendering a wealth of information
on aggregate shapes and sizes.
In order to investigate the microscopic properties of the individual aggregates or compartments
constituting the surfactant systems toposelective methods of spectroscopy have
been applied in this study. Focusing on the microscopic dynamics, three complementary
methods will be discussed. First, the self-diffusion coefficients of individual components
have been distinguished by FT-NMR pulsed gradient spin echo experiments. This technique
has been applied to characterize the diffusion of micelles with solubilized hydrocarbons and
residual mobilities in cubic phases (ringing gels) that are formed by aggregation of swollen
micelles.
An alternative method for tracing the motion of micellar aggregates is their selective
labelling by photochromic dye probes. Diffusion coefficients are determined by monitoring
the time dependence of laser induced gratings using the forced Rayleigh scattering (FRS)
technique. In this way, the influence of alcohol content on the diffusion of a system forming
multilamellar vesicles has been investigated. In a further study, the influence of solubilized
water on the diffusion of reverse micelles in the AOT/octanol/water system and diffusion in
the corresponding bicontinuous gel phase have been studied.
A fascinating aspect of surfactant systems resides in the fact that within the same chemical
system one-dimensional aggregates (rod-like micelles), two-dimensional bilayers, and
space-filling continuous structures may be formed. This aspect of dimensionality has been
probed by studying the time-dependence of fluorescence, as induced by lipophilic quencher
molecules diffusing within dimensionally restricted regions of space.
The second part of this study was devoted to the spectroscopic probing of biopolymers.
Transitions between right and left-helical conformations of oligonucleotide duplexes
were monitored by Raman and 2D NMR techniques.
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
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20 Spectroscopic Probes of Surfactant Systems and Biopolymers
The conditions for the applicability of surface enhanced Raman spectroscopy for studies
of oligo and polynucleotides has been assessed. In particular, surface enhanced resonance
Raman spectroscopy (SERRS) has been used to study the binding of chromophores to
DNA strands at probe concentrations as low as 10 –6 M.
20.2 Diffusion in surfactant systems
Three different spectroscopic techniques have been used to determine molecular and micellar
diffusion coefficients in surfactant systems. Each of these methods is probing a different
aspect of molecular mobilities. First, the NMR pulsed gradient spin echo (PGSE) method is
used to determine composition-induced changes in the diffusion coefficients of micelles and
inverse micelles in ternary surfactant systems. In particular, differences between the diffusion
coefficients in micellar and cubic phases are monitored. Second, the diffusion of absorbing
dye probes in light-induced gratings is monitored by the method of FRS, providing
information on the structure of multilamellar vesicles. Third, the dynamics of fluorescent
probes due to diffusing quenchers is investigated, rendering information on the dimensionality
of diffusion pathways and hence on the structure of lyotropic mesophases.
20.2.1 Structural characteristics of micellar solutions, cubic phases,
and multilamellar vesicles from NMR self-diffusion measurements
Ternary systems consisting of surfactant, hydrocarbon and water exhibit a rich variety of
phases. With increasing fraction of the hydrophobic components, frequently a sequence of
structural changes of the type
micelles ) hexagonal phase ) lamellar phase ) inverse hexagonal phase )
inverse micelles
is observed [1]. Additional to the mentioned phases, the occurrence of cubic phases has recently
been discovered [2, 3] which include both bicontinuous structures [4] and cubic or
disordered packages of micellar aggregates. Of particular interest are the so-called ringing
gel phases [5], which exhibit a metallic sound upon mechanical excitation, and have been
extensively investigated within the Sonderforschungsbereich 213.
In our experiments, we have focused attention on a homologous series of ternary systems
consisting of alkyl-dimethylaminoxide surfactants, C n H 2n+1 (CH 3 ) 2 N + O – (C n DMAO), a linear
hydrocarbon (C m H 2m+2 ) or a cyclic hydrocarbon (C m H 2m ), and water as the solvent.
A systematic investigation of the concentration dependence in the system C 14 DMAO/C 6 H 12 /
402

20.2 Diffusion in surfactant systems
H 2 O [6] showed that water constituted the continuous phase in both, the micellar solution and
the ringing gel phase. In the latter, the diffusion coefficients of hydrocarbon and surfactant
were lower by one and two orders of magnitude, respectively, as compared to the micellar solution.
Thus, we are dealing with an aggregated network where the residual mobilities are caused
by exchange of surfactant and solute molecules between neighbouring micelles in the gel.
In order to investigate whether the mentioned results are generic properties of the
mentioned class of ringing gels, a series of micellar solutions and of gel samples was prepared
where the water weight fraction was fixed to 70 % and 57.50 %, respectively, varying
the surfactant and hydrocarbon chain lengths. Self-diffusion coefficients have been determined
by the stimulated echo experiment described by Tanner [7]. Details of the experiment
as well as the exact composition of the samples with the corresponding phase diagrams of
the ternary surfactant systems have been given recently [8].
The influence of the surfactant chain length on the self-diffusion of the solvent (here
D 2 O) is shown in Fig. 20.1. For both, the micellar phase (rhombs) and the cubic phase
(squares), the solvent diffusion coefficient is seen to rise with the alkyl chain length of the
surfactant. In this series, the hydrodynamic radius of the micelles is known to increase with
the surfactant length. Hence, at constant weight fraction of the micelles, the system
C 16 DMAO/C 6 H 12 /D 2 O contains fewer but larger micelles, as compared to C 12 DMAO/
C 6 H 12 /D 2 O. As these micelles may be considered as obstacles for the diffusion of water the
observed trend is then easily interpreted in terms of a standard model [9].
Figure 20.1: Diffusion coefficient of water in ternary C n DMAO/C 6 H 12 /D 2 O systems, as a function of
the surfactant chain length n. Rhombs: micellar phase; squares: ringing gel phase. Details of the compositions
are given by Panitz et al. [8].
The self-diffusion coefficient of the solubilized hydrocarbon was determined by Fourier
transformation of the second half-echo [8]. In the micellar phase an interesting trend is
observed (Fig. 20.2): the diffusion coefficient first increases with the hydrocarbon chain
length up to m = 10, then a pronounced decrease for the larger dodecane and tetradecane solutes
follows.
Light scattering experiments by Oetter and Hoffmann [5] have shown that the hydrodynamic
radius of the micelles decreases (together with the total capacity for solubilization)
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20 Spectroscopic Probes of Surfactant Systems and Biopolymers
Figure 20.2: Influence of the hydrocarbon chain length m on the micellar self-diffusion coefficient D in
ternary C 14 DMAO/C m H 2m+2 /D 2 O surfactant systems. For m = 6, cyclohexane C 6 H 12 has been chosen as
the solute.
as the chain length of the hydrocarbon solute increases from C 6 H 12 to C 10 H 22 . This result
agrees with the increase in the diffusion coefficient for the same series in Fig. 20.2.
In order to account for the subsequent decrease in the diffusion coefficient for longer
hydrocarbon chains we have to recall that the C 14 DMAO surfactant by itself forms rod-like
micelles in the binary solution in water. Due to the addition of a critical amount of hydrocarbon
these rods are transformed into spherical micelles. As mentioned above, the solubilization
capacity is rapidly decreasing with the size of the solute. Thus, for very long hydrocarbons
the solubility limit is reached prior to induction of the rod ) sphere transition. In fact,
for the C 14 DMAO/C 14 H 30 /D 2 O system the amount of solubilized hydrocarbon in the investigated
sample was only slightly above the critical concentration required for the mentioned
transition. In such a situation small dynamic deviations from an average spherical shape of
the micelles are expected to occur. Thus a lower diffusion coefficient is observed for this
system – and to a lesser extent also for the homologous system containing C 12 H 26 – which
is attributed to shape fluctuations of the micelles [8].
The NMR investigations have been complemented by studying the solvent diffusion in
multilamellar vesicles in a related system. When the ionic surfactant C 14 H 29 N + (CH 3 ) 3 Br –
(CTAB) and the cosurfactant n-C 6 H 13 OH are added to the above-mentioned C 14 DMAO/
H 2 O system vesicles and lamellar phases are formed [10]. The aim of this study was to
complement the rheological measurements in these systems, where a decrease of both, the
storage modulus G' and the yield stress, with increasing salinity of the water had been
found.
Results are shown in Fig. 20.3 for three systems containing NaCl (c = 0, 10, 100 mM).
The measured signal may be attributed to extravesicular water. The diffusion coefficient of
the solvent D 2 O increases by about 50 % upon the addition of NaCl (c = 10 mM), in agreement
with a lowering of the yield stress observed in the rheological measurements [10].
Using the equation D H2 O = D o (1 – F v ), where D o is the diffusion coefficient of free
water and F v is the volume fraction of the micelles, an increase of D H2 O appears to be accompanied
by a decrease in F v . Assuming a constant total surface area, a decrease of F v
must be due to deformation of the vesicles from the maximum volume spheres.
404

20.2 Diffusion in surfactant systems
Figure 20.3: Diffusion coefficient of the solvent D 2 O in a system consisting of C 14 DMAO (c = 90 mM),
CTAB (c = 10 mM), and n-C 6 H 13 OH (c = 220 mM), which forms multilamellar vesicles. The dependence
on diffusion time D is investigated. Circles: no salt added; rhombs: NaCl (c = 10 mM); squares:
NaCl (c = 100 mM).
Increasing the NaCl concentration to 100 mM gives rise to a further strong increase in
the diffusion coefficient. Interestingly, the apparent coefficient depends now on the diffusion
time set by the experiment, i. e. the time t between the two gradient pulses. Such a dependence
is standardly interpreted as being due to restricted diffusion. Both, the absolute increase and
the diffusion time dependence, hint to a further deformation of the multilamellar vesicles into
planar lamellar stacks or aggregates, which are partially ordered within the NMR tube.
20.2.2 Probing of mobilities in multilamellar vesicles by forced
Rayleigh scattering
Recently, increasing use has been made of laser-induced gratings for visualizing static and
dynamic properties of gases and condensed phases [11]. In FRS two coherent laser beams
are made to interfere in the volume of the sample. In our case that was a surfactant solution
doped with a chromophoric dye probe. Excitation of the absorber molecules in the regions
of high light intensities of the interference pattern gives rise to a spatial modulation of both,
the absorption and of the refractive index of the sample. The lattice constant L of this grating
is determined by the wavelength l and by the angle y between the two beams (Fig. 20.4),
L ˆ l=2 sin…y=2† :
…1†
The hologram induced in this manner may be probed by a second laser of wavelength
l', which is diffracted by the grating (Fig. 20.4). For thick holograms it is essential that the
grating is probed at an angle j for which the Bragg condition (Eq. 2) for diffraction on the
planes within the sample is fulfilled (Fig. 20.4),
2 L sinj ˆ m l 0 …2†
405

20 Spectroscopic Probes of Surfactant Systems and Biopolymers
ϕ
ka
kb
q
Λ
Figure 20.4: FRS experimental setup. Beam (1) is the writing laser gated by a Kerr cell (5) between
crossed polarizers (4, 7). The probe laser (10) is diffracted by the grating in the sample (11); the diffracted
order is isolated by an iris diaphragm (13), and recorded by a pin diode (15). The Bragg condition
for reading out the grating in a thick hologram is illustrated on the right-hand side.
In order to monitor the diffusion of the probe molecules the sample is illuminated for
a short time interval gating the laser with a Kerr cell (Fig. 20.4). The length of the excitation
pulse must be carefully matched to the intrinsic relaxation time of the photochromic probe.
After terminatio of excitation the grating is washed out by diffusion of both, excited and unexcited
dye molecules. Hence, the diffraction efficiency of the grating is diminished and the
intensity I (t) of the diffracted probe laser beam decreases according to the equation
I …t† ˆ…A e t=s ‡ B† 2 ‡ C 2 :
…3†
The decay constant s is influenced by both, the intrinsic relaxation time s dye of the
probe (i. e. the time constant of relaxation of the photoisomer produced) and the diffusion
constant D, according to the equation
s 1 ˆ s 1
dye ‡ Dq2 ;
…4†
where q =2p/L. Hence, if the grating constant L is varied via the angle of intersection y
(Fig. 20.4), a plot of s –1 vs. q 2 will yield the desired diffusion coefficient as the slope.
This procedure is illustrated in Fig. 20.5, using as an example the diffusion of the hydrophobic
dye methyl red (2-carbohydroxy-4'-dimethylamino-azobenzene) in simple linear chain
aliphatic alcohols. This photochromic probe molecule is transformed from the trans into the
cis form by irradiation with the Ar + laser wavelength of 514.5 nm. The lifetime s dye varies between
about 20 ms and 1 s, depending on the medium and its viscosity. The dependence of
the diffusion coefficient on the viscosity is clearly seen from the different slopes in Fig. 20.5.
The power of the method will be illustrated by two applications. First, we investigate
the influence of the alcohol content on the quaternary system for which the formation of
multilamellar vesicles has been discovered by Hoffmann et al. [10]. Second, diffusion in the
water-in-oil microemulsion and in the bicontinuous cubic gel phase of the AOT/octanol/
water system is investigated.
The multilamellar vesicle system was investigated without the addition of salt. The
surfactant mixture was held constant at 90 mM C 14 DMAO and 10 mM C 14 H 29 N + (CH 3 ) 3 Br – ,
406

20.2 Diffusion in surfactant systems
Figure 20.5: Evaluation of diffusion coefficients from angle-dependent FRS measurements. The grating
decay rate s –1 is plotted against q 2 (Eq. 4). The diffusion of methyl red in simple linear chain alcohols
was investigated.
with the hexanol (C 6 OH) concentration being varied. Methyl red was added in a concentration
of 7610 –5 M. Polarization microscopy clearly shows [12] that the dye molecules are attached
to the vesicles and hence are probing diffusion processes within the vesicles.
The results presented in Fig. 20.6 comprise diffusion measurements both, in the micellar
L 1 phase (c (C 6 OH) = 0, 30, and 60 mM) and in the L a phase (c (C 6 OH) = 150 mM). The
diffusion coefficient is largest for c (C 6 OH) = 0 M; it drops rapidly upon the addition of
30 mM of hexanol as rod-like micelles are formed. Interestingly, the diffusion coefficient increases
again if the alcohol concentration is doubled. This behaviour has been successfully
interpreted [12] by noting that increasing the hexanol concentration from 30 to 60 mM
causes a shortening of the rod-like micelles, which according to the theory of Doi and Edwards
[13] gives rise to an increase in the diffusion coefficient.
Figure 20.6: Influence of the hexanol concentration on the diffusion coefficient of methyl red in the
C 14 DMAO/CTAB/C 6 H 13 OH system (see text).
407

20 Spectroscopic Probes of Surfactant Systems and Biopolymers
In the L a phase (c (C 6 OH) > 150 mM in Fig. 20.6), the diffusion coefficient of methyl
red is found to be low and approximately constant. Freeze-fracture microscopy has revealed
that the system consists of densely packed multilamellar vesicles (Hoffmann 1995). Their
average size (1 µm) is smaller than the spacing of the light-induced grating (15–25 µm),
but some large vesicles with sizes up to 30 µm are present as well. Hence, the observed decay
of the FRS signal is due to the exchange of dye molecules between surfactant bilayers
on the one hand and due to diffusion within giant vesicles on the other hand. Variation of
the alcohol content within the L a phase changes the number of the vesicles but does not alter
the qualitative nature of the diffusional processes, thus understanding the constancy of the
measured diffusion coefficient.
The section is concluded by reporting results for the AOT/octanol/water system. The
water-soluble dye congo red – a structurally related photochromic azo dye [12] – has been
used as a probe. Compositions corresponding to two cross sections through the ternary phase
diagram [14] are investigated. Details are given by Hahn et al. [12].
First, a series of four samples within the region of water-in-oil (w/o) microemulsion
phase L 2 was investigated (Fig. 20.7 a). The diffusion constant of the probe (residing within
the water phase) is seen to decrease strongly with the water content, to stay constant above a
water concentration of 50 wt%. This behaviour is interpreted in terms of the structural models
of Scriven [15] and of Fontell [1]. The microemulsion is thought to undergo continuous
a)
b)
Figure 20.7: Diffusion coefficients in the AOT/octanol/water system determined by FRS. a) variation of
the water content within the L 2 phase; b) transition from the L 2 into the I 2 phase induced by decreasing
the octanol weight fraction.
408

20.2 Diffusion in surfactant systems
structural changes with increasing water content, i. e. swelling of inverse rod-like micelles
and formation of long narrow water channels within which the diffusion of the dye probe
would be strongly impeded. As mentioned above, the spacing of the light-induced grating
amounts to 15–25 µm. Over such distances, diffusion within the water channels may be excluded.
Thus, the observed decay is ascribed to motion of the aggregates as a whole, which
is slowed down as the channels are swelling due to the addition of water.
The transition from the L 2 into the cubic I 2 phase was investigated by varying the octanol
content (Fig. 20.7b). The sample containing 20 wt% octanol corresponds to an L 2
phase, whereas the three samples with 10 and 14 wt% of octanol, respectively, are cubic gels
(I 2 phase). The diffusion coefficient is distinctly higher in the gels, however, the increase
with respect to the w/o microemulsion appears to be a gradual one, rather than a stepwise
change. The highest value recorded (5610 –11 m 2 s –1 ) is still four times smaller than the diffusion
coefficient of congo red in water. The observed behaviour would not be consistent
with the model of a gel consisting of closely packed, aggregated micelles. Rather, it suggests
a bicontinuous structure of the gel, in which water channels are opening up with decreasing
octanol content. This result is in agreement with the characterization of the gel by Gradzielski
et al. [16], who found that the I 2 phase is characterized by higher long-range order.
The mentioned description is also consistent with the structural models developed by Fontell
[1] for related systems.
20.2.3 Dimensionality of diffusion in lyotropic mesophases from fluorescence
quenching
Fluorescence quenching has been successfully used in surfactant research to determine aggregation
numbers in micellar systems [17, 18]. Fluorescence quenching in a homogeneous
phase leads to the well-known monoexponential decays but not if quencher diffusion is limited
to restricted regions in space (e. g. the lipophilic regions in a surfactant system). It has
early been realized [19] that the precise shape of the time dependence could be used to determine
the dimensionality of the space available for diffusion. The probability for diffusive
encounter between a fluorescent molecule and a given quencher decreases exponentially
with time for three-dimensional diffusion. Whereas the integrated probability of encounter
approaches unity in one or two dimensions the time dependence is distinctly different for
each case.
The aim of the present study was to demonstrate this concept within a single surfactant
system forming both, one-dimensional aggregates (rod-like micelles) and a two-dimensional
a-lamellar phase. The C 14 DMAO/C 7 H 15 OH/H 2 O system was chosen for this purpose because
it forms in addition to the above-mentioned phases two distinct L 3 mesophases of different
viscosity [20]. The structure of these cubic phases has been the subject of intense investigations.
Freeze fracture electron microscopy demonstrated their bicontinuous character [21].
In our study, the C 14 DMAO concentration was held constant at 100 mM, while the
heptanol concentration was varied between 0 and 200 mM in order to realize compositions
that correspond to the mentioned phases. With view to comparability of earlier studies in
rod-like micelles [22], pyrene (c =10 –5 M) was chosen as the fluorescent probe and benzo-
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20 Spectroscopic Probes of Surfactant Systems and Biopolymers
phenone as the quencher diffusing exclusively within the lipophilic regions of the surfactant
system. Details of the experiment and of the data reduction procedure have been described
in the thesis work of Wiesner [23] and Meyer [24].
Experimental results for a micellar phase consisting of rod-like micelles and of an a-
lamellar phase in the same system are compared in Fig. 20.8. Fluorescence decay curves are
presented as logarithmic plots. The deviations from exponentiality in the micellar system
(Fig. 20.8a) are evident as all traces are curved even for long times except for the system
containing no quencher (top trace). Evidently, the decay is faster in the a-lamellar phase
(Fig. 20.8b) when comparing runs at the same quencher concentration. This result meets the
intuitive expectation that the approach of quenchers from all sides within a lamellar plane
should give rise to a faster decay as compared to one-dimensional random walk within a cylindrical
or rod-like micelle.
a) b)
Figure 20.8: Fluorescence decay of pyrene in the C 14 DMAO/C 7 H 15 OH/H 2 O system. a) c (C 7 H 15 OH) = 0 M,
rod-like micelles; benzophenone quencher (top to bottom: c = 0, 0.1, 0.3, 0.5, 0.7 mM).
b) c (C 7 H 15 OH) = 62 mM, a-lamellar phase; benzophenone quencher (top to bottom: c = 0, 0.1, 0.3, 0.5,
0.7 mM).
Corresponding representative results for the L 3 phase (Fig. 20.9) show again a distinctly
non-exponential decay. In view of the poorer signal-to-noise ratio, obtained with this
sample, an extensive data set was acquired with this sample.
For the analysis the analytic result of Almgren et al. [22], of Alsins and Almgren [25]
for 1D diffusion, and the model of Owen [19] for 2D diffusion have been used. Their equations
were implemented in a fitting routine which performed, for a given sample, a global
analysis of the entire data set recorded at different quencher concentrations.
Prior to deriving values of the diffusion coefficients, it was checked whether the quality
of the non-exponential decays recorded was sufficient to warrant a distinction of one and
two-dimensional diffusion. In fact it was found that the use of the inappropriate model (e. g.
2D diffusion for the rod-like micelles) resulted in considerably higher mean square deviations,
thus demonstrating the significance of the fits. Technically, the intrinsic fluorescence
decay time of pyrene was derived from the curve without added quencher, the quenching
rate constant k q was adjusted as a global parameter and an average diffusion constant was
then derived using the known quencher concentrations [24].
410

20.2 Diffusion in surfactant systems
Figure 20.9: Fluorescence decay of pyrene in the L 3 phase of the C 14 DMAO/H 2 O system (c (C 7 H 15 OH)
= 190 mM). Temperature was held constant at 25 8C. Concentration of benzophenone quencher (top to
bottom: c = 0, 0.3, 0.6, 0.67, 0.75, 0.83, 1.0, 1.33 mM).
The selection of the appropriate model for the L 3 phase deserves some comment. According
to the accepted model [21,26], it consists of a bicontinuous structure featuring inner
and outer water channels separated by surfactant bilayers. An idealized structure resembling
a Gaussian minimal surface with positive and negative curvature at each point has been proposed
[27]. The real structure probably corresponds to a multiply branched, irregular analogue
of the Gaussian surface. The decisive feature for the present analysis is that the surfactant
molecules are arranged in a (positively and negatively curved) surface in space, such
that the application of the 2D model of diffusion is appropriate, see below.
Results for the diffusion coefficients derived from the fits are compiled in Tab. 20.1.
Surveying the results we see that the diffusion coefficient of the benzophenone quencher is
2–3 times higher in the two investigated lamellar phases than in the micellar phases. At first
sight, this result seems to contradict the higher viscosity of the a-lamellar phase. However,
we have to consider that we are dealing with the motion of the comparatively large benzophenone
molecule, containing two phenyl rings, within the lipophilic region consisting of
the surfactant alkyl chains. A simple model suggests that a preferred orientation of the plane
of the benzophenone molecule is perpendicular to the rod axis and that diffusional motion
parallel to the rod axis is severely impeded. In contrast, within a surfactant bilayer there are
many more ways for the translational motion of benzophenone within the lipid region, so
that the higher diffusion coefficient obtained for the latter phase may be understood.
Most interesting is the result that the diffusion coefficient in the L 3 phase, 6.7610 –11 m 2
s –1 , is significantly smaller than the one in the a-lamellar phase, although we have argued that
diffusion in the L 3 phase should be essentially two-dimensional. Resolution of this apparent contradiction
comes from a recent theoretical analysis by Anderson and Wennerström [4] who
showed that the diffusion coefficient of a random walker on a Gaussian minimum surface should
be only about 2/3 of its value on a plane. In view of this result, which strictly applies only for an
idealized regular structure, the results in Table 20.1 are fully supporting the presently accepted bicontinuous
model of the L 3 phase.
411

20 Spectroscopic Probes of Surfactant Systems and Biopolymers
Table 20.1: Quencher diffusion coefficients in the C 14 DMAO/C 7 H 15 OH/H 2 O system.
C 7 H 15 OH concentration/mM Diffusion coefficient D/10 –11 m 2 s –1 Phase and model
0 4.9 ± 2.2 rod-like micelles
1D diffusion
20 3.0 ± 0.9 rod-like micelles
1D diffusion
62 12.4 ± 5.5 lamellar phase
2D diffusion
170 11.6 ± 2.4 lamellar phase
2D diffusion
190 6.7 ± 3.4 L 3 phase
modified 2D diffusion
20.2.4 Summary of results
In Section 20.2 the use of three spectroscopic methods has been exemplified for obtaining
information on diffusion coefficients in surfactant systems. Regarding the micellar phases
both, NMR self-diffusion measurements and FRS, have been successfully used to monitor
changes in the micellar diffusion coefficients resulting from size and shape changes. This information
complements the results from static and dynamic light scattering experiments in a
most valuable manner.
Two structurally different types of cubic phases have been clearly distinguished by the
diffusion measurements. In the C n DMAO/hydrocarbon/water ringing gels which consist of
aggregated micelles, residual mobilities resulting from exchange processes between the micelles
have been characterized by NMR. On the other hand, the increase in long-range order
and the opening of water channels in the bicontinuous AOT/octanol/water gels was evidenced
by the increased diffusion coefficient of the water soluble congo red dye probe, monitored
by FRS.
For the quaternary C 14 DMAO/CTAB/hexanol/water system NMR was used to investigate
salt-induced changes in the volume fraction occupied by the multilamellar vesicles, as
well as deformation and shape alterations. In the same system, FRS yielded information on
changes in diffusion coefficients upon the transition from rod-like micelles to the vesicular
phase, induced by increasing the alcohol concentration.
For the C 14 DMAO/heptanol/water system monitoring the time dependence of fluorescence
due to diffusing quenchers resulted in a clear distinction of one and two-dimensional
diffusion. For the L 3 phase an experimental confirmation of a recent theoretical model for
diffusion on a Gaussian minimal surface has been achieved.
412

20.3 Vibrational spectroscopy and conformational analysis of oligonucleotides
20.3 Vibrational spectroscopy and conformational analysis
of oligonucleotides
Three projects have been pursued, with the aim of providing spectroscopic tools for the research
of Section 19. First, conformational changes in oligodeoxyribonucleotide duplexes
were monitored by Raman spectroscopy and by 2D NMR techniques. The second aim was
to assess the potential of surface enhanced Raman spectroscopy (SERS) for the characterization
of oligonucleotides. Third, the interaction of intercalating and groove binding dye
probes with DNA strands has been investigated by resonance Raman spectroscopy.
20.3.1 Spectroscopic characterization of right and left-helical forms
of a hexadecanucleotide duplex
Changes between the right and left-helical forms of DNA are important for understanding
control mechanisms of replication [28]. In particular, Jovin et al. [29] have characterized the
structural parameters of Z DNA and have indicated Raman bands that might be used as indicators
for the B , Z transition.
Our first goal was to assess the sensitivity of normal Raman spectroscopy for monitoring
the right-to-left conformational transition. Defined substrates for this investigation, i. e.
the hexadecanucleotides d (CG) 8 and d(m 5 CG) 8 , have been synthesized by Weber et al. [30].
Duplexes were obtained in the right-helical B conformation from the synthesis, transforming
them into the left-helical Z conformation by exposure to high salt concentration
(c (NaCl) & 3M).
Distinct differences between the Raman spectra of the two forms are evident in
Fig. 20.10. In particular, the phosphodiester vibration of the backbone at a frequency of
824 cm –1 is characteristic for the C 2' -endo pucker of the ribose rings in the B conformation.
In the spectrum of the Z conformation, this band is downshifted to 786 cm –1 and overlaps a
cytosine ring breathing vibration.
Further pronounced changes concern vibrations within the guanine ring system, which
is oriented more towards the exterior surface of the double helix in the Z conformation because
of the C 3' -endo-syn conformation at the ribose rings. The ring breathing vibration is
downshifted from 680 to 612 cm –1 upon the B ) Z transition. In addition, several ring
stretching vibrations of guanine in the 1300–1360 cm –1 region are affected (Fig. 20.10).
2D NMR spectroscopy is an established tool for elucidating the three-dimensional
structure of biopolymers in solution. We have used 2D NOE spectroscopy as an alternative
technique for monitoring the B , Z conformational transition. The procedure is exemplary
illustrated in Fig. 20.11 for the oligonucleotide d (m 5 CG) 8 . As mentioned above, the orientation
of guanine relative to the ribose ring is distinctly different for the right and left-helical
forms. Consequently, there are in the B conformation strong cross peaks between the guanine
proton GH 8 and the ribose proton GH2'. In contrast, for the Z conformation, the H 8
413

20 Spectroscopic Probes of Surfactant Systems and Biopolymers
Figure 20.10: Raman spectra of the deoxyribonucleotide duplex d(CG) 8 .
Top: B conformation, c = 39 mM; bottom: Z conformation, c = 24 mM.
Figure 20.11: NOESY spectrum of d(m 5 CG) 8 in the Z conformation (medium: c (MgCl 2 ) = 10 mM).
Experimental conditions have been given in Ref. [30].
414

20.3 Vibrational spectroscopy and conformational analysis of oligonucleotides
proton of the base and the GH1' proton of the sugar are spatially close. The spectrum demonstrates
that from the presence of the respective cross peak the conformation present under
the given conditions can be unambiguously inferred [30].
20.3.2 SERS spectra of deoxyribonucleotides
For SERS spectroscopy of biomolecules [31], the use of small sample volumes is mandatory.
For this purpose, two spectroelectrochemical cells were constructed in which the liquid sample
is held in a thin cylindrical volume between a silver rod electrode and the optical access
window. The design of the cells (from glass or teflon) and the oxidation-reduction cycle, used
for the required roughening of the silver, have been described in detail by Zimmermann [32].
The power of the technique is illustrated by the SERS spectrum of a mononucleotide,
5'-adenosine-monophosphate (5'-rAMP). The top spectrum in Fig. 20.12 (left) was recorded
with a concentration of 1.5 mM 5'-rAMP in phosphate buffer, with the electrolyte concentration
adjusted to 100 mM. In particular, attention is drawn to the excellent signal-to-noise
ratio achieved at a millimolar concentration.
The high salinity conditions that are often used to induce a transition into left-helical conformations
in oligo or polynucloetides were critically considered for the envisaged application.
Figure 20.12: SERS spectra of mononucleotides excited at 530.9 nm with the power of 29 mW and recorded
in a spectroelectrochemical cell using a resolution of 7 cm –1 [33].
Left diagram: 5'-rAMP (1.5 mM), top: NaCl (84 mM), KCl (16 mM), U Ag/AgCl = –0.5 V; bottom:
NaCl (3.3 M), KCl (80 mM), U Ag/AgCl = –0.6 V.
Right diagram: top: 5'-rCMP (2 mM), phosphate buffer (2 mM);
bottom: 5'-rCMP (9 mM), phosphate buffer (9 mM), NaCl (0.1 M), U Ag/AgCl = –0.3 V.
415

20 Spectroscopic Probes of Surfactant Systems and Biopolymers
Therefore, the SERS spectrum of 5'-rAMP was also recorded in the presence of NaCl (3.3 M).
The obtained high quality spectrum (Fig. 20.12, lower trace) demonstrates that SERS spectroscopy
is feasible in the media that are typically used to prepare and study the Z conformation.
Corresponding experiments with 5'-cytidine-monophosphate (5'-rCMP) showed an
optimum adsorption potential of –0.3 V vs. Ag/AgCl (Fig. 20.11, right). Because the silver
surface is positively charged at this potential, chloride and phosphate ions as well as impurities
from the solution are adsorbed much more strongly to the electrode surface as compared
to the potential of –0.5 V where the 5'-rAMP adsorption had been found to be an optimum
(left). Therefore, the sensitivity in the 5'-rCMP, recorded at a concentration of 2 mM, was
considerably lower as compared to the 5'-rAMP spectrum [33].
An attempt to record the SERS signals of the oligonucleotide d(CG) 8 resulted in an
uninformative spectrum (Fig. 20.13, top). In spite of the comparatively high total concentration
(2 mg/ml & 3 mM), the signal-to-noise ratio is poor and only a few vibrations that are
characteristic for d(CG) 8 are detected. In the bottom trace, the normal Raman spectrum recorded
at a 10 times higher concentration is reproduced from Fig. 20.10 for comparison.
Figure 20.13: SERS spectrum of d(CG) 8 (2 mg/mL), excited at 530.9 nm with the power 29 mW. Conditions:
phosphate buffer (0.17 M), NaCl (0.33 M), U Ag/AgCl = –0.1 V. In the bottom trace, the normal
Raman spectrum of d(CG) 8 is reproduced from Fig. 20.10 for comparison.
Several reasons may be given for the unsatisfactory sensitivity of the SERS spectrum
of d(CG) 8 . A comparatively more positive potential (–0.1 V vs. Ag/AgCl) had to be applied
in order to promote adsorption of the negatively charged exterior of the double helix
on the silver electrode and to avoid destabilization of the duplex [31]. At this potential,
however, the binding of d(CG) 8 is severely competed by adsorption of phosphate ions
from the buffer and chloride ions from the electrolyte, as shown by corresponding test experiments
with 5'-rCMP. From these experiments we conclude that only at very low buffer
and electrolyte concentrations favourable conditions might be found for recording sensitive
SERS spectra.
416

20.3 Vibrational spectroscopy and conformational analysis of oligonucleotides
20.3.3 Studies of chromophore-DNA interaction by vibrational spectroscopy
The binding of dye molecules onto DNA strands is an important process in biochemistry
[28]. Depending on its nature, this interaction may result in mutagenic, carcinogenic, and cytotoxic
properties of the chromophore so that specific dyes have been suggested for the use
as antitumor pharmaceuticals. Furthermore DNA binding dyes have been extensively used
as stains in biochemical fluorescence microscopy work.
In the present project we have investigated whether surface enhanced resonance Raman
spectroscopy (SERRS) could be used to study various types of DNA-chromophore interactions.
In this technique, a silver colloid is added to the solution containing the molecule
to be studied [34]. First, we had to test whether the complex, between dye and DNA, would
remain unaltered upon adsorption to the silver colloid surface which is a necessary condition
for a valid application of the method. A second system specific question concerns the useful
information that can be extracted from frequency shifts and relative intensity alterations in
the vibrational spectrum of the DNA-bound chromophore [35–37].
As a first example, the complex of acridine orange (AO) with calf thymus DNA has
been studied [38]. The stability of the complex upon adsorption on the silver colloid was
confirmed by absorption spectroscopy (Fig. 20.14, right). In solution the absorption maximum
at 504 nm (trace b) of AO in the complex is red shifted as compared to 495 nm (trace
a) of the spectrum of the free AO molecule. The analogous red shift is maintained when the
complex is added into a solution containing the silver colloid (trace d).
Highly sensitive SERS spectra have been recorded at the remarkably low AO concentration
of 2.5610 –6 M (Fig. 20.14, left). The spectra of the free (trace a) and of the DNAbound
dye (trace b) appear to be quite similar and only small changes in relative band intensities
are detected in the difference spectra (trace c). A careful analysis has been based on a
Figure 20.14: Interaction of AO with calf thymus DNA. Left diagram: SERRS spectrum of the free
AO (a) and of the AO-DNA (b) complex, both excited at 476.2 nm with a power of 25 mW. The top
trace (c) represents the difference between the normalized spectra (b)-(a). Right diagram: absorbance
spectra of AO (2.5610 –6 M); aqueous solution (a), complex with calf thymus DNA (b), free AO adsorbed
on silver colloid (c), complex with calf thymus DNA adsorbed on silver colloid (d).
417

20 Spectroscopic Probes of Surfactant Systems and Biopolymers
complete assignment of the rich vibrational spectrum [38]. The observed alterations are
compatible with a geometry in which the DNA double helix is adsorbed parallel to the silver
surface [31] with the plane of the intercalated dye molecule approximately perpendicular to
the helix axis.
As a second example, the groove binding dye Hoechst 33 258 [39] was investigated.
The SERRS spectrum has been recorded at a concentration of 10 –6 M. It was analyzed in
detail and assigned using normal coordinate analysis [32]. The influence of protonation on
the binding to the silver surface was studied and an adsorption geometry with the two-benzimidazole
rings approximately parallel to the silver surface was inferred.
Besides intercalation and groove binding the ionic interaction of charged reagents
with the exterior surface of the double helix is a third mode of molecular interaction with
DNA strands. In this context, platinum(II) complexes have been intensely investigated with
respect to their cytostatic properties. In the classic representative dichlorodiamin-platinum(II)
or cisplatinum the binding to DNA is initiated by dissociation of a chloride ligand. In
this context, we have investigated a series of novel dichlorobis(cycloalkylamin)Pt(II) complexes
[40], in which the size of the cycloalkyl ring (C n H 2n–1 -) of the amine ligands was varied
from n = 3 to n = 8 [41]. The aim of the study was to detect influences of the ligand
size on the Pt-Cl bond strength in the complexes. But analysis showed that it is fairly constant
throughout the investigated series. The pronounced differences in pharmaceutical activity
seem to be mainly caused by the increasing lipophilicity of the higher membered rings.
20.3.4 Summary of results
Vibrational spectroscopy and, in particular, Raman scattering have been used to elucidate selected
interactions of DNAs. The B , Z conformational transition of oligo-DNA duplexes
and the interaction of the intercalating dye AO with calf thymus DNA have been explicitly
analyzed in the preceding sections.
Surface enhancement of the Raman signals due to adsorption of the analyte onto
either silver electrodes or colloids provides large gains in sensitivity in suitable cases. Limitations
are implicit in the requirement of adsorption on the silver surface. In the case of negatively
charge DNA double helices this leads to unfavourable competition with the binding
of phosphate ions from the buffer at positive surface potentials and to desorption at negative
surface potentials. Therefore, for intact double helices it seems to be difficult to take full advantage
of the enhancement, demonstrated for the smaller building blocks.
Very intense Raman signals may be obtained if the enhancement, due to addition of
silver colloids, is acting in combination with resonant enhancement of the Raman scattering
cross section by close-lying electronic transitions. This favourable situation is found for
many important dye molecules that have been shown to interact with DNA in a specific
manner. The present results on intercalating dyes demonstrate the high potential of SERRS
for studying chromophore-DNA interactions.
418

20.4 Related projects carried out within the framework of the Collaborative Research Centre
20.4 Related projects carried out within the framework
of the Collaborative Research Centre
The projects discussed in Section 20.2 have been centred around the spectroscopic probing
of the structure and dynamics of surfactant systems and lyotropic mesophases, in particular
hydrocarbon gels. Three related investigations not yet discussed in detail due to limitations
in space shall be briefly mentioned. These are the study of the conformation of surfactant
molecules by surface enhanced Raman spectroscopy, the development of a lipophilic dye
probe for surfactant systems, and an outlook onto a different class of reaction gels created
by hydrolysis of metal alkoxide precursors.
The conformation of several surface-active molecules, polymers (alkyl-trimethylammonium
surfactants, polyvinyl alcohol), and of model compounds such as choline has been
studied by SERS [42, 43]. In particular, information on the geometry of their binding to the
surface of silver colloids was obtained. Special methods for the synthesis of monodisperse
colloids have been developed [44].
The microscopic polarity in the interior of micellar aggregates is of intrinsic interest
in surfactant research. A well-known approach to this information is the use of solvatochromic
dye probes. For these chromophores, the wavelengths of absorption and emission (and
hence the Stokes shift) depend on the environment in which they are dissolved. A large variety
of dye molecules exhibiting this property has been tested with surfactant systems. However,
inspection reveals that most of these dyes contain ionic groups and, therefore, are
either water-soluble or tend to bind to the hydrophilic exterior of micellar aggregates.
Within the present project, we have synthesized and tested [23] the lipophilic dye
probe 4-dioctylamino-anthra-thiadiazol-1,2-dion (TDA), for the structure see Fig. 20.15. As
a measure for the solvatochromic properties of the compound, the Stokes shift of various
O
O
N
Oc
N
Oc
N
S
Figure 20.15: Stokes shift as a function of solvent orientational polarization df for TDA. The structure
of the molecule is shown on the right-hand side. Solvents: hexane (1), toluene (2), CCl 4 (3), dioxane
(4), dibutylether (5), chlorobenzene (6), diethylether (7), ethylacetate (8), 1-chlorobutane (9), tetrahydrofuran
(10), CH 2 Cl 2 (11), dimethylsulfoxide (12), dimethylformamide (13), acetonitrile (14).
419

20 Spectroscopic Probes of Surfactant Systems and Biopolymers
simple solvents is presented in Fig. 20.14. It shows a remarkable range of variation from
about 500 cm –1 to about 2000 cm –1 . Solvent polarity is characterized in Fig. 20.15 by the
orientational polarization df, which includes the effects of both, static and dynamic polarization
of the solvent [45]. Linearity in the plot of the Stokes shift vs. df, as expected according
to Lippert [46], shows that TDA is a promising candidate for polarity probing in surfactant
systems.
The fluorescence lifetime of TDA is highest (10–12 ns) in hydrocarbon solvents and
decreases to 0.6 ns in polar solvents, such as acetonitrile. A plot of the decay rate against
the dimensionless parameter E T N [47], which characterizes solvent polarity, has been given by
Wiesner [23]. In contrast to the pronounced polarity dependence the lifetime is found to depend
only weakly on the viscosity of the environment.
Chromophore doping has been used not only as a spectroscopic probe but also in order
to get non-linear optical properties for Langmuir-Blodgett films. The aim of the latter
project was to monitor changes in molecular order and orientation when a spreaded film of
arachidic acid (AA) is transferred from the water surface to a solid substrate in the Langmiur
trough. For this aim three dyes of the phenylethenyl-pyridinium and of the benzaldehyde-hydrazone-type
were dissolved in the AA film. Then surface second harmonic generation
(SHG) was measured before and after transferring the films to a glass slide [48].
In the context of this study both, SHG [49] and enhanced Raman scattering [50], from
multilayer samples have been studied. These substrates, which consist of a silver island film
separated from a silver mirror by a dielectric spacer layer, excel by remarkable reflectivity
and absorption properties that have been accounted for by appropriate modelling [51].
Polymers doped with triazene chromophores were used for sensitizing non-absorbing
polymers, such as PMMA, for excimer laser ablation at 308 nm [52]. This project, which
provided a strong link with the polymer-related activities described in Chapter III, resulted
in the development of a new class of photosensitive polymers containing the triazene
(–N=N–N–) functional group in the main chain [53].
This Section is concluded by briefly mentioning structural investigations on a different
type of gels, which were generated by network formation during hydrolysis of metal alkoxide
precursors. Mixed oxides are materials of high technological interest and the sol-gel technique
[54] is one of the most promising routes for preparing homogeneous materials at low
process temperatures. In the present context, the use of these mixed oxide gels as catalysts
or catalyst supports is of particular interest.
Depending on the drying method employed, mixed oxides derived from the sol-gel
process are designated either as xerogels (solvent evaporation) or as aerogels (supercritical
drying). The properties of TiO 2 /SiO 2 -xerogels to be used as catalyst supports were studied
by Schraml-Marth et al. [55], using vibrational spectroscopy. Walther et al. [56] have characterized
the connectivity of vanadia and silica in V 2 O 5 /SiO 2 -xerogel catalysts by 29 Si
MASNMR.
Mixed oxide aerogels composed of titania, vanadia, and niobia have been shown to be
highly active catalysts for the selective catalytic reduction (SCR) of nitric oxides with ammonia.
These gels were characterized by FTIR, Raman, photoelectron spectroscopy, and secondary
ion mass spectrometry [57–59].
420

20.5 Remarks and acknowledgements
20.5 Remarks and acknowledgements
Spectroscopic tools are one of the links for our investigations of two seemingly different
classes of materials, namely the surfactant systems and the biological polymers. Specific
conclusions for each of the individual projects are presented in Sections 20.2.4 and 20.3.4,
respectively. We now want to focus on some features common to both activities.
In the studies of biopolymers, emphasis has been given to the development of methods
that are sufficiently sensitive to detect small toposelective changes in large molecules often
only available in small quantities. Vibrational and NMR spectroscopy have been successfully
used to characterize conformational changes in oligonucleotides and chromphore-DNA interactions.
Particular emphasis was devoted to the use of surface enhanced and resonant Raman
scattering techniques.
Structural characterization also formed the basis for the investigations of surfactants
(SERS, dye probes) and of gel systems (NMR and vibrational spectroscopies). Here the
main challenge was the monitoring of the dynamics of molecules and aggregates constituting
the various investigated lyotropic mesophases. NMR self-diffusion measurements of micelles
and solvents, diffusion of dye probes in laser induced gratings by FRS, and time-dependent
fluorescence techniques were combined to unravel the mobilities of different entities.
As in these mesoscopic systems a variety of motional processes is occurring simultaneously
on vastly different time scales, the use of different spectroscopic techniques, delivering
complementary information, was mandatory for a consistent interpretation of the
fascinating dynamics of these systems.
The results reported in this chapter have been obtained by the dedicated effort of past
and present members of the group. The corresponding publications, mentioning the names
of all persons involved, have been quoted in the respective sections. With regard to the
NMR self-diffusion measurements the author is particularly indebted to J.-C. Panitz, K.L.
Walther, Th. Schaller, and A. Sebald. FRS experiments have been carried out by G. Rehder
and Ch. Hahn. Fluorescence decay by diffusing quenchers was studied by J. Wiesner,
G. Meyer, and M. Klenke.
The investigations of biopolymers and of DNA-chromophor interactions have been advanced
by F. and B. Zimmermann. We are grateful to Th. Lippert for his valuable contributions.
Members of the team involved in the other projects have been mentioned in
Section 20.4.
This spectrum of activities has only been made possible thanks to the fruitful interaction
with colleagues in joint projects within the Sonderforschungsbereich. In particular, the
author would like to express his gratitude to H. Hoffmann and M. Sprinzl for numerous discussions.
Many of the reported investigations have been stimulated by their suggestions. We
are indebted to C.D. Eisenbach for his support in developing the dye probes and to O. Nuyken
for a fruitful collaboration on the subject of photopolymers.
421

20 Spectroscopic Probes of Surfactant Systems and Biopolymers
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54. C.J. Brinker, G.W. Scherer: Sol-Gel Science, Academic Press, San Diego, (1990)
55. M. Schraml-Marth, B.E. Handy, A. Wokaun, A. Baiker: J. Non-Cryst. Solids, 143, 93 (1992)
56. K.L. Walther, A. Wokaun, A. Baiker: Mol. Phys., 71, 769 (1990)
57. U. Scharf, M. Schneider, A. Baiker, A. Wokaun: J. Catal., 149, 344 (1994)
58. M. Schneider, M. Maciejewski, S. Tschudin, A. Wokaun, A. Baiker: J. Catal., 149, 326 (1994)
59. M. Schneider, U. Scharf, A. Wokaun, A. Baiker: J. Catal., 150, 284 (1994)
423

21 Energy Transport by Lattice Solitons in a-Helical
Proteins
Franz-Georg Mertens, Dieter Hochstrasser, and Helmut Büttner
21.1 Introduction
For about 40 years the question of energy transport in muscle proteins has gained considerable
attention. The biological energy quantum of 0.42 eV is given by the hydrolysis of adenosine
triphosphate (ATP). It is assumed that this energy is transported practically without
loss and is eventually used for the contraction of muscle fibers. The soliton concept can
give an elegant answer to this question because here the dispersion of the energy can be prevented
by non-linear effects.
The fibers of striated muscles in vertebrates contain many myofibrils consisting of sarcomers.
Each sarcomer consists of parallel-running thick and thin filaments. The basic mechanism
for the muscle contraction consists in a sliding of the thin filaments relative to the
thick ones [1]. This sliding can be described by phenomenological models consisting of a sequence
of molecular processes [2, 3].
The thick filaments consist of myosin molecules which resemble rods with a diameter
of about 40 Å and a length of about 1600 Å. The myosin consists of two polypeptide chains
forming a double a-helical structure. At one end there are globular heads where the ATP hydrolysis
takes place.
In each helix there are three chains of peptide groups coupled by hydrogen bonds.
Since 1973 Davydov [4] developed a quantum theory for solitons on these hydrogen bonded
chains. The idea is that the energy from the ATP hydrolysis leads to an excitation of the
amide-I vibration in the first peptide groups at one end of the chain. This vibration has an
energy of about 0.205 eV and an electric dipole moment of 0.30 debye which is directed approximately
along the helix axis. By the dipole-dipole interaction the next peptide groups on
the chain can be excited, and so on. However, in this way the energy would not be transported
but only dispersed. The essential point in Davydov’s assumption is that the vibrational
excitations are coupled non-linearly to a deformation of the hydrogen bonds. If the coupling
exceeds a certain threshold, solitons can be formed which are solutions of a non-linear
Schrödinger equation. Further assumptions must be made in order to explain how the solitons
eventually produce the relative sliding between thick and thin filaments.
Davydov’s model has been refined or modified by several authors, which we do not discuss
here because the stability of the solitons seems questionable. Both, thermal [5] and
424 Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

21.1 Introduction
quantum fluctuations [6], reduce the lifetime such that it may not be large enough for the biological
energy transport. (However, for one of the modified models the stability against thermal
fluctuations seems to be better [7].) Moreover, some doubts have appeared whether the
non-linear Schrödinger equation can be derived properly from Davydov’s Hamiltonian [8].
In 1984 Yomosa [9] proposed an alternative model which is much simpler than Davydov’s
model because only the hydrogen bonds are involved in the energy transport. The peptide
groups are rigid, i. e. they are represented by single masses which are coupled by
strongly non-linear hydrogen bonds. Thus the energy is transported by lattice solitons. This
has several advantages:
a) The conditions for the occurrence of solitary waves on a one-dimensional lattice are
very weak [5]. This means that every realistic interaction potential can be used, e. g. Lennard-Jones
or Morse potentials. Solitons in the mathematical sense exist only for a completely
integrable model, the Toda lattice, but for simplicity we will generally use the term soliton.
b) Lattice solitons are non-topological, i. e. there is no energy gap. Thus the whole energy
of a soliton can be converted into the mechanical work for the muscle contraction [9]. By
contrast, Davydov solitons show an energy gap and only the kinetic part of the energy can
be converted.
c) Using lattice solitons, a molecular interpretation can be given [9] for the phenomenological
rowing boat model [3] which describes the relative sliding between thick and thin filaments.
d) Molecular dynamics simulations by Perez and Theodorakopoulos [11] have shown that
even at room temperature lattice solitons are very stable against thermal fluctuations. Moreover,
the lattice solitons are more stable than Davydov solitons if collisions between the two
types of solitons are considered [11].
e) Quantum mechanics is necessary only for the initial condition. The idea is that the energy
quantum of 0.42 eV, released by the ATP hydrolysis, produces impulsively a pulse-like
compression of the hydrogen-bonded lattice. According to the inverse scattering theory [12]
an arbitrary initial pulse develops into a finite number of solitons plus a radiative background
(phonons). This result holds for integrable classical systems. However, at least for
one integrable quantum lattice model soliton-like excitations have been found. For the quantum
Toda lattice in the strong-coupling regime there is a branch of excitations with a dispersion
curve (energy vs. momentum) which is practically identical to that of the classical solitons,
though the ground state shows large quantum fluctuations [13]. In fact, a quantum
Toda lattice with parameters appropriate for the a-helix [9] can be shown to be in the
strong-coupling, i. e. semiclassical, regime [14]. Reference [13] used the Bethe ansatz. This
method has been complemented recently by a semiclassical quantization of the periodic
Toda chain with practically identical results in the limit of an infinite chain [15].
f) The reasoning of point (a) also holds for quantum lattice models with realistic interaction
potentials. Such a model can always be replaced in a good approximation by a corresponding
classical model with a renormalized potential [16], which fulfills the weak conditions
for the existence of solitary waves. In the case of the Toda lattice the form of the potential
remains unchanged and only the parameters are renormalized.
425

21 Energy Transport by Lattice Solitons in a-Helical Proteins
For the above reasons lattice solitons are very good candidates for the energy transport.
In this paper we generalize Yomosa’s model [9] in two ways:
1) The peptide groups are no longer rigid. For simplicity we consider only one internal degree
of freedom (Section 21.2) which leads already to interesting new features for the solitons
(Section 21.4). The generalization to more degrees of freedom is straightforward
(Section 21.7).
2) Contrary to Yomosa, we do not work in the continuum limit. In fact, discreteness effects
turn out to be decisive. These effects are first taken into account by applying the quasicontinuum
approach (QCA) of Collins [17]. We use a rederivation of the approach in Fourier
space [18] which offers several advantages (Section 21.3). However, at least for a part of the
relevant energy range, the QCA is not sufficient (Section 21.5). Therefore we apply an iterative
method [18] where the accuracy of taking into account the discreteness effects can be
increased as much as necessary (Section 21.6). Finally, we discuss the energy loss of the
solitons due to the emission of optical phonons (Section 21.6). The results of the various
sections are always checked by computer simulations.
21.2 The model
Following Yomosa [9], Perez, and Theodorakopoulos [11] we consider a one-dimensional
lattice of hydrogen-bonded peptide groups. The non-linear interactions between neighbouring
peptide groups are described by a suitable potential V, e. g. a Toda potential or a Lennard-Jones
potential, with parameters from the literature (Section 21.5).
In contrast to Refs. [9, 11] we do not neglect the internal vibrations of the peptide
groups. In a first step we describe each group by two masses M 1 and M 2 coupled by a linear
interaction with eigenfrequency O 0 . The resulting Lagrangian is
L ˆ X
1
2 M 1 A _ 2 n ‡ 1 2 M 2 B _ 2 n VA … n‡1 B n †
n
1
2 mO2 0 … B n A n † 2 ; …1†
where m is the reduced mass; A n and B n are the displacements from the equilibrium positions
of M 1 and M 2 of the n th peptide.
The neglection of anharmonic terms for the internal vibrations is justified by the fact
that the covalent bonds within the peptide groups are considerably stronger than the hydrogen
bonds between the groups. Thus the relative displacements for the internal motion,
D n ˆ … A n B n †=a ; …2†
are expected to be much smaller than the relative displacements,
426

21.2 The model
j n ˆ … A n‡1 B n †=a ; …3†
of the hydrogen bonds.
D n and j n are defined in units of the equilibrium distance a of neighbouring hydrogen
bonds, times in units of O 0 –1 , and energies in units of mO 0 2 a 2 . After this scaling we write the
non-linear interaction in the form aV (j 2 n ), where the harmonic part of V has the form 1 2 j2 n.
The dimensionless parameter a measures the strength of the non-linear interaction compared
to the linear one. In this notation the equations of motion are
j n ‡ a dV 1
…
dj n 1 ‡ m mD n ‡ D n‡1 † ˆ 0 …4†
D n ‡ D n
a
1 ‡ m m dV ‡ dV
ˆ 0
dj n j n 1
…5†
with the mass ratio m = M 1 /M 2 .
As D n appears only linearly, it can be eliminated which leads to a fourth order equation
in time
d 2
dt 2
j n ‡ j n ‡ a dV
dj n
‡ c 2 m a 2 dV
dj n
dV
dj n‡1
dV
ˆ 0
dj n 1
…6†
with
c 2 m ˆ
m
…1 ‡ m† 2 ˆ m
M ;
…7†
where M = M 1 + M 2 is the total mass.
Linearization (dV/dj n % j n ) yields the dispersion curves of acoustic and optical phonons
o 2 ‡ a
…q† ˆ1
2
The sound velocity is
(
16a
1 1
q ) 1=2
…1 ‡ a† 2 c2 m sin2 : …8†
2
c s ˆ
r
a
c m :
1 ‡ a
…9†
427

21 Energy Transport by Lattice Solitons in a-Helical Proteins
21.3 Quasicontinuum approximation
For the diatomic chain with non-linear nearest-neighbour interactions a standard decoupling
technique in the continuum limit [19] can be used and yields acoustic pulse-type solitary
waves and optical envelope-type solitary excitations [20]. These calculations are rather involved,
even more for our model which has not only alternating masses but also alternating
interactions. We prefer not to apply this method because there are two general problems:
a) Only polynomial interaction potentials can be used; usually the realistic potentials are
expanded up to the fourth order, assuming that the anharmonicities are small.
b) In the standard continuum approximation the difference operator is expanded up to a
certain order which can lead to an ill-posed Cauchy problem [21]. This difficulty occurs because
the dispersion due to the discreteness of the system is not taken into account consistently.
For the monoatomic chain both problems have been overcome by the QCA of Collins
[17] for solitary waves and periodic modes. Here the difference operator is inverted instead
of expanded. Rosenau [21] developed a still more general approximation scheme which reduces
to the result of Collins in the case of solitary waves.
For the diatomic chain Collins [22] considered only pulse-like solitary waves. We are
not interested here in the optical solitons because for the energy transport the pulse solitons
are much better candidates:
1) They are supersonic, in contrast to the optical ones.
2) The mechanism needed for the conversion of the energy into a shortening of muscle fibers
seems to be simple only for pulse solitons [9].
Since the QCA overcomes the above-mentioned problems we now apply it to our diatomic
model with alternating interactions. However, we use a rederivation of the QCA in
Fourier space [18]. This formulation is much simpler than the original one [17] and can
easily be generalized from the monoatomic chain to our model.
We are interested in solitary waves
j n …t† ˆj…n ct† ˆj…z† …10†
with velocity c, satisfying decaying boundary conditions. Because of these conditions the
Fourier transform exists,
~j…q† ˆ R1 dz e iqz j…z† ;
1
…11†
and analogously ~ D (q) for D (z). Moreover, we define the force
428

21.3 Quasicontinuum approximation
F…z† ˆa
dV
dj…z†
…12†
and its Fourier transform ~ F (q).
The equations of motion now read
c 2 q 2 ~j ‡ ~F
1
1 ‡ m eiq ‡ m ~ D ˆ 0 ; …13†
c 2 q 2 ‡ 1 ~D
1
1 ‡ m e iq ‡ m ~F ˆ 0 : …14†
The elimination of ~ D yields
c 2 q 2 1 c 2 q 2
~j…q† ˆ 4c 2 q
m sin2 c 2 q 2 F…q† ~ ; …15†
2
which can also be obtained directly from Eq. 8.
According to Ref. [18] the QCA now consists in the following procedure. We write
Eq. 15 as
A…q† ~j…q† ˆ ~F…q†
…16†
and expand A (q) in a Taylor series for |q|

21 Energy Transport by Lattice Solitons in a-Helical Proteins
with
V ef f ˆ aV…j†
1
2 a 0 j 2 ; …21†
where we have set V (0) = 0.
Because of c 2 m ^ 1/4, a 2 in Eq. 19 is always positive and can be interpreted as an effective
mass. Equation 20 has pulse-like, solitary solutions if there is a range of j for which
V eff (j) ^ 0, where the equality must hold at the boundaries of this range. For the considered
intermolecular potentials we have V eff (j) ^0 for a negative range j 1 ^j^0, with a 0 >0
and a < a 0 (because V % j 2 /2 for small j). These conditions lead to
c s c c m
…22†
which means we get supersonic, compressional solitary waves with amplitude j 1 . The meaning
of the upper limit c m will be discussed in the next Section.
The shape j (z) of the pulses is obtained by the integration of Eq. 20:
z…j† ˆ
Z j
dj 0
p : …23†
2V ef f …j 0 †=a 2
j 1
For the Toda or Lennard-Jones potential this integral can only be calculated numerically.
However, if we use an expansion of these potentials up to the fourth order
V…j† ˆ1
2 j2 2bj 3 ‡ 2gj 4 ; …24†
where 9 b 2

21.4 Velocity range for the quasicontinuum approach
~D…q† ˆm ‡ e iq
1 ‡ m
c 2 q 2
~j…q† : …28†
4c 2 m sin2 …q=2† c 2 q2 In order to be consistent with the approximations leading to Eq. 17, we expand Eq. 28
to order q 2 and perform an inverse Fourier transformation, which yields
D…z† ˆa 0 j…z†
1
1 ‡ m j0 …z†‡
1
21‡ … m†
a 0 c 2
m
12c 2 j 00 …z† : …29†
21.4 Velocity range for the quasicontinuum approach
There are restrictions for the velocity of the solitons, both in principle and for technical reasons.
In Eq. 17 the function
A…q† ˆ
c 2 q 2 …1 c 2 q 2 †
4c 2 m sin2 …q=2† c 2 q 2 …30†
has been expanded in a Taylor series with the implication that the Fourier transform ~j (q) of
the solitary wave is negligible for |q|6 q c . Here q c is the radius of convergence, defined by
the first non-trivial solution of
2c m sin q ˆ cq :
2
…31†
In Fig. 21.1 this condition is visualized as the intersection of the straight line cq with
the dispersion curve o m (q) =2c m |sin (q/2)|. Going back to the original units, o m can be
identified as the dispersion of a monoatomic chain of masses M = M 1 + M 2 with a linear interaction
with coupling constant mO 2 0. For this reason we see now that the upper limit c m for
the soliton velocity in Eq. 22 results from the linear interaction in our diatomic model with
alternating linear and non-linear interactions. For the usual diatomic model there is no upper
limit for the velocity [22].
In practice, i. e. for solitons with a finite width, the condition (Eq. 22) must be tightened
to
c s

21 Energy Transport by Lattice Solitons in a-Helical Proteins
Figure 21.1: Phonon dispersion curves. o + (q): optical, o – (q): acoustic, o m (q): monoatomic (from
Eq. 31). The intersections with the straight line o = cq are discussed in Sections 21.4 and 21.6. Parameters:
a = 0.6, m =2,c/c s = 1.3.
range of soliton velocities satisfying both conditions in Eq. 32 we must demand a P 1. In
fact, this is fulfilled for our model since the hydrogen bonds between the peptide groups are
considerably weaker than the covalent bonds within these groups (a is the ratio of the coupling
strengths, see Section 21.2). In Section 21.5 we will give estimates for a.
So far we have discussed only the limitations which result from the finite radius of
convergence for the Taylor series of A (q). Moreover, we must test whether the truncation
(Eq. 17) behind the second term of the series is justified. For a given mass ratio m, i. e. for
given c m , we choose a velocity c satisfying Eq. 32 and estimate the q range for which the
truncated expansion agrees well with A (q). Then we choose a potential V and calculate the
solitary wave j (z) which belongs to c. Its Fourier transform ~j (q) must be negligible outside
the above-mentioned q range. The results are given in the next Section.
21.5 Solitary waves for realistic parameter values
The lattice constant a for the H-bonded peptide chains in the a-helix is about 4.5 Å [9]. As a representative
value for the eigenfrequencies of a peptide group we choose O 0 = 3.11610 14 s –1
from the amide-I vibration. The total mass M = M 1 + M 2 corresponds to the mass of a peptide
group plus an average residue in a muscle protein, which gives together about 100 proton
masses [9, 10]. The mass ratio m = M 1 /M 2 is kept as a free parameter which is varied between 1
and 10 (our results are invariant under the transformation m 7 ! 1/m).
We model the hydrogen bonds between neighbouring peptide groups by a suitable
non-linear interaction, e. g. a Toda potential with parameters fitted to an ab initio self-consistent-field
molecular-orbital calculation for an H bond in a formamide dimer [9]. In our dimensionless
units this corresponds to
432

21.5 Solitary waves for realistic parameter values
aV T …j† ˆ a
b 2 exp… bj†‡bj 1
…33†
with b = 18 and a = 0.00123/c m 2 . Note that in Section 21.2 the interaction was introduced in
the form aV, where V = j 2 /2 for j?0. With these parameters the sound velocity is
c s % 4900 m/s for 1 ^m^10.
As a second example we take a Lennard-Jones potential with parameters fitted to the
equilibrium distance a and the bond energy [11, 23], which gives
V LJ …j† ˆ 1
1 2
72 …1 ‡ j† 12 …1 ‡ j† 6 ‡ 1
and a = 0.000811/c m 2 . The corresponding sound velocity is about 4000 m/s for 1^m^10.
For fixed mass ratio m, i. e. for fixed c m , we choose a velocity c in the range of Eq. 32
calculating the corresponding solitary wave j (z) and its Fourier transform ~j (q). A first test
shows that ~j (q c )/~j (0) is indeed negligible for a large range of velocities (about 30% above
c s ), as expected from the discussion of the radius of convergence q c in Section 21.4.
However, this large velocity range is considerably reduced by the second test described
in Section 21.4. For the potentials and parameter values used here the QCA is valid for velocities
c which do not exceed the sound velocity by more than about 5 to 10%. Eventually
we perform a final test by comparing them with the results of a computer simulation. If we
take Eq. 23 and Eq. 29 as initial conditions and integrate the difference-differential Eqs. 4
and 5 numerically then the time evolution of a single pulse (Fig. 21.2) shows that the QCA
solution is a good approximation. Only very small oscillations (phonons) appear immediately
after the start. After a while these phonons are left behind and the pulse travels without
changing its shape (Section 21.6). Figure 21.3 shows the scattering of two solitons.
…34†
Figure 21.2: Computer simulation of time-evolution of a single pulse for a chain of 200 unit cells with
m = 1 and the Toda potential (Eq. 33). As initial condition a QCA solution with c/c s = 1.05 is used.
After these tests we turn to the essential question whether the solitons of the QCA can
transport enough energy. Figure 21.4 shows the energy E of a single soliton as a function of
c/c s , for Toda and Lennard-Jones potentials. E must be compared with the biological energy
quantum of 0.42 eV (Section 21.1). Yomosa [9] assumed that these quanta set the initial
conditions for the lattice solitons. Naturally, an arbitrary initial condition generally produces
several solitons plus an oscillatory background. Knowing that nature usually works fairly ef-
433

21 Energy Transport by Lattice Solitons in a-Helical Proteins
Figure 21.3: Collision of two QCA solitons, same parameters as in Fig. 21.2.
Figure 21.4: Soliton energy vs. velocity. QCA results for Toda (solid line) and for Lennard-Jones interaction
(dashed line). Iterative solutions for Toda (x) and for Lennard-Jones interaction (y).
fectively, and in order to be on the safe side, let us assume that the energy of one quantum
essentially goes into one or two solitons. Then we see from Fig. 21.4 that we need a velocity
of at least 1.2c s for which the QCA clearly is no longer valid. Naturally, this velocity is only
a rough estimate that depends on the energy unit mO 2 0 a 2 , i. e. on our choice of O 0 . But in
any case we need a new method in order to handle the case of larger velocities. This is treated
in the next Section.
21.6 Iterative method and stability
With increasing velocity and energy the solitary waves become narrower and their width can
be in the order of the lattice constant. In this case the QCA does not hold. And it would not
help to take more terms of the Taylor expansion of A (q) in Eq. 16 into account. The resulting
higher order differential equations cannot be integrated like Eq. 17. Moreover, even the
infinite Taylor series cannot fully represent A (q) because of the finite radius of convergence.
Fortunately, an iterative method [18] has been developed for the monoatomic chain.
Here the accuracy of taking into account the discreteness effects is increased systematically.
In the case of the Toda lattice the iteration converges to the exact one-soliton solution.
434

21.6 Iterative method and stability
This method can also be applied to the pulse-type solitary waves of our diatomic
model. Our basic Eq. 15 can be written in the form
~j…q† ˆA 1 …q† ~F …q† ;
…35†
which already suggests an iteration because ~F (q) is the Fourier transform of F (q), (Eq. 12).
However, instead of Eq. 35 we use a slightly different form which will turn out later to be
more convenient. We first split the linear part of the force (Eq. 12) from the non-linear part,
denoted by G,
F …j…z†† ˆ aj…z†‡G……j…z†† : …36†
Then we insert Eq. 36 into Eq. 15 and isolate ~j
a c 2 q 2 o 2 m
~j…q† ˆ
…q†
~G…q†
c 2 q 2 o 2 ; …37†
‡ …q† c2 q 2 o 2 …q†
where o + (q) and o m (q) are the dispersion curves from Eq. 8 and Eq. 31, respectively.
Similar to the monoatomic case [18], an iteration for Eq. 37 would converge only to
the trivial solution j (z) : 0. This can be prevented by keeping ~j (0) constant during the
iteration [18]. This condition leads to the elimination of c from Eq. 37 and thus to a new
iteration procedure
with
a c 2 i
~j i‡1 …z† ˆ
q2 o 2 m …q†
~G
o 2 ‡ …q† c
2
i q 2 o 2 i …q† …38†
…q†
c 2 i q2
c 2 i ˆ c 2 s
~j…0†‡ ~G i …0†
~j…0†‡c 2 s =c2 m ~ G i …0† :
…39†
This implies c s

21 Energy Transport by Lattice Solitons in a-Helical Proteins
c i changes during the iteration and converges towards a value which usually is considerably
lower than the initial one. By the way, this behaviour is qualitatively similar to a computer
simulation. When starting with an approximate solution as initial condition, an adaptation
to the lattice by the emission of phonons is observed. The resulting solitary wave has a
lower velocity (and energy) than the initial wave. For the potentials and parameters, used
here, a sufficient accuracy is achieved after at most 8 iterations. As a test, these results were
used as input for a computer simulation. Contrary to the QCA result for a velocity outside
of the validity range (Fig. 21.5a), the shape of the pulse remains unchanged, no adaptation
to the lattice is observed (Fig. 21.5b). The soliton character of the pulses is also seen clearly
in scattering experiments, even in the highly discrete regime (Fig. 21.6). Figure 21.4 shows
the soliton energies. They are considerably higher than the QCA results for the same velocities.
Figure 21.5: Computer simulations for initial conditions with c/c s = 1.28 for the QCA (a) and the iterative
solution (b), using the Toda potential and m =1.
However, a complete convergence cannot be achieved, i. e. an exact solitary wave solution
probably does not exist. In fact, contrary to the monoatomic case [18], our iteration procedure
(Eq. 38) shows a pole, namely at the solution q 0 of
c i q ˆ o ‡ …q† :
…41†
The existence of the pole means that Eq. 38 makes sense only if ~G (q) and ~j (q) are
negligible for |q| 6 q 0 . In practice a cut-off is necessary in each step of the iteration.
Contrary to the finite convergence radius q c which limits the validity range of the
QCA (Section 21.4), the occurrence of the pole is not a technical problem but is connected
with a physical effect. Peyrard et al. [24] showed by computer simulations and theoretical investigations
that a solitary wave with velocity c cannot be stable if the line cq has an intersection
with one of the phonon branches o (q). The solitary wave permanently looses energy
by the emission of phonons with frequency spectrum centred around o(q 0 ) where q 0 is the
intersection point. Due to the energy loss the velocity c decreases gradually and the width of
the wave increases. Thus the discreteness effects which are responsible for the phonon emission
become smaller and smaller. Eventually the solitary wave is practically stable because
the energy loss is negligible.
This scenario holds for our model, too. The condition (Eq. 41) corresponds to the intersection
of c i q with the optical phonon branch (Fig. 21.1). In a computer simulation two
very different time scales appear. Starting with any initial condition including the QCA re-
436

21.7 Conclusion
Figure 21.6: Computer simulation of the collision of two very narrow solitons (c/c s = 1.9) from the
iterative solution for the Toda potential and m =1.
sult the above-mentioned adaptation to the lattice only takes a rather short time (Fig. 21.5a).
But this adaptation does not take place when the result from our iterative method is used as
initial condition (Fig. 21.5 b). Contrary to the adaptation process, the emission of optical
phonons occurs for a much longer time, which in principle goes on forever, while the solitary
wave disappears asymptotically.
However, this effect is extremely small when a-helix parameters are used. The situation
can be illustrated by drawing Fig. 21.1, now using the a-helix parameters of
Section 21.5. As a is very small there is a wide gap between the acoustic and optical phonon
branches and the intersection point q 0 is far outside of the first Brillouin zone (e. g. q 0 % 5p
for c % 2 c s ). Therefore the above condition for the convergence of the iteration, i. e. ~j (q 0 )
and ~G (q 0 ) are negligible, is indeed very well fulfilled and the energy loss due to emission
of optical phonons is negligible.
21.7 Conclusion
Lattice solitons remain good candidates for the explanation of the energy transport in the
a-helix if an internal vibration mode of the peptide groups is taken into account. Our theory
can easily be generalized for a lattice with a basis of more than two atoms, e. g. three masses
representing a peptide group N–C=O. The neglection of anharmonicities for the internal vibrations
is the essential point which allows the elimination of coordinates in the equations of
motion. This neglection is justified because the covalent bonds within the peptide groups are
considerably stronger than the hydrogen bonds between the peptides.
Discreteness effects are very important. They are partially incorporated by the quasicontinuum
approximation. However, for the parameters of the a-helix, a systematic implementation
of the discreteness is necessary, which is achieved by an iterative method. The
convergence of this method is limited in principle by an instability of the solitons due to the
emission of optical phonons. However, this effect turns out to be negligible.
In our opinion mainly two questions remain:
437

21 Energy Transport by Lattice Solitons in a-Helical Proteins
a) Stability of solitons
The coupling between the three hydrogen bonded chains of an a-helix possibly reduces or
destroys the stability of the solitons on the chain. So far this problem has not yet been
tackled analytically. Molecular dynamics simulations, which include transversal fluctuations,
do not show stable solitons [25]. On the other hand, we have performed simulations for a
Toda chain that branches into two Toda chains with different parameters [26]. Here we
choose the parameters of the first branch so that the interactions become strong and nearly
linear, representing the covalent bonds between the hydrogen-bonded chains. The parameters
of the second branch are chosen similar to the original chain, namely representing the weak,
very non-linear hydrogen bonds. The simulations show that a soliton, which is initiated on
the original chain, prefers to go into the second branch. Very little energy is lost via the first
branch, unless the energy of the soliton is very small. This is an interesting result, but naturally
it does not prove the stability of solitons on the very complicated structure of three
coupled chains.
b) Detection of solitons
In contrast to the topological solitons, it was not clear for a long time whether non-topological
lattice solitons yield a clear-cut signature in the dynamic form factor S (q,o), which can
be measured by inelastic neutron scattering. Recently combined Monte Carlo molecular
dynamics simulations yielded only one peak in S(q,o) for the thermal equilibrium and the
soliton and phonon contributions could not be distinguished [27]. However, if certain nonequilibrium
configurations are used as initial condition for the molecular dynamics, additional
small peaks on the high-frequency side of the main peak appear which can be identified
with lattice solitons. Therefore, the question now is whether the ATP hydrolysis produces
such initial conditions.
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26. D. Hochstrasser: PhD thesis, Universität Bayreuth (1989)
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de Physique, Les Ulis and Springer, Berlin, (1995)
439

IV
Appendix
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1

22 Documentation of the Collaborative Research Centre 213
Chairmen
Prof. Dr. Markus Schwoerer
Prof. Dr. Heinz Hoffmann
22.1 List of Members
Name Institute Membership
Blumen, A. Experimentalphysik 1987–1991
Büttner, H. Theoretische Physik 1984–1995
Dormann, E. Experimentalphysik 1984–1990
Eisenbach, C.D. Makromolekulare Chemie 1988–1995
Faulhammer, H. Biochemie 1984–1989
Fesser, K.** Theoretische Physik 1984–1995
Friedrich, J. Experimentalphysik 1984–1989
Friedrich, J. Experimentalphysik 1993–1995
Haarer, D. Experimentalphysik 1984–1995
Höcker, H. Makromolekulare Chemie 1984–1986
Hoffmann, H. Physikalische Chemie 1984–1995
Kalus, J. Experimentalphysik 1984–1992
Kiefer, W. Experimentalphysik 1984–1986
Kramer, L. Theoretische Physik 1984–1995
Krauss, H.L. Anorganische Chemie 1984–1992
Lattermann, G. Makromolekulare Chemie 1989–1995
Laubereau, A. Experimentalphysik 1984–1995
Mertens, F.G. Theoretische Physik 1987–1995
Nuyken, O. Makromolekulare Chemie 1988–1992
Pobell, F. Experimentalphysik 1987–1995
Richter, W. Experimentalphysik 1990–1995
Rösch, P. Struktur und Chemie der Biopolymere 1993–1995
* “Promotion” supported by the Sonderforschungsbereich 213
** “Habilitation” supported by the Sonderforschungsbereich 213
Macromolecular Systems: Microscopic Interactions and Macroscopic Properties
Deutsche Forschungsgemeinschaft (DFG)
Copyright © 2000 WILEY-VCH Verlag GmbH, Weinheim. ISBN: 978-3-527-27726-1
443

22 Documentation of the Collaborative Research Centre 213
Name Institute Membership
Schmidt, M. Makromolekulare Chemie 1993–1995
Schwoerer, M. Experimentalphysik 1984–1995
Seilmeier, A. Experimentalphysik 1993–1995
Spiess, H. W. Makromolekulare Chemie 1984–1986
Sprinzl, M. Biochemie 1984–1995
Strohriegl, P. Makromolekulare Chemie 1989–1995
Wokaun, A. Physikalische Chemie 1987–1995
Rentsch, S. Optik- und Quantenelektronik 1992–1995
University of Jena
22.2 Heads of Projects (Teilprojektleiter)
22.2.1 Projektbereich A: Gemeinsame Einrichtungen
A 1 FTIR-Gerät Krauss 1984–1992
A 5 Rasterelektronenmikroskopie Haarer 1987–1995
A 6 Transmissionselektronenmikroskopie Eisenbach 1989–1995
22.2.2 Projektbereich B: Festkörper
B 1 ENDOR in Diacetylenkristallen: Schwoerer 1984–1986
die molekulare Elektronik der topochemischen Reaktion
B 2 UV-Holographie und das Wachstum von Schwoerer 1984–1989
Mikrostrukturen in Diacetylen-Einkristallen
B 3 Photoleitung und Redox-Photochemie Molekularer Haarer 1984–1989
und hochpolymerer Festkörper
B 4 Disubstituierte Diacetylene mit besonderen Dormann 1984–1990
di- und pyroelektrischen Eigenschaften
B 5 Kommensurable und inkommensurable Büttner 1984–1986
strukturelle Übergänge in Charge-Transfer-Kristallen
B 6 Nichtlineare Anregungen in konjugierten Polymeren Fesser 1984–1995
B 7 Untersuchung elementarer Anregungen mit Kalus 1984–1992
hochausgelöster inelastischer Röntgenstreuung
444

22.2 Heads of Projects (Teilprojektleiter)
B 8 Elektron-Phonon-Wechselwirkung und elektronische Büttner 1990–1995
Korrelation in quasi-eindimensionalen Charge-
Transfer Systemen
B 9 Lochbrennspektroskopie an excitonischen Friedrich 1987–1989
Zuständen in Gläsern
B 10 Spektroskopie und Kalorimetrie an polymeren Haarer/Pobell 1987–1995
und nicht-kristallinen Substanzen
B 11 Hierarchische Modelle zum Ladungs- und Blumen 1987–1991
Massentransport in ungeordneten Medien
B 12 Lochbrenn-Spektroskopie in Polymeren: Haarer 1990–1995
Elektrische Feldeffekte und Druckeffekte
B 13 Nichtlineare Optik in Makromolekülsystemen: Schwoerer 1990–1995
Materialien und Bauelemente
B 14 Lichtinduzierte spektrale Diffusion von Richter 1990–1995
Farbstoffmolekülen in glasartigen Matrizen
B 15 Lochbrennspektroskopie an excitonischen Friedrich 1992–1995
Zuständen von Aggregatketten
22.2.3 Projektbereich C: Funktionale Systeme – Mizellen, Oberflächen
und Polymere
C 1 Viskoelastische Tensidlösungen Hoffmann 1984–1995
C 2 Lyotrope nematische und cholesterische Mesophasen Hoffmann 1984–1995
C 3 Flüssigkristalline Haupt- und Seitenkettenpolymere Höcker 1984–1986
C 4 Stabilität und Selektion in den strukturbildenden Kramer 1984–1995
Instabilitäten von flüssigen Kristallen
C 5 Struktur und Reaktivität von koordinativ Krauss 1984–1990
ungesättigten Oberflächenverbindungen
C 6 CARS-Spektroskopie an fluoreszierenden Materialien Kiefer 1984–1986
C 7 Darstellung und Untersuchung von Oligomeren Höcker 1984–1986
und Polymeren mit definierter Struktur und
besonderen Eigenschaften
C 8 NMR-Untersuchungen der molekularen Bewegung Spiess 1984–1986
und der molekularen Ordnung in festen Polymeren
C 9 Zeitaufgelöste Schwingungsspektroskopie Laubereau 1984–1995
an Polymeren mit ultrakurzen Laserimpulsen
C 10 Zeitaufgelöste toposelektive Spektroskopie von asso- Laubereau 1984–1986
ziierten Flüssigkeiten mit ultrakurzen Laserimpulsen
C 12 Spektroskopische Charakterisierung flüssigkristalliner Wokaun 1987–1995
Tensidphasen und lichtempfindlicher Polymere
C 13 Untersuchungen zur Konformation und Segment- Nuyken 1988–1992
beweglichkeit von Polymeren
C 14 Neue flüssigkristalline Modellsubstanzen und Polymere Lattermann 1988–1995
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22 Documentation of the Collaborative Research Centre 213
C 15 Photoleitende Polymere Strohriegl 1988–1995
C 16 Segmentierte Poyurethan-Elastomere mit und ohne Eisenbach 1988–1991
Wasserstoffbrückenbindungen: Synthese von Modellsystemen,
Struktur, Morphologie und Eigenschaften
C 17 Spektrales Verhalten von Photochromen Eisenbach 1988–1995
und Fluorophoren in Polymeren und deren
Verwendung als Molekülsonden
C 18 Brummgele Hoffmann 1990–1995
C 19 Spin-Korrelation bei magnetischen Lipidschichten Mertens 1990–1995
C 21 Toposelektive Reaktionen zur Darstellung und Schmidt 1993–1995
Charakterisierung supramolekularer Strukturen
C 22 Molekulare Verstärkung von Polymeren Eisenbach 1993–1995
durch Polydiacetylen-Propfcopolymere
C 23 Untersuchung des Energietransfers in Copolymeren Seilmeier 1993–1995
durch ultraschnelle Thermometrie nach Anregung
im mittleren Infrarot
22.2.4 Projektbereich D: Biopolymere
D 1 Optische Spektroskopie an DNA-Daunomycin Haarer 1984–1990
Intercalationskomplexen
D 2 Stark-Effekt an optisch anisotropen photochemischen Friedrich 1984–1986
Löchern: Antennenpigmente in organischen Gläsern
D 3 Elektronenspinresonanz-Spektroskopie mit Faulhammer 1984–1989
Proteinen und Nukleoprotein-Komplexen
D 4 Magnetische Kernresonanzspektroskopie spezifisch Sprinzl 1984–1995
markierter Ribonukleinsäuren, Ribonukleoproteinkomplexe
und Oligodesoxynukleotide
D 5 Spektroskopische Reportergruppen zur Untersuchung Sprinzl 1984–1995
der Struktur und Dynamik der Protein-Nukleinsäure-Wechselwirkung
D 6 Strukturelle Untersuchungen an Oligonukleotiden Wokaun 1987–1995
und DNA-Assoziationsverbindungen
D 7 Gittersolitonen auf Polypeptid-Ketten in Proteinen Mertens 1987–1989
D 8 Die Struktur von Makromolekülassoziaten in Lösung Rösch 1993–1995
am Beispiel von Protein-Nukleotid Komplexen
Ye 1 Untersuchung photoleitender Polymerthiophene auf Rentsch 1992–1995
der Pikosekunden- und Subpikosekunden Zeitskala
Yw 1 Untersuchung photoleitender Polythiophene Laubereau 1992–1995
auf der Piko-Subpikosekunden-Zeitskala
mit optischen und elektrischen Methoden
446

22.3 Guests
22.3 Guests
(Guests with a duration of stay longer than 2 weeks)
Name Acad. grad. Institute Year
Allakhverdiev, K. Prof. Academy of Sciences, Baku 1991/95
Aranson, I. Dr. Uni Jerusalem, Israel 1990/92
1994
Arnold, L. Dr. Insitute of Organic Chemistry. 1989/90
Czech. Acad. of Sc., Prague 1991/93/94
Bachvarov, I. Dr. Uni Sofia, Griechenl. 1993
Bara, M. Dr. Université Paris 1985/86
Bartusch, G. Dipl.-Chem. TU Dresden 1990/91
Boesch, R. Dr. Universität Dijon 1992
Brezesinski, G. Dr. Martin-Luther-Universität Halle Wittenberg 1989
Buka, A. Prof. Inst. f. Physics, Budapest 1989/90/91
1992/93/94
Caceres, M. Dr. Centro Atomico, Bariloche 1990/92
Cameron, J.H. Dr. Heriot-Watt University Edinburgh 1995
Chen, S.H. Prof. Massachusetts inst. of Technology, USA 1987/88
Chigrinov,V. Prof. Organic Intermediats & Dyes Inst., Moscow 1992/94
Danielius, R. Dr. Vilna, Litauen 1989
Delev,V. Dr. Baschkirische Uni, Ufa 1994
Dobrov,V. Dr. State University Moscow 1994
Eber, N. Dr. Academy of Sciences Budapest, Hungarian 1994/95
Favorova, O. Dr. Soviet Academy of Sciences, Moscow 1989
Fidy, J. Dr. Inst. of Biophysics Budapest 1994
Foltynowicz Poznan 1991
Gadonas Dr. Universität Vilnius 1989/90
Gaididei, Y. Prof. Academy of Sciences Kiev, Ukraine 1994
Gammel, J.T. Dr. National Laboratory, Los Alamos 1990
Getautis,V. Dr. Uni Kaunas, Litauen 1994
Golov, A. Dr. Academy of Sciences Chernogolovka, Moscow 1994
Gouvèa, M.E. Dr. Universidade Federal de Minas Gerais, Brasilien 1993/94
Grazulevicius, J. Dr. Technicol University Kaunas 1992/95
Herenyi, L. Institut of Biophysics University of Budapest 1994
Hiltrop, K. Dr. University Paderborn 1987
Imae, T. Prof. Nagoya University Japan 1991
Imrich, J. Dr. P.J. Safarik University Kosice 1994
Ivanov, B. Dr. Institute for Metical Physics Kiev, Ukraine 1994
Jonak, J. Dr. Institute of Molecular Biology 1988
Czech. Acad. of Sc., Prague
Joshi, R.L. Université Paris VII 1986
447

22 Documentation of the Collaborative Research Centre 213
Name Acad. grad. Institute Year
Kasperczyk, J. Dr. Zawiercie, Polen 1989/90
1991/92
Katunin,V. Dr. Inst. of Nuclear Physics, Soviet Acad. of Sc., 1993/95
St. Petersburg
Kauffmann, H. Prof. Universität Wien 1991
Kharlamov, B. Dr. Academy of Sciences, Moscow 1992/94
Kikas, J. Dr. Inst. of Physics, Tartu 1994/95
Kim, H. S. Prof. University of Seoul Südkorea 1991
Kohler, B. Prof. University California 1984/85/
88/90
Korrovits,V. Dr. Academy of Sciences, Tartu 1988
Kotomin, E. Dr. Universität Riga 1990/91
Kozar, T. Dr. Academy of Sciences Kosice Slovakia 1993
Krekhov, A.P. Dr. Baschkirische Uni,.Ufa 1993/94
1995
Kumar, A. Dr. Calcutta-University, Indien 1988/89/90
Kurlat, D.H. Prof. University Buenos Aires Argentinien 1987
Li, W. Dr. 1991/92
Lin, T.L. Prof. Universität Taiwan 1991
Mainz
Matyjaszewsky, K. Prof. Carnegia-Mellon-Uni. Pittsburgh, USA 1991
Mikhin, N. Dr. Academy of Sciences Kharkov, Ukrainian 1995
Miller, C.A. Prof. Rice University Dep. Houston Texas, USA 1989/95
Mistriotis, A. Dr. Universität Kreta 1991
Möhle, L. Dr. Leuna-Werk, Merseburg 1990/91
Muthukumar, M. Prof. University of Massachusetts 1994
Nawrot, B. Dr. Technical University Poznan Poland 1994/95
Nesrullajev, A.N. Prof. Universität Baku 1989
Nikogosyan, D.N. Prof. Inst. of Spectroscopy 1993
Academy of Sciences, Moscow
Noolandi, J. Prof. Xerox Research Centre of Canada 1991/92
Ofengand, J. Prof. Roche Inst. of Mol. Biology Nutley, New Jersey 1988
Ollikainen, O. Academy of Sciences, Tartu 1991
Personov, R. Prof. Inst. of Molecular Spectroscopy, Dep. Moscow 1987/91
Planer-Kühner, G. MPI f. Polymerforschung Pushchino, Russian 1990/91
Quèmarais, P. Prof. LEPES – CNRS, Grenoble 1994/95
Ragunathan,V. Dr. Bangalore, Indien 1989
Rebane, A. Dr. Academy of Sciences, Tartu 1987/88/92
Renge, I. Dr. Academy of Sciences, Tartu 1991
Reshetnikova, L. Dr. Inst. of Mol. Biology Acad. of Sciences, Moscow 1990/91
Riecke, H. Prof. Uni Northwestern, USA 1991/94
Rieckhoff, K. Dr. Simon Frazer University North Vancouver 1987/90
448

22.3 Guests
Name Acad. grad. Institute Year
Rötger, A. Dr. Universitè Fourier, Grenoble 1994
Ruckenstein, E. Prof. University Buffalo, USA 1985/86
Rudinger, J. Dr. Centre National de la Recherche Scientifique, 1992/93
Strasbourg
Salaev, F. Prof. Academy of Sciences of Baku, Aserbaijan 1992
Santa, I. Dr. Academy of Sciences Budapest, Ungarn 1990/91
Sarbak, Z.M. Dr. Universität Poznan, Polen 1989
Schott, M. Prof. Université Paris VII 1985
Schneider, A. Martin-Luther-University 1991
Inst. of Biochemistry, Halle-Wittgenstein
Sédlak, E. University Kosice 1994
Serra i Albet, A. Dr. University Barcelona 1987
Shalaby, G.A. Faculty of Science 1995
Minoufia University Shebin El-kom, Ägypten
Shchipunov,Y.A. Prof. Academy of Sciences Vladivostok, Russia 1993
Shirokov,V. Dr. Academy of Sciences 1991
Sigler, P. B. Prof. Dep. of Molecular Biophysics and Biochemistry, 1995
Yale University, New Haven
Silber, M. Dr. Passadena, Caltech, USA 1991
Smrt, J. Dr. Inst. of Organic Chemistry 1988/89
Czech. Acad. of Sc., Prague 1990
Sobkowski, M. Dr. Inst. of Bioorganic Chemistry, 1991
Polish Acad. of Sc., Poznan
Spirin, A. Prof. Inst. of Protein Research, Academy of Sciences, 1989
Pushchino, Russia
Stasko, A. Prof. Technical University, Bratislava, CSFR 1990/91
Steinberg, S. V. Dr. Inst. of Molecular Biology 1991/92
Soviet Acad. of Sc., Moscow 1993/94
Stoylov, S.P. Prof. Academy of Sciences Sofia, Bulgarien 1993/94/95
Suzuki, H. Dr. NTT Opto-Electronic Laboratories Tokai-Mure, 1989/92/93
Naka-Cun1
Svitova, T. Prof. Academy of Sciences, Chem. Dep., Moscow 1994
Szkaradkiewicz, K. University of Poznan 1993/97
Tamori, K. Academy of Sciences, Tokyo 1990
Tezak, D. Dr. University Zagreb, Croatia 1988
Thelakkat, M. Dr. NSS College, Manjeri, 1995
University of Calicut, India
Trommsdorff, P. Prof. University of Grenoble 1990
Tunkin,V. Dr. University Moscow 1991/92
Ueda, T. Dr. University of Tokyo 1989
Vainer,Y. Dr. Academy of Sciences Moscow 1993/94
Valiente-Martinez, M. Dr. Universidad de Alcala de Henares, Spanien 1992
449

22 Documentation of the Collaborative Research Centre 213
Name Acad. grad. Institute Year
Vitukhnovski, A. Dr. Academy of Sciences Moscow 1990
Vodopyanov, K Dr. Inst. of General Physics Moscow 1990/92
Wada, Y. Prof. University of Tokyo 1988/93
Wysin , G. Dr. Kansas State University 1990/91
Xu, J. Chengdu University, China 1990
22.4 Co-workers
Name Acad.grad Institute Project Period
Adam, D. Exp.Physik B3 1991/92/93/94
Aechtner, Dr. Exp.Physik C9/10 1984/85/86/89
Aggarwal, K. Dr. Makro.Chem. C13 1989
Ahmadian, R. Dr.* Biochem. D5 1989/91/92
Amberger, E. Dr.* Anorg.Chem. C5 1989/90/91
Ambrosch, A. Makro.Chem.. C14 1990/91/92
Angel, G. Dr.* Exp.Physik C9 1984–89
Angel, M. Dr.* Phys.Chem. C1/C2 1984/85/86
Angel, S. Dr.* Phys.Chem. C2 1987/88/89
Angstl, R. Dr.* Exp.Physik B2 1985–89
Bächer, R. Dr.* Phys.Chem. C2 1986/87/88
Bachmann, S. Biochem. D4/D5 1987/88/89
Bara, M. Exp.Physik B1 1985/86
Baranowski, D. Theo.Physik B8 1990–94
Barnickel, P. Dr.* Phys.Chem. D6 1989/91
Barth, K. Dr.* Exp.Physik B14 1991/92/93/94
Bauer, F. Exp.Physik B13 1994/95
Bauer, H.-D. Dr.* Exp.Physik B2/B13 1985/86/88–91
Beck, M. Biochemie D4/D5 1985–89
Beginn, Ch. Dr.* Makro.Chem. C15 1993
Beginn, U. Makro.Chem. C14 1993
Beikmann, M. Biochemie D4/D5 1993/94/95
Bettenhausen, J. Makro.Chem. C15 1993/94/95
Bieger, P. Dr.* Biochem. D5 1984–89
Bittl, Th. Biopolymere D8 1993
Blank, J. Biochemie D5 1991/92
450

22.4 Co-workers
Name Acad.grad Institute Project Period
Blechschmidt, B. Dr. Biochemie D5 1994/95
Blumen, A. Prof. Exp.Physik B11 1987–91
Bodenschatz, E. Dr.* Theor.Physik C4 1985–89
Bojer, B. Anorg.Chem. A1/C5 1987/88/89
Bratengeier, K. Exp.Physik C9 1984/85/86
Breinl, W. Dr.* Exp.Physik D2 1986
Bronold, F. Theo.Physik B6 1991/95
Buchwald, E. Exp.Physik B13 1995
Büttner, H. Prof. Theor.Physik B6/B8 1984–1995
Burger, A. Dr.* Phys.Chem. C1 1987/88/89
Burkhardt,V. Dr.* Makro.Chem. C13 1989/90/91/92
Caceres, O. Dr. Theor.Physik C4 1990
Carron, K. Dr. Phys.Chem. D6 1986/87/88
Cuc, T. T. Dr. Makro. Chem. C3/C7 1984/85
Dahinten, T. Exp.Physik C23 1993/94
Decker, W. Theor.Physik C4 1991/92/93
Deeg, M. Theor.Physik B8 1993/94/95
Denninger, G. Dr.** Biochemie/Exp.Phys. D3/B4 1984–89
Denzner, M. Anorg.Chem. C5 1988/89
Diemer, E. Phys.Chem. C18 1988–92
Dirnberger, K. Makro.Chem. C17 1993/94/95
Döge, B. Phys.Chem. C1 1986/87/88/89
Dörfler, S. Biochemie D4/D5 1994/95
Dohlus, R. Dr.* Exp.Phys. C9 1986/87
Domes, H. Dr.* Exp.Phys. B3 1984–89
Dormann, E. Prof. Exp.Phys. B4 1984–1990
Dreßel, U. Exp.Phys. B4 1988/89
Ebert, G. Dr.* Phys.Chem. C1 1986/87/88/89
Eisenbach, C.D. Prof. Makro.Chem. C16/C17 1988–1995
Ernst, U. Anorg.Chem. A1/C5 1985–89
Esquinazi, P. Dr. Exp.Phys. B10 1988/89
Faulhammer, H. Dr. Biochemie D3 1984–1989
Fehn, T. Exp.Phys. B13 1992/93
Fehske, H. Dr.** Theor.Phys. B8 1990–1995
Feile, M. Exp.Phys. B4 1988/89
Feng, Q. Dr.* Theor.Phys. C4 1990/91
Fesser, K. Prof.** Theor.Phys. B6 1984–1995
Feyerherd, B. Makro.Chem. C14 1992
Ficht, K. Dr.* Makro.Chem. C17 1990/91/92
Fickenscher, M. Exp.Phys. C9 1985/86/89/90
Fiebig, A. Exp.Phys. B4 1986–90
Fischer, K. Dr.* Makro.Chem. C17 1986/87/88/89
451

22 Documentation of the Collaborative Research Centre 213
Name Acad.grad Institute Project Period
Fischer, W. Dr.* Biochemie D4 1984/85
Flöser, G. Dr.* Exp.Phys. D1 1984/85/86/88
Förster, Ch. Dr.* Biochemie D4/D5 1990/91/92/93
Franzke, D. Dr.* Phys.Chem. C12 1987/88/89/91
Friedrich, J. Prof. Exp.Phys. B9/B15 1984–1989
1993–1995
Frosch, H. Theor.Phys. B5 1984/85
Gafert, J. Exp.Phys. B15 1994/95
Gagel, R. Dr.* Exp.Phys. C9/10 1986–91
Gammel, T. Theor.Phys. B8 1990
Gebhardt, H. Exp.Phys. B4 1988
Gebhardt-Singh, E. Dr.* Bioochemie D5 1985/86
Geissinger, P. Dr.* Exp.Phys. B12 1990–94
Geist, F. Exp.Phys. B7 1991/92
Giering, T. Exp.Phys. B12 1994/95
Gläser, D. Exp.Phys. A6 1986/87/88/89
Gollner, G. Makro.Chem. C14 1988/89
Gotschy, B. Exp.Phys. B2 1987
Gradl, G. Dr.* Exp.Phys. D2 1987/88/89
Gradzielski, M. Dr.* Phys.Chem. C18 1988/89/90/91
Graener, H. Dr.** Exp.Phys. C9 1986/87/88/89
Grillenbeck, N. Biochemie D4/D5 1988–92
Grottenmüller, R. Makro.Chem. C21 1993/94/95
Grotz-Green, C. Anorg.Chem. C5 1988/89
Gruner-Bauer, P. Dr.* Exp.Phys. B4 1986–91
Güttler, W. Dr. Exp.Phys. A3 1984–89
Haarer, D. Prof. Exp.Phys. A5/B3/B12 1984–1995
Häfner, W. Dr. Exp.Phys. C6 1984–89
Haegel, F.-H. Dr.* Phys.Chem. C1/D6 1984–88
Häfner, W. Phys.Chem. D6 1986/87/88/89
Härtl, P. Dr.* Biochemie D3 1985/86/87/88
Hahn, Ch. Phys.Chem. D6 1995
Halbritter, A. Biochemie D4/D5 1987/88/89
Hammon, W. Dr.** Anorg.Chem. C5 1986
Hecht, E. Phys.Chem. C18 1993/94/95
Heindl, D. Exp.Phys. B4 1986/87
Heitz, T. Makro.Chem. C3 1986
Henglein, F. Phys.Chem. D6 1988/89
Herold, M. Exp.Phys. B13 1992/93/94
Herrmann, F. Dr.* Biochemie D8 1994/95
Hertel, G. Dr.* Phys.Chem. C2 1986/87/88/89
Hirschmann, R. Dr.* Exp.Phys. B9 1987/88/89
452

22.4 Co-workers
Name Acad.grad Institute Project Period
Hochstraßer, D. Dr.* Theor.Phys. D7 1986/87/88/89
Höcker, H. Prof. Makro.Chem. C3/C7 1984–1986
Hoff, H. Dr.* Makro.Chem. C17 1988/90/91
Hoffmann, H. Prof. Phys.Chem. C1/C2/C18 1984–1995
Hoffmann, K. Dr.* Anorg.Chem. C5 1986
Hofmann, J. Makro.Chem. C22 1993/94/95
Hofmann, M. Biochemie D4/D5 1985/86
Hofmann, M. Dr.* Exp.Phys. C9 1992/93/94/95
Hofmann, W. Dr.* Exp.Phys. B7 1984–88
Hoffmüller, P. Biochemie D4/D5 1995
Hopfmüller, H. Exp.Phys. C6 1984/85/86
Huber, G. Dr.* Phys.Chem. C1/C2 1984–89
Hübner, J. Exp.Phys. B13 1993/94/95
Hüser, B. Makro.Chem. C8 1984/85/86
Hums, E. Dr. Anorg.Chem. C5 1984/85
Illner, J.- Ch. Dr.* Phys.Chem. C1 1991–95
Juarez, M. de L.T. Dr.* Theor.Phys. C4 1988/89
Kaiser, M. Dr.* Theor.Phys. C4 1987/88/89
Kalus, J. Prof. Exp.Phys. B7 1984–1992
Kanischka, G. Makro.Chem. C3/C7 1984/85/86
Kaul, H. Dr.* Exp.Phys. B2 1988/89
Kiefer, W. Prof. Exp.Phys. C6 1984–1986
Klenke, M. Phys.Chem. C12 1992/93/94/95
Knöchel, F. Makro.Chem. C7 1986
Köhler, G. Dr.* Exp.Phys. B11 1987–92
Köhler, M. Exp.Phys. B15 1995
Köhler, W. Dr.* Exp.Phys. D2 1984–88
Kohles, N. Exp.Phys. C9 1984/85/86/87
König, R. Dr. Exp.Phys. B10 1988–93
Krämer, U. Dr.* Phys.Chem. C2 1987/88/89
Kramer, L. Prof. Theor.Phys. C4 1984–1995
Krapf, M. Exp.Phys. B7 1990/91
Krauss, H.L. Prof. Anorg.Chem. A1/C5 1984–1992
Kreutzer, R. Dr. Biochemie D4/D5 1990/91/92
Kuang, W. Theor.Phys. C4 1986/87
Lattermann, G. Dr. Makro.Chem. C14 1989–1995
Laubereau, A. Prof. Exp.Phys. C9/10/Yw1 1984–1995
Lenk, M. Biochemie D4/D5 1991–95
Lenz, U. Dr.* Phys.Chem. C1 1988/89/90/92
Lieberth, M. Exp.Phys. B14 1990/91
Li, J. Exp.Phys. B10 1993/94
Limmer, St. Dr. Biochemie D4 1990–94
453

22 Documentation of the Collaborative Research Centre 213
Name Acad.grad Institute Project Period
Lindenberger, F. Exp.Phys. C20 1991/1992
Löbl, M. Dr.* Phys.Chem C1/C2 1984/85
Löw, A. Biochemie D3 1989
Lorenz, W. Exp.Phys. B10/B13 1990/91/92
Luding, St. Exp.Phys. B11 1990/91/92
Maier, H. Exp.Phys. B10 1992/93/94/95
Martins, J. M. N. Makro.Chem. C14 1994/95
Materny, A. Exp.Phys. B2 1987/88
Matuschek, D. Exp.Phys. B11 87/88/89/90
Mertens, F.G. Prof. Theor.Phys. C19/D7 1987–1995
Meyering-Vos, M. Dr. Biochemie D4/D5 1995
Merkel, K.-R. Dr.* Anorg.Chem. A1/C5 1985/86
Meyer, G. Dr.* Phys.Chem. C12 1988/89/92
Meyer, H. Dr.* Exp.Phys. B3 1989/90/91/92
Mirke, M. Makro.Chem. C21 1993/94/95
Morys, P. Prof. Dr. Anorg.Chem. C5 1986/87/88/89
Müller, K.-P. Dr.* Exp.Phys. B10 1986–91
Müller, R.G. Dr.* Makro.Chem. C15 1988/89
Müller-Horsche, E. Dr.* Exp.Phys. B3 1984/85
Müller-Nawrath, R. Dr. Exp.Phys. B1 1984/85/86
Munkert, U. Dr.* Phys.Chem. C2 1995
Nattler, G. Exp.Phys. C9 1988/89
Nawrot, B. Dr. Biochemie D4/D5 1994
Nefzger, H. Dr.* Makro.Chem. A6 1986/87/88/89
Nittke, A. Exp.Phys. B10 1993/94
Nomayo, M. Phys.Chem. D6 1993/94
Nuyken, O. Prof. Makro.Chem. C13 1988–1992
Oetter, G. Dr.* Phys.Chem. C1 1986/87/88
Ott, G. Dr. Biochemie D4 1988–1994
Paap, H.-G. Theor.Phys. C4 1985/86
Pecher, U. Theor.Phys. B8 1990/91/92/95
Pesch, W. Prof. Dr. Theor.Phys. C4 1984–88
Peter, M. Dr.* Biochemie D5 1986/89
Platz, G. Prof. Dr. Phys.Chem. C1/C2 1984/85
Pöhlmann, E. Exp.Phys. B4 1985/86
Pöhlmann, T. Makro.Chem. C13 1990/91/92
Pößnecker, G. Dr.* Phys.Chem. C2 1987/88/89/90
Pobell, F. Prof. Exp.Phys. B10 1987–1995
Pschierer, H. Exp.Phys. B15 1993/94
Purucker, H.-G. Dr.* Exp.Phys. C9 1987/88/92
Raible, Ch. Exp.Phys. B13 1991/92
Rauscher, A. Dr.* Phys.Chem. C1 1987–92
454

22.4 Co-workers
Name Acad.grad Institute Project Period
Rehage, H. Dr.** Phys.Chem. C1/C2 1984–89
Rehberg, I. Dr. Theor.Phys. C4 1988/89
Reichl, J. Dr.* DipL.Phys. B5 1984–86/88/89
Reichstein, W. Exp.Phys. A5 1985–89
Reiser, Ch. Dr.* Biochemie D5 1987–91/93
Rentsch, S. Doz., Dr. Exp.Phys. Ye1 1992–1995
Renner, H. Dr.* Phys.Chem. C2 1994/95
Reul, St. Dr.* Exp.Phys. B10 1991/92
Ribbe, A. Dr.* Makro.Chem. C17 1992
Richter, W. Dr. Exp.Phys. B14 1990–1995
Riederer, W. Dr.* Anorg.Chem. C5 1986/87/88/89
Robisch, P. Phys.Chem. C2 1988/89
Roder, S. Exp.Phys. B7 1986
Rösch, P. Prof. Struktur und Chem. d. Biop. D8 1993–1995
Röthlein, P. Exp.Phys. B7 1990
Röttger, J. Makro.Chem. C3/C7 1984/85/86
Rothemund, P. Phys.Chem. C1 1986/87
Schellenberg, P. Dr.* Exp.Phys. B15 1994
Schellhorn, M. Dr.* Makro.Chem. C14 1990/91/92/93
Schießwohl, M. Biochemie D4 1990/91/92
Schirmer, N. Dr.* Biochemie D5 1991/92
Schmelzer, U. Dr. Exp.Phys. B7 1986/87/88/89
Schmerbeck, S. Dr.* Anorg.Chem. C5 1984/85/86
Schmid, W. Dr.* Exp.Phys B2/B13 1994
Schmidt, H. Anorg.Chem. A1 1986/87/88/89
Schmidt, St. Makro.Chem. C13/C14 1990/92
Schmidt, M. Prof. Makro.Chem. C21 1993–1995
Schmutzler, M. Exp.Phys C9 1985/86
Schnabel, E. Phys.Chem. C1 1986/87
Schneider, J.M. Makro.Chem. C15 1992
Schnitzer, H.-J. Theor.Physik C19 1993/94/95
Schnörer, H. Dr.* Exp.Phys. B11 1987–92
Schörner, H. Makro.Chem. C15 1986/87/88/89
Schultes, H. Dr.* Exp.Phys. B4 1984/85/86/87
Schulz, S. Dr. Phys.Chem. C1 1990/91/92
Schumann, Ch. Anorg.Chem. A1 1990/91/92/93
Schwandner, B. Dr.* Phys.Chem. C1 1984/85/86
Schwarz-Schultz Dr. Biochemie D4 1985/86/87
Schwoerer, M. Prof. Exp.Phys. B1/B2/B13 1984–1995
Seidler, L. Dr.* Biochemie D5 1984/85/86
Seilmeier, A. Prof. Exp.Phys. C23 1993–1995
Sesselmann, R. Dr.* Exp.Phys. D1 1987/88
455

22 Documentation of the Collaborative Research Centre 213
Name Acad.grad Institute Project Period
Siebenhaar, B. Dr.* Anorg.Chem. C5 1987/88/89
Spiess, H.W. Prof. Makro.Chem. C8 1984–1986
Sprinzl, M. Prof. Biochemie D4/D5 1984–1995
Sühler, G. Phys.Chem. C1/C2 1993/94/95
Staufer, G. Dr.* Makro.Chem. C14 1986/87/88/89
Stöcklein, W. Exp.Phys. B1 1985
Strohriegl, P. Dr.** Exp.Phys. B2/B4 1984–89
Struller, B. Dr.* Phys.Chem. C2 1986/87/88/89
Sum, U. Dr.* Theor.Phys. B6 1984–89
Terskan-Reinhold, M. Makro.Chem. C17 1992/93/94
Thaufelder, H. Dr.* Makro.Chem. C16/17 1986–91
Thom, W. Theor.Physik C4 1987/88
Thurn, H. Dr. Phys.Chem. C1 1987
Thurn, R. Exp.Phys. C6 1984–89
Trapper, U. Theor.Phys. B8 1995
Treiber, M. Theor.Phys. C4 1992//93/94/95
Troeger, P. Dr.* Exp.Phys. C10 1984/85
Überla, H. Dr.* Theor.Phys. B5 1984/85/86
Ulbricht, W. Dr. Phys.Chem. C1/C2 1984–89
Völkel, A. Theor.Phys. C19 1990/91/92/93
Vogtmann, T. Dr.* Exp.Phys. B13 1987/88/89/92
Voit, J. Dr.** Theor.Phys. B8 1986/87/88/89
Voit, S. Exp.Phys. B13 1990
Vollstädt, K.-U. Exp.Phys. B13 1990/91/92
Vornlocher, H.-P. Biochemie D4/D5 1992/93
Voss, D. Exp.Phys. B3 1995
Waas,V. Dr.* Theor.Phys. B8 1986/87/88/89
Wagner, R. Dr.* Exp.Phys. B2 1987–91
Walther, K.L. Dr.* Phys.Chem. C12 19898/89
Wanka, G. Phys.Chem. C2 1988/89
Weber, R. Dr.* Phys.Chem. C2 1984/85/86
Weber, S. Dr.* Theor.Phys. B5/B8 1984–89
Wefing, St. Makro.Chem. C8 1984/85/86
Weidlich, K. Dr.* Exp.Phys. C9 1994/95
Weiss K. Dr.** Anorg.Chem. A1 1986/87/88/89
Weigel, J. Phys.Chem. C12 1993/94
Weißhaar, M. Dr. Biochemie D4/D5 1987/88/89
Wiesner, J. Dr.* Phys.Chem. C12 1986–90
Wietasch, H. Makro.Chem. C14 1994/95
Winter, H. Exp.Phys. B4 1987/88/89
Wittenbeck, P. Phys.Chem. C12 1988/89
Wokaun, A. Prof. Phys.Chem. C12/D6 1987–1995
456

22.5 International Cooperation
Name Acad.grad Institute Project Period
Wolf, M. Dr.* Theor.Phys. B6 1989–94
Wolff, P. Anorg.Chem. C5 1987/88/89
Wolfrum, K. Exp.Phys. C9 1986/87/92
Wunderlich, I. Dr.* Phys.Chem C1/C2 1984–89
Yamaguchi, Y. Phys.Chem. C1 1993/94/95
Ye, T. Dr.* Exp.Phys. C9 1987–91
Zahn, M. Anorg.Chem. C5 1986/87/88/89
Zimmermann, F. Dr.* Phys.Chem. D6 1988–93
Zimmermann, H. Exp.Phys. B7 1984/85
Zimmermann, W. Dr* Exp.Phys. C4 1984–89
Zollfrank, J. Dr.* Exp.Phys. B9 1987/88
Support of young scientists
* “Promotion” supported by the Sonderforschungsbereich 213
** “Habilitation” supported by the Sonderforschungsbereich 213
and about 200 diploma theses
22.5 International Cooperation
A6
Prof. Hashimoto, Kyoto University, Kyoto, Japan
Dr. H.G. Schmeler, Miles Inc. Pitsburgh, PA, USA
B3
NTT, Tokyo, Japan
Universität Wien, Österreich
B6
Los Alamos National Laboratory T-11, USA
Prof. Y. Ono, Japan
B8
Dr. Bishop, LANL, USA
Los Alamos National Laboratory, Theoretical Group, Los Alamos, NM, USA
B10
MIT, Boston, USA
Prof. Yu Kogan, Dr. Burin, Kurchatov Institut, Moskau, Rußland
Dr. R. Nava, Universidad Central de Venezuela, Caracas
457

22 Documentation of the Collaborative Research Centre 213
B12
Prof. R. Silbey, MIT Cambridge, USA
Prof. Skinner, Dept. of Chemistry, University Wiscousin, USA
C1/C2/C18
Prof. Dr. J.S. Candau, Universite Louis Pasteur, Strasbourg, Frankreich
Dr. M.E. Cates, Cavendish, Laboratory Cambridge, Großbritannien
Prof. Dr. C.A. Miller, Rice-University, Houston USA
Prof. Dr. D. Langevin, Universite Paris VI, Paris, Frankreich
Dr. K. Esumi, Nagoya University Tokyo, Japan
Dr. Kell Mortensen, Risö Laboratorium, Dänemark
Doz. Dr. P. Stern, Akad. d. Wissenschaften Tschechien
Dr. D. Roux, CNRS Pessac, Frankreich
C4
Weizmann Institut, Dept. of Physics, Israel
Group de Physique de Solides, Orsay, Frankreich
Dept. of Physics, University California, Santa Barbara, USA
Dept. of Physics, Hebrew-University, Jerusalem, Israel
Dept. of Physics, CALTECH, Passadena, USA
Dept. of Physics, Northwestern University, Evanston, USA
Centro Atomico, Bariloche, Argentinien
Prof. G. Ahlers, Santa Barbara, USA
Prof. A. Chuvyrov, Ufa, Russland
Prof. V. Steinberg, Rehovot, Israel
Dpto. Estructura y Constiuyentos de la Materia, Facultät de Fisica, Universität Bacelona, Spanien
C10
Laser Research Center,Vilnius, Litauen
C12
ETH Zürich
Prof. F.R. Aussenegg/Univ. Doz. Dr. A. Leitner, Institut für Experimentalphysik, Graz, Österreich
Prof. A. Baiker, Chemieingenieurwesen und Industrielle Chemie, ETH Zürich, Schweiz
PD. Dr. A. Leitner, Institut für Experimentalphysik, Universität Graz, Österreich
C13
Slowakische Techn. Hochschule, CSFR
C14
Dr. A.M. Levelut, Universite Orsay, Frankreich
Dr. B. Gallot, Laboratoire des Materiaux Organiques, CNRS, BP 24,Vernaison, Frankreich
C16/17/22
Prof. Blackwell, Case Western Reserve University, Cleveland, Ohio, USA
Prof. Mac Knight, University of Massachusetts, Amherst, USA
Dr. J. Noolandi, Xerox Research Centre Canada, Missisauga, Ontario, Canada
Prof. Newkomo, Univ. of South Florida, Tampa, Florida
458

22.6 Funding
C19
Center of Nonlinear Studies, Los Alamos, National Laboratory, Los Alamos USA
Servece National des Champes Intenses, Grenoble, Frankreich
C21
N. Pogodina, Institute of Physics, Universite St. Petersburg, Russia
D4/D5
Prof. Dr. A. Redfield, Brandeis University, USA
Prof. Dr. O. Uhlenbeck, University of Boulder, USA
Prof. Dr. A. Spirin, Academy of Science, Russia
Prof. Dr. M. Boublik, Roche Institute of Molecular Biology, Nutley, New Jersey, USA
Prof. Dr. B. F. Clark, Aarhus, Denmark
Prof. Dr. J. Nyborg, Aarhus, Denmark
Prof. Dr. O. Lavrik, Russia
Prof. Dr. L. Bosch, Leiden University, The Netherlands
Prof. Dr. P. Sigler,Yale University, USA
Prof. Dr. J. Heinfeld, USA
Dr. M. Makinen, USA
Dr. L. Arnold, Czech Republic
Prof. Dr. L. Spremulli, USA
Prof. Dr. R. Giegé, France
Prof. Dr. A. Barciszewski, Poland
D6
Prof. Dr. H. Eicke, Institut f. Phys. Chemie, Universität Basel, Schweiz
D8
Prof. K. Kirschner, Schweiz
Prof. A. Yaniv, Israel
Prof. M. Breitenbach, Österreich
Dr. M. Auer, Österreich
22.6 Funding
The Collaborative Research Centre 213 was supported by grants of the Deutsche Forschungsgemeinschaft
totalling DM 28814200 in the period 1984–1995.
459