(DPPC) vesicles - dirac

A characterization of 

the alcohol-induced structural changes of 

unilamellar dipalmitoyl-phosphatidylcholine 

(DPPC) vesicles 

• UNIVERSITAS ROSKILDENSIS • 

IN TRANQUILLO MORS • IN FLUCTU VITA 

Bitten Plesner and Thomas Hecksher 

Physics supervisor: Dorthe Posselt 

Chemistry supervisor: Peter Westh 

Master Thesis Dissertation 

Roskilde University, Denmark 

February 2007

Abstract 

Title: “A characterization of the alcohol-induced structural changes of unilamellar 

dipalmitoyl-phosphatidylcholine (DPPC) vesicles” 

The structural changes of unilamellar dipalmitoylphosphatidylcholine (DPPC) vesicles 

in excess water (with a radius of about 60 nm ± 20%) induced by pentanol, hexanol 

and heptanol were studied. The working hypothesis was that the alcohol would saturate 

the lipid vesicles, making the system go from the L β ′ phase to the L βI phase. The 

point of saturation and partition coefficient could then be found by isothermal titration 

calorimetry (ITC). The transition temperature between the L βI phase and L α phase could 

be determined by differential scanning calorimetry (DSC) and the structural differences 

of these two phases, notably the bilayer thickness, could be studied by small angle X-ray 

scattering (SAXS). The interdigitated phase of lipid vesicles is well-documented with a 

lot of different inducers, but the combination of DPPC and pentanol, hexanol or heptanol 

has until now been unexplored (by SAXS). 

The partition coefficients were all lower, but still at the same order of magnitude as the 

values stated in the literature. The transition temperature was found to decrease as 

alcohol was added to the lipid suspension. The phase transition temperature decreased 

from 41 ◦ C (pure DPPC) to (34 ± 1) ◦ C for DPPC and pentanol, (29 ± 1) ◦ C for DPPC 

and hexanol and (35 ± 1) ◦ C for DPPC and heptanol. The SAXS spectra modelling 

analysis revealed nearly the same values for the lipid bilayer thicknesses for all of the 

measurements in the order of 34-38 Å. Adding alcohol mainly seemed to decrease the 

bilayer thickness slightly independent of the temperature. And for all alcohols there are 

slight tendencies towards a lower bilayer thickness as the temperature is increased. With 

respect to the presence of interdigitation our results are inconclusive, we can neither 

reject nor confirm for any of the alcohols that our low temperature measurements are in 

fact in the L βI phase. According to the literature and our ITC measurements we should 

have enough alcohol to induce the phase, but the results from the modelling on the SAXS 

data do not allow such a clear-cut conclusion. 

The system behaved more complex than initially expected. The DSC measurements 

indicate a stable system with a well-defined phase transition, but the ITC and the SAXS 

measurements did not. It is still unclear as to why the samples are very sensitive to the 

thermal history after adding the alcohol. But using our sample preparation method, we 

do get systems which give reproducible SAXS-spectra. The modelling of these spectra 

did not completely account for all the features seen in the spectra. They indicate that a 

large fraction of the system still has vesicle structure and a smaller fraction of the system 

form larger crystal-like aggregates, whose structures are yet unknown to us.

Resumé 

Titel: “Karakterisering af alkoholinducerede strukturelle ændringer i unilamellare dipalmitoylfosfatidylcholine 

(DPPC) vesikler” 

Den strukturelle ændring af unilamellare dipalmitoylfosfatidylcholine (DPPC) vesikler, 

med en radius omkring 50 nm, tilsat pentanol, hexanol eller heptanol er undersøgt. Arbejdshypotesen 

var at alkoholen ville mætte vesiklerne og derved få systemet til at gå 

fra L β ′-fasen til L βI -fasen. Mætningspunktet og partitionskoefficienten kunne findes med 

isotermisk titreringskalorimetri (ITC). Faseovergangstemperaturen mellem L βI -fasen og 

L α -fasen bestemmes med differentiel scanningskalorimetri (DSC) og den strukturelle 

forskel mellem disse to faser, især bilagstykkelsen, er undersøgt med småvinkelrøntgenspredning 

(SAXS). Den interdigiterede fase af lipidvesiklerne er veldokumenteret med 

en række forskellige inducere, men kombinationen af DPPC og pentanol, hexanol eller 

heptanol har hidtil været uudforsket (med SAXS). 

Partitionskoefficienterne var alle lavere, omend af samme størrelsesorden, end litteraturværdierne. 

Faseovergangstemperaturen faldt fra 41 ◦ C (ren DPPC) til (34 ± 1) ◦ C 

for DPPC og pentanol, (29 ± 1) ◦ C for DPPC og hexanol og (35 ± 1) ◦ C for DPPC og 

heptanol. Analysen af SAXS modeleringen afslørede ensartede bilagstykkelser for alle 

målinger i størrelsesordenen 34-38 Å. Tendensen ved tilføjelse af alkohol var en anelse 

tyndere bilagstykkelse uafhængigt af temperaturen. Og for alle alkoholer var tendensen 

en anelse tyndere bilagstykkelse når temperaturen steg. Vores resultater er ikke entydige 

i forhold til tilstedeværelsen af interdigitering, vi kan, for ingen af de benyttede alkoholer, 

hverken be- eller afkræfte at lavtemperaturmålingerne rent faktisk finder sted i 

den interdigiterede fase. På baggrund af litteraturværdier og ITC-målingerne burde der 

være nok alkohol til stede til at inducere fasen, men resultaterne fra modelleringen af 

SAXS-dataene lægger ikke op til en entydig konklusion. 

Systemet opførte sig mere komplekst end først antaget. DSC-målingerne indikerede et 

stabilt system med en veldefineret faseovergang, mens ITC- og SAXS-målingerne indikerede 

et mere ustabilt system. Det er stadig uklart, hvorfor prøverne er så følsomme 

overfor den termiske historie efter alkoholtilsætning. De rå spektrer indikerer at en stor 

del af systemet har vesikelstruktur og en mindre andel af systemet danner store krystallignende 

aggregater, hvis struktur endnu er ukendt for os.

Acknowledgements 

A number of people deserves acknowledgements for their assistance during the process 

of making this thesis. 

First of all the supervisors; Associate Professor of physics Dorthe Posselt for patient 

guidance, instruction to making the SAXS measurements and helpful discussions on the 

experimental work, and Professor Peter Westh for rewarding supervision, fruitful discussions 

on the lipid-alcohol system and the interpretation of the ITC experiments as well 

as helpful instructions on the DSC and ITC experiments. 

We would also like to thank a number of people at and around IMFUFA: 

Laboratorian Oda Brandstrup - for her helpfulness in the X-ray laboratory. 

Engineer Ib Høst Pedersen - for rapidly fixing miscellaneous lab problems and manufacturing 

a special device for the cuvette in order to make the temperature calibration. 

Assistant engineer Ebbe Hyldahl Larsen - for manufacturing a sample holder for a 

heater/shaker, fixing a leakage in the cuvette as well as a number of other matters. 

Assistant engineer Torben Steen Rasmussen - for helping with the Kratky camera set-up. 

IT guru Heine Larsen - for linux help and miscellaneous IT help in general. 

Ph.D. student Ulf Rørbæk Pedersen - for helping with the Matlab programs and sample 

preparation procedure and for numerous discussions about the lipid-alcohol system. 

Cand. scient, Ph.D. Christa Trandum for very rewarding discussions on lipid-alcohol 

interactions and the use of ITC as well as DSC. 

Cand. scient, Ph.D. Brian Igarashi - for helping with the introductory DSC experiments. 

Cand. mag. Casper Gaarde Madsen - for English proof reading. 

Cand. scient, Ph.D. Kristine Niss - Reading and comments. 

Cand. scient Gitte Margrethe Jensen - Reading and comments. 

Ph.D. student Bjarke Skipper Petersen - for sharing food, beer, being good company 

and participating in numerous jam-sessions. 

The computer lab.: Ditte Gundermann, Neslihan Sağlanmak, Martin Graversgaard 

Nielsen and others - for being good company and for many cookies, cakes, candy and 

fruit (not knowing they shared). 

Software used 

Matlab [48] (KDE linux) - data processing, modelling. 

Kile [35] (KDE linux) - a L A TEXfrontend. 

Basic Support for Cooperative Work (BSCW) [4] - online version control of shared documents. 

Grace [19] (KDE linux) - a 2d graph plotting tool. 

Xfig [86] (KDE linux) - a drawing tool. 

GNU Image Manipulation Program (GIMP) [18] (KDE linux, Windows). 

Roskilde, February 19th 2007 

Thomas Hecksher and Bitten Plesner.

Reading guidelines 

The authors of this report are evaluated on different premises. For Bitten this is an 

inter-disciplinary chemistry and physics thesis. For Thomas this is just a physics thesis. 

Certain parts of the report are therefore dedicated as Bitten’s chemistry thesis and are 

solely the work of Bitten. This applies to section 3.1, 4.1 and appendix E. The rest of 

the report is a result of the joint effort of Bitten and Thomas. 

The reader is urged to read the The course of events appendix at page 111, but maybe 

not before reading the report. We chose to write this course of events primarily for the 

purpose of other fellow students to make use of our experiences and reflections during 

the work of this thesis. 

If the reader is familiar with the experimental techinques isothermal titration calorimetry, 

differential scanning calorimetry and small angle X-ray scattering, the sections 3.1, 3.2 

and 3.3 can be skipped. 

Throughout chapters 4 and 5 references will be made to the Matlab source code appendix 

showing the actual implementations of the models. 

Throughout the report we use the phrases low temperature and high temperature referring 

to measurements at 25 ◦ C and 45 ◦ C, repectively. 

Certain abbreviations are used over and over again throughout the report. These include 

isothermal titration calorimetry (ITC), differential scanning calorimetry (DSC), small angle 

X-ray scattering (SAXS), multi lamellar vesicle (MLV), unilamellar vesicle (ULV), dipalmitoylphosphatidylcholine 

(DPPC), dimyristoylphosphatidylcholine (DMPC). Equally, 

references to the various phases of phospholipid vesicles are made constantly. The abbreviations 

for the frequently referred phases are indicated in table 1. 

Abbreviation 

L α 

L βI 

L β ′ 

P β ′ 

Phase 

Liquid crystal 

Interdigitated 

Gel 

Rippled (gel) 

Table 1 Some abbreviations of the phases of phospholipid vesicles perturbated by n-alcohols. 

Most of the experimental results obtained in this thesis are placed in the appendices due 

to the readability of the thesis, not because the experimental work should be put in the 

background. 

The concentrations are stated differently according to the given system. The concentration 

of lipid is stated as a weight ratio, m lipid/ mwater , and the concentration of alcohol in 

a lipid suspension is stated as a mole fraction, n lipid/ nalcohol . Here n alcohol refers to total 

amount of alcohol in the system, i.e. inside the bilayer and outside in the bulk water. 

Otherwise the concentrations are stated in mol per litre M. 

The target group of this thesis is primarily physics students and chemistry students with 

a special interest in biophysics and biochemistry.

Contents 

Contents 

vii 

I Introduction and Theory 1 

1 Introduction 3 

1.1 Phospolipids and membranes . . . . . . . . . . . . . . . . . . . . . . . . . 3 

1.2 Phases and properties of the vesicles . . . . . . . . . . . . . . . . . . . . . 5 

1.3 Additives - interactions with small amphiphilic molecules . . . . . . . . . 7 

1.4 Interdigitated phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 

1.5 Aim of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 

1.6 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 

2 Scattering Theory 17 

2.1 X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 

2.2 The classical (particle) scattering experiment . . . . . . . . . . . . . . . . 20 

2.3 X-Ray scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 

2.4 Bragg’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 

II Experiments and Results 31 

3 Experimental Techniques 33 

3.1 ITC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 

3.2 DSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 

3.3 SAXS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 

3.4 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 

4 Results 55 

4.1 ITC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 

4.2 DSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 

4.3 SAXS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 

III Analysis 75 

5 Modelling 77 

5.1 Overview of the models used . . . . . . . . . . . . . . . . . . . . . . . . . 77 

5.2 Instrumental smearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 

5.3 Symmetric (1d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 

vii

viii 

Contents 

5.4 Symmetric (4g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 

5.5 Asymmetric (1d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 

5.6 Spherically symmetric (3d) . . . . . . . . . . . . . . . . . . . . . . . . . . 87 

5.7 Multilamellar vesicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 

5.8 Crystalline structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 

5.9 Results and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 

6 Discussion 99 

7 Conclusion 103 

Bibliography 105 

A The course of events 111 

B Theory 115 

C Fourier transform 117 

D Experimental series 121 

E ITC 123 

F DSC 129 

G SAXS 133 

H Modelling results 135 

H.1 Pure DPPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 

H.2 Pentanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 

H.3 Hexanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 

H.4 Heptanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 

I Matlab source code 157 

Index 183

Part I 

Introduction and Theory

1 Introduction 

The behaviour and characterization of cell membranes are ongoing topics in the reseach 

field of biochemistry and biophysics. The cell membrane consists primarily of proteins 

and lipids but it also contains a variety of other biological molecules. Among other 

purposes the membrane serves to separate the intracellular components from the extracellular 

environment and regulate the exits and entries of the cell. The intra membrane 

morphology has an effect on the transmembrane processes and hence molecules influencing 

the membrane morphology and their mutual interactions are of great interest. 

Unilamellar phospholipid membranes are used as a model system for ordinary cell membranes 

in vivo, which aside from phospholipids also consist of different kinds of lipids, 

proteins and cholesterol. 

1.1 Phospolipids and membranes 

A lipid consists of a hydrophilic polar head group, and a hydrophobic nonpolar tail. The 

polar head group can be either charged or neutral (zwitterion or without charge), and 

the tail consists of one or two hydrocarbon chains. Lipids are poorly soluble in water but 

they are highly soluble in organic solvents. Most phospholipids are derived from glycerol, 

a three-carbon alcohol, and consist of a glycerol backbone, two fatty acid chains, and 

phosphorylated alcohol - these phospholipids are called phosphoglycerides, see figure 1.1. 

The natural fatty acid chains usually contain an even number of carbon atoms, in animal 

cells most frequently 14-24 [17], and may be saturated or unsaturated. The length and 

degree of unsaturation of the chains have a strong effect on membrane fluidity. 

R 

PHOSPHAT 

G 

L 

Y 

C 

E 

R 

O 

L 

FATTY ACID 

FATTY ACID 

Figure 1.1 The basic structure of a phospholipid. It consists of a phosphat group to which a 

chemically functional group, R, is bound, a glycerol group and two identical or two different 

fatty acids. R can e.g. represent choline, ethanolamine or serine. 

When the phospholipids are dissolved in water their water-insoluble and primarily 

hydrophobic nature forces them to assemble rapidly [66]. Those biological phospholipids 

which have two long alkyl chains per molecule will primarily aggregate to form bilayers, 

they do seldom form spherical micelles 1 . In excess water the micelles are dispersed 

1 The nonpolar portion of the molecule has a fixed molecular volume and a maximal length, this will 

3


(c w >40%) and the lipid bilayers are said to be fully hydrated. 

The hydrophobic force is the most significant thermodynamically driven force. This 

so-called force arises from the unfavourable constraints placed on water as it packs around 

a nonpolar hydrocarbon. The force drives the system to adopt a configuration which 

minimizes the contact between water and the nonpolar portions of the lipid. The van der 

Waals force which represents short, weak attractive forces between adjacent hydrocarbon 

chains is another stabilizing factor. The attraction results from interactions between 

polarizable electrons (induced dipoles) and compared to the hydrophobic force, the van 

der Waals force is a relatively minor stabilizing factor [17]. Forming a bilayer has different 

advantages including the possibility of forming a spherical bilayer vesicle which compared 

to disks or flat sheets is highly favourable due to the lacking contact by water at the edge. 

By forming a vesicle the edge is eliminated, the sphere is also favoured by being smaller 

and, hence, entropically preferable to very large bilayer sheets. The bilayers most often 

form multilammelar “onion-layered-shaped” vesicles with a certain repeat distance. A 

single phospholipid membrane consists of a lipid bilayer where the hydrophilic part of the 

phospholipids is situated towards the water, and the hydrophobic part, the acyl chains, 

is inside the membrane as shown in figure 1.2. The drawing in the lower right corner of 

figure 1.2 illustrates a single bilayer. The head group of the lipid is shown as circles, the 

tails as curved lines. The grey lamella represents the hydrophobic part and the white 

lamella represents the hydrophilic part. 

Multilamellar vesicle 

Unilamellar vesicle 

100 nm 

Bilayer normal 

5 nm 

Hydrophilic 

Hydrophobic 

Hydrophilic 

Figure 1.2 A multilamellar vesicle (MLV) and a unilamellar vesicle (ULV). The multilamellar 

vesicle consists of several bilayers. The unilamellar vesicle consists of only one bilayer. The 

drawing in the lower right corner illustrates a single bilayer. The head group of the lipid is 

shown as circles, the tails as curved lines. The grey lamella represents the hydrophobic part 

and the white lamella represents the hydrophilic part. The denoted lengths can vary, but the 

values for the bilayer thickness and vesicle diameter denoted here are characteristic for the ULV 

DPPC vesicles used in this project. 

A multilamellar vesicle (MLV) can contain up to a thousand bilayers [32]. A unilamellar 

vesicle (ULV) consists, as the name reveals, of only a single bilayer and is frequently 

called a liposome. 

determine the maximal radius of a spherical micelle as well as the number of molecules that can fit 

into the micelle. The optimal surface area required by the polar headgroup must also be considered. 

For biological phospholipids the area per molecule in a hypothetical spherical micelle is much larger 

that the optimal value for headgroup packing and, hence, these amphiphiles do not form stable 

spherical micelles [17].

1.2 Phases and properties of the vesicles 5 

A lot of work has been done on different phospholipids and their response to changes 

in temperature, pressure, concentration and chemical solution [5], [14], [38], [53], [70], 

[78]. 

In this work we use the phospholipid di-palmitoyl-sn-phosphatidyl-choline, DPPC. 

This phospholipid consists of two identical fatty acids, each the length of 16 C-atoms, a 

phosphatidyl group and a choline group. The chemical structure of DPPC is shown in 

figure 1.3. 

CH 3 

CH 2 

H 2 C CH 3 

H 2 C 

O 

C 

O 

H 2 C 

H 2 C 

C 

O 

H 

CCH 2 

O 

O 

H 2 C 

O 

P 

O 

O 

H 3 C 

H 3 C 

N 

CH 2 

CH 3 

Figure 1.3 DPPC is chemically known as 1,2-dipalmitoyl-sn-glycero-3-phosphocholine or 1,2- 

dihexadecanoyl-sn-glycero-3-phosphocholine. The numbers 1, 2 and 3 refer to the position 

of the fatty acids and the phosphatidyl-choline group on the glycerol. In this case the two 

identical fatty acids are placed next to each other and the phosphatidyl-choline group is placed 

at the end of the glycerol. The nitrogen is positively charged, whereas one of the oxygens of the 

phosphat is negatively charged. This makes the headgroup of the phospholipid a zwitterion. 

1.2 Phases and properties of the vesicles 

All lipid bilayers show a first-order phase transition associated with an increased conformational 

freedom for the lipid fatty acyl chain [42]. This process is designated to the 

”melting” of the hydrocarbon chains. The chain melting temperature differs from phospholipid 

to phospholipid and is determined by a balance of competing factors. Below 

this phase transition temperature, the membrane is gel like, and above the transition 

temperature, the membrane is fluid, liquid crystalline like. The gel phase is denoted L β ′ 

and the liquid crystalline phase is denoted L α . The phase most relevant for comparison 

with biological systems is the L α phase [12], [75].


Tristram-Nagle & Nagle [78] have done a great deal of work on these phospholipid 

systems including phase transitions. For DPPC they find that the largest phase transition 

enthalpy of MLV is the transition between the L β ′ and L α phase - ∆H ∼ 36 kJ/mol. 

Less than half of that amount is due to disordering of the hydrocarbon chains. The rest 

can be accounted for by volume expansion as one can see in figure 1.4. 

Figure 1.4 A graph showing the volume per lipid (DPPC) vs. temperature for DPPC bilayers 

in excess water (experimental data). [32] 

Besides the above mentioned phases the system can also exist in the low temperature 

subgel L c phase, and a phase between the gel and liquid phase called ripple phase, 

denoted P β ′. Therefore two other phase transitions (in the temperature range 0-60 ◦ C) 

exists, the subgel-gel, and the gel-ripple transitions, the latter is also called pre-transition. 

The subgel-gel transition exhibits hysteresis, but apparently has a ”true” transition temperature 

at 18 ◦ C for DPPC [42]. 

The disordered chains in the L α phase are characterized by gauche conformations, 

which are favoured entropically compared to the highly ordered all-trans chain conformation 

in the gel state. The gauche conformations are characterized by being energetically 

unfavourable compared to the trans conformation, where the CH 2 -groups in the acyl 

chain are situated as far away from each other as possible [13], [17]. The two conformations 

are illustrated by a Newman projection 2 in figure 1.5. 

Numerous techniques indicate that in the gel phase, the hydrocarbon chains of saturated 

diacyl phospholipids are predominantly in the all-trans conformation [17]. 

Sackmann refers to experiments which have shown that the main transition is characterized 

by an abrupt increase in the number of gauche conformations, n g . The concentration 

of gauche conformations is defined as x g ≡ ng 

n , where n is the number of CH 2 

segments in the chain. The concentration increases from x g = 0.07 to x g = 0.4 upon 

going to the L α phase [66]. 

In the gel phase the acyl chains of DPPC are tilted about 30 ◦ with respect to the 

bilayer normal, effectively increasing their cross-sectional area to be compatible with that 

of the headgroup; the chains remain in the all-trans configuration. The tilt is a result of 

2 The Newman projection views a carbon carbon chemical bond from front to back, front carbon as 

a dot and back carbon as a circle. This type of representation makes it easy to assess the torsional 

angle between two bonds, one at each carbon atom.

1.3 Additives - interactions with small amphiphilic molecules 7 

H 

CH 3 CH3 H 

CH 3 

H 

CH 3 

CH 3 

H 

H 

H H 

CH 3 

H H 

H 

g + = t= g − = 

H 

H 

Figure 1.5 The Newman projection diagram of the gauche conformations, g + and g − , and trans 

conformation, t. Adjusted from Gennis [17]. 

the rather large headgroup of the DPPC molecule [66]. The headgroup of DPPC requires 

a minimum packing of 50 Å 2 . In the L α phase the introduction of gauche configurations 

increase the effective chain cross section to about 60-70 Å 2 [17]. Katsaras & Gutberlet 

[32] report the interfacial area of DPPC in the L α phase to be 68-71Å 2 . Hence, in the L α 

phase, the hydrocarbon chains are not tilted. 2 H NMR indicates that the thickness of 

the hydrocarbon domain of DPPC is 35 Å in the L α phase, compared to 45 Å expected 

if the chains were all-trans and oriented along the bilayer normal. The shorter distance 

is due to gauche configuration and the chains are basically aligned and perpendicular to 

the plane of the bilayer and are not coiled [17]. 

Bilayer thickness 

A common definition of bilayer thickness is D B = 2VL 

A , where V L is the volume of a lipid 

molecule in the bilayer and A is the interfacial area per lipid; V L has been measured 

accurately (relative uncertainty of 0.2%) by a number of groups [54]. The thickness 

of lipid bilayers (vide infra for a discussion on various definitions of thickness) is an 

important structural quantity for discussing the incorporation of intrinsic membrane 

proteins [32]. As mentioned the bilayer membranes are organized assemblies with a 

sheet-like structure, they typically have a thickness about 5 nm as shown in figure 1.6. 

There is a large uncertainty in the interfacial area A per lipid (DPPC), different methods 

yield different results. Neutron scattering suggests A = 58Å 2 , and X-ray scattering 

suggests A = 71Å 2 [32], however most of the experiments referred in the literature point 

towards a value of about 65Å 2 [53]. 

1.3 Additives - interactions with small amphiphilic molecules 

Generally, the molecules added to the bilayer system can be categorized by being entirely 

hydrophilic (sugars, salts, DNA, polyethylene glycol), mostly hydrophobic (steroids like 

cholesterol and androsten) or amphiphilic (fatty acids, alcohols). It turns out that additives 

”which are hydrophilic or amphiphilic will affect the phase behaviour in most 

dramatic ways”.[32, page 165] 

The literature shows several examples of phospholipids and their interactions with 

small amphiphilic molecules under different thermodynamical circumstances [10], [43], 

[47], [49], [50], [63], [68], [70], [80], [83], [84]. An example of an amphiphilic molecule 

is alcohol. The role of phospholipids and their interactions with especially alcohols has 

been of great interest due to the fact that alcohol has an anaesthetic effect [16].


V L 

∼ 5 nm D B = 2·VL 

A 

V L 

Figure 1.6 The bilayer thickness of a unilamellar vesicle. V L is the volume of one phospholipid 

and D B represents the lipid bilayer thickness. A is the interfacial area per. DPPC. The value 

of V L has been measured very accurately, but there is a great uncertainty in the determination 

of a value for A. The value varies from 58 Å 2 , measured by neutron diffraction, and 71 Å 2 , 

measured by X-ray scattering [32]. 

Alcohols 

The chemical term alcohol covers any organic compound in which a hydroxyl group, the 

hydrogen-oxygen group, is bound to a carbon atom, which in turn is bound to other 

hydrogen and/or carbon atoms. A sub-group of alcohols is the n-alcohols, which consist 

of the hydroxyl group and an acyl chain. Hexanol is an example of an n-alcohol, see 

figure 1.7. 

HO 

H 2 C 

H 2 C 

H 3 C 

Figure 1.7 The chemical structure of the n-alcohol hexanol. Hexanol consists of an hydroxyl 

group and a 6-carbon atom chain. 

CH 2 

CH 2 

CH 2 

The alcohol can be described as amphiphilic. Because of its dipol the OH -group 

is characterized as hydrophilic and the nonpolar properties of the acyl chain account 

for its hydrophobia. The alcohol gets more hydrophobic as the acyl chain length increases. 

When pertubating the phospholipid membrane the alcohols are arranged with 

the hydrophobic group towards the inner part of the membrane and the hydrophilic part 

towards the bulk [39].

1.4 Interdigitated phase 9 

1.4 Interdigitated phase 

At high temperatures the liquid L α phase is stable. At low temperatures, various structures 

are found, dependent on temperature and alcohol concentration. At low concentrations, 

we find the transition from the highly ordered subgel phase, L c , via the gel phase, 

L β ′, to the rippled phase, P β ′. In all these phases (see figure 1.9) the tails have a tilt 

with respect to the bilayer normal. At high concentrations of alcohol the rippled phase 

disappears, and the interdigitated phase, L βI , is formed, in which the tails do not show 

a tilt [39]. Except for the interdigitated L βI phase, these observations are similar to the 

observations of membranes without alcohols. 

The hydrophobic thickness of an ordinary bilayer is approximately twice the length 

of the hydrophobic tails of a phospholipid, in the interdigitated phase the hydrophobic 

thickness is reduced to the sum of the length of the hydrophobic tails of a single 

phospholipid and the alcohol [39]. This is illustrated in figure 1.8. 

Gel phase 

Interdigitated phase 

Alcohol concentration 

Figure 1.8 The gel phase and the interdigitated phase. In the gel state the hydrophobic 

thickness is approximately twice the length of the hydrophobic tails of a phospholipid, in 

the interdigitated phase the hydrophobic thickness is approximately reduced to the sum of 

the length of the hydrophobic tails of a single phospholipid and the alcohol. Modified from 

Kranenburg et al. [39]. 

A wide variety of amphiphilic compounds are capable of inducing the interdigitated 

L βI phase in saturated like-chain phosphatidylcholines [10], [47], [50], [63], [68], [70]. The 

longer the lipid chain, the more energitically favourable the interdigitated state becomes 

[69]. The ability of a molecule to induce the interdigitated phase does not depend on 

the presence or absence of a formal charge while molecules that are totally hydrophobic 

such as hexane and benzene do not induce the interdigitated phase [69]. DPPC does not 

stably interdigitate without an inducer [14]. Methanol through heptanol are known to 

induce interdigitation (in DSPC, a 18 carbon-chain lipid). From octanol (and above) the 

alcohols do not induce the interdigitated phase [39]. The molecule must displace water 

from a particular location in the interfacial region and its nonpolar moiety cannot extend 

too deeply into the bilayer interior [50]. 

In the case of short amphiphilic molecules whose non-polar moieties are not as long 

as the DPPC hydrocarbon chains, this would potentially cause voids between chains in 

the bilayer interior. Since the energy of formation of holes in hydrocarbons is large the 

chains must eliminate the voids. To do this the chains could either bend cooperatively 

or else interdigitate. The lowest energy phase is the interdigitated phase [50]. 

Another explanation of interdigitation could be, that the chain-end methyl groups are 

slightly recessed from the neighboring headgroups to form an amphipathic cavity. Small 

amphipathic molecules are thought to bind within the cavity and shield the hydrophobic


methyl chain ends from water. As expected for a cavity of defined geometry, and as 

observed for many biological processes, there is a sharp cutoff in the ability of longer 

alcohols to stabilize interdigitation [14]. 

Absorption of alcohol in a cell membrane can induce significant changes in the structure 

of the membrane. These structural changes of the membrane can influence the 

conformation of proteins or other structures embedded in the membrane. Several experiments 

and simulations on phospholipids have shown that alcohol can induce the 

interdigitated phase, one of these experiments is shown in figure 1.9. 

Figure 1.9 Schematic representation of the phase diagram of a phosphatidylcholine/alcohol 

mixture as a function of the alcohol concentration and temperature. From Kranenburg et al. 

[39]. 

The analysis by Simon et al. of the interdigitated phase shows that the partition coefficients, 

see section 1.4, for the anaesthetics are larger for the interdigitated gel phase 

than for the usual bilayer gel phase. They claim that this is due to the fact that the 

interdigitated phase increases the number of acyl chains per head group at the bilayer/water 

interface by a factor of 2, and thus there are more “sites” where amphiphilic 

anaesthetics e.g. alcohol can bind to the membrane [69]. Most of the work done on 

n-alcohol as an inducer has been done on the short chain alcohol ethanol [1], [25], [37], 

[55], [62]. Kaminoh et al. [31] have shown that short-chain alcohols (with anaesthetic 

potency 3 ) decrease the temperature in the main transition of phospholipid membranes 

between the solid (rippled, P β ′) and liquid (L α ) states. Using longer chain alcohols, the 

potency of the alcohols to decrease the transition temperature increases. But when the 

alcohols reach a certain length and when used in low concentrations they start to elevate 

the transition temperature. The switch from depression to elevation coincides with the 

3 In the late 19th century it was postulated that anaesthetics potency is related to the solubility of the 

anaesthetics in an oil phase [69].


cut-off length 4 . Thus, a correlation of anaesthetic potency with the phase transition of 

lipid membranes is to be expected [31]. 

Biphasic effect 

Short chain alcohols have two different effects on the transition from the low temperature 

gel phase to the high temperature liquid phase, depending on concentration [63], [62], 

[74], [47]. At low concentration of alcohol the main phase transition temperature shifts to 

a lower temperature, whereas at high concentration this transition temperature shifts to 

a higher temperature compared to a pure liquid bilayer. This effect is called the biphasic 

effect [39], see figure 1.10. 

Long chain alcohol (10 − ) 

Long chain alcohol (5 − 9) 

Short chain alcohol (1 − 4) 

Phase transition temperature 

Concentration 

Figure 1.10 An illustration of the main phase transition temperature as a function of the alcohol 

concentration. At low concentration the short chain alcohols (1-3) decrease the main phase 

transition temperature, at high concentration the short chain alcohols increase the main phase 

transition temperature. The long chain alcohols (4-9) only decrease the main phase transition 

temperature, whereas the long chain alcohols (10-?) only increase the main phase transition 

temperature. Adapted from Rowe [62]. 

Kaminoh et al. [31] argue that butanol to octanol depress the main phase transition 

temperature, decanol is biphasic, depressing the temperature at low concentration and 

elevating at high concentrations. Dodecanol does not have any effect on the transition 

temperature, tridecanol and tetradecanol elevate the transition temperature. The low 

concentration biphasic effect results from the increasing disorder of the lipid tails which 

leads to a lower transition temperature. The high concentration biphasic effect, however, 

results from the more tightly packed interdigitated phase, resulting in an increase of the 

main phase transition temperature [39]. The longer chain alcohols (above pentanol) show 

no biphasic effect [47]. 

4 Tamura et al. [74] explain the cut-off effect by relating it to the anaesthetic potency. The anaesthetic 

potency increases with increasing carbon chain length up to about C 10 -C 12 , where the anaesthetic 

activity suddenly disappears. The cutoff chain length coincides with the chain length where the effect 

on the phase transition temperature of DPPC membrane changes from depression to elevation, see 

figure 1.10.


Partition coefficient 

It is clear that partitioning of solutes between the membrane and the aqueous phase is a 

very important aspect of the interaction of solutes with membranes. However, determining 

the partition coefficients experimentally is still not a very common practice, primarily 

due to the experimental difficulties [88]. 

A partition coefficient is a measure of differential solubility of a compound in two 

solvents. For this system the partition coefficient describes the ratio of the solute concentration 

in the lipid bilayer and in the bulk water, and its value increases with increasing 

chain length of the alcohol, caused by the hydrophobic effect. 

The partition coefficient is usually defined in terms of mole fractions. The mole 

fraction of a component in a mixture is the relative proportion of the specific molecules 

in the mixture, by number of molecules. For each component, i, the mole fraction x i is 

the number of moles n i divided by the total number of moles in the system n. 

x i = n i 

n , 

where n = ∑ j 

n j (1.1) 

The sum is over all components, including the solvent in the case of a chemical 

solution. 

The partition coefficient is then defined as 

K x = X a,l 

X a,w 

(1.2) 

where the numerator is the mole fraction of alcohol in the lipid bilayer, denoted “a,l”, 

and the denominator is the mole fraction of alcohol in the solution, denoted “a,w”. Experimentally 

it is very difficult to measure the concentration of alcohol in the membrane 

directly. Still using isothermal titration calorimetry the so-called solvent-null method 

Zhang & Rowe [88], Rowe et al. [65] and Suurkuusk & Singh [73] have measured values 

for the partition coefficient of the partitioning of hexanol and pentanol into DPPC. 

Simulations can be used as as supplement to experiments as a way of finding an 

estimate of the mole fraction. Kranenburg et al. have made mesoscopic simulations on 

the interdigitated phase of DSPC in order to determine the partition coefficient and 

the fraction of alcohol needed to induce the interdigitated phase. From these simulations 

it is found that a ratio of 1:2(X a,b = 0.33) for pentanol:DSPC, hexanol:DSPC and 

heptanol:DSPC, respectively, is needed to induce interdigitation [39]. This ratio is significantly 

lower than the ratios obtained by different experimental methods, see table 1.1, 

which explains why the ratio obtained by simulations is not used further in this work. 

The free concentration of the alcohol is higher i.e. the partition coefficient is lower 

for the interdigitated phase, L βI , than for the liquid, crystalline phase, L α , even though 

the L βI phase is found at lower temperature than the L α phase, see figure 1.9. In the 

literature there is disagreement about whether the partition coefficient increases with 

increasing temperature or is independent of temperature [73]. 

Löbbecke & Cevc [47] have experimentally demonstrated an exponential relation between 

the threshold concentration for inducing the interdigitated phase and the number 

of carbon atoms in the alcohol. They find that the threshold ratio decreases by a factor 

3 for each CH 2 added to the n-alcohol. It should be stressed that in order to make such 

correlations it is very important to know the concentration of lipid in the solution. The 

threshold concentration is highly dependent on the concentration of lipid in the solution. 

The difference between alcohol and lipid ratio is listed in table 1.1. The variations in


reported ratios are presumed to be a result of the limitations of the different experimental 

methods. 

Method Alcohol n l /n a 

Fluorescence [63] 

DSPC conc: 0,13 mM Ethanol 1:453846 

(MLV) Propanol 1:1407 

Butanol 1:569 

Pentanol 1:138 

Hexanol 1:123 

Heptanol 1:50 

Optical [74] 

DPPC conc: 1,01 mM Ethanol 1:480 

(MLV) Butanol 1:31 

Hexanol 1:2(1,79) 

Heptanol 1:2(1,9) 

DSC [47] 

DPPC conc: 5 mM Ethanol 1:220 

(MLV) Pentanol 1:7 

Hexanol 1:2 

Fluorescence [47] 

DPPC conc: 5 mM Ethanol 1:220 

(ULV) Pentanol 1:12 

Hexanol 1:10 

ITC [88] 

DPPC conc: 70.8 mM Butanol 1:5 

(MLV) 

Table 1.1 Different methods of measuring the threshold mole ratio for inducing interdigitation in 

DPPC or DSPC vesicles. n l and n a are the amount of matter for lipid and alcohol, respectively. 

And it is important to notice that the alcohol threshold ratio is highly dependent on the 

lipid concentration. The lower the lipid concentration is, the more alcohol per lipid is 

needed to induce the interdigitated phase. This observation can be explained by the fact 

that a higher concentration of alcohol is needed to saturate the larger amount of the bulk 

water.


1.5 Aim of the thesis 

The aim of this thesis is to investigate and characterize the structural differences between 

the alcohol induced interdigitated lipid phase, L βI , and the liquid fluid phase, L α , of 

the phospholipid dipalmitoyl-phosphatidylcholine and elucidate the differences associated 

with the use of 3 different n-alcohols. 

Investigations of the structural changes of DPPC ULVs associated with the presence of 

large amounts of pentanol, hexanol or heptanol with SAXS are not well-described in the 

literature. In fact, we have after a comprehensive search not found preceding examples 

of these experiments in the literature of this field of research. The SAXS experiments on 

this system have not been reported before and will contribute to the understanding of 

lipid and alcohol interactions. 

Our working hypothesis is that the alcohol will saturate the lipid vesicles, making the 

system go from the L β ′ phase to the L βI phase. The point of saturation and partition 

coefficient can then be found by isothermal titration calorimetry (ITC). The transition 

temperature between the L βI phase and L α phase can be determined by differential 

scanning calorimetry (DSC) and the structural differences of these two phases, notably 

the bilayer thickness, can be studied by small angle X-ray scattering (SAXS). 

1.6 Method 

Isothermal titration calorimetry is used for the purpose of getting a value for the threshold 

alcohol concentration needed to induce the interdigitation as well as elucidating the 

difference between using pentanol, hexanol and heptanol as inducer. Especially hexanol 

is well-described in the literature, still the role of heptanol as an inducer remains to be 

demonstrated for this specific lipid. The method used is the so-called solvent null method 

introduced by Zhang & Rowe [88]. This method provide the opportunity for calculating 

the partition coefficient, and on this basis it is possible to suggest the minimum alcohol 

to lipid ratio needed to induce the interdigitated phase. 

On the basis of the threshold values found in the literature and by use of ITC, 

samples of alcohol and DPPC in adequate ratios are prepared. Differential scanning 

calorimetry (DSC) is used in order to get the phase transition temperature between the 

apparent L βI phase and the L α phase as shown in figure 1.9. In addition the appearance 

and position of the peak from the DSC scan provide information about the transition 

enthalpy and the influence of the added alcohols on both the enthalpy and the phase 

transition temperature. 

The SAXS spectra obtained on either side of the L βI -L α phase transition temperature 

are used to get structural information about the electron density bilayer profile of the 

samples and thus a measure for the bilayer thickness. By using and taking existing 

models further estimates for the bilayer thickness are obtained. The bilayer thickness 

estimates, the temperature at which they are obtained and the related lipid to alcohol 

ratio would then provide the basis for conclusions regarding the structural appearance 

of the L βI and L α phase, respectively. 

The structure of the report 

The report is divided into three main parts, “Introduction and Theory”, “Experiments 

and Results” and “Analysis”. The first part, chapters 1 and 2, consists of an introduction 

to the biochemical system, the aim of the thesis and the scattering theory used. The

1.6 Method 15 

second part, chapters 3 and 4, consists of a description of the experimental techniques 

and the sample preparation as well as the results obtained with the three different techniques. 

The third and final part, chapters 5, 6 and 7, consists of both the analysis of the 

experimental results and the conclusion.

2 Scattering Theory 

This chapter follows, more or less, the structure of [3] where gradually more complex 

scattering objects are treated - but other sources have been used as well, mostly [76, 81]. 

The content of the chapter is primarily selected in order to give a theoretical foundation 

for the modelling of the SAXS data. But we also try to make a paedagogic presentation for 

our target group in which the cases of scattering from one and two electrons, respectively, 

are explained in great details. And it reflects, at some degree, how we have come to 

understand the scattering theory. 

2.1 X-rays 

Although the X-ray phenomenon was observed by J. Hittorf, N. Tesla and H. Hertz 

before W.C. Röntgen, he is credited as the discoverer of X-rays - since he was the first 

to publish in the late nineteenth century (1895). He saw yellow-green light was flickering 

from a (fluorescent) plate near his Geisler discharge tube, from which no visible light 

could escape. He found that pieces of paper and wood inserted between the plate and 

the tube did not cast a shadow, but that metal indeed did. Since then, in the twentieth 

century, the theory of X-rays has evolved and X-rays have found many applications. X- 

rays were found to be scattered by gases and to give crystal diffraction patterns (1912). 

Later, complex molecular structures such as the structure of DNA (1953) was obtained 

by X-ray diffraction. And surely, the X-rays have had a tremendous impact on diagnostic 

medicine and therapy. In the 1990s an X-ray observatory was built to monitor violent 

processes in the distant universe, such as “stars being torn apart by black holes, galactic 

collisions, and novas, neutron stars that build up layers of plasma that then explode 

into space” [85, X-rays]. Now, in the beginning of the twentyfirst century, scientists and 

engineers are working on the X-ray laser for assembling nano-robots. 

The wavelengths of X-rays are in the order of Å= 10 −10 m. In quantum mechanics 

light is quantized as photons, and the dispersion relation, E = pc, and the De Broglie 

wavelength p = h λ 

gives the relation between the photon energy (E) and photon wavelength 

(λ) 

λ = hc 

E 

= 

12.398 keV 

E 

(2.1) 

where h is Planck’s constant, p is the momentum of the photon and c is the speed of 

light in vacuum. 

Interaction with matter 

X-rays interact with matter (mostly the bound electrons of the atom) in different ways. 

The most important interactions are absorption, diffraction and reflection. In the case 

of photoelectric absorption the photon annihilates and the bound electron is ejected 

into the continuum of energy states. The free electron can be ejected in any direction, 

which for a macroscopic object means energy loss in terms of heat. There is no classical 

17


interpretation of the absorption. The electron vacancy can be filled by another bound 

electron if it is stimulated by, e.g., a photon with an energy equal to the energy gab 

between the states and this is called fluorescent emission. If the photon energy exceeds 

the energy gab, a less tightly bound electron may be expelled, and is then called an 

Auger electron emission. 

In the case of linear absorption the intensity, I(z), of the beam is reduced by the same 

fraction for every small distance, dz, travelled into the medium dI = −µI(z) where µ is 

dz 

the linear absorption coefficient. The solution is I(z) = I 0 exp(−µz), and the coefficient 

is approximately proportional to Z 4 , Z being the sum of electrons in the atom. This 

makes it excellent for imaging (X-ray images) due to the large contrast mainly between 

tissue and bone in the body. 

Diffraction and reflection of a beam are explained by media with different refractive 

index and are the classical interpretations of the scattering process. 

Sources 

The state of the art X-ray source is a synchrotron, which is a particular type of cyclic 

particle accelerator in which the magnetic field (to turn the particles so they circulate) 

and the electric field (to accelerate the particles) are carefully synchronized with the 

travelling particle beam. Synchrotrons were developed to study high-energy particle 

physics [85], but they are used as X-ray sources as well - in fact some synchrotrons are 

built only with this purpose in mind. In the synchrotrons the electrons pass so-called 

wigglers or undulators and emit very intense X-rays. 

The sources have been improved since the fixed X-ray tube. The rotating anode, 

1st, 2nd and 3rd generation of the synchrotrons seriously improved the brilliance of the 

beam, and the 3rd generation is approximately 10 12 times more brilliant than the fixed 

tube. The brilliance of an X-ray beam is determined by several aspects: 

⊲ The number of photons per second (more is better) 

⊲ The collimation of the beam - how much it spreads out in milli-radian (less is 

better) 

⊲ The spectral distribution (monochromatic is better) (0.1 % relative bandwith) 

⊲ The source area (mm 2 ) (less is better) 

Photons per second 

Brilliance ∼ 

(mrad) 2 (mm 2 [3] (2.2) 

source area)(0.1% bandwidth) 

For the experimental part of this thesis we have used an X-ray tube as source, which we 

discuss in section 3.3 on page 41. 

The plane wave representation 

X-rays are transverse waves, which can be represented by the monochromatic plane wave 

vector k which is pointing in the direction of the travelling wave perpendicular to the 

electric E and magnetic B fields. It has the magnitude |k| = k = 2π λ 

, λ is the wavelength 

of the photon. 

Ẽ I (r, t) ≡ E 0 exp(i(k · r − ωt))ˆx (2.3) 

where ω is the angular frequency, E 0 is the amplitude of the incident electric field Ẽ I 

which is a function of time, t, and position, r. The complex notation (exp(iα) = cos(α)+ 

i sin(α)) is used and the physical electric field is the real part of the complex number 

E = Re(Ẽ) which is implied for the rest of the chapter.

2.1 X-rays 19 

Coherence and incoherence 

Two X-ray beams are coherent if they are parallel and monochromatic. In reality this 

idealization is not fulfilled which means that the two beams at some places will make 

constructive and destructive interference. Non-parallel beams cause transverse coherence 

and polychromatic beams cause longitudinal coherence. The longitudinal coherence 

length, l L , is the length between constructive and destructive interference, thus 2l L is the 

length between constructive and constructive interference. If one beam has a wavelength, 

λ, and the other has a slightly smaller wavelength, λ ′ ≡ λ − δλ, then 

2l L = Nλ = (N + 1)λ ′ ⇒ N = 

1 

λ/λ ′ − 1 

(2.4) 

where N is the number of wave crests of λ in order to make constructive interference 

with the other wave (with wavelength λ ′ ). This gives the longitudinal coherence length 

l L = 

λ 

2 (λ/λ ′ − 1) = λλ ′ 

2 (λ − λ ′ ) ≈ λ2 

2 δλ 

ifλ ≈ λ ′ (2.5) 

a) Longitudinal coherence length, l L 

θ 

2l L =N λ 

A 

λ 

λ − δλ 

b) Transverse coherence length, l T 

B 

B 

A 

θ 

R 

2 l T 

D 

λ 

S 

Figure 2.1 Longitudinal coherence length, l L, and transverse coherence length, l T . a) Two plane 

waves are emitted in the same direction. The two waves are out of phase by a factor of π after 

the distance l L. b) Two waves with the same wavelength are emitted from the ends of a finite 

sized source of a width D, in this work equivalent to the width of the beam. 

If a perfect crystal is used for collimation in a synchrotron then δλ λ 

means the longitudinal coherence length is about 5µm. 

≈ 10−5 , which 

The transverse coherence length, l T , is the length between constructive and destructive 

interference. Both beams have the same wavelength λ but propagate in slightly different


directions, θ, e.g. due to the finite size of the entrance slit of the X-ray source. Travelling 

the distance l T from the point S along the wavefront of wave A one reaches the point 

where the wave A is out of phase with the wave B. Proceeding the distance l T along 

the wavefront the waves are in phase again, as 2l T θ = λ, see figure 2.1. If the waves 

propagate from the same source but with a distance D from each other, the distance 

from the point S to the source is R, then 

tan(θ) = λ = D 2l T R ⇒ l T = λR 

2D 

(2.6) 

Thus, the coherence length gives an upper limit on the distances between the scattering 

objects. If the scattering centers are separated by distances greater than the coherence 

length then we add the intensities rather than the amplitudes. 

For the Kratky camera the distance to the sample from the source is about 5cm and the 

beam width is about 1cm which gives a transverse coherence length about 3.9Å as the 

wavelength is about 1.55Å. 

2.2 The classical (particle) scattering experiment 

Impact parameter ρ 

p 0 

Axis 

Figure 2.2 Classical scattering of a point particle with initial momentum p 0 and an impact 

parameter ρ (figure from Taylor [76, p. 44]). 

In the classical scattering experiment a projectile is fired at a target. We can measure 

or, otherwise, somehow determine the incident and scattered momentum of the projectile, 

but we cannot measure the microscopic details of the event. In particular we cannot 

measure the impact parameter, ρ, which is defined as a vector with a direction perpendicular 

to an axis chosen to go through the target and has the length magnitude from 

the axis to the actual trajectory of the projectile (see figure 2.2). We cannot measure the 

impact parameter, because no source is truly point like. We always have a cross sectional 

area of the beam of projectiles. The problem is then to extract as much information as 

possible about the target given these limitations. 

The outcome of a single experiment shows us if the projectile has hit the target or not. 

The projectile has a different momentum if it has hit the target, and it is undeviated and 

has the same momentum if it has missed the target. As the experiment is repeated many 

times with the same incident momentum and random impact parameters, it makes sense 

to talk about the average number of incident projectiles per unit area (n i ) perpendicular 

to the incident beam. The number of scattered projectiles (N s ) by the cross sectional 

area of the target (σ) is then 

N s = n i σ (2.7) 

This actually allows us to determine the σ since we, in principle, can count n i and 

N s . And in this case N s is the total count of scattered projectiles. If you only count

2.3 X-Ray scattering 21 

the scattered projectiles at a certain solid angle, N s (∆Ω), usually, it is only a part of σ, 

which is the source of those projectiles. Thus, σ is a function of ∆Ω: 

N s (∆Ω) = n i σ(∆Ω) (2.8) 

[8]. 

In the case of a target with a cross-sectional area larger than the incident beam, 

the scattered intensity is proportional to the incident intensity. And in the case of a 

target with a cross-sectional area smaller than the incident beam, the scattered intensity 

is proportional to the incident flux[3, p. 261]. In the latter case the differential cross 

section can be defined as 

dσ (Number of X-rays scattered per second into ∆Ω) 

≡ (2.9) 

dΩ (Incident flux)(∆Ω) 

The total scattering cross section is found by integrating equation 2.9 over all of space. 

The differential cross section is perhaps the most important quantity in scattering theory 

since it is the meeting point between theory and experiment. 

2.3 X-Ray scattering 

In a scattering experiment the incident beam consists of either light, X-rays, neutrons 

or other probes and is directed at the sample. In the case of X-ray scattering most of 

the photons pass through the sample undeviated and a few are scattered once from the 

scattering volume. Mainly there are three types of scattering experiments in which you 

get different information about the sample: 

⊲ measuring the dependence on angle of the average scattered intensity (’static scattering’) 

yields structural information. Shapes and internal structure of individual 

particles in a dilute system, and spatial, positional correlations in a concentrated 

system. 

⊲ analysis of the time dependence of fluctuations in the scattered radiation yields dynamic 

information. The Brownian movement and/or how the shape/configuration 

fluctuate in time. 

⊲ measuring the absolute magnitude of the scattered intensity (time or frequency 

average) yields mass or molecular weight of the scattering objects. 

Weak scattering (also called kinematical scattering, which essentially means a gentle 

probe that does not destroy or alter the target) is the case when the sample is a perturbation 

to the incident beam, which means the Born approximation holds. But the weak 

interaction also means a weak signal, which is then compensated by an intense beam. 

As the time of interaction for the X-rays with the sample is small compared to the motion 

time of the particles within the sample, the measurements are series of ”snapshots” 

which are averaged. 

In this X-ray scattering section, we have chosen a structure similar to [3], in which 

the complexity of the scattering objects are gradually increased, starting with just one 

electron, then treating two electrons, the electrons in an atom, a molecule, a small crystal 

and a lipid bilayer. 

One free electron 

The electric field, E I (t ′ ) = E 0 exp(iωt ′ )ˆx, of the incident plane wave travelling along the 

ẑ axis is the driving force. Therefore, the equation of motion for the free electron (placed


at origo) is 

m e a(t ′ ) = −eE I (t ′ ) (2.10) 

where m e and e is mass and charge of the electron, respectively. The oscillating electron 

can be considered an accelerated dipole, ¨p, if the amplitude of the oscillation, |E I | is 

much smaller than the distance to the point of observation, r, see figure 2.3. 

x 

n 

θ 

r 

E 

B 

R 

ṗ . 

Ψ 

z 

ϕ 

y 

Figure 2.3 The accelerated dipole, ¨p, located at origo, radiates a spherical wave. At this 

snapshot the observation point, r, the propagation vector, ˆk, the radiated electric field, E R, 

and the radiated magnetic field, B, are sketched. (figure from [3, p. 269]) 

The observation is performed at time t, which is r / c later, so t = t ′ + r / c . The radiated 

electric field, E R (r, t), is derived from Maxwell’s equations along the following path: the 

current density J −→ the retarded vector potential A −→ the magnetic field B −→ the 

electric field E = cB × ˆr. The details of this derivation are explained in [3, p. 267-271] 

and the result is 

E R (r, t) = 1 

4πǫ 0 

1 

c 2 r (¨p(t′ ) × ˆr) × ˆr = 1 

4πǫ 0 

−e 

c 2 r (a(t′ ) × ˆr) × ˆr (2.11) 

since ¨p(t ′ ) = d2 

dt ′2 p = −ea(t ′ ). 

Isolating a(t ′ ) in (2.10) inserting that into (2.11) then the radiated field in terms of 

incident field becomes 

E R (r, t) = 

e 2 

4πǫ 0 m e c 2 1 

r (E I(t ′ ) × ˆr) × ˆr (2.12) 

And since E I (t) = E 0 exp(iω(t − r / c ))ˆx = E 0 exp(iωt − k · r)ˆx the radiated field 

becomes 

( 

e 2 ) ( ) 

E0 exp(i(ωt − k · r)) 

E R (r, t) = 

4πǫ 0 m e c 2 (ˆx × ˆr) × ˆr (2.13) 

r


or written as the ratio between radiated and incident 

( ) 

|E R (r, t)| exp(−ik · r) 

E 0 exp(iωt) = (b T) 

(− cos(ψ)) (2.14) 

r 

where b T is the Thomson scattering length 

b T = 

e 2 

4πǫ 0 m e c 2 = 2.82 × 10−5 Å (2.15) 

It is worth noting the sign of the last term, which means that the radiated beam is π 

out of phase with respect to the incident beam. This is the same as the E R is pointing 

opposite to ¨p along the x-axis in figure 2.3. cosψ is valid when the measuring point is 

in a plane parallel to the incident plane of polarization. Otherwise, the factor is 1 if the 

measuring point is in a plane perpendicular to the incident plane of polarization, since 

full acceleration is seen from the observation point. 

The intensity, I, is the absolute square of the scattered electric field amplitude 

I ∝ |E| 2 (2.16) 

which means that the ratio between the scattered intensity, I S , and the incident intensity, 

I I is 

I S 

= |E R| 2 r 2 ∆Ω 

I I |E I | 2 (2.17) 

σ I 

where r 2 ∆Ω is the area of the detector and σ I is the cross sectional area of the incident 

beam. 

Combining (2.17) and (2.14) gives the differential cross sectional area of Thomson 

scattering 

dσ 

dΩ = 

I S 

(I I /σ I )∆Ω = |E R| 2 r 2 

|E I | 2 = b 2 T cos 2 ψ = b 2 TP (2.18) 

where P is the polarization factor for the intensity. The Thomson cross section can then 

be found by integration 

∫ ∫ 

σ T = dσ = b 2 T cos 2 (ψ)dΩ = 8π 3 b2 T = 0.665 barn (2.19) 

where 1 barn = 10 −24 cm 2 is a typical measuring unit for cross sections. Hence, the cross 

section for Thomson scattering is a constant and does not depend on energy. 

In the case of an unpolarized incident beam, the polarization vector can point along 

any direction in the x-y plane, and the polarization factor is averaged to be 1 / 2 (1+cos 2 ψ). 

In order to summarize, the polarization factor, P , for the Thomson differential cross 

section is 

⎧ 

1 measuring point in a plane perpendicular 

⎪⎨ 

to the incident plane of polarization 

P = cos 2 ψ measuring point in a plane parallel 

(2.20) 

to the incident plane of polarization 

⎪⎩ 1/ 2 (1 + cos 2 ψ) an unpolarized source 

For small angle scattering P ≈ 1.


At this point we may wonder; how on Earth can we detect the scattering, when the 

intensity ratio is in the order of 10 −24 just a few centimeters away from the electron? A 

part of the answer to that question is that the amount of electrons in a sample is in the 

order of Avogadro’s number, but it is not the whole answer, since the arrangement of the 

electrons is crucial to whether or not the intensity becomes measurable. 

Two free electrons 

The electron seems to be a structureless particle, hence the most simple structure must 

be two electrons. If the first electron is placed at origo, the scattered waves from this 

electron are out of phase. The two free electrons are situated at a fixed distance r between 

each other and the displacement vector is r. The phase lag (behind) of the incident wave 

between the two electrons are φ in = −k · r (marked as z on figure 2.4), and the phase 

lag (ahead) of the scattered wave between the two electrons are φ out = k ′ ·r 1 . This gives 

the combined phase lag between the incident and scattered wave of the second electron 

φ = φ in + φ out = −k · r + k ′ · r = −q · r, which defines the scattering vector, 

q ≡ k − k ′ (2.21) 

which is sometimes, suggestively, written as the momentum transfer vector q ≡ k−k ′ . 

The geometry of figure 2.4 shows that for Thomson scattering |k ′ | = |k| 

|q| = |k| sin(θ) + |k ′ | sin(θ) = 2|k| sin(θ) = 4π λ 

sin(θ) (2.22) 

In the case of Compton scattering |k ′ | < |k| equation (2.22) does not hold. Looking at 

one particular q is the same as looking at one particular angle. The wavelength of the 

X-ray is inversely proportional to the scattering vector. 

k’ 

r 

q 

k 

z 

k’ 

2θ 

λ 

k 

Figure 2.4 a) The vertical straight lines represent the plane wave crests each separated by λ and 

the wave vector k indicates the direction of propagation. The electron in the middle is placed 

at origo and the other electron is displaced by r. b) The two plane wave vectors are placed in 

a suggestive way, such that the definition of the scattering vector q becomes apparent. The 

scattering angle is defined as 2θ such that the sine term in equation (2.22) becomes θ. 

Ignoring the polarization, the form factor, f , for two electrons separated by r is 

f two electrons (q) = −b T exp(i(q · 0)) − b T exp(i(q · r)) = −b T (1 + exp(i(q · r))) (2.23) 

1 We could equally have defined it as φ = q · r and we drop the minus as many textbooks do.


which in the case of parallel q and r vector gives the intensity 

I(q) ‖ = 2b 2 T(1 + cos(qr)) (2.24) 

and an orientational average over all different angles gives 

( 

〈I(q)〉 or. = 2b 2 T (1 + 〈exp(i(q · r))〉) = 2b2 T 1 + sin(qr) ) 

qr 

(2.25) 

where the intermediate steps of 

〈exp(iq · r)〉 orientation = sin(qr) 

qr 

(2.26) 

are skipped, see Als-Nielsen & McMorrow [3, p. 111]. This shows that the arrangement 

of the electron is crucial to the diffraction pattern. Now, it’s important not to forget that 

q is a function of both angle as well as λ, which means that for a polychromatic beam 

the detected intensity I for one particular q gets contributions from different angles. 

Secondary scattering occurs when a photon is scattered by an electron and afterwards 

hit another electron, and is detected subsequently. The detected energy and position of 

this electron would contribute to the spectrum with a so to speak misleading information 

about the geometry of the system. 

4 

q and r parallel 

Orientational average 

3 

Intensity / b T 

2 

2 

1 

0 

0 0.5 1 1.5 2 

q / [2π / r] 

Figure 2.5 Two diffraction patterns of two electrons placed the distance r from each other but 

with different angles in respect to the incident beam. The solid line represents the diffraction 

pattern where r is parallel to q and the intensity given by equation (2.24). The dotted line 

represents the diffraction pattern where r is randomly oriented and an average of all angles 

between r and q and the intensity is given by equation (2.25). 

There is a periodic variation in the calculated intensities, reflected in the scattered 

waves being either out of phase (minimum) or in phase (maximum). By measuring the 

intensities as a function of the scattering vector q the diffraction pattern is obtained, and 

when subsequently fitting r the structure of the two-electron system is obtained.


When extending the model to comprising more than two electrons, all the elastic 

scattering amplitudes must be added up and 

∑ 

N 

f many electrons (q) = −b T exp(iq · r j ) (2.27) 

where r j is every displacement vector between the electrons. 

Defining the form factor, equation 2.27, is the first step in the direction of developing 

form factors for more complex systems such as atoms, molecules and finally the lipid 

bilayer. 

An atom 

Depending on its atomic number the atom contains a number of electrons. The electrons 

are represented by wavefunctions ψ(r) and the electron propability density ρ(r) = |ψ(r)| 2 

and the propability ρ(r)dr of finding an electron in the volume element dr. The form 

factor of an atom, f atom , is then defined as the contributions to the scattering field from 

each volume element 

j=1 

f atom (q) = 

= 

∫ 

ρ(r)exp (iq · r)dr (2.28) 

V 

{ 

Z if |q| → 0 

(2.29) 

0 if |q| → +∞ 

The limit for the form factor as |q| → 0 equals the total number of electrons in 

the atom, Z, as the phase factor approaches zero. And the limit for the form factor as 

|q| → +∞ equals zero as there is destructive interference between the waves scattered 

from the different electrons in the atom and the radiation waves, as the latter are small 

compared to the atom (for further explanation, see Als-Nielsen & McMorrow [3, p. 112]). 

The right hand side of equation 2.28 is recognized as a Fourier transform, hence the 

form factor is the Fourier transform of the electron distribution ρ of the given sample. 

A molecule 

Upon defining the form factor for an atom comes the definition of the form factor for a 

molecule containing a number of atoms. The molecule form factor is composed of atomic 

form factors, fj atom (q), located at r j . 

f molecule (q) = 

N∑ 

j=1 

f atom 

j (q)exp (iq · r j ) (2.30) 

The atomic form factor differs for each of the atoms in the molecule, and in order 

to determine the atomic arrangement of the molecule one has to have an idea of the 

structure before interpreting the measured data. Molecules with the same number of 

atoms can differ radically in structure. 

In case of measuring geometric properties of e.g. large biomolecules it is most often 

necessary to dissolve these in an appropriate solvent. The electron density is relative to 

the solvent 

∆ρ(r) = ρ(r) − ρ solvent (2.31)


where ρ(r) is the actual electron density at r. In order to correct for the contribution 

from the solvent a so-called background spectrum is obtained. This spectrum specifies the 

contributions from the solvent and other possible contributors within the experimental 

setup. 

This relative electron density gives rise to the possibility of contrast variation, a topic 

of which neutron scattering is especially usefull, since the scattering length of neutrons 

depends strongly on the isotope. The scattering length for deuterium, e.g., is much larger 

than hydrogen, whereas the X-ray scattering lengths for these two isotopes are the same. 

A small crystal 

In this context a crystal is defined as a crystalline material which is periodic in space [3]. 

The crystal structure is specified as a lattice of points in space, reflecting the symmetry 

of the crystal and a unit cell structure, referring to which atoms to associate with each 

lattice sites. The scattering amplitude for a crystal is then the product of the unit cell 

structure factor and the lattice sum 

f crystal (q) = f basis (q) ∑ r l 

exp (i(q · (r l + r))) (2.32) 

= 

Basis 

{ }} { 

f basis (q)exp(iq · r) ∑ r l 

exp(iq · r l ) 

} {{ } 

Lattice sum 

(2.33) 

where r l = n 1 a 1 + n 2 a 2 + n 3 a 3 is the lattice vector with a as the basis in real space, 

and r is the position vector of the unit cell structure relative to the lattice. The unit cell 

structure factor f basis could be an atom, a molecule, or possibly multiple combinations 

of both. The lattice sum usually consists of a huge amount of terms which only blow up 

as q takes on special positions in q-space (also called the reciprocal space). This happens 

when q · r l = 2π × integer and is called the Laue condition 

q = g = ha ∗ 1 + ka∗ 2 + la∗ 3 (2.34) 

where g is the reciprocal lattice vector, h, k, l are the Miller indices, and a ∗ is the basis 

vector in the reciprocal space. So, for every lattice in real space you can construct a lattice 

in reciprocal space which gives constructive interference 2 . Conversely, if the scattering 

objects in the sample are dispersed (no characteristic lengths) there is no structure factor 

- it is unity. 

For every h, k, l-plane in real space there is a lattice point in the reciprocal space. 

The distances between the planes are given by 

d hkl = 

2π 

|g hkl | 

(2.35) 

As an example consider a hexagonal structure with the lattice vector 

⎛ √ ⎞ ⎛ 

3a/2 

r n = n 1 a 1 + n 2 a 2 + n 3 a 3 = n 1 

⎝ a/2 ⎠ + n 2 

⎝ −√ ⎞ ⎛ 

3a/2 

a/2 ⎠ + n 3 

⎝ 0 0 

0 

0 

c 

⎞ 

⎠ (2.36) 

2 The Laue condition is equivalent to Bragg scattering.


with the two lattice constants a and c (constructed such that |a 1 | = |a 2 | = a and |a 3 | = c). 

The reciprocal lattice basis becomes 

a ∗ 1 = 

a ∗ 2 = 

a ∗ 3 = 

2π 

a 1·(a 2×a 3) a 2 × a 3 = 

2π 

a 1·(a 2×a 3) a 3 × a 1 = 

2π 

a 1·(a 2×a 3) a 1 × a 2 = 

⎛ 

4π 

√ ⎝ 

3a2 c 

⎛ 

4π 

√ ⎝ 

3a2 c 

⎛ 

4π 

√ ⎝ 

3a2 c 

ac/2 

√ 

3ca/2 

0 

−ac/2 

√ 

3ac/2 

0 

0 

0 

√ 

3a 2 /2 

⎞ 

⎠ = 2π a 

⎞ 

⎠ = 2π a 

⎞ 

⎠ = 2π c 

⎛ 

⎝ 1/√ 3 

1 

0 

⎛ 

⎞ 

⎝ −1/√ 3 

1 

0 

⎛ ⎞ 

⎝ 0 0 

1 

⎠ (2.37) 

⎞ 

⎠ (2.38) 

⎠ (2.39) 

and the reciprocal lattice vector is 

⎛ 

g hkl = ha ∗ 1 + ka ∗ 2 + la ∗ 3 = 2π ⎝ 

√ 

3(h − k)a 

−1 

(h + k)a −1 

lc −1 

⎞ 

⎠ (2.40) 

with the length 

|g hkl | = √ g hkl · g hkl = 

√ 

4π 2 ( ( √3(h−k) 

a 

√ 

= 2π 

) ) 

2 ( 

+ 

h+k 

) 2 ( 

a + l 

) 2 

c 

h 2 +k 2 −hk 

(a/2) 2 

(2.41) 

+ l2 

c 2 (2.42) 

The lattices in direct space and reciprocal space are sketched in figure 2.36. 

If one suspects hexagonal rods (l = 0), one should look for q-peaks with the following 

distances 

2π 

|g hk0 | 

|g 100 | = 

√ ( ) 

h 2 +k 2 −hk 

(a/2) 

+ 02 

2 c 2 

2π√ ( 

1 2 +0 2 −0 

(a/2) 2 + 02 

c 2 ) = √ h 2 + k 2 − hk = √ 1, √ 3, √ 4, √ 7, √ 9, · · · (2.43) 

for different combinations of h and k in terms of the first peak, i.e. the first peak is 

|q ′ | = √ 1|g 100 |, the next is |q ′′ | = √ 3|q ′ | and the next is |q ′′′ | = √ 4|q ′ | and so forth. 

2.4 Bragg’s law 

Fulfilling the requirements for the Laue condition and referring to the relations in equation 

2.22 Bragg’s law is obtained. 

nλ = 2d sin(θ) (2.44) 

where n is an integer and d is the repeat spacing of the lattice planes. Combining (2.44) 

with (2.22) gives the Bragg plane distance 

d = 2π 

|q ′ | n (2.45)

d 11 

d 10 

g 11 

2.5 Summary 29 

g 01 

g 10 

Figure 2.6 The figure on the left-hand side shows a schematic drawing of the hexagonal type II. 

The characteristic repeat distances are denoted d 10 and d 11 where d 10 = 2π 

2π 

|g 10 

and d11 = . 

| |g 11 | 

The figure on the right-hand side shows the diagram of the reciprocal lattice. The diffraction 

planes are perpendicular to the reciprocal lattice vectors g 10 and g 11, the basis vectors are 

shown in red. Both figures adapted from [24]. 

k 

k’ 

d 

θ 

θ 

Figure 2.7 Bragg scattering, adapted from [3]. 

where n is an integer. 

Laue condition is not true for Compton scattering, which gives a smoothly varying 

background. For bound electrons with initial momentum the Compton cross section gives 

the momentum distribution. The Compton scattering length is approximately 137 times 

larger than the Thomson scattering length. 

2.5 Summary 

Experimentally, both Thomson and Compton scattering are observed, but the classical 

theory can only account completely for Thomson scattering. 3 Even so, there are several 

reasons why the classical treatment is valid and sometimes sufficient:[81] 

3 Please refer to appendix B, page 115, for a short outline of the quantum mechanical treatment of the 

scattering process.


⊲ Correct wave mechanical treatment shows that the sum of intensities from Thomson 

and Compton scattering is close to the sum of individual classical electron 

intensities. 

⊲ The polarization is given correctly for both Thomson and Compton scattering (the 

polarization is the classical representation of the quantum mechanical spin). 

⊲ Plane waves can be superpositioned in order to get a normalizable wavepacket. 

In the elastic scattering theory the incident beam is considered a plane wave which 

makes the electrons of the sample oscillate and radiate spherical waves. The diffraction 

pattern then depends on the arrangement of the electrons. The diffraction is governed 

by a reciprocal law: The scattering angle becomes smaller as the dimensions of the 

scattering object become larger with respect to the wavelength of the incident beam. If 

the system consists of many electrons it makes sense to talk about the electron density 

and replace the sum with an integral. The intensity is thus integrated and by convention, 

the system is divided into identical subsystems (form factors), and the arrangement of 

these unit cells gives rise to terms labelled structure factor. Apart from the polarization 

factor the integrated intensity sometimes gives rise to additional terms which depend on 

the scattering angle, θ. This is referred to as the Lorentz factor, but the textbooks on 

X-ray scattering ([3, p. 146], [81, p. 44], [45, p. 206]) leave vague impressions on what 

exactly the definition of the Lorentz factor is. They all agree that it depends highly 

on geometry. [3] states that for a small crystallite proper integration over k ′ and θ in 

the case of slightly incoherent incident and/or scattered beam a factor of 4 1 

sin(2θ) ≈ 1 q is 

added to the differential cross section. According to [45] the same expression is added as 

a result of rotating the crystal with a constant angular velocity. Both [45] and [81] refer 

to the angel-dependent terms as the polarization-Lorentz factor (or correction). 

4 Valid approximation for small angles.

Part II 

Experiments and Results

3 Experimental Techniques 

This chapter is intended to introduce the experimental techniques used. We have chosen 

to review the experimental equipment thoroughly as the instrumental setup often is one 

of the keys to understanding the potentials and limits of the conducted experiments. 

Especially with regard to the SAXS experiments the experimental setup has a significant 

impact on the evaluation of the obtained data and the further modelling process. 

3.1 ITC 

Isothermal Titration Calorimetric (ITC) is a very powerful tool to determine thermodynamic 

parameters of various binding processes by directly measuring the heat absorbed 

or generated during the binding processes. ITC has the great advantage that all thermodynamic 

quantities such as stoichiometry, binding constants, Gibbs free energy change, 

enthalpy change, and entropy change can be obtained at the same time from only one 

experiment [71]. 

The ITC experiments were carried out using two different calorimeters, a VP-ITC 

instrument and a MSC-ITC instrument from both Microcal, Inc. (Northampton, MA). 

The data were primarily analysed using ORIGIN software from Microcal, Inc., Amherst, 

MA. Both instruments are ultrasensitive isothermal calorimeters and use a cell feedback 

network to differentially measure and compensate for heat produced or absorbed between 

the sample and reference cell. The temperature difference between the two cells 

is measured thermoelectrically while another device measures the temperature difference 

between the cells and the jacket, see figure 3.1. When a chemical reaction in the sample 

cell occurs, heat is either generated or absorbed, resulting in a temperature difference 

between the two cells. 

If an exothermic reaction occurs, the temperature of the sample cell increases upon an 

injection from the syringe. This causes a decrease in the feedback power to the sample 

cell for the purpose of maintaining an equal temperature between the two cells. The 

opposite occurs in case of an endothermic reaction; the feedback power increases in order 

to keep the temperature constant. 

Measuring the partition coefficient 

Zhang & Rowe [88] have used ITC to measure the heat of binding and the partition coefficient 

of n-butanol into multilamellar vesicles of DPPC. Zhang & Rowe [88] demonstrate 

that ITC can be used to give a direct measure of the X a,b , see equation 1.2, associated 

with the phase transition from the gel phase, L β ′, to the interdigitated phase L βI by 

applying the so-called solvent null method. 

Applying the method to the investigated system in this work would yield additional 

information about the phase diagram, see figure 1.9. By use of this method it would be 

possible to suggest an alcohol to lipid ratio for the phase transition from the L β ′ to the 

L βI . Besides this ratio the partition coefficient, the free energy of transfer, the enthalpy 

33


Alcohol in 

reaction cell 

Lipid/alcohol in 

stirring syringe 

∆T 1 

∆T 2 

Jacket 

feedback 

Sample cell 

feedback 

(a) Diagram of the ITC cell and the 

syringe. For the experiments the 

lipid/alcohol suspension is placed in 

the injection syringe and the alcohol 

solution is placed in the ITC sample 

cell. The syringe rotates during the 

experiment. The end of the syringe 

has been adapted to provide continuous 

mixing in the ITC cell. The 

plunger is computer-controlled and injects 

precise volumes of the lipid/alcohol 

solution. 

(b) A diagram of the heat flow between 

the reference cell and the sample 

cell. During the experiment the 

coinshaped cells are kept at a constant 

temperature and ∆T 1 , the temperature 

difference between the sample 

and reference cell, is kept at a constant 

value. ∆T 2 is the temperature 

difference between surroundings and 

the reference cell. The two cells are 

connected to the outside by narrow 

tubes through which the lipid/alcohol 

suspension is injected and subsequently 

the syringe is immersed. 

These cells are kept in an adiabatic 

environment. Adapted from [51]. 

Figure 3.1 Schematic drawing of the ITC setup. 

of transfer and the entropy of transfer can be calculated for each experimental series. 

These experiments should hence confirm that the ratio of the lipid/alcohol suspensions 

used in the SAXS experiments was high enough to induce the interdigitated phase. 

The method can be used to determine the concentration of free alcohol at any alcohollipid 

composition. In this work the solvent-null method has been used to examine the 

pentanol/hexanol/heptanol-DPPC vesicle system. The interactions between DPPC and 

the three different alcohols were studied by placing the lipid suspension in the syringe 

and the alcohol solutions in the sample cell. This was preferred over placing lipid in the 

cell and alcohol in the syringe, as alcohol has a very large heat of dilution whereas the 

heat of dilution of the lipid solution is negligible [21], [88]. 

The principle behind the method is that when the alcohol concentration in the cell 

exceeds the free alcohol in the alcohol-DPPC solution being injected, the measured signal 

is correlated with the resultant partitioning of alcohol to the vesicles, and an endothermic 

signal is measured. Similarly if the alcohol concentration in the cell is below the free

3.1 ITC 35 

concentration, the heat signal correlates to the release of bound alcohol from the lipid 

vesicles, and an exothermic signal is measured. Therefore, when the heat signal is zero, 

there is no net binding/release and the alcohol concentration in the cell equals the free 

alcohol concentration in the syringe. 

The raw data are plotted as the heat flow in µcal/sec as a function of time in seconds, 

causing the raw data to consist of a series of peaks of heat flow, each peak referring 

to an injection. Integrating these peaks with respect to time gives the total heat effect 

per injection. The profile of the plot of these effects as a function of the molar ratio 

can be used in the analysis of the thermodynamic parameters of the interaction under 

investigation. Pentanol stock solution and DPPC ULVs were premixed and kept shaken 

at 850 rpm at 52 ◦ C for at least 48 hours prior to using. The DPPC-pentanol total 

molar ratio was approximately equal to 2:1. The pentanol solution and the pentanollipid 

suspension were degassed prior to being loaded into the 1.3 mL reaction cell and 

the 100 µL syringe, respectively. After equilibrating the system to 25 ◦ C, 4 injections of 5 

µL are titrated into the cell. Hexanol stock solution and DPPC ULVs were premixed and 

kept shaken at 850 rpm at 52 ◦ C for at least 48 hours prior to using. The DPPC-hexanol 

total molar ratio was approximately equal to 3:1 for all the experimental series. The 

hexanol solution and the hexanol-lipid suspension were degassed prior to being loaded 

into the 1.3 mL reaction cell and the 100 µL syringe, respectively. After equilibrating the 

system to 25 ◦ C, 4 injections of 5 µL are titrated into the cell. Heptanol stock solution 

and DPPC ULVs were premixed and kept shaken at 850 rpm at 52 ◦ C for at least 48 

hours prior to using. The DPPC-heptanol total molar ratio was approximately equal 

to 6:1. The hexanol solution and the heptanol-lipid suspension were degassed prior to 

being loaded into the 1.3 mL reaction cell and the 100 µL syringe, respectively. After 

equilibrating the system to 25 ◦ C, 4 injections of 5 µL are titrated into the cell. 

A typical example of the ITC experiments is shown in figure 3.2. 

C cell 

> C syringe 

C cell 

< C syringe 

21 

20 

20.5 

19.5 

19 

µcal/sec 

20 

µcal/sec 

18.5 

19.5 

18 

17.5 

19 

0 500 1000 1500 

Time (sec) 

17 

0 500 1000 1500 

Time (sec) 

(a) An endothermic reaction, indicated 

by the peaks pointing upwards. The free 

concentration of hexanol in the syringe 

is lower than the hexanol concentration 

in the cell, see section 3.1. 

(b) An exothermic reaction, indicated by 

the peaks pointing downwards. The free 

concentration of hexanol in the syringe 

is higher than the hexanol concentration 

in the cell, see section 3.1. 

Figure 3.2 Raw ITC data from two scans of 10.5 mM hexanol in 45.5 mM DPPC suspension in 

the syringe and a hexanol concentration in the cell of 1 mM and 10 mM, respectively.


3.2 DSC 

Differential Scanning Calorimetry (DSC) is a measuring technique to determine e.g. 

specific heat capacity, phase transition temperatures or the associated change in specific 

enthalpy. 

The scan rate ( T) ˙ is constant during the experiment and heat flowing ( ˙Q) into the 

sample is measured as a function of temperature (T ). During a first order transition the 

sample temperature remains constant during the whole transition, still the heating power 

is continuously supplied. The isobaric heat capacity ( dH 

dT 

) at this point is (infinitely) high 

and results in a jump on a (T, ˙Q) graph. But since the scan rate is not infinitely low, 

generally, the system does not have enough time to equilibrate and the jump becomes 

a slope or peak instead, resulting in the characteristic DSC scan-graphs, where the heat 

flow is plotted as a function of the temperature. 

The DSC scans were run prior to the SAXS experiments to get an idea of where 

to find the phase transition temperature for the interdigitated gel phase to the liquid, 

crystalline phase transition. Two different DSCs have been used for the experiments, a 

scal-1 microcalorimeter produced by Scal Co, Pushchino, Russia, and a DSC7 produced 

by Perkin Elmer, USA. 

The compensating DSC 

In general there are three different types of DSC instruments; heat-exchanging calorimeters, 

temperature-changing calorimeters and compensating calorimeters [22]. Amongst 

the compensating calorimeters the power compensating and the heat compensating DSC 

are found. The name ”compensating” refers to the fact that the temperature changes of 

the sample due to e.g. a phase transition are compensated by adding or eliminating an 

equally high, well-known amount of energy in the form of heat or electric energy of the 

opposite sign. In an ideal compensating DSC each ∆T signal appearing between sample 

and reference sample would immediately be compensated by a corresponding change in 

the heating power [27]. 

The adiabatic scanning calorimeter 

The scal-1 microcalorimeter is a differential adiabatic 1 scanning calorimeter with a cylindertype 

measuring system. In this calorimeter the heating power is supplied according to a 

given, preset temperature program, and the compensation of heat is measured with the 

aid of electric energy [27]. The measuring system, the two sample containers, with the 

temperature sensors determine the temperature inside the system. When the measuring 

system is ideally symmetrical, equally high heat-flow rates flow into the cells; the differential 

temperature signal is then zero. If this steady-state equilibrium is disturbed by a 

sample transition, a differential signal is generated which is proportional to the difference 

between the heat-flow rates flowing to the sample and to the reference. Heat losses are 

minimized by adapting the temperature of the surroundings as well as possible to the 

temperature of the sample. In practice the heating power (the feed heat) is often given 

and the resulting heating rate (determined by the temperature difference between the 

two cells) is measured. Heat losses are minimized by the adiabatic jacket [22]. 

In the calorimeter the reference and the sample calorimetric cells are made of glass, 

since glass has a high chemical resistance to the majority of chemicals, and the calorimeter 

is designed for studying liquids of various compositions. The internal and external 

1 adiabatic refers to a prevention of the heat exchange between measuring system (sample) and surroundings 

[7].

3.2 DSC 37 

thermostat shield and the cold thermostat are cylindrically shaped, made of aluminium 

alloy and placed in a hermetic housing, see figure 3.3. 

Sample cell 

Reference cell 

External thermostatic shield 

Internal thermostatic shield 

Temperature difference sensor 

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Cold thermostat 

Heating elements for internal 

and external shield 

Heating element for calorimetric cells 



Figure 3.3 A schematic drawing of the scal-1 microcalorimeter. The reference and sample 

cells are made of glass and they are placed in a low-melting alloy. The internal and external 

thermostat shield and the cold thermostat have the shape of cylinders, are made of aluminium 

alloy and placed in a hermetic housing. Adapted from Senin et al. [67] 

Before the measurements begin the sample and the reference are degassed for 5 minutes 

prior to being aspirated into the syringe. The syringe containing the 500 µL aqueous 

lipid-alcohol suspension is immersed into the cell. The reference cell is filled with degassed 

water. The cells are filled at room temperature and the calorimeter starts the 

cooling process. When the temperature in the cells is about 6 ◦ C, the instrument switches 

automatically to controlled heating. The internal and external thermostatic shields will 

have the same temperature as the cells. The heating power of the two cells is set by 

a voltage source, and this power is proportional to the rate of their controlled heating. 

The rate of the controlled heating is linear through the whole temperature range. The 

temperature difference between the two cells is controlled by the temperature difference 

sensors, and a compensation regulator distributes the measured current via other thermopiles 

in order to keep the temperature difference between the two cells at a minimum. 

The compensating current is detected and used in the further processing. When the 

temperature in the cells reaches the preset value, the heaters of the shields are switched 

off and the instrument turns to uncontrolled cooling. The shields and the cells are cooled 

to the temperature of the cold thermostat, which is kept at a temperature of ∼ 0 ◦ C. 

When the temperature of the internal thermostatic shield is 6 ◦ C, the controlled heating 

is switched on again [67]. 

The instrument can be programmed to measure several identical scans successively, 

which makes it possible to check the precision of the scans as well as a possible time 

dependence. The measurements are performed at a pressure of 3 atmospheres and a scan 

takes approximately 3 hours 2 . Using different scanrates result in different baselines, see 

2 At a scanrate of 1 K/min.


appendix F. 

Power compensating DSC 

The DSC7 is a disk type power compensating DSC with an isoperibolic 3 mode of operation. 

The measuring system consists of two platinum-iridum alloy microfurnaces, which 

contain one temperature sensor and two heating elements. Both furnaces are positioned 

in an aluminium block kept - thermally decoupled - at a constant temperature, the 

measuring range extends from -90 ◦ C 4 to 725 ◦ C. 

Into each of the two furnaces a small pan of aluminium containing the sample and 

the reference, respectively, is mounted. The sample pans are made of aluminium as 

they need to have a very high thermal conductivity. Using aluminium pans lower the 

measuring area from 725 ◦ C to lower than 660 ◦ C, the melting temperature of aluminium. 

The microfurnaces are be covered with lids of platinum-iridium. 

Sample 

Reference 

* 

* * * * 

* * * 

o o o o o 

o o o o 

P T T P 

S S R R 

T(t) − program 

P − control 

T − amplicification 

m P T 

Figure 3.4 A schematic drawing of a power compensating DSC. Each furnace is provided with 

heating elements and a temperature sensor. φ m is the heat flow and P is the power. Adapted 

from Hemminger [22]. 

A control circuit supplies the same heating power to both furnaces in order to change 

their mean temperature according to the preset temperature-time program. In case of 

ideal symmetry (the sample and reference are similar) the temperature of both furnaces 

is the same. When a sample reaction takes place, the symmetry breaks and causes a 

temperature difference between the two microfurnaces. 

This temperature difference is at the same time the measured signal and input signal of 

a second proportional control circuit, which tries to compensate the reaction heat-flow 

rate by increasing or decreasing an additional heating power. The compensating heating 

3 isoperibol refers to constant temperature surroundings with the temperature of the measuring system 

possibly differing from this. 

4 the block could reach -175 ◦ C if it was cooled with fluid nitrogen.

3.2 DSC 39 

power is proportional to the measured temperature difference. The time integral over 

the compensating heating power is proportional to the heat, which was consumed or 

released in the sample[22]. This information is sent to an output device, a computer, 

and results in a plot of the heat flow between the reference and sample cell as a function 

of temperature. Whether the peak is above the baseline, positive, or below the baseline, 

negative, depends on whether it is an endothermic or exothermic process. 

35 

DPPC : hexanol 

Scanrate: 5 K / min 

Heat flow [mW] 

30 

25 

20 

10 20 30 40 50 60 

Temperature [°C] 

Figure 3.5 A DSC scan of DPPC with hexanol, ratio 1:2. Scanrate 5 K/min. Perkin Elmer. 

Two separately controlled electrical heaters are connected to each microfurnace. Heating 

block A supplies both cells with a constant heat flow, depending on the scanrate. If 

the content of the two cells is identical then a plot of the temperature as a function of 

the feed heat would result in a straight line without any peaks [22], and heating block 

B will not contribute to the total heat supply to the cells. If the content of the cells 

differ, then heat block A will supply both cells with the constant specific amount of heat 

defined by the chosen scanrate and during e.g. a phase transition in the sample cell heat 

block B will supply the sample cell with heat equivalent to the heat of transition. The 

two heating blocks, A and B, are decoupled and serve basicly different purposes. 

Sample 

Reference 

* * 

* * * * 

* 

A 

B 

* 

* 

T 

O O 

O O O 

O O 

A 

B 

O O 

O O 

Figure 3.6 A rough sketch of the heating system i the DSC7, Perkin Elmer. 

For the DSC7 instrument small pans measuring 5 mm across are used. 30 µL of the 

sample is placed in the sample pan by using a pipette, a lid is placed on top of the pan


and the lid is pressed down the pan using a specially designed pan-sealer, see figure 3.7. 

Figure 3.7 A photo of the lid, the pan before and after the sealing. During the sealing the outer 

edges of the lid and the pan are cut off before being used in the DSC7. 

Afterwards the pan is placed in the sample cell. The reference consists of a matched 

aluminium sample pan containing distilled water. It is placed in the reference cell of the 

instrument. A scan takes between 5 minutes and 30 minutes depending on the chosen 

scanrate. During the experiments the pressure has not been regulated, the experiments 

have been performed at ambient pressure. In appendix F there is a figure showing the 

baselines at different temperatures.

3.3 SAXS 41 

3.3 SAXS 

The Kratky camera is the most widely used long-slit block camera. The most characteristic 

feature of the camera is the block collimation. The original camera was designed by 

Otto Kratky, hence the name Kratky-camera, in the 1950s and redesigned in the 1970s 

[15]. The Kratky camera at IMFUFA, see figure 3.8, is a modified Compact Camera 

produced by M. Braun-Graz Systems Gmbh. The measuring equipment is situated in 

a thermostatically controlled, lead isolated room with a constant room temperature of 

20 ◦ C. This camera is well-suited for measurements on randomly oriented molecules and 

objects in solution, and covers a range of scattering vectors q down to 0.025 Å −1 . 

Figure 3.8 A schematic drawing showing the most important components of the Kratky compact 

camera. From the detector to the X-ray tube, the camera measures about 55 cm, (drawing 

adapted from [11]). 

The camera 

The camera is mounted on top of the source, the X-ray tube. The camera head, beam 

length and beam width can be adjusted easily when calibrating by turning the knobs 

shown in figure 3.8. To reduce air scattering the chamber is evacuated (P < 0.3 Torr 

≈ 2 mPa). As a safety mechanism the shutter only opens when the pressure is low. 

This ensures that the lead glass plate on top of the camera is mounted air-tight after a 

possible maintenance operation or adjustment carried out inside the chamber. A beam 

stop is situated just in front of the detector to protect it from the direct beam. 

Source 

The source is a sealed X-ray tube (Phillips Fine line) which is cooled by circulating water. 

The X-ray generator (Phillips PW 1729) is for our measurements provided with a voltage 

of 40 kV and current of 40 mA. As the cathode is heated, the electrons are vapourized 

and accelerated towards the copper anode, called the target. The electrons interact 

with the target in two ways in order to create X-rays: bremsstrahlung and fluorescence. 

Bremsstrahlung (or “braking radiation”) occurs as the electrons are decelerated in the 

electric fields of the nuclei. The energy loss is emitted as energetic photons, which form a


continuous spectrum. The other source of X-rays is fluorescence. The incoming electrons 

excite the bound electrons of the target. When they decay, photons are emitted. This 

radiation spectrum consists of the characteristic lines of copper. The copper spectrum 

shows primarily radiation from the two electron transitions n = 2 → n = 1, (K α , λ = 1.55 

Å), n = 3 → n = 1, (K β , λ = 1.41 Å), see figure 3.9. 

Log (Intensity) 

K α 

K β 

Bremsstrahlung 

Energy 

Figure 3.9 The low energy radiation K α originates from the transition from the L-shell to the 

K-shell of the copper atom, whereas the high energy radiation K β originate from the transition 

from the M-shell to the K-shell. 

Monochromator 

Generally, the intensity of a beam decreases as it passes through an absorbing medium. 5 

The absorption spectrum of nickel has a top between Cu-K α and Cu-K β , thus inserting 

the nickel filter dampens the Cu-K β peak strongly, and the Cu-K α peak lightly, resulting 

in a nearly monochromatic beam. 

Block collimation 

In the camera, the primary beam is guided by three bodies, two blocks and one edge 

[40]. The edge is situated as an entrance slit, S, in front of the X-ray tube and controls 

the height of the beam. The entrance slit, the middle block, B 1 , and the second block, 

B 2 , also called the bridge, have to be accurately aligned with respect to the centerline, 

H, and in the direction perpendicular to the paper. If this is done correctly there will 

be no parasitic (secondary) scattering above the centerline, H [8]. The block collimation 

principle of the camera is shown in figure 3.10. 

In figure 3.10(c) and figure 3.10(d) incorrect setups of the Kratky camera are shown. In 

figure 3.10(c) B 2 is placed too low, resulting in scattering by edge k 2 which illuminates 

region S, whereas in figure 3.10(d) B 2 is placed too high, resulting in parasitic scattering 

above the primary beam. In the correct setup of the Kratky camera, the primary beam, p, 

forms a triangle shaped intensity distribution in the vertical direction in the registration 

plane, R. 

5 In a linear, spatially isotropic, absorbing medium the intensity as a function of distance is I(x) = 

I 0 exp[−µx], where I 0 is the initial intensity (at x = 0) and µ is the linear absorption coefficient (as 

explained in Chapter 2).

3.3 SAXS 43 

(a) A schematic view of the collimation system. 

The bridge B 2 is put on top of the 

side-bars of the U -shaped body the central 

part of which is the middle block B 1 . 

(b) The correct arrangement of the block 

collimation camera. The section along the 

middle axis of the camera, perpendicular to 

the plane of the primary-beam. The vertical 

scale is multifold stretched compared to 

the horizontal one. S is the entrance slit, 

B 1 and B 2 are the two blocks used for collimation. 

R is the plane of registration and 

p is the primary beam. 

(c) Incorrect Kratky setup as the second 

collimation block, O 2 , is too low. 

(d) Incorrect Kratky setup as the second 

collimation block, O 2 , is too high. 

Figure 3.10 The principle of block collimation. Note that right and left is reversed in 

comparison to figure 3.8. (All four schematic drawings are from [40]) 

The block collimation increases the intensity of the beam, but it also increases the smearing 

(see figure 5.2) of the beam. 6 

Sample holder 

The sample holder, shown in figure 3.11, is placed inside the middle of the Kratky camera. 

It is temperature controllable (heated from below) and holds the cuvette (glass tube) with 

the sample perpendicular to the beam. 

The sample holder can be adjusted by turning the knobs. It is possible to adjust the 

cuvette both perpendicular and parallel to the beam (height and tilt angle). The best 

arrangement of the sample is found by adjusting and subsequently measuring. As the 

6 Smearing of the beam is one of the biggest problems using line focusing cameras instead of pinhole 

cameras, the topic will be discussed further in section 5.2.


Figure 3.11 A schematic drawing of the sample holder (from [11]). 

largest intensity is found, the sample holder is kept at this position during the experiments. 

The dimensions of the glass tube and its holder are shown in figure 3.12. 

Figure 3.12 The dimensions of the cuvette (the sliding gauge is in millimetres). The lids of 

both ends can be unscrewed in order to inject the sample fluid. 

Detector 

The detector is a MBraun Position Sensitive Detector which detects both energy and 

position of the scattered photons. The detector consists of a platinum wire encased in a 

chamber filled with an argon/methane mixture kept at about 7 to 8 bar. As the photons 

enter the chamber, they ionize the gas atoms, and the electrons from this ionization 

process are accelerated towards the anode, which consists of wires kept at a potential of 

about 3.6 kV relative to ground. 

In the close vicinity of the anode wire the electrons gain enough energy to start an 

electron avalanche process, resulting in an induced charge distribution on the cathodes, 

which amplify the charge signal [87]. The size of this signal is proportional to the energy 

of the photon [2]. The distribution of the energy of the incoming photons and the energy 

window used is shown in figure 3.13. 

The position of the photon is detected by a backgammon technique, see figure 3.14, 

in which this technique and the main components of the detector are shown. 

In linear counters the anode wire usually has a high resistance and the position of 

the event can e.g. be determined by the division of the charge collected at the two ends. 

The detector systems form an RC circuit and due to the dependence of the resistance on

3.3 SAXS 45 

2 

1.5 

Energy spectrum 

Lorentz distribution 

Counts per second 

1 

0.5 

0 

200 300 400 500 600 700 

Channel number 

Figure 3.13 The energy spectrum for the multilamellar DPPC measurement. The vertical 

lines frame indicates the energy window used. The red line is a Lorentz distribution, 

L = 1.53 · [( ch.nr.-459 ) 2 + 1], in which 459 specifies the location of the peak of the distribution. 

92 

the position the rise time of the signals depends on position. By appropriate electronic 

processing of the signals the position can be determined and the energy can be obtained 

from the sum of the two signals [8]. 

Calibration 

q-calibration 

When performing a scattering experiment the outcome is the scattering intensity as a 

function of the detector’s channel numbers. To convert the detector channel number into 

a q-vector value, a q-calibration is made. The scattering intensity is measured along the 

anode wire and picked up in 1024 channels. The calibration is made with silver behanate 

(AgBH), chosen because of its well-defined repeat distance, 58.380 Å [26]. As seen in 

figure 3.15 there is a repeating pattern in the SAXS spectrum of silver behanate. 

Combining this information with the measurements of the direct beam and Bragg’s 

law, equation 2.44, it is possible to find the q-value as a function of the channel number. 

Reading the value for the direct beam and the observed Bragg peaks gives 

q(c) = 7.914 · 10 

−4 Å−1 

ch.no. · c − 0.185Å−1 (3.1) 

where q can be found as a function of the channel number. This calibration is used 

in the further data processing.


(a) Subfigure a) shows a schematic drawing 

of the non-encoding cathode (1), the 

anode wire (2) and the encoding cathode 

(3), and subfigure b) shows the backgammon 

cathode. The broken circle indicates 

a spot of the induced charge. 

(b) A schematic diagram of the electronic 

setup of the detector. 

Figure 3.14 An overview of the electronic equipment inside the detector. From [87]. 

3.5 x 104 Channel number 

3 

2.5 

2 

Intensity (a.u) 

1.5 

1 

0.5 

0 

−0.5 

0 200 400 600 800 1000 1200 

Figure 3.15 The measured intensity of scattering from AgBH.

3.3 SAXS 47 

Temperature calibration 

The sample is heated from below, which leads to a temperature gradient from the heater 

to the sample, and by calibrating the system a connection between the temperature set 

by the machine to the absolute temperature of the sample is found. 

The temperature calibration was done with a negative temperature coefficient (NTC) 

(semiconductor) resistor, of which the resistance was calibrated as a function of the 

absolute temperature with an error of 0.1 K using a Ametek D55SE 7 . The temperature 

dependence (given a certain temperature range) of the resistance of the semiconductor is 

a constant times a Boltzmann factor with a well-known activation energy, E ∼ 0.297 eV 

(suggested by the manufacturer), see figure 3.16. 

( ) E 

R(T) = R ∞ exp 

(3.2) 

k B T 

R(T) of NTC resistor 

12 

10 

Calibration of NTC resistor 

Fit: R(T) = 0.105 · exp( 0.295 / (k B 

T)) 

Resistance [kΩ] 

8 

6 

4 

2 

0 

20 30 40 50 60 

Absolute temperature [°C] 

Figure 3.16 The resistance of the NTC resistor as a function of the absolute temperature. 

It is worth noting that, unlike metals, the resistance of the semiconductor decreases 

as the temperature increases (hence the name, NTC). This is because the effect of an 

increased amount of charge carriers in the conduction band dominates the effect of electrical 

resistance due to the scattering of charge carriers by the ion lattice - the increase 

of charge carriers giving a positive (and the latter giving a negative) contribution to the 

electrical conductivity. 

The NTC resistor was placed inside the cuvette and connected to an Ohm-meter 

outside the Kratky camera. Different calibration measurements were performed, varying 

the content of the cuvette and with or without evacuated chamber. The cuvette contained 

either air or light-fluid silicone oil. Light-fluid silicone oil was chosen over water, as we 

did not dare to test insulatory properties of the coating of the NTC resistor and the 

7 Jofra Instruments, DK-3520 Farum


wiring. We waited about 30 minutes between the measurements and it seemed enough 

to obtain a steady state (settled temperature). 

Temperature calibration, T(T S 

) 

60 

Absolute temperature [°C] 

50 

40 

30 

20 

Airfilled cuvette 

Fittet line (linear regression): 

T = 0.847T S 

+ 45.0 K 

Cuvette with silicone oil 

Fittet line (linear regression): 

T = 0.794T S 

+ 60.5 K 

20 30 40 50 60 70 

Set temperature [°C] 

Figure 3.17 Temperature calibration with evacuated chamber and a cuvette filled with either 

air or silicone oil. The correlation coefficient for □ is 0.9999754, and the correlation coefficient 

for ∗ is 0.9999929. 

The correlation between the absolute and the set temperature was found to be linear, 

see figure 3.17. The figure shows that the presence of silicone oil causes a smaller slope 

because of a higher thermal conductivity. 

Matlab Code 

☞ page 180 T(T S ) air = 0.847T S + 45.0 K 

T(T S ) oil = 0.794T S + 60.5 K 

The difference between the set temperature and the measured temperature is especially 

distinct at high temperatures where a set temperature of 50 ◦ C corresponds to an 

actual temperature of about 44 ◦ C.

3.4 Sample preparation 49 

3.4 Sample preparation 

First, here is a short version of the experimental procedure for preparation of ULV. Later 

in this section some of the steps in the procedure are elaborated. 

Preparation of MLVs The lipid powder 8 is weighed directly into the glass vial (8 mL) 

and boiled, lukewarm milli q water 9 is added. A 5 weight percentage (w%) is 

endeavoured. The sample holder is sealed with parafilm. Using an Eppendorf 

Thermomixer comfort the sample is consistently shaken and held at a constant 

temperature, 10 degrees above the phase transition temperature, for two hours. 

The liquid sample is now usually homogeneous and milk-white, if not so the sample 

is exposed to ultrasound for 10 minutes also at 10 degrees above the phase transition 

temperature. The ultrasound procedure was seldom required. 

Preparation of ULVs, extruding By following the directions given in the “Lipex Extruder” 

manual the extruder is assembled. A combination of filters is used, first 

a polycarbonate filter, then a drain filter, then a polycarbonate filter and finally 

another drain filter. The filters are wet by milli q water and water is extruded 

three times at a pressure of 20 bar before the sample is extruded. The sample is 

extruded 10 times at 20 bar, the glass vial and the pipette are changed during the 

last three extrusions, for further explanations see later in this section. 

Refining the sample, centrifugation After the extrusions the sample is centrifuged. The 

sample is rotated by 19.500 rounds per minute at a temperature of 20 ◦ C for 20 

minutes. After the centrifugation the sample is separated into 3 phases, the top 

phase is water, the middle phase is unilamellar vesicles and the bottom phase is 

multilamellar vesicles. By use of a pipette the two top phases are transferred to 

a glass vial. In [59, p. 70] it was found that this procedure removes MLVs which 

have survived the extrusion such that the number of scattered photons from MLVs 

are halved, approximately. 

Concentration of lipid suspension 

For the gravimetric determination of the lipid concentration 4 small metal pans (diameter 

of 5 mm) are numbered and weighed 4 times each. Each of the 4 times 4 weights are 

noted. A drop of the ULV-solution is placed at one of the cups and a stop watch is 

started and the weight is noted every 15 seconds for two minutes. This procedure is 

repeated for the last three cups. Finally the 4 cups are placed in an oven at 50 ◦ C. After 

a couple of days the 4 dehydrated cups are weighed 4 times each, again. The weights are 

noted. By plotting these data it is possible to determine the lipid concentration. The 

linear correlation between the time and the weight of the lipid drop is extrapolated to 

zero, see figure 3.18. The weight at t=0 is equivalent to the weight of the lipid drop. 

By combining these informations the concentration of the lipid suspension is found. 

Even if a lipid concentration of 5 % is endeavoured, the centrifugation process makes it 

impossible to end up with the same lipid concentration in each experimental series. The 

concentrations vary from 1.62 (first experimental series) to 4.59, the average standard 

being around 3.4, see appendix D. 

The aqueous, pure lipid sample is kept in a container, shaken constantly at 10 degrees 

above the phase transition temperature until the alcohol is added. A SAXS experiment 

is performed with the pure lipid sample. 

8 Purchased from Avanti Polar Lipids Inc. www.avantilipids.com 

9 Arium® 611 Ultrapure Water Systems from Sartorius AG


52.2 

Raw data 

y=-0.006 (mg/sec)x+52.21 (mg) 

52 

m [mg] 

51.8 

51.6 

51.4 

50 100 

Time [sec] 

Figure 3.18 An example of the determination of the gravimetric determination of the lipid 

concentration. The weight is plotted as a function of time. The measurement after 15 seconds 

might not be as precise as the other 7 measurements as it takes 10-20 seconds for the weighing 

machine to stabilize. 

Adding alcohol 

The amount of alcohol added depends on the experiment performed, although the lipid 

to alcohol ratio is the same, two alcohol molecules to one lipid molecule. This ratio was 

chosen on the basis of the literature values in table 1.1. The alcohol to lipid ratios are 

highly dependent on the lipid concentration, the lower the lipid concentration the higher 

alcohol to lipid ratio is needed to induce the interdigitation. As to the concentrations 

of lipid used in this work only one experiment can be used in strict comparison. Zhang 

& Rowe [88] use a lipid concentration of 70.8 mM and report interdigitation from the 

n-alcohol butanol at a 1:5 total molar ratio. Using this information along with the 

correlation reported by Löbbecke & Cevc [47], see section 1.4, the DPPC to pentanol 

total molar ratio should be more than 1:1.7. And as less and less alcohol is needed to 

induce interdigitation the longer the n-alcohols are, a fixed total molar ratio of 1:2, DPPC 

to alcohol, was used. 

Different amounts of alcohol are needed for the pentanol, hexanol and heptanol experiments 

respectively. The procedure is the same, though. An Eppendorf tube is placed 

on the weight and by using a syringe a specific amount of alcohol 10 is added to the side 

of the tube and the weight is noted. Afterwards the tube is filled with the sample and 

the final weight is noted. The tube is sealed with parafilm. 

The Eppendorf tube with lipid and alcohol is placed in the tempered shaker and is 

shaken for at least 48 hours before the SAXS experiment is performed. The temperature 

of the shaker is subsequently set to 52 ◦ C, 10 ◦ C above the main phase transition 

temperature. 

10 Purchased from Aldrich-Chemie, D-7924 Steinheim


Vesicle extrusion and Polydispersity 

The extrusion procedure primarily involves pushing a solution of large MLVs through 

polycarbonate membrane filters several times. The polycarbonate filters have roughly 

cylindrical pores with a diameter of 100 nm and a length of 6 µm. By pushing the MLVs 

through the pores, they break up into smaller vesicles. The extrusion process can be 

explained in terms of blowing bubbles through pores, see figure 3.19. 

A 

P 1 

B 

P 1 

P 2 

P 0 

l 

d 

C 

P 0 

Figure 3.19 (A) Multilamellar vesicles are extruded through the polycarbonate membrane filters 

resulting in unilamellar vesicles whose size is comparable to the pore size, see figure 3.20. l is 

the length and d is the diameter of pores in the polycarbonate filter. (B) A small fragment 

of the vesicle is pulled into the pore. This fragment of the vesicle has the same radius as the 

pore, whereas the large fragment outside the pore has a radius comparable with the original 

radius of the vesicle. (C) Schematic diagram of the vesicle extrusion. The force due to the 

applied pressure is balanced by a force caused by line tension around the neck of the vesicle. 

The difference between P 0 and P 1 is the applied pressure, P 2 is the pressure inside the vesicle. 

Adapted from [58]. 

Patty & Frisken [58] have modelled the vesicle size (radius), R v as a function of pore 

size (radius), R p and the extrusion pressure, P . 

√ √ 

Rp γ l 

R v (P) = A 

2P + B or R v γl 

= a + b (3.3) 

R p 2PR p 

where γ l is the lysis tension and A and B (or a and b) are fitting parameters. The model 

agrees with their experimental results, which unfortunately is restricted to 1-palmitoyl- 

2-oleoyl-sn-glycero-3-phosphatidylcholine (POPC). POPC has asymmetric tail lengths, 

one of 16 and one of 18. Nevertheless, these results were used since it was not possible 

to find similar results for DPPC (but the reader should keep this in mind). For POPC 

they find a = A = 0.61 ± 0.04, b = B R p 

= 1.09 ± 0.02 and γ l = (7.4 ± 0.04) mN/m. 

The vesicle size does not change much as the extrusion pressure is increased from 20 

to 30 bar 

R v (P = 20 bar) ≈ 604 Å and R v (P = 30 bar) ≈ 593 Å (3.4) 

using equation 3.3 and R p = 50 nm. In our preparation procedure, we have certainly 

stayed within this range, trying to stay at 20 − 21 bar and only higher pressure when we


Figure 3.20 The average size and polydispersity of the vesicles in a sample with a 5% lipid-water 

mass ratio calculated from the model in equation 3.3 and a reported 20% standard deviation 

[58]. 

had trouble extruding (when nothing happened as we tried to extrude). The polydispersity 

of the vesicle size was determined by light scattering [58], and Patty & Frisken 

[58] find a standard deviation of about 20% for the smaller pore diameters (50 nm) and 

about 30% for the larger pore diameters (200 nm). According to the LIPEXExtruder 

Manual [44] the vesicle size should be ±20% of the filter pore size. This was verified by 

light scattering on their test system, egg phosphatidylcholine (EPC) vesicles 11 . But the 

manual does not say anything about the extrusion pressure used. 

If all ULVs are dispersed, the characteristic distance, d, between two vesicles can be 

estimated from 

d ULV = 3√ v = 3 √ 

V 

N V 

(3.5) 

where v is the volume per vesicle, i.e. the fraction between the total volume V of the 

sample (lipid and water) and the number of vesicles N V . N V must be the ratio between 

the total number of lipid molecules, N l , and the number of lipid molecules per vesicle, 

N mpv 

N V = N l 

N mpv 

(3.6) 

11 The fatty-acid composition of egg lecithin prepared by the cadmium chloride method has about 30% 

of C16, 55% of C18 and 15% of C20 and C22 acids; approximately 39% and 45%, respectively, of the 

total acids were saturated [41].


d 

d 

1.6 µm 

Figure 3.21 The characteristic distance between two ULVs in a sample with a 5% lipidwater 

mass ratio, calculated using equation 3.9 with the following values: A l = 50 Å 2 [32], 

r ULV = 60 nm (see estimate, equation 3.4), m l = 255 mg (measured), V = 5.4 mL (measured), 

N A = 6.0221 · 10 23 mole −1 [85], M l = 734.1 kg/mole [32]. It is about 14 times larger than the 

vesicle diameter. The number of vesicles are about N V = 1.1 · 10 12 using equation 3.6. 

with a characteristic radius r ULV . The number of lipid molecules are 

N l = n l N A = m l 

M l 

N A (3.7) 

where n l , N A , m l and M l are the amount of lipid molecules, Avogadro’s number, the 

total lipid mass and the molar mass of the lipid, respectively. The number of molecules 

per vesicle can be estimated from the total (of a single) vesicle surface area, A, and the 

area pr. lipid molecule, A l , 

N mpv = A A l 

= 2 · 4πr2 ULV 

A l 

(3.8) 

assuming that the inner and outer surface areas are roughly the same. Combining 3.5, 

3.6, 3.7 and 3.8 gives 

√ 

8πr 2 

d ULV = 3 ULV V M l 

(3.9) 

N A m l A l 

In figure 3.21 the distances are sketched. This is of course a rough estimate, but there is 

no doubt that d ULV is much larger than the coherence length of the Cu − K α X-rays.

4 Results 

In this chapter we present the results obtained with ITC, DSC and SAXS. The results 

in section 4.2 and 4.3 are denoted with a series number, the details about this specific 

experimental series can be found in appendix D. 

4.1 ITC 

High sensitive isothermal calorimetry has become more and more popular for studying 

heats of reaction observed upon the incorporation or partitioning of molecules into membranes 

[34]. There are several examples in the literature of the use of ITC in determining 

different thermodynamical properties of the lipid-alcohol system [65], [73], [82], [83], [88]. 

Experiments 

Eight different kinds of experiment series were conducted, two with pentanol, five with 

hexanol and one with heptanol. All the experiments were conducted at 25 ◦ C. The samples 

differed mutually in the lipid/alcohol ratio. A common characteristic of all the series 

was that at least 10 different runs were made per series varying the cell concentration 

of the cell, gradually approaching and passing the free concentration of alcohol in the 

syringe. Several introductory experiments were carried out prior to the eight experiment 

series represented. Blank injections of pure ULV of DPPC were made to measure the 

heat of dilution. The resulting heat release was insignificant compared to the heat of 

dilution from the alcohols. 

After conducting an experiment the heat of each peak is integrated to find the enthalpy 

originating from each injection. The integrated heat from the second injection is 

used in the further data processing. From each experiment the outcome was consequently 

one set of simultaneous data of the enthalpy and the concentration of alcohol in the cell. 

The integrated heats of reaction are plotted as a function of the alcohol concentration in 

the cell, and a linear function is used to describe the correlation. An example of such a 

correlation can be seen in figure 4.1. 

The data of all the eight series have been processed, and the results are found in 

appendix E. The calculated free alcohol concentration, C alc,free , equivalent to the intersection 

of the graph with the axis of abscissa, is used to calculate the molar partition 

coefficient. The amount of alcohol in the lipid is calculated on the basis of the total concentration 

of alcohol in the lipid/alcohol suspension and the free alcohol concentration: 

C alc, total = C alc,free + C alc,lipid 

The concentration of alcohol in the lipid, C alc,lipid , is then found by using the injected 

volume, called V lipid : 

55

56 Results 

250 

66.6 mM pentanol in 45.5 mM lipid 

200 

cal/mole of injectant 

150 

100 

50 

0 

−50 

−100 

Raw data 

y=30.6 x − 1600 

−150 

50 52 54 56 58 60 62 

Alcohol concentration in reaction cell/ mM 

Figure 4.1 The integrated heat contributions as a function of the concentration of alcohol in the 

reaction cell. The free alcohol concentration, equivalent to the intersection of the graph with 

the axis of abscissa, is found to be 53.9 mM. The total alcohol concentration in the syringe is 

66.6 mM. The concentration of alcohol in the lipid is then 66.6 − 53.9 = 12.7mM. Each point is 

equivalent to an experimental series. 

C alc,lipid = C alc,total − C alc,free 

C alc,lipid = n alc,lipid 

V lipid 

n alc,lipid = C alc,lipid · V lipid (4.1) 

The amount of alcohol in the lipid is found as the concentration of alcohol in the lipid 

suspension and the injected volume is known. The amount of lipid, n DPPC , free alcohol, 

n alc,free , and water, n water , is calculated in the same way: 

n DPPC = C DPPC · V lipid 

n alc,free = C cell · V lipid 

n water = V lipid · ρ water 

18.02 

The factor 18.02 is the molar mass of water. n alc,free is the concentration of alcohol 

in the cell. The partition coefficient can then be calculated, since it is defined as: 

K x = X a,b 

X a,w 

(4.2)

4.1 ITC 57 

where X a,b is the mole fraction of alcohol in the lipid bilayer given by 

n alc,lipid 

X a,b = 

n alc,lipid + n DPPC 

and X a,w is the mole fraction of alcohol in water given by 

X a,w = 

n alc,free 

n alc,free + n water 

The partition coefficients for all the experiments are listed in table 4.1. The figures 

marked by stars refer to experiments with samples which have been kept at room 

temperature for 24 hours. 

n-Alcohol Lipid K x Lipid:Alcohol 

1 Pentanol (28.7 mM) DPPC (53.5 mM) 261 8.5:1 

2 Pentanol (66.6 mM) DPPC (45.5 mM) 225 3.5:1 

3 Hexanol (10.5 mM) DPPC (45.5 mM) 367 18:1 

4 Hexanol (15.1 mM) DPPC (54.3 mM) 2553* 4.8:1 

5 Hexanol (21.6 mM) DPPC (54.3 mM) 529 6.7:1 

6 Hexanol (22.6 mM) DPPC (54.3 mM) 749/ 3546* 5.3/3*:1 

7 Hexanol (31.3 mM) DPPC (53.5 mM) 986 3:1 

8 Heptanol (7.3 mM) DPPC (45.5 mM) 2166/1713* 9.4/10.3*:1 

Table 4.1 The first column represents the used alcohol and the related concentration in the 

syringe, the second column represents the lipid concentration, the third column the calculated 

partition coefficient and the fourth column represents the lipid to alcohol ratio within the lipid 

membrane. The literature values are found on page 59. 

There are two main issues related to the calculated partition coefficient, the phase 

dependence and the alcohol-chain dependence. 

During the experimental process some characteristic features of the different lipidalcohol 

systems were discovered. All the samples were kept at 52 ◦ C until used for the 

experiments. Normally, an experiment series were conducted within 10 hours, the results 

were reproduceable within this period of time. While the experiments were carried 

out, the samples were kept at room temperature. After keeping the samples at this 

temperature for 24 hours, some of the experiments were conducted again. These ITC 

results differed systematically from the results obtained the previous day. 

The ITC experiments for pentanol and hexanol revealed that the free concentration 

of alcohol in the lipid/alcohol suspension had decreased, entailing that a clearly larger 

amount of alcohol had partitioned into the lipid vesicles after these have been kept at 25 ◦ C 

for 24 hours. Hence for these measurements a larger partition coefficient was calculated 

(marked by (*) in table 4.1 and E.2). The significantly larger partition coefficient could 

be a sign of evaporation of the free alcohol from the sample or a sign of the appearance 

of another lipid phase. According to Zhang & Rowe [88] the partition coefficient for the 

L βI phase is lower than the partition coefficient for the L β ′ phase and much lower than 

the partition coefficient for the L α phase, see table 4.2. The values for K x in general 

demonstrate that the partition coefficient for the liquid crystalline phase is significantly 

larger than for the three other phases. It is a reasonable assumption that the lipid vesicles 

are primarily in the interdigitated phase when the main experiments are conducted. 

This is also reflected in the measured partition coefficients, which are similar to the

58 Results 

values found in the literature. There is most likely a certain equilibrium between the 

amount of vesicles in the interdigitated phase and the amount of vesicles in the ripple 

phase, dominated by the interdigitated phase. After being kept at room temperature 

for 24 hours the part of vesicles in the L β ′ phase have increased. As the partition 

coefficient for alcohol into vesicles of this type is higher than the partition coefficient 

for the interdigitated phase, the total measured partition coefficient consequently will be 

higher. 

Löbbecke & Cevc [47] allowed the samples of DPPC and alcohol to equilibrate at 

room temperature for more than a week, at least, to make sure the lipids were in the 

interdigitated phase. In our work a DSC scan was run on samples, which had been 

equilibrating for a week. These scans were not a source of concern as the scans did not 

differ from the scans obtained the previous week. For this reason the samples have only 

been equilibrating for 48 hours. 

The ITC results for the heptanol samples kept at room temperature for 24 hours 

differed from those obtained with pentanol and hexanol samples. These results indicated 

that the free heptanol concentration in the heptanol/DPPC suspension increased and 

hence the partition coefficient for the DPPC/heptanol system decreased. Still the partition 

coefficient for heptanol into DPPC is much larger than the comparable experiments 

with hexanol and pentanol. But the lower partition coefficients obtained after keeping 

the samples at room temperature for 24 hours indicate a difference in the interactions 

between this alcohol and DPPC compared to the lipid interactions with pentanol and 

hexanol. This could indicate a quite long equilibration time for heptanol into the DPPC 

vesicle. Only one example of comparable experiments with lipids and heptanol have 

been found in the literature [30]. These measurements were conducted on multilamellar 

vesicles and in the liquid crystalline phase, both differing from the conditions used in 

this work. Nevertheless the values are of the same order of magnitude. The partition 

coefficients are in accordance with the partition coefficients found in the literature, see 

table 4.2. The values for the partition coefficient obtained in this work are lower than 

those obtained by Kamaya et al. [30], primarily owing to the fact that the partition coefficients 

are obtained in the L α phase and the fact that they use MLVs as opposed to 

ULVs. Using MLVs instead of ULVs gives rise to a larger partition coefficient [73] 

In general the partition coefficient for a specific phospholipid system increases with 

increasing alcohol length [73]. This tendency is also reflected in the results obtained in 

this work. The partition coefficients averaged increase by a factor 3 as a CH 2 group 

is added to the n-alcohol. This observation is in relation to values for the partition 

coefficient obtained by Katz & Diamond [33], Kamaya et al. [30] and Ly & Longo [46]. 

These groups obtained values for the partition coefficient of DMPC, SOPC, and DPPC, 

respectively. All series of partition coefficients increased approximately by a factor 3 as 

the chain length of the n-alcohols was increased by a methyl group. These relations all 

follow Traube’s Rule 1 as it is interpreted by Ly & Longo [46]. Ly & Longo [46] state that 

the difference between measuring values for e.g. propanol, butanol and pentanol roughly 

follows Traube’s Rule of a factor 3 indicating that alcohol partitioning actually can be 

described by Traube’s rule. The increase in partition coefficient with alcohol chain length 

can also be related to the solubility of the alcohol in water. The longer the alcohol, the 

lower the solubility and the higher the hydrophobia. 

1 Traube’s Rule predicts that for each CH 2 group in an alcohol molecule, three-times-lower alcohol 

concentration will be required to reach the same interfacial surface tension between a hydrophobic 

and water/alcohol solution [77]

4.2 DSC 59 

Lipid n-Alcohol Phase T ( ◦ C) K x Method Reference 

DPPC Hexanol L β ′ (-) 100 Solvent-null Janes [29] 

DPPC Hexanol L β ′ (-) 1000 Solvent-null Janes [29] 

DPPC Hexanol L α (-) 3500 Solvent-null Janes [29] 

DMPC Hexanol L α 25 1342 Solvent-null Suurkuusk [73] 

DPPC Butanol L βI 15 70 Solvent-null Zhang [88] 

DPPC Butanol L β ′ 35 138 Solvent-null Zhang [88] 

DPPC Butanol L α 50 180 Solvent-null Zhang [88] 

DPPC Hexanol L α 45 839 Solvent-null Rowe [65] 

DPPC Pentanol L α 41 278 T c depression Kamaya [30] 

DPPC Hexanol L α 41 963 T c depression Kamaya [30] 

DPPC Heptanol L α 41 4060 T c depression Kamaya [30] 

Table 4.2 Partition coefficients for some relevant alcohol-lipid systems reported in the literature. 

K x is the mole fraction partition coefficient. 

Increasing the length of the partitioning alcohol cause a practically constant decrease 

in the absolute value of ∆∆G ◦ of around 3 kJ per one carbon chain, when the lipid/alcohol 

system is investigated in the same lipid phase [30]. This decrease in free energy 

is also seen in this work, by increasing the n-alcohol length from pentanol to hexanol 

an average increase in ∆G of 2.4 kJ per carbon chain is found. Going from hexanol to 

heptanol an increase in the free energy of 3.0 kJ is found, see table E.2. 

The partition coefficients for DPPC and hexanol is expected to be larger than the 

partition coefficients for DMPC and hexanol as DPPC is more hydrophobic or lipophilic 

than DMPC [73]. 

4.2 DSC 

In the literature there are several examples of how DSC is used to measure the phase 

transition temperature between the interdigitated phase, L βI , and the liquid, crystalline 

phase, L α [47], [63], [88]. The specific phase transition temperature differs according to 

the lipid and alcohol as a result of which a DSC scan has been run prior to each SAXS 

experiment. It is important to know the phase transition temperature, as the SAXS 

instrument must be set correctly in order to measure in either the interdigitated or the 

fluid phase of the lipid. Directly from the shape of the DSC scan it is possible to determine 

whether the expected phase transition has been reached and measured. When the alcohol 

threshold concentration is reached one can see that the small “shoulder” (referring to 

the pretransition) in the DSC scan disappears. The interdigitated phase is induced at 

the same concentration as the pretransition disappears [79]. The width of the peak is 

another basis for qualitatively comparison, as presence of enough alcohol to induce the 

interdigitated phase is reflected in a broader peak [88]. In simulations it is found that 

the main phase transition temperature does not depend on the alcohol concentration if 

the interdigitated phase is reached [39]. Experimentally it is found that increasing the 

alcohol concentration at a distinct temperature can cause the interdigitated phase to coexist 

with either the ripple or the gel phase, depending on temperature. This co-existence 

is probably due to an inhomogeneous distribution of the alcohol in the bilayer, where 

the alcohol molecules aggregate in certain regions cause small domains of interdigitation 

[39].

60 Results 

DSC on water 

Several experiments on the DSC7 instrument were conducted with water in both pans. 

These experiments were made in order to estimate the uncertainty of the heat flow and 

find out whether there was a correlation between the baseline and used scanrate. As the 

DSC experiments with the lipid/alcohol system are carried out at different scanrates, 

this correlation would be important in the further data processing. All the DSC scans 

on water were conducted after a so-called burn-off, where the furnaces were heated to 

600 ◦ C to burn off all impurities in the furnaces. 

DSC on water in both sample and reference cells 

Source data 

Linear fit to baseline above settling interval 

80 

80 

Heat flow / mW 

60 

40 

5 °C / min 

10 °C / min 

15 °C / min 

20 °C / min 

25 °C / min 

30 °C / min 

35 °C / min 

40 °C / min 


60 

40 

5 °C / min 

Linear fit to 5 °C / min 

20 °C / min 


40 °C / min 


20 

20 

0 

10 20 30 40 50 60 

Temperature / °C 

0 

10 20 30 40 50 60 


Residual between source data and linear fit 

1 

SDOM of the heat flow 

as a function of the scanrate 


2 

1 

0 

-1 

-2 

20 30 40 50 

5 °C / min 

20 °C / min 

40 °C / min 


SDOM of heat flow / mW 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

SDOM 

Linear fit to SDOM 

0 

0 10 20 30 40 

Scanrate / °C pr. minute 

Figure 4.2 The four graphs show different aspects of DSC performed on water in both sample 

and reference cells. The upper-left figure shows that the settling interval increases as the 

scanrate increases. The baselines are rising as function of the temperature and they seem to 

be less straight as the scanrate increases. This is indeed the case as shown in the upper-right 

corner and the lower left corner in which just three scanrates are treated (in order not to overfill 

the graphs). The upper-right graph shows the linear fit above the settling interval and the 

lower-left show the residual between the source data and the fit. The lower-right graph shows 

that the SDOM of the heat flow increases as the scanrate increases, see table 4.3. 

From the residuals between the source data and the linear fit in figure 4.2 a more or less 

systematic deviation is found. This deviation is clearly not dependent on the scanrate 

as the same correlation is found for 3 different scanrates. We can not explain the reason 

for the systematic deviation. 

Looking at the low temperature end of the scan in figure 4.3 one can recognize the 

same settling interval as seen in the water experiment, see figure 4.2. Even though the

4.2 DSC 61 

Scanrate, ◦ C/min 5 10 15 20 25 30 35 40 

SDOM of heat flow, mW 0.19 0.28 0.46 0.54 0.65 0.71 0.71 0.81 

Table 4.3 The estimated uncertainties of the heat flow. Extrapolation to 0 ◦ C/min gives an 

uncertainty of 0.15 mW as indicated in the lower right corner in figure 4.2. 

DSC on MLV (DPPC) 

with different scanrates 


40 

20 

5 °C / min 

10 °C / min 

15 °C / min 

20 °C / min 

25 °C / min 

30 °C / min 

40 

40 45 50 

10 20 30 40 50 60 


Transition temperature as a function of scanrate 

Transition time as a function of scanrate 

48 

0.6 

Transition temperature / °C 

46 

44 

42 

Transition temperature 

estimated peak 

Linear fit, Temp = 

0.27 minute · Scanrate + 

41 °C 

Transition time / minutes 

0.5 

0.4 

0.3 

0.2 

0.1 

Transition time estimated from 

transition temperature interval 

40 

0 5 10 15 20 25 30 


0 

0 5 10 15 20 25 30 


Figure 4.3 The top figure shows the source data of DSC measurements performed on multilamellar 

DPPC vesicles with a concentration of (5.12 ±0.02) %. The transition temperature and 

time are then plotted against the scanrate in the two lower graphs. The transition temperature 

decreases as the scanrate decreases, and the transition temperature vs. the scanrate seems to be 

linear and gives a “true” transition temperature at about 41 ◦ C for an infinitely slow scanrate.

62 Results 

transition temperature decreases as the scanrate decreases, the transition time actually 

shows an increasing tendency as the scanrate decreases (one should not be fooled by the 

narrower peaks, since the transition time is transition temperature interval divided by 

the scanrate). The smaller temperature gradients for low scanrates apparently slow down 

the transition process. The system does not seem to reach a state of equilibrium after 

the main transition, which is indicated by the wobbly post-transition baselines - opposed 

to the much nicer baselines before the transition in figure 4.3 (on the left). The expected 

pretransition is not convincingly detected. 

DSC error sources 

There were a lot of error sources. Some of which occur randomly, and some do not. 

The three worst error sources were lack of thermal equilibrium, lack of thermal contact 

between the heater and the pans and water condensating inside the apparatus. It was 

hard to tell which error source was to blame in the specific case. The presented data is 

only a small portion of the measurements performed. And due to the error sources, a lot 

of data was rejected. We have not had an objective criterion on when to reject data from 

our DSC measurements. It was almost a reversed situation. We have had an objective 

criterion on when to accept data: our results had to be reproducible. 

The error sources possibly include: 

• Pollution from the dry air fed into the sample holder chamber 

• Water condensating on top of the sample holder (and other places) 

• Pollution inside the sample holder 

• Mechanical deformation of the pans 

Enthalpy of transition 

Several DSC scans were conducted on the different lipid/alcohol solutions varying the 

alcohol concentration. The pure DPPC main phase transition was also measured. The 

data processing of these scans served two purposes, determining the phase transition 

temperature and calculating the phase transition enthalpy. The former is used with 

regards to the setting of the SAXS instrument and the latter is primarily used as a basis 

for comparison with comparable values found in the literature. 

This value for the phase transition temperature is in accordance with values reported 

in the literature. Löbbecke & Cevc [47] find the main transition enthalpy of DPPC to 

be 43 kJ/mole. According to Löbbecke & Cevc [47] the phase transition temperature 

decreases with increasing alcohol concentration and the enthalpy of transition decreases 

at low alcohol concentration and increases with high alcohol concentration, relative to 

the main transition enthalpy of pure DPPC. This tendency was also recognized in this 

work, but as the measurements were primarily conducted to find the phase transition 

temperature, the enthalpies have not been calculated systematically. 

The enthalpies presented in figure 4.4 are all found by determining the area between 

the peak and the baseline, principle shown in figure F.3, and the results are represented 

graphically in figure 4.3. The location of the baseline and the peak have been estimated 

and as these are determined subjectively, they will contribute to the uncertainties of 

the final integral. The calculated enthalpies should be constant and independent of the 

scanrate, and should not decrease with increasing scanrate.

4.2 DSC 63 

50 

Main transition enthalpy 

Transition enthalpy/ kJ/mol 

45 

40 

35 

30 

25 

Raw data 

y=−0.82x+45.3 

20 

0 5 10 15 20 25 30 

Scanrate/ °C per minute 

Figure 4.4 The main transition enthalpy of DPPC as a function of scanrate. As seen on figure 

4.3 the sizes of the peaks are varying with scanrate in which case the enthalpies of transition 

also vary with scanrate. The “true” phase transition enthalpy is found by extrapolation to 

45 ± 2 kJ/mole. 

Temperature of transition 

The scans show that the phase transitions decrease when alcohol is added to the lipid 

solution. This behaviour is also reported in the literature [47], [64]. The phase transition 

temperature decreases with increasing alcohol concentration (results from the scal-1 

microcalorimeter, not shown). In a pure DPPC lipid solution the phase transition temperature 

is 41 ◦ C, see figure 4.3. By adding pentanol the temperature decreases, and the 

measured phase transition temperature is around 35 ◦ C. Adding hexanol decreases the 

phase transition temperature even more drastically, and the measured temperature is 

around 29 ◦ C, see figure 4.5. For heptanol the phase transition temperature is around 

35 ◦ C. The phase transition temperatures are found by reading the top position of the 

peak. The uncertainties are estimated to be ±1 ◦ C. This value is based on the results 

from several performed DSC scans. 

From the scan in figure 4.5 one can see that no pretransition has been detected 

and that the peaks have widened compared to the pure DPPC main phase transition. 

Both observations indicating the presence of an interdigitated phase. The enthalpy of 

transition has been calculated for the four scans represented in figure 4.5. The enthalpy 

has been calculated as the integral of the peak, see figure F.3. The four scans all had 

a phase transition enthalpy of 62-65 kJ/mole. This enthalpy is about 20 % higher than 

the values reported in the literature [47]. Still the enthalpy is known to increase with 

increasing concentration of alcohol in the lipid. 

A test was made to investigate whether the lipid/alcohol suspension suffered damage 

when being heated and cooled rapidly, which is the case when conducting a DSC scan. 

This was at the same time a test to check whether the phase transition temperature 

increased or decreased when the sample was repeatedly heated and cooled. 12 successive 

scans were conducted and plotted on top of each other, see figure 4.6. 

The DSC scans reveal that the DPPC/hexanol suspension is quite stable. The sus-

64 Results 

DPPC : pentanol 


12 


14 


11.5 

13 


11 

10.5 


12 

10 

11 

9.5 

10 

10 20 30 40 50 60 


10 20 30 40 50 60 




24 


35 


23 


22 

21 


30 

25 

20 

19 

10 20 30 40 50 60 


20 

10 20 30 40 50 60 


Figure 4.5 DSC scans for series 9, showing DPPC with pentanol and hexanol, respectively. 

The upper-left figure shows DPPC and pentanol, ratio 1:2, there is a sharp peak at 34 ◦ C. The 

upper-right figure shows DPPC and pentanol, ratio 1:2, there is a sharp peak at 35 ◦ C. The 

lower-left figure shows DPPC and hexanol, ratio 1:2, there is a sharp peak at 28.5 ◦ C. The 

lower-right figure shows DPPC and hexanol, ratio 1:2, there is a sharp peak at 29.5 ◦ C. The two 

“ghost-peaks” in the last part of the scans of pentanol are most likely due to impurities in the 

apparatus. These DSC scans have been obtained by using the Perkin-Elmer DSC7 instrument. 

pension responded similarly to being heated, even after being heated to 80 ◦ C and cooled 

to 10 ◦ C several times. The position of the peak and hence the phase transition temperature 

does not change throughout the 12 scans, indicating that the phase transition from 

the L βI phase to the L α phase is fast, reversible and that the transition temperature is 

stable. The same results apply for the samples with pentanol and heptanol, see appendix 

F.

4.2 DSC 65 

12000 


10000 

9500 

scan 1 

scan 12 

8000 

mV 

9000 

mV 

6000 

8500 

38 38.5 39 


4000 

2000 

0 

36 38 40 42 44 46 48 50 


Figure 4.6 12 successive scans are plotted on top of each other. There is practically no 

difference between the 12 scans and the location of the peak does not differ. A lipid to hexanol 

ratio of 1:17, w = 0.05% and a scanrate of 1K/min was used. There is an average standard 

deviation of 174.3 relative to the first scan. These scans have been obtained by using the scal-1 

microcalorimeter.

66 Results 

4.3 SAXS 

Introductory experiments 

Several introductory experiments were made varying different parameters i.e. the sample 

preparation procedure 2 in order to elaborate on the sensitivity of the system. During 

these experiments it was found that there was no difference in the obtained spectra 

when keeping the alcohol-containing samples shaking at 32 ◦ C instead of at 52 ◦ C. The 

first series of the SAXS experiments consisted of experiments conducted at 4 different 

temperatures - two experiments below and two experiments above the phase transition 

temperature. The two experiments obtained in the apparent L βI phase are practically 

indistinguishable just like the two experiments obtained in the L α phase, see figure 4.7. 


Serie 7 

8 


Serie 7 

8 

25 °C 

35 °C 

40 °C 

45 °C 

Intensity [a.u.] 

6 

4 

2 


6 

4 

2 

0 

0 0.1 0.2 0.3 0.4 0.5 

q [Å -1 ] 

0 

0 0.1 0.2 0.3 0.4 0.5 

q [Å -1 ] 

(a) The two datasets obtained at 

25 ◦ C and 30 ◦ C are practically indistinguishable. 

(b) The two datasets obtained at 

40 ◦ C and 45 ◦ C are practically indistinguishable. 

Figure 4.7 The raw data (with subtracted background) from the SAXS experiments with series 

07, DPPC:hexanol 1:2. The temperatures are chosen given the DSC results. 

The results led to the conclusion that one experiment in each lipid phase was sufficient. 

Besides wearing the Kratky camera and its associated equipment lesser and saving time 

on the experiment per se, the decision also had the advantage that the sample was exposed 

to X-rays for a shorter period of time, causing less possible damage to the vesicles. The 

results also show a noticeable difference between the SAXS spectra obtained in the two 

phases, reflecting the assumption of a measurable structural difference between the two 

phases, see also figure 4.8. 

Matlab Code 

☞ page 157 

In order to get measurements with enough counts to obtain good statistics the SAXS 

experiments last 18 hours. To see if there is any temporal development this has been 

divided into three measurements of six hours each. In figure 4.9 these three spectra from 

the same SAXS experiment are represented. The photon counts of the three spectra are 

added and divided by the total measuring time (equivalent to averaging the intensities). 

The averaged intensity is then normalized (see equation 5.32) for the modelling. 

2 For further details see appendix D.

4.3 SAXS 67 

DPPC : Pentanol 

Series 15 

10 

DPPC : Hexanol 

Series 12 

8 

25°C 

45°C 

8 

45 °C 

25 °C 


6 

4 

2 


6 

4 

2 

0 

0 0.1 0.2 0.3 0.4 0.5 0.6 

q [Å -1 ] 

0 

0 0.1 0.2 0.3 0.4 0.5 0.6 

q [Å -1 ] 

Figure 4.8 Spectra obtained at 25 ◦ C and at 45 ◦ C. 

Normally we have been doing two measurements in a row, leaving the sample to be 

exposed to radiation for 36 hours. This long period of exposure will unquestionably 

have an effect on the sample. We have shown that there is not detectable effect within 

the first 18 hours. But during the experimental work with the Kratky camera we have 

experienced more than once that an amount of lipid vesicles had aggregated in the glass 

tube just where the beam passes through the sample holder when the sample had been 

exposed to radiation for more than 36 hours. This shows that the sample can be affected 

by long-term beam exposure. 

8 


Series 14 

6 

6 hours 

12 hours 

18 hours 


4 

2 

0 

0 0.1 0.2 0.3 0.4 0.5 

q [Å -1 ] 

Figure 4.9 Three spectra obtained at the same temperature and recorded during 0-6, 6-12 and 

12-18 hours of measuring, respectively. The spectra are averaged and used in the following data 

processing. 

The three spectra in figure 4.9 are practically indistinguishable, demonstrating that there 

are no significant changes within the sample during the time of the 18 hour long experiment, 

and justifying the averaging. The stability of the structure also indicates that the 

beam does not alter the chemistry of the sample.

68 Results 

The SAXS spectra used for the data modelling are all obtained by measuring 3 times 6 

hours at different temperatures, starting at the low temperatures. The DSC experiments 

already indicated that the system did not exhibit notable hysteresis going from the high 

temperature phase to the low temperature phase, but we checked that there was no 

difference in starting the experiment in the low temperature area instead of the high 

temperature area. Two experiments with the same sample were made to verify this. The 

first experiment started in the low temperature range going to the high temperature 

range, denoted “upscan”, the second experiment started in the high temperature area 

going down to the low temperature area, denoted “downscan”. There was no difference 

between the two experiments. The results are represented in appendix G. These results 

are in good agreement with the results obtained with DSC. 

Background 

In the data processing it is necessary to correct for non-system (air, solvent and cuvette) 

scattering called the background. One could argue that this actually is a part of the 

modelling, but since this correction is made before the description of spectra characteristics 

we decided to include it in this chapter. The background is measured with a 

sample of pure water as shown in figure 4.10. Water is chosen over a water/n-alcohol 

solution since the large partition coefficients suggest that most of the alcohol is in the 

bilayer. Even so, the bilayer will eventually get saturated and there will be a considerable 

amount n-alcohol in the solvent. This was realized too late (after the modelling) and 

we hope it does not compromise our conclusions. Throughout the whole experiment the 

Intensity / counts pr. second 

0.5 

0.4 

0.3 

0.2 

0.1 

Measured 

Background 

Measured with subtracted background 

0 

0 0.1 0.2 0.3 0.4 0.5 

q / Å -1 

Figure 4.10 The measured, background and measured with subtracted background intensity 

spectrum. The sample is DPPC vesicles perturbated by hexanol, series 14 at 45 ◦ C. The 

background is almost constant as a function of q. Both sample and background measurement 

lasted 18 hours and had a photon count N of 1.3 · 10 7 and 6.8 · 10 6 , respectively.

4.3 SAXS 69 

lipid/alcohol sample absorb a far greater number of photons than the water sample. At 

high q-values, q = 0.4Å −1 to q = 0.5Å −1 , the modelled I(q) is nearly zero, and hence 

the transmission, T , is calculated at high q-values 

∫ q2 

q 

T = 

1 

I measured (q)dq 

∫ q2 

(4.3) 

q 1 

I solvent (q)dq 

The transmission calculated with equation 4.3 from q 1 = 0.45 Å −1 to q 2 = 0.49 Å −1 

is found to usually be about T = 1.2. The intensity as function of the q values of an 

experiment used in the data processing is then, 

I sample (q) = I measured (q) − T · I solvent (q). (4.4) 

The transmission is calculated during the data processing of each spectrum and hence, 

the transmission varies slightly from experiment to experiment. 

Matlab Code 

☞ page 157 

8 

Intensity / a.u. 

7 

6 

5 

4 

3 

2 

7.5 

7 

6.5 

6 

5.5 

5 

1 

0 

0.025 0.05 0.075 0.1 0.125 0.15 

0.1 0.2 0.3 0.4 0.5 

q / Å -1 

Figure 4.11 A sample of DPPC vesicles perturbated by hexanol, series 14 at 45 ◦ C. The 

absolute uncertainty of the photon count is √ N and the relative uncertainty is √ N 

= √ 1 

N N 

(= 0.94% as a minimum relative uncertainty for this particular case). 

The spectra used 

One of the imposed conditions of using a spectrum in the modelling process was that the 

spectrum could be reproduced. The experimental progress was delayed repeatedly due to 

several unexpected difficulties with the Kratky camera, primarily. Among the difficulties 

was the break down of the air condition in the lead-isolated room, numerous software 

breakdowns and leaks in the cuvette. Still, the spectra for all six different experiments 

was reproduced, see figure 4.12. For larger versions of the graphs the reader is referred 

to appendix H, page 135.

70 Results 


25 °C 

8 


45 °C 

8 

Series 14 

Series 15 

Series 13 

Series 15 


6 

4 

2 


6 

4 

2 

0 

0 0.1 0.2 0.3 0.4 0.5 

q [Å -1 ] 

0 

0 0.1 0.2 0.3 0.4 0.5 0.6 

q [Å -1 ] 

10 


25 °C 

10 


45 °C 

8 

Series 12 

Series 15 

8 

Series 14 

Series 12 


6 

4 


6 

4 

2 

2 

0 

0 

0 0.1 0.2 0.3 0.4 0.5 0.6 

q [Å -1 ] 

0 0.1 0.2 0.3 0.4 0.5 0.6 

q [Å -1 ] 

10 

DPPC : Heptanol 

25 °C 

10 


45 °C 

8 

Series 10 

Series 15 

8 

Series 10 

Series 15 


6 

4 


6 

4 

2 

2 

0 

0 0.1 0.2 0.3 0.4 0.5 0.6 

q [Å -1 ] 

0 

0 0.1 0.2 0.3 0.4 0.5 0.6 

q [Å -1 ] 

Figure 4.12 The raw data from the SAXS experiments with all the alcohols used. The spectra 

have been normalized according to equation 5.32 - we could not expect the intensities to be the 

exact same, since the refining process leaves varying amounts of vesicles and thus scattering 

centers from series to series.

4.3 SAXS 71 

Some features characterize the raw data plots. All the spectra display an envelope curve 

reflecting the main structure of the lipids in the suspension, the unilamellar vesicles. The 

spectrum for unilamellar vesicles are found in figure 4.13. The envelope curve is fairly 

10 

DPPC 

Unilamellar vesicles, 25 °C 

DPPC 

Unilamellar vesicles, 45 °C 

8 

8 


6 

4 


6 

4 

2 

2 

0 

0 0.1 0.2 0.3 0.4 0.5 

q [Å -1 ] 

0 

0 0.1 0.2 0.3 0.4 0.5 0.6 

q [Å -1 ] 

Figure 4.13 SAXS spectra of unilamellar vesicles of DPPC, measured at two different temperatures. 

smooth and does not display the indent around q = 0.05Å −1 as opposed to the spectra 

obtained with DPPC and alcohol. Besides the main feature, the envelope curve, at least 

three peaks, see close-up in figures 4.14, on the graphs are found at low q-values for low as 

well as high temperature measurements. On the raw data for the low temperature measurements 

of heptanol/DPPC these peaks are not as pronounced. A common feature of 

all the low temperature measurements compared to the high temperature measurements 

is the fact that the second peak is moving towards higher q-values as the temperature is 

raised. The first peak, however, is practically not moving as the temperature is raised, 

except from the pentanol/DPPC measurements where the peak moves towards higher 

q-values as the temperature increases. For the low temperature pentanol/DPPC and 

hexanol/DPPC measurements one can also find a very broad peak around q = 0.4Å −1 . 

The peak is also found at the pure ULV DPPC low temperature measurement, see figure 

4.13, but it is not found in any of the high temperature measurements. 

The location of the peaks is also differing. For the low temperature pentanol/DPPC 

measurements the first peak is located at lower q-values than for the high temperature 

measurements with the same alcohol. The opposite apply to the location of the second 

peak. The third peak does not change location as the temperature increases. For the 

hexanol/DPPC measurements the peaks do not change location with temperature. For 

the heptanol/DPPC measurements only the second peak re-locates, it moves to higher 

q-values as the temperature increases. The graphs flatten differently depending on the 

experimental temperature. The low temperature measurements for pentanol/DPPC and 

hexanol/DPPC seem to flatten faster than the high temperature measurements. The low 

temperature and high temperature heptanol/DPPC measurements do not seem to differ 

in this way. 

As earlier mentioned an imposed condition in using a spectrum in the further data 

modelling was that it was reproducible. In figure 4.12 examples of low temperature 

hexanol/DPPC spectra are represented, but two other spectra of low temperature hexanol/DPPC 

are worth mentioning. There are several characteristic features in the spectra

72 Results 

8 


45 °C 

Series 13 

7.5 


7 

6.5 

6 

5.5 

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 

q [Å -1 ] 

10 


25°C 

9 

Series 12 


8 

7 

6 

5 

4 

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 

q [Å -1 ] 


25 °C 

9 

Series 15 


8 

7 

6 

5 

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 

q [Å -1 ] 

Figure 4.14 The main three peaks displayed at low q-values. The values in table 4.4 are found 

by reading the q-values in a close-up.

4.3 SAXS 73 

Low temperature 

High temperature 

1. Peak 2. Peak 3. Peak 1. Peak 2. Peak 3. Peak 

[Å −1 ] [Å −1 ] [Å −1 ] [Å −1 ] [Å −1 ] [Å −1 ] 

Pentanol 0.038 (14) 0.093 (14) 0.105 (14) 0.043 (13) 0.088 (13) 0.106 (13) 

0.038 (15) 0.090 (15) 0.108 (15) 0.043 (15) 0.089 (15) 0.106 (15) 

Hexanol 0.0432 (12) 0.09 (12) 0.106 (12) 0.0425 (12) 0.088 (12) 0.106 (12) 

0.0417 (14) 0.089 (14) 0.1061 (14) 0.0425 (14) 0.091 (14) 0.106 (14) 

Heptanol 0.043 (15) 0.082 (15) 0.106 (15) 0.0434 (15) 0.0908 (15) 0.106 (15) 

0.0415 (10) 0.085 (10) 0.0434 (10) 0.0908 (10) 0.106 (10) 

Table 4.4 The location of the three peaks. The values are found by reading from close-ups of 

the spectra, see figure 4.14. The numbers in the brackets refer to the series. The figures in the 

brackets refer to the series. 

10 


25 °C 

8 

Series 9 

Series 13 


6 

4 

2 

0 

0 0.1 0.2 0.3 0.4 0.5 0.6 

q [Å -1 ] 

Figure 4.15 The raw spectra of hexanol/DPPC obtained at 25 ◦ C. There is a certain repeatdistance 

between the peaks, ∆q≃0.12 Å −1 , equivalent to a distance of around 52 Å. 

of the low temperature measurements of hexanol/DPPC series 9 and series 13. Unlike 

the spectra in figure 4.12 these spectra seem to demonstrate the presence of multilamellar 

vesicles as there is a certain repeat distance, ∆q≃0.12 Å −1 , between the peaks. The first 

and largest peak is located at high q-values like the other low temperature hexanol/DPPC 

measurements.

74 Results 

MAX-Lab measurements 

In late January an opportunity came by to do measurements on a synchrotron 3 . After 

preparing the samples, one pure DPPC, one with hexanol and one with heptanol, as 

described in section 3.4, the samples were kept at room temperature until the measurements 

were performed, that is 24 to 48 hours after leaving the thermomixer. The spectra 

obtained from the synchrotron measurements differ from the spectra obtained with the 

Kratky camera. The three peaks at low q-values were not detected, but the envelope 

curve was the same as what we see. Except from the thermal history (being kept at 

room temperature), the samples have been treated like all the other samples used for 

this work. 

3 Performed by Dorthe Posselt at the MAX-lab in Lund, Sweden

Part III 

Analysis

5 Modelling 

This chapter contains an overview of the models used, introductions to each of the models 

and a summary of the modelling results. Please refer to appendix H, page 135, for an 

exhaustive list of model graphs and fitted parameters. 

Before we begin, it is enlightening to take one step back and look more generally at the 

modelling process. Often, physical modelling can be expressed as 

ˆKx = y (5.1) 

where ˆK is a model operating on some modelling parameters x giving the output y, 

which can be compared with the measured data obtained by experiments. The direct 

problem would be to evaluate y given ˆK and x, but it leaves two other indirect problems: 

to determine ˆK or x (given x and y, or, ˆK and y, accordingly). Both types of problems 

are found in the various disciplines of physics. 

Direct problems are often well-posed, which means that ([85, well-posed problem]) 

• a solution exists, 

• the solution is unique, and 

• the solution depends continuously on the data, in some reasonable topology. 

Conversely, the indirect problems are often ill-posed and hence lack some or all of the 

mentioned properties (existence, uniqueness and stability). 

An example of a well-posed problem could be to determine the temperature field, y, 

after a period of time given Fourier’s law of heat conduction, ˆK, with specified initial 

conditions, x. By contrast the backwards heat equation, deducing an initial temperature 

field from the final field is ill-posed in that the solution, x, is highly sensitive to changes 

in the final data y [85, well-posed problem], and the solution might not be unique. So, 

in this particular case both the uniqueness and stability conditions are violated. 

Actually our indirect problem is to determine both ˆK and x given y, since we have to 

find a suitable electron distribution model and, then, determine the parameters of that 

specific model. 

5.1 Overview of the models used 

The elastic scattering theory is the basis of the models. It states that the differential 

cross section depends on the density and the arrangement of the electrons as described 

in chapter 2. The form factor (f bilayer model ) is proportional to the Fourier transform of 

the relative electron density distribution, ρ(r): 

f bilayer model (q) ∝ F{ρ(r)} ≡ 

1 

(2π) 3/2 ∫u.c. 

ρ(r)exp(−iq · r)d 3 r (5.2) 

77

78 Modelling 

where q is the momentum transfer vector and the unit cell (u.c.) is the bilayer of a 

vesicle. 

Several aspects of the experimental setup, including the cuvette glass, air and solvent 

scattering should be taken take of, since the background has been subtracted (see section 

4.3, page 68). But other aspects such as secondary scattering and inelastic scattering are 

completely disregarded. 

Looking at one particular q-value, the scattered intensity measured gets contributions 

from all over the scattering volume. This smearing of the intensity spectrum is based on 

the geometry of the experimental setup (see section 3.3, page 41) and is explained further 

in section 5.2. This correction is very important as the Kratky camera has a slit-shaped 

beam, hence it has been applied to all of the models used: 

∫ 

I smeared (q) = P(r)I p (q ′ )d 3 r (5.3) 

V 

where V is the scattering volume and I p (q ′ ) is the scattered intensity from a point shaped 

beam and P(r) is the intensity profile of the beam. 

Although we try to keep the amount of MLVs to a minimum, both ULVs and MLVs are 

suspended in the solvent. And since MLVs scatter more than ULVs the sample does 

not have to be polluted with many MLVs in order to see their presence on the intensity 

spectrum. Therefore, we include MLVs in our modelling. We assume that the vesicles 

are dispersed in the sample so no net interference occurs between the vesicles, see figure 

3.21. The intensity from each of the vesicles can then be added 

I p (q) ∝ I ULV (q) + A b I MLV (q) (5.4) 

where the factor A b is the relative amount of scattered photons by MLVs in the sample. 

The models representing the electron density are formulated in direct space and are 

Fourier transformed to the inverse space to comprise the form factors of the model. The 

models are based on Gaussian functions, which have the advantage of having analytical 

Fourier transforms. This is not an idea of ours - others have used Gaussian functions 

before us: [6], [9], and see figure 5.1. It does make sense to use such coarse-grained 

functions to represent the density of electrons of large assemblies of molecules since the 

time of exposure during the measurement is about 18 hours - all dynamics on smaller 

time scales are completely smeared out in time 

I ULV (q) ∝ 〈|f bilayer model (q)| 2 〉 and I MLV (q) ∝ 〈|f bilayer model (q)| 2 S(q)〉 (5.5) 

where 〈···〉 is the time average of all the vesicles (the time average will be implied throughout 

the whole chapter) and S(q) is the structure factor [56], [89]. 

We have used 6 models in our analysis of the SAXS data as they all contribute in 

different ways to the understanding of the modelled system. Clearly the 6 models attend 

to slightly different features in the interpretation of the SAXS spectra and hence confirm 

or invalidate the features characterizing the alcohol perturbated DPPC vesicles. We have 

applied all models to data of both low and high temperature phases in order to see which 

model was suited to which phase. 

sym and sym-mlv are similar, the latter differs from the former by correcting for the 

presence of MLVs. 3d and 3d-mlv share the same similarity. The symmetric model sym

5.2 Instrumental smearing 79 

Figure 5.1 Bilayer electron density profile and a propability profile of the bilayer constituents 

[53]. (a) shows the probability density distribution functions for the different components of 

the bilayer. (b) shows the electron density profile, the dotted line from simulations and the 

solid line from X-ray studies. (c) shows two volumetric pictures. 

is the most commonly used model for describing the electron density profile. Taking 

this one dimensional model further to three dimensions lead to the symmetric three 

dimensional model 3d-sym. The 3d-sym model fits the spectra as if the scattering object 

were a spherical vesicle also taking into account the distribution of vesicle radius. 

The number of parameters differ from model to model. The sym and 3d-sym models 

have 4 parameters, the asym and sym-4g models have 5 parameters and the two models 

which also correct for the presence of multilamellar vesicles, sym-mlv and 3d-mlv both 

have 6 parameters. Choosing the number of fitting parameters is a tricky balancing act. 

The more fitting parameters the broader the parameter space yielding a greater the risk 

of finding a local minimum than a global minimum. But more fitting parameters might 

also do the difference in finding a good fit for your raw data as there simply are more 

parameters to turn on so to speak. The general guideline is that the more parameters 

the less certain is the fitted values of the single parameter. We have tried to keep the 

numbers of parameters low in order to get a better determination of the single parameters 

as especially one of these parameters, z h is crucial in our analysis of the SAXS spectra. 

In table 5.1 we have summarized the number of parameters used, the names we use as 

references and the conceptual basis for each of the models. In the following sections we 

describe each of the models (and their motivations) and present the results in the end of 

the chapter. 

5.2 Instrumental smearing 

The scattered intensity measured (at one particular q value) gets contributions from all 

over the scattering volume, hence the name smearing. Compared to pinhole collimation 

the block collimation of the Kratky camera provides a higher intensity beam, but one 

has to take into account that photons arriving to a certain point at the detector can

80 Modelling 

Name Parameters Conceptual basis 

sym 4 symmetric bilayer electron profile 

with three Gaussian functions 

sym-4g 5 symmetric bilayer electron profile 

with four Gaussian functions 

asym 5 asymmetric bilayer electron profile 

sym-mlv 6 symmetric bilayer electron profile 

and scattering from both ULV and MLV vesicles 

3d 4 spherically symmetric bilayer electron profile 

3d-mlv 6 spherically symmetric bilayer electron profile 

and scattering from both ULV and MLV vesicles 

Table 5.1 A summary of the models applied. 

Figure 5.2 A schematic drawing of 2-dimensional smearing due to the slit-shaped beam (adapted 

from [57]). 

originate from a range of scattering angles, see figure 5.2. This corresponds to a range 

of q-values and the measured intensity at the detector becomes 

Matlab Code 

☞ page 174 

∫ 

∫ 

I smeared (q) = P(r)I p (q ′ )d 3 r ≈ 

V 

P(x)I p (q ′ )dx (5.6) 

where V is the scattering volume and I p (q ′ ) = √ q 2 + x 2 is the scattered intensity from 

a point shaped beam and P(x) is the horizontal propability density profile of the beam. 

Here, we ignore the smearing along the y and z axes, since the beam is much wider than 

the height of the beam and the depth of the sample, see figure 5.3. 

It should be noted that the axis of abscisses is different in figure 5.3(a) and 5.3(b). 

The beam width is much larger than the beam height, justifying that we only consider 

smearing from the width of the beam.

5.3 Symmetric (1d) 81 

350 

3000 

300 

2500 

250 

2000 

G(x) [a.u.] 

200 

G(y) [a.u.] 

1500 

150 

1000 

100 

500 

50 

0 

0.1 0.2 0.3 0.4 0.5 0.6 

x [Å −1 ] 

0.05 0.1 0.15 0.2 0.25 0.3 

y [Å −1 ] 

(a) The intensity profile along the 

horizontal axis, x. The measured 

beam is the spots, the fitted Gaussian 

is the line. The fitting parameters 

are a=346.60, b=0.31127 and 

c=0.14681 

(b) The intensity profile along the 

vertical axis, y. The measured 

beam is spots, the fitted Gaussian 

is the line. The fitting parameters 

are a=2398.9, b=0.13386 and 

c=0.0056663 

Figure 5.3 Two Gaussian functions have been used to model the beam G(x) = a·exp 

“− 

”. 

(x−b)2 

c 2 

In order to find the beam profile in both the horizontal (x axis) and vertical (y axis) direction a 

measurement of the direct beam is performed by rotating the detector and using a filter instead 

of a beam stop. 

5.3 Symmetric (1d) 

One of the most commonly used models for the bilayer electron density profile is a 

symmetric lipid bilayer electron density profile in which the bilayer is divided into the 

three areas: two head group parts and one tail part [6], [9], see figure 5.1. In this 

one-dimensional model there is no representation of actual vesicles, but the bilayers are 

represented by infinitely large sheets which have no preferred orientation - the effect of 

the membrane curvature is therefore neglected in this model. 

Compared to the electron density of the solvent (primarily water), the electron density 

is larger for the head group and smaller for the acyl chains (tails). The electron density, 

ρ(z), of these three areas can be regarded as a sum of Gaussian functions. A normalized 

Gaussian function is used to describe each of the three areas 

where 

ρ(z) = ∑ n 

A n 

√ 

2πσ 

2 n 

exp 

(− (z − z n) 2 ) 

2σ 2 n 

(5.7) 

√A n 

is the amplitude, z 2πσ 2 n is the distance from the one headgroup to the middle 

n 

of the bilayer and σ n is the spread. The Fourier transform of 5.7 is 1 

F{ρ(z)} = ∑ n 

A n 

2 

( 

exp − 1 ) 

2 q2 σn 

2 exp(iqz n ) (5.8) 

where A n , z n and σ n are parameters, which can be determined by a fitting procedure. 

1 The derivation is found in appendix C.

82 Modelling 

In detail, the three parts are 

ρ sym (z) = ρ head group inside (z) + ρ head group outside (z) − ρ tail (z) (5.9) 

where 

ρ h inside = √ ρ0A exp 2πσ 2 

h 

ρ h outside = √ ρ0A exp 2πσ 2 

h 

(− (z−z h) 2 

2σh 

2 

(− (z+z h) 2 

2σ 2 h 

( ) 

ρ tail = − √ ρ0 

exp − z2 

2πσ 2 2σ 2 

t 

t 

) 

) 

(5.10) 

Matlab Code 

☞ page 168 

The first two parts of equation 5.9 and hence 5.10 refer to the electron density of the head 

group, cf. the label h, whereas the last group describes the tail region, labelled t. The 

label inside and outside only refer to the fact that the two headgroups are situated on the 

inside and outside of the bilayer of the vesicle, respectively. This naming convention goes 

for all of the models even though it really does not make sense in the one-dimensional 

models. ρ 0 has the dimension length times the electron density, z h and σ have the 

dimension length and A is dimensionless. z h represents half the distance between the 

areas with high electron density, the head groups of the phospholipids, and therefore z h 

can be used as a measure for the thickness of a lipid monolayer. σh 2 is the variance of the 

two first Gauss functions in equation 5.9 and refers to the propagation of electrons along 

the normal of the headgroup. σt 2 is the variance of the last Gaussians in equation 5.9 and 

refers to the propagation of electrons along the normal of the tails. The dimensionless 

parameter A specify the weighing of the three Gaussians. 

By using equation 5.8 and equation 5.9 the symmetric form factor becomes 

f sym (q) = 1 2q 

( 

Aexp 

( 

− 1 2 q2 σ 2 h 

) 

cos(qz h ) − 1 2 exp(q2 σ 2 t ) 

) 

(5.11) 

The factor of 1/q is added to mimic the orientational averaging of the sheets, which 

automatically appears in the three-dimensional model in section 5.6. Please note that 

this is what others apply to the intensity (which then is squared, 1/q 2 ) and refer to as 

the Lorentz factor. 

The model is shown in figure 5.4. The intensity was calculated as the absolute square 

of the form factor (equation 5.5) and the smeared intensity was calculated using equation 

5.6 with the beam profile in figure 5.3. 32 steps were used in the numerical integration 2 

of the smearing for this model and all of the other models. Above the beam stop cutoff 

(at about 0.025 Å −1 ) there was no significant difference using more than 32 steps as 

indicated by the dotted line with 10000 steps in figure 5.5. 

2 Using the built-in Matlab function trapz.

5.3 Symmetric (1d) 83 

Figure 5.4 The form factor and both point scatterer and smeared intensity for the symmetric 

bilayer electron density profile model. For the form factor, equation 5.11 was used with the 

parameters A = 0.5, z h = 17 Å, σ h = 4 Å and σ t = 6 Å. 

Figure 5.5 The smeared intensity for the symmetric bilayer electron density profile model with 

varying amount of steps in the numerical integration done in equation 5.6.

84 Modelling 

5.4 Symmetric (4g) 

Matlab Code 

☞ page 173 

On the basis of the electron density profiles of the interdigitated phase found in the 

literature, e.g. [60], [80], we have expanded the symmetric (1d) model to consist of 4 

Gaussians, two Gaussians representing the tail region and two Gaussians representing the 

head region, see figure 5.6. The form factor for the sym-4g model is based on equation 

5.8 and is a sum of 4 contributions. 

( ( ) 

f sym-4g (q) = 1 

2q 

Aexp − q2 σh 

2 

2 

(exp(−iqz h ) + exp(iqz h )) 

( ) 

) 

− exp − q2 σ 2 t 

2 

(exp(−iqz t ) + exp(iqz t )) 

z t represents the distance from the bilayer centre to the peak of the Gaussians representing 

the tail region. Presuming that the phospholipid chains do interdigitate the tail 

region of the electron density profile will differ from the profile represented in figure 5.1. 

The electron density will increase in the tail region and the distance from the two head 

groups will decrease. The sym-4g model takes into account that the electron density 

probably will increase in the middle of the lipid bilayer. 

Figure 5.6 The form factor and both point scatterer and smeared intensity for the symmetric 

bilayer electron density profile model with 4 Gauss functions. For the form factor, equation 

5.12 was used with the parameters A = 1.5, z h = 20 Å, σ h = 4 Å, z t = 3.5 Å and σ t = 2 Å. The 

intensity was calculated as the absolute square of the form factor and the smeared intensity 

was calculated using equation 5.6 with the beam profile in figure 5.3.

5.5 Asymmetric (1d) 85 

5.5 Asymmetric (1d) 

Studies of lipid bilayers have indicated that the electron density profile is not completely 

symmetric. Brzustowicz & Brunger [9] have shown an asymmetric electron density profile 

of the lipid bilayer through model fitting and direct calculation of the form factor of 

unilamellar vesicles of SOPS. Hirai et al. [23] have also observed bilayer asymmetry in 

a mixture of small unilamellar DPPC vesicles with monosiaganglioside. Brzustowicz & 

Brunger [9] use SOPS over DPPC arguing that SOPS has a larger difference in scattering 

length compared with buffer and that the lipid for this reason is easier to do experiments 

on. SOPS differs primary from DPPC by having two different acyl chains, one with a 

length of 14 C-atoms and one with the length of 18 C-atoms, in addition DPPC contains 

a choline group whereas SOPS contain a serine group. To describe the asymmetric 

electron density profile the two head group Gaussians must be different. Brzustowicz & 

Brunger [9] find that the electron density profile of the two headgroups are described by 

the below-mentioned parameters 

ρ headgroup inside (z) = 

ρ headgroup outside (z) = 

ρ tail = 

) 

(1.18 ± 0.03)exp 

(− (z+(22.4±0.42)Å)2 

2((4.87±0.13)Å) 2 ) 

(2.78 ± 0.04)exp 

(− (z+(19.6±0.46)Å)2 

2((2.11±0.06)Å) 2 

( 

− exp 

) 

z 2 

2((7.76±0.07)Å) 2 

(5.12) 

(5.13) 

(5.14) 

We have tried to use these constants, leaving the same four parameters as in the case of 

the symmetric model. But this fitted poorly to our data and we decided to discard the 

constants and we have developed another model to take the asymmetry into account. In 

our asymmetric model we added an amplitude parameter to the symmetric model leaving 

the spread of the head groups the same 

ρ headgroup inside (z) = 

ρ headgroup outside (z) = 

ρ tail = 

√A 1 

exp 2πσ 2 

h 

√A 2 

exp 2πσ 2 

h 

( 

− exp 

(− (z−z h) 2 

2σh 

2 

(− (z+z h) 2 

z 2 

2σ 2 t 

2σ 2 h 

) 

) 

) 

(5.15) 

(5.16) 

(5.17) 

Using equations 5.8, 5.9 and 5.15 gives the form factor of the asymmetric model 

f asym (q) = 1 ( ( 

exp − q2 σ 2 ) 

( 

h 

(A 1 exp(−iqz h ) + A 2 exp(iqz h )) − exp − q2 σ 2 )) 

t 

2q 2 

2 

(5.18) 

with the five parameters A 1 , A 2 , z h , σ h and σ t . Figure 5.7 shows the form factor, intensity 

from both point scatterer and smeared. The intensity was calculated as the absolute 

square of the form factor and the smeared intensity was calculated using equation 5.6 

with the beam profile in figure 5.3. 

Matlab Code 

☞ page 169

86 Modelling 

Figure 5.7 The form factor and both point scatterer and smeared intensity for the asymmetric 

bilayer electron density profile model. For the form factor, equation 5.18 was used with the 

parameters A 1 = 1, A 2 = 1.5, z h = 17 Å, σ h = 4 Å and σ t = 6 Å.

5.6 Spherically symmetric (3d) 87 

5.6 Spherically symmetric (3d) 

Since the bilayers are shells it would be tempting to make a spherically symmetric model, 

since you could be lucky to capture the features that the curvature is responsible for. We 

developed the model ourselves, but later realized that Brzustowicz & Brunger [9] had 

used it before us. 

I 3d (q) = 

∫ ∞ 

0 

P(R)I 3d (q, R)dR (5.19) 

where P(R) is the propability density of a vesicle in the sample having a radius between 

R and R + dR. We have estimated P(R) in section 3.4 and intensity for a vesicle with 

radius R is 

I 3d (q, R) = |f 3d (q, R)| 2 (5.20) 

In the fitting procedure equation 5.19 was evaluated with 64 steps in the (trapezoidal) 

numerical integration. Below 64 steps the graph becomes noticeably wobbly, and there 

is no difference between using 64 and more (256) steps. Even so, the intensity graph of 

this model does not become as smooth as the one-dimensional models. 

Again, the form factor is the Fourier transform of the electron density distribution 

Matlab Code 

☞ page 172 

f 3d (q, R) ∝ F{ρ(r, R)} ≡ 

1 

(2π) 3/2 ∫u.c. 

ρ(r, R)exp(−iq · r)d 3 r (5.21) 

since the integrand is independent of the azimuthal angle φ, you get 

and equation 5.21 becomes 3 

f 3d (q, R) ∝ 

d 3 r = 2πr 2 sinθ dθ dr and q · r = qr cosθ 

= 

= 

= 1 q 

∫∫ 

1 

ρ(r, R)exp(−iqr cosθ)2πr 2 sin θ dθ dr (5.22) 

(2π) 3/2 ∫ 

1 ∞ ∫ π 

√ ρ(r, R)r 2 exp(−iqr cosθ)sin θ dθ dr (5.23) 

2π 

0 

∫ 

1 ∞ 

√ 

2π 

√ 

2 

π 

0 

∫ ∞ 

0 

0 

ρ(r, R)r 2 2 sin(qr) 

qr 

dr (5.24) 

ρ(r, R)r sin(qr)dr (5.25) 

if the electron density function ρ(r, R) is spherically symmetric. One has to pick a 

function, which makes the integral converge. It is important to notice the factor 1 q is 

independent of which ρ(r, R) we choose. In the case of a Gaussian 4 the integration over 

r becomes[9] 

f 3d (q, R) = σ ( 

r 

q exp − q2 σr 

2 ) (R 

sin(qR) + σ 

2 

2 

r q cos(qR) ) (5.26) 

Matlab Code 

☞ page 171 

3 R h i 

π 

0 exp(−iqr cos θ)sinθ dθ = exp(−iqr cos θ) π 

iqr = exp(iqr)−exp(−iqr) 

0 

iqr 

exp(iψ)−exp(−iψ) 

2i 

4 ρ(r, R) = 

1 √ 2πσ 2 r 

and d exp(−iqr cos θ) = exp(−iqr cos θ)iqr sin θ 

dθ ” 

exp 

“− (r−R)2 

2σ 2 r 

= 2 sin(qr) 

qr , since sin(ψ) =

88 Modelling 

in which the peak of the spherically symmetric Gauss function is at R, and the spread 

is σ r . 

The electron density profile used in the symmetric (1d) model has been the basis 

for the symmetric (3d) model, as the sym-1d model was the 1d model, which fitted the 

spectra best. The underlying idea of extending the sym-1d model from 1 dimension to 3 

dimensions was to take into account the variation in vesicle radius. 

The form factor of the 3d model differs notably from the other form factors. Imagining 

it is was possible to make a sample with vesicles of only one radius then the intensity 

would look like the intensity in figure 5.8. 

Figure 5.8 The form factor and both point scatterer and smeared intensity for the spherically 

symmetric bilayer electron density profile model. The form factor was calculated using equation 

5.26 with the parameters A = 0.5, z h = 17 Å, σ h = 6 Å, σ t = 6 Å and R = 600 Å, i.e. a 

single vesicle radius. The intensity was calculated using equation 5.19 (with a vesicle radius 

distribution) and the smeared intensity was calculated using equation 5.6 with the beam profile 

in figure 5.3. 

5.7 Multilamellar vesicles 

During the sample preparation most of the multilamellar vesicles have been removed, 

but even if there is only a small percentage left in the final solution it is visible in the 

SAXS experiment. Unlike ULV the MLV Bragg-scatter much more due to the lamellar 

repeat distance. In modelling the SAXS data the presence of MLV has to be taken into 

account. Still assuming no interference between the vesicles in the solution the equation 

describing the measured intensity from the ULV is added the measured intensity deriving 

from the scattering from MLV. The total measured intensity is then 

I p (q) ∝ I ULV (q) + A b I MLV (q) (5.27)

5.7 Multilamellar vesicles 89 

In figure 5.9 the result of a SAXS experiment on a suspension containing multilamellar 

vesicles is shown. This spectrum clearly differs from the spectra represented in figure 4.12 

by displaying several distinct peaks. These peaks are found with the same ∆q between 

them, which indicates the repeating structure characterizing the multilamellar vesicles. 

By using equation 2.45 the distance between the two lipid bilayers is found. 

40 

Multilamellar vesicles 

25 °C 

Raw data 

Structure factor 

30 


20 

10 

0 

0 0.1 0.2 0.3 0.4 

q [Å -1 ] 

Figure 5.9 The raw data from a SAXS experiment on MLV of DPPC. The distance between 

the two peaks is ∆q = (0.1003 ± 0.0005)Å −1 , corresponding to a distance of d = 2π = 62.6 Å. 

∆q 

This distance is equivalent to a lamellar repeat distance [53]. 

The structure factor used in the description of the intensity contribution from the 

scattering of multilamellar vesicles can be approximated as a sum of Gaussians 

S(q) = ∑ A n 

√ exp 

(− (q − nq 0) 2 ) 

n=1 2πσ 

2 

b 

2σb 

2 (5.28) 

The fitted structure factor in figure 5.9 is a sum of three Gaussians 

1 

S(q) = √ 

(exp 2πσ 2 

b 

(− (q−q0)2 

2σ 2 b 

+ 0.1Aexp 

) 

+ Aexp 

(− (q−3q0)2 

2σ 2 b 

(− (q−2q0)2 

) ) 

2σ 2 b 

) 

(5.29) 

Matlab Code 

☞ page 174 

where A = 0.22, q 0 = 0.1003Å −1 and σ b = 0.0128Å. 

There were some experimental difficulties related to the SAXS experiments on the multilamellar 

vesicles at 25 ◦ C. Normally a spectrum was obtained by letting the Kratky 

camera measure for 6 hours in a row, but with a sample of the multilamellar vesicles of 

DPPC this was not possible. After a short period of time (half an hour) the vesicles were 

no longer randomly distributed within the suspension. They fell down the bottom of the 

glass tube. A way to circumvent this problem would be to dilute the lipid suspension, but 

then the experimental time would have to be increased. Instead the cuvette was taken 

out of the camera every hour and shaken until the suspension was milk-white again. The 

experiment was subsequently resumed and repeating this procedure over a period of 10 

hours a spectrum of MLV of DPPC was obtained. The experiment with the same sample

90 Modelling 

at 45 ◦ C, however, was carried out without a hitch. The fitted structure factor is plotted 

with the raw data in figure 5.10 

30 

Multilamellar vesicles 

45 °C 

25 

Raw data 

Structure factor 


20 

15 

10 

5 

0 

0 0.1 0.2 0.3 0.4 

q [Å -1 ] 

Figure 5.10 The raw data from a SAXS experiment on MLV of DPPC. The distance between 

the two peaks is ∆q = (0.096 ± 0.0005)Å −1 , corresponding to a distance of d = 2π =65.5 Å. 

∆q 

This distance is equivalent to a lamellar repeat distance [53]. 

The fitted structure factor in figure 5.10 is a sum of two Gaussians 

Matlab Code 

1 

☞ page 174 S(q) = √ 

(exp 

(− (q − q 0) 2 ) 

+ Aexp 

(− (q − 2 · q 0) 2 )) 

2πσ 

2 

b 

2σ 2 b 

2σ 2 b 

(5.30) 

where A = 0.3, q 0 = 0.096Å −1 and σ b = 0.0159Å. 

There is a danger of modelling the structure factor as an intensity and not an amplitude. 

The intensity is by definition a positive quantity whereas the amplitude can be negative 

as well as positive. Modelling the intensity does not take into account the possible negative 

amplitudes of structure factor and hence their contribution to the total scattering 

intensity. 

5.8 Crystalline structure 

None of the models could capture the feature of the small peaks at low q-values, see 

table 4.4. The peaks are possibly Bragg reflections (which we unfortunately realized 

after the modelling) and we have tried to match them with the Bragg peak of hexagonal 

structures, since this have been observed in perturbated lipid systems as well as high 

temperature (above 70-80 ◦ C, the H II phase) lipid systems before - the existence of the 

H II phase is well documented [20], [36], [61]. 

In section 2.3 page 27 the relation between q-values and the location of the peak in 

a hexagonal lattice structure is described. The relation between the q-values in the 

hexagonal structure does not apply conclusively to the location of the peaks in the spectra 

obtained in this work. Only if the first peak q ′ was located at 0.053 Å −1 instead of around 

0.042 Å −1 the next two peaks actually fits q ′′ = √ 3q ′ and q ′′′ = √ 4q ′ , reasonably. But the 

difference is too great 0.053Å−1 −0.042Å −1 

≈ 26% to be within experimental uncertainty. 

0.042Å −1 

Then one or more of the peaks could refer to scattering from the hexagonal rods (l ≠ 0)

5.8 Crystalline structure 91 

Figure 5.11 The point scatterer intensity and smeared intensity for the symmetric bilayer 

electron density profile model (sym-mlv) with multilamellar correction. The intensity was 

calculated in equation 5.27 and 5.5 using the parameters A = 0.5, z h = 17 Å, σ h = 4 Å, 

σ t = 6 Å, A b = 0.05 and q 0 = 0.1Å −1 . The smeared intensity was calculated using equation 5.6 


with a characteristic length. Another explanation of one of the peaks could be the 

presence of multilamellar vesicles. As seen in figure 5.9 and 5.10 the presence of MLVs 

gives rise to a distinct peak around q = 0.1Å −1 . And, furthermore, the distances could 

be larger than what we can observe - with a peak hidden by the beam stop. Nevertheless, 

none of these explanations could account for the peaks to indicate a hexagonal lattice 

structure. Actually, the small peaks at the spectra between q = 0.1Å −1 and q = 0.2Å −1 

could be indicative of a more complicated structure than the presence of the hexagonal 

lattice. Similar peaks are found in Katsaras & Gutberlet [32, page 258] as an example 

of the presence of the complex cubic phase. 

Retrospectively, we should have tried to separate the smooth envelope curve and the 

peaks and model them individually. If the models were developed so they could account 

for the peaks and the envelope curve individually then the models used would probably fit 

a lot better. It would also assess the value of the bilayer thickness with less uncertainty. 

Developing a model which could fit the peaks would partly give information about the 

underlying structure giving rise to the peaks, partly indicate how much of the total 

structure were in fact vesicles and how much were crystalline structure.

92 Modelling 

Form factor 

Intensity from point scatterer 

Intensity (a.u.) 

Electron density (a.u.) 

-40 -30 -20 -10 0 10 20 30 40 

z / Å 

0.1 0.2 0.3 0.4 0.5 

q / Å -1 

Figure 5.12 The point scatterer intensity and smeared intensity for the spherical symmetric 

bilayer electron density profile model (3d-mlv) with multilamellar correction. The intensity 

was calculated in equation 5.27 and 5.5 using the parameters A = 0.5, z h = 17 Å, σ h = 4 Å, 

σ t = 6 Å, A b = 0.02 and q 0 = 0.1Å −1 . The smeared intensity was calculated using equation 5.6 


5.9 Results and summary 

The quality of the fits is evaluated by a function, sse (sum of squares due to error) based 

on the least-squares method, which minimizes the sum of the squares of the divergence: 

sse(x) = ∑ i 

[I measured (q i ) − I model (x, q i )] 2 (5.31) 

Matlab Code 

☞ page 177 

Here (q i , I measured (q i )) is the i’th point of measurement and I model (x, q i ) is the intensity 

calculated by the model, which is a function of q i and the fitted parameters, x. The 

value of sse determines the quality of the fit (lower the value → better the fit). Please 

note, that both I measured and I model are normalized before the subtraction such that 

∫ q2 

I(q)dq = 1 (5.32) 

q 1 

where q 1 and q 2 are the lower and upper limit on our SAXS spectra. 

We have used an altered version 5 of the fitting function in the MATLAB add-on package, 

nlinfit, which uses a Gauss-Newton algorithm to minimize the sse function. The 

method is an effective way to solve non-linear least-squares problems. It calculates the 

gradient from partial derivatives and moves towards minimum. However in using this 

method you might end up in one of the, possibly many, local minima and not the global 

minimum. 

5 We have modified the script in order to fit our needs.

5.9 Results and summary 93 

We do not only use the sse as criterion for a good model fit. We require a reasonable 

correlation between the parameters as well. Combining this with simply looking at a 

plot of the modelled curve with the raw data determined whether the fitting procedure 

should be repeated. Besides looking at the model fitted with the raw data and the sse 

value the parameters from the model should also be realistic. Some constraints were 

embodied in the fitting procedure making sure e.g. that the calculated amount of MLVs 

in the sample was not negative. Most of the evaluations of the model parameters were 

based on subjective judgements. The values for the bilayer thickness, 2z h , the deviation 

of the head group, σ h , and tail, σ t , as well as z t (sym-4g model) should all be positive. 

z h should be between 5 Å and 30 Å, the deviations between 1 Å and 15 Å and z t (sym-4g 

model) should be between 0 and 5 Å. It is important to stress that these ranges are all 

based on either experience or literature values. 

Generally a model fit can be discarded if there is an unrealistic correlation between the 

model parameters. First of all the deviations of the tail and head regions should not 

differ significantly from each other, and the values for the deviations should be lower 

than the values for z h . If the deviations are higher than z h then the electron density 

profile would have much broader peaks, and the peak (z h ) would be determined with 

high uncertainty. 

The result of a good fit selection process (to narrow our analysis down) is listed in table 

5.2. Please refer to appendix H (p. 135) for a complete list of model graphs and fitted 

parameters. One of the good model fits can be found in figure 5.13. This fit shows one 

of the best fits to the sym-mlv model with an sse of only 8.6. 

Figure 5.13 also demonstrates the great difference in the modelled electron density 

profile compared to the relatively small difference of the fit to the measured data. The 

model fits plotted with the raw spectrum clearly display that none of the models take the 

distinct peaks at low q-values into account. The best fits are from the sym-mlv model 

and 3d-mlv. Even though these two models have almost the same fitting parameters 

the electron density profiles differ. The bilayer thickness directly read from the electron 

density profile is lower for the 3d-mlv model than for the sym-mlv model. 

In appendix H it is important to notice the absurdly large parameter uncertainties especially 

related to the sym-4g model, e.g. for the fit of heptanol, high temperature, series 10 

the sse is 16.9, but the value for z h is 0.2Å±8000Å. A large parameter uncertainty means 

that the model is insensitive to changes of that parameter (in the particular minimum 

found). 6 These parameters have been excluded from the further analysis. 

The bilayer thickness (2z h ) is an average of billions of vesicles and you have to take both 

z h and σ h into account as they interdepend. The criteria for the selection of z h -values 

are sse values under 40, and a reasonably low deviation of the head group Gaussian 

σ h . From the models listed in table 5.2 the z h values are found. The parameter values 

(z h and σ h ) are listed in table 5.3. sym-mlv has the best over-all fit and the parameter 

values (z h and σ h ) are listed i table 5.4. The values in the two tables show that adding 

alcohol to the sample does not change the z h or σ h values much taking the uncertainties 

into account. They also show that most of the fits overlap in their confidence intervals 

between the low temperature and high temperature which means we cannot argue that 

there is a difference in the bilayer thickness for the two phases. 

6 The uncertainties are calculated on the basis of partly the residuals, partly the Jacobian (the Jacobian 

is the partial derivatives of each the parameters (and for each of the data points) with respect to the 

sse).

94 Modelling 


Measured 

Sym model, sse=17.3 

Asym model, sse=18.8 

Sym-mlv model, sse=8.6 

3d model, sse=17.2 

3d-mlv model, sse=8.6 

Sym-4g model, sse=16.9 

0.1 0.2 0.3 0.4 0.5 

q / Å -1 

Electron density / a.u. 

-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 

Figure 5.13 The fit of DPPC with heptanol, high temperature (45 ◦ C), Series 10 

Generally, the models with MLV correction fit the data best (lowest sse), this goes for 

the pure DPPC samples as well as the samples added alcohol. But these models tend 

to have an increased deviation for the tail region, σ t . Another general and interesting 

result is that the high temperature measurements are fitted better than the low temperature 

measurements. This could partially be explained as a consequence of the samples 

containing more multilamellar vesicles in the high temperature phases than in the low 

temperature phases. It seems to be the case and is indicated by the generally larger A b 

values. 

Evaluating the models individually, sym-mlv had the best fits overall. The 3d-mlv model 

fitted the pentanol and hexanol data best, and the sym-4g model fitted hexanol data 

best. The fact that the one-dimensional model with correction for the presence of MLVs, 

sym-mlv, overall fits the data best can indicate that most of the system do consist of 

symmetric lipid bilayers. It also indicates that you need to take the presence of MLVs 

into account and that all the MLVs have not been removed during the sample preparation


Low temperature High temperature 

Pure sym (20) 

sym-mlv (10.2) sym-mlv (4.8) 

3d (20.3) 

3d-mlv (10.3) 3d-mlv (4.8) 

sym-4g (20.1) 

Pentanol sym (13) 

asym (12.5) 

sym-mlv (10.4/15.3) 

3d (12.8) 

3d-mlv (12.8) 3d-mlv (10.4/15.3) 

Hexanol sym (33.5) sym (18.5/15.8) 

asym (39) asym (23.1) 

sym-mlv (13.9/14.8) 

3d (33.8) 3d (18.5/15.8) 

3d-mlv (13.9/14.8) 

sym-4g (33.3) sym-4g (18.5/15.8) 

Heptanol sym (41) 

asym (18.8) 

sym-mlv (9.58) sym-mlv (8.61/18) 

3d (42) 

3d-mlv (15/35.7) 3d-mlv (8.58/18) 

sym-4g (41.6) 

Table 5.2 The models used for interpreting the SAXS data. The sse values are listed in 

brackets. Low and high temperature refer to 25 ◦ C and 45 ◦ C, respectively. 

Low temperature (25 ◦ C) High temperature (45 ◦ C) 

Alcohol z h / Å σ h / Å z h / Å σ h / Å 

None 19.2 ± 0.8 6.6 ± 0.5 (5) 18.8 ± 0.2 3.8 ± 2 (2) 

Pentanol 18.2 ± 0.9 6.7 ± 0.7 (3) 16.5 ± 0.4 4.5 ± 0.4 (4) 

Hexanol 18.1 ± 1.3 5.8 ± 0.8 (5) 17.6 ± 1.3 4.2 ± 0.7 (11) 

Heptanol 19.4 ± 0.5 5.3 ± 0.5 (6) 18.1 ± 0.3 3.8 ± 0.3 (5) 

Table 5.3 The averaged values for z h and σ h from the different models. The figures in brackets 

refer to the number of models used for the averaging. 

Low temperature (25 ◦ C) High temperature (45 ◦ C) 

Alcohol z h / Å σ h / Å z h / Å σ h / Å 

None 17.9 ± 0.4 7.7 ± 0.4 (1) 18.8 ± 0.2 3.8 ± 0.2 (1) 

Pentanol 18 ± 5 6 ± 2 (1) 16.5 ± 0.4 4.5 ± 0.4 (2) 

Hexanol 16 ± 2 10 ± 2 (1) 18.0 ± 0.5 4.0 ± 0.4 (2) 

Heptanol 17.1 ± 0.5 8.0 ± 0.5 (1) 18.0 ± 0.4 3.7 ± 0.4 (2) 

Table 5.4 The averaged values for z h and σ h from the sym-mlv model. The figures in brackets 

refer to the number of data sets used for the averaging.

96 Modelling 

procedure. 

Additionally, based on prior experiences with modelling electron density profiles of similar 

systems [59], we found it necessary to extend the models to take into account the presence 

of MLVs. The sym-mlv and 3d-mlv are two models which overall fit the best, this stresses 

the importance of embedding the structure factor in the modelling. 

For the asym model the parameters A 1 , A 2 and z h are often stated with great uncertainty. 

For the spherical symmetric model the parameter σ t is generally stated with large 

uncertainties. When extending the models to correct for the presence of MLVs the fitted 

parameters z h , σ h and σ t from the “non”-extendend model are often changing upon the 

new fitting procedure. Still the parameters from fitting the models sym-mlv and 3d-mlv 

are frequently very similar. Also for the asym model it is noted that mostly there are no 

difference in the values of the two amplitudes, and this indicates that the lipid bilayer is 

relatively symmetrical. The fitted values from the asym model are overall fairly similar 

to those obtained with the sym model. 

The asymmetric model asym has been used for similar systems previously [9], [59] with 

good results. This model accounts for possible asymmetry in the bilayer, a feature one 

could imagine being more pronounced as the amount of partitioned alcohol increased. 

However, the modelling results did far from convincingly detect the presence of any 

asymmetry within the lipid bilayer. 

This alcohol induced interdigitated phase of DPPC has not been modelled previously, 

but based on electron density profiles in the literature, [60], [80], describing the ethanol 

induced interdigitated phase we have developed the sym-4g model. The results from the 

sym-4g model did not support the hypothesis that the spectra obtained at low temperature 

were interdigitated, at least not if taking the electron profiles presented by [60], [80] 

as being valid. 

Regarding the sensitivity of the parameters we have not conducted a sensitivity analysis 

per se. Still from the uncertainties of the parameters given in the fitting process some 

conclusions regarding the sensitivity of the parameters can be made. The modelling 

results from especially the sym-4g model and the asym state values of the different fitting 

parameters with absurdly large uncertainties. These uncertainties indicate that the 

overall fitting of the spectrum and the sse value is not much affected by changing this 

specific parameter. Mainly the large uncertainties centre on the amplitudes (asym) and 

the value for z h and z t (sym-4g). 

The comparable values found in the literature mostly relate to the bilayer thickness, 

which for pure DPPC depends on temperature - the thickness decreases with increasing 

temperature. Within the temperature range used in this work, the value for 2z h should 

change from 44.2 Å (20 ◦ C) to 38.3 Å (50 ◦ C) [53]. In general the thickness of a lipid bilayer 

decreases as the concentration of alcohol is increased [47], [59]. If the interdigitated phase 

is obtained the bilayer thickness is decreased significantly from 2z h = 42 Å to 2z h = 30 Å 

(DPPC and ethanol, [68]). The bilayer thickness of the interdigitated phase of DPPC 

depends on the alcohol used, the longer the chain, the larger z h , [1]. This can be explained 

by referring to the fact that the thickness of an interdigitated bilayer is the sum of the 

the lipid chains and the alcohol chains, hence when lengthening the alcohol, the sum 

of alcohol-chain and lipid chain will increase. By increasing the used alcohol with a 

methylgroup the fully interdigitated lipid bilayer thickness would increase by about 1Å 

[1].


2z h Lipid phase Inducer Reference 

42/30 Å L β ′/L βI Ethanol Simon & McIntosh [68] 

62/49 Å L β ′/L βI Ethanol Vierl et al. [80] 

63/49 Å L β ′/L βI Ethanol Nambi et al. [55] 

43/30 Å L β ′/L βI Ethanol Simon et al. [69] 

49/41 Å L β ′/L βI Ethanol Adachi et al. [1] 

45/19 Å L β ′/L βI Ethanol Mou et al. [52] 

30 Å L βI Chlorpromazine McIntosh et al. [50] 

30 Å L βI not mentioned Veiro et al. [79] 

44.2Å L β ′ None Nagle & Tristram-Nagle [53] 

38.8 Å L α None Nagle & Tristram-Nagle [53] 

Table 5.5 Chosen z h from the literature, DPPC and ethanol. 

The bilayer thicknesses in table 5.5 are varying from 42-63 Å in the L β ′ and from 

19-49 Å in the L βI . It should be noted that all the SAXS spectra have been obtained 

with DPPC and ethanol as an inducer. Therefore the bilayer thicknesses are not directly 

comparable to the modelled thicknesses in this work. The values from the literature reveal 

great variation in the obtained values for the distance between the two head groups, still 

there is a trend to measuring lower values for the lipid bilayer thickness in the L βI phase. 

There is an overlap of 7 Å between values for the L βI phase and the L β ′ phase, within 

this range it is impossible to determine whether the lipid is in one or the other phase. 

The modelled values for the bilayer thickness obtained in this work are just below this 

overlapping range, both for the pure and perturbated vesicles.

6 Discussion 

Are the vesicles alcohol saturated? 

The ITC experiments give us ambiguous results and hence no clear-cut determination of 

whether the vesicles are alcohol saturated or not. The alcohol actually getting into the 

bilayer, however, is supported by the SAXS results which indicate the greater part of the 

system still has vesicle structure. Thus, we believe the heat exchange does dominantly 

stem from alcohol molecules moving into and out of the bilayer. And, if the temporal 

change of the partition coefficients can be explained by water vapourization the ITC 

experiments show that the vesicles are saturated after 24 hours - and still are after two 

days. However, the evidence is not that compelling as we do not know anything about 

the spooky sub-system. 

Are the vesicles interdigitated? 

The DSC scans have three indications of the lipid bilayers being interdigitated: the main 

phase transition temperature decreases with increasing alcohol concentration, the peak 

broadens (and has a higher enthalpy) and the small peak indicating the pretransition is 

not detected. Thus the DSC results lead us to believe that we have reached the L α -L βI 

phase transition. The SAXS spectra show a small but apparently unmistakable difference 

between the low and high temperature measurements. Still, the modelling of the SAXS 

spectra did not allow us to conclude that the bilayer thicknesses had changed notably. 

The modelling results show that adding alcohol do influence the structure of the lipid 

vesicles. This is primarily reflected in the difference in the obtained spectra from the pure 

DPPC suspension and the suspension with DPPC and alcohol. From the modelling of 

the lipid bilayer thickness no clear-cut conclusions could be made regarding the specific 

lipid phase. Still, the results of the modelling, i.e. the electron density profile as well 

as the values for z h , show that adding alcohol indeed do have an impact on the lipid 

membrane. We cannot rule out the possibility of domain interdigitation. The generally 

low z h values actually support this hypothesis. An average of interdigitated and noninterdigitated 

vesicles could possibly produce the SAXS spectra we see. The modelling 

results reflect an average over all the vesicles in the suspension, and one could argue that 

a part of the vesicles was interdigitated and another part was not fully interdigitated. In 

the modelling we did not take the different bilayer thicknesses into account, still from the 

modelled electron density profiles, appendix H, it looks like there is a slight distribution 

of the lipid head group - head group length, as the spread (σ h ) is quite large. This 

distribution could cover up the presence of two different lipid bilayer thicknesses. 

Another issue in the discussion of whether we have obtained the L βI phase concerns 

whether we have added enough alcohol to the lipid suspension. The total molar lipid 

to alcohol ratio was 1:2 for all the alcohols used. According to the values found in the 

literature, and found by ITC, this should be enough to induce the interdigitated phase. 

99

100 Discussion 

As we have not found examples of experiments with this specific lipid concentration 

in the literature (most experiments are performed with lower lipid concentration) we 

needed to estimate the total molar lipid to alcohol ratio in this work. We chose the same 

total molar lipid to alcohol ratio for all the alcohols even though we knew the partition 

coefficients are higher for longer n-alcohol. 

The enthalpy of transition calculated from the DSC results is generally higher (around 

20%) than the literature values. This could indicate that the transition enthalpy might 

cover more than just the transition from the L βI phase to the L α phase. As mentioned 

earlier the SAXS spectra display peaks which could be interpreted as the presence of a 

more complex structure than the L βI phase and the L α phase. A transition within this 

structure could contribute to the phase transition enthalpy. As we have not determined 

the structure for the possibly crystalline aggregates, we cannot comment on the size of 

the contribution to the phase transition enthalpy. 

Where is the alcohol? 

By letting the system settle for 48 hours at 10 ◦ C above the main phase transition temperature 

before beginning the experiments we expect the alcohol to be evenly distributed 

throughout the sample during measurements. And we do certainly not expect a situation 

with completely alcohol-free vesicles and alcohol-saturated vesicles side by side. However, 

we do not have any evidence to support these assumptions, so how can we tell where 

the alcohol is? The modelling of the SAXS data should give us a hint, but none of the 

models distinguished themselves in fitting the L βI phase or the L α phase better. Still, 

from the experiments in this work the only indicator of where the alcohols are located 

in the membrane is the modelled electron density profiles. By comparing the electron 

density profile of the pure lipid bilayer to the profile of the perturbated lipid bilayer one 

can see that the spread of the head group region is lower in the modelled profile of the 

pure lipid bilayer than in the perturbed lipid bilayer. This shows that the location of 

the head groups is more well-defined for the pure lipid bilayer. Adding alcohol might not 

result in a distinct decrease in the lipid bilayer thickness (insofar as our modelling) but it 

definitely has an effect on the location of the phospholipid head group. This is reflected 

in the broadening of the spread of the Gaussians representing the lipid head groups. The 

broadening of the head group could also be interpreted as a sign of the location of the 

alcohol, i.e. the alcohol could be placed with the hydrophobic tail along the fatty acid 

chains of the phospholipid and the hydrophilic head group could be located next to the 

much larger phosphatidylcholine group. The acyl chain of the alcohols is fairly long and 

it would be energetically favourable for the alcohol to place its tail down the lipid membrane. 

For lipid membranes perturbed by a lower concentration of alcohol, this location 

has been demonstrated using MD-simulations [59]. 

For the three n-alcohols used there are three different solubilities in water, decreasing 

values with increasing acyl chain length. When the concentration of alcohol in the 

lipid suspension approaches this concentration, the risk of unstabilizing the lipid-alcohol 

system increases. For both hexanol and heptanol this limit is formally crossed, still more 

alcohol can be dissolved in a lipid suspension than in pure water as the alcohol partitions 

into the membrane and in doing so it does not contribute to the bulk concentration of 

alcohol. If a larger amount of alcohol is added (than we have used) the alcohol would 

probably aggregate and phase separate from the lipid suspension but there is a risk that 

the lipid molecules of the vesicles simply will be dissolved in a solution containing alcohol 

and water. This is not the case in this work as the SAXS spectra show signs of distinct

structures within the lipid suspension. Even so, by working with such a large alcohol 

concentration the possibility increases that the lipid molecules form other structures than 

bilayers. We could not have avoided this risk as the lipid concentration had to be at least 

5 % in order to get a reasonable signal to noise ratio in the SAXS experiments, and we 

had to add enough alcohol to be certain that the interdigitated phase could be obtained. 

101

102

7 Conclusion 

With respect to interdigitation our results are inconclusive, we can neither reject nor 

confirm for any of the alcohols that our low temperature measurements are in fact in 

the L βI phase. According to the literature and our ITC measurements we should have 

enough alcohol to induce the phase, but the results from the modelling on the SAXS 

data do not allow such a clear-cut conclusion. 

The system behaved more complex than initially expected. The DSC measurements 

indicate a stable system with a well-defined phase transition, but the ITC and the SAXS 

measurements did not. It is still unclear as to why the samples are very sensitive to the 

thermal history after adding the alcohol. But using our sample preparation method, we 

do get systems which give reproducible SAXS-spectra. The modelling of these spectra 

did not completely account for all the features seen in the spectra. They indicate that a 

large fraction of the system still has vesicle structure and a smaller fraction of the system 

form larger crystalline aggregates, whose structures are yet unknown to us. 

Numerical results and trends 

The transition temperature was found to decrease as alcohol was added to the lipid 

suspension. The phase transition temperature decreased from (41 ± 1) ◦ C (pure DPPC) 

to (34 ±1) ◦ C for DPPC and pentanol, (29 ±1) ◦ C for DPPC and hexanol and (35 ±1) ◦ C 

for DPPC and heptanol. The threshold ratio for inducing the low temperature phase was 

found to be lower than the ratios stated in the literature. The SAXS spectra modelling 

analysis revealed nearly the same values for the lipid bilayer thicknesses for all of the 

measurements in the order of 34-38 Å. Adding alcohol mainly seemed to decrease the 

bilayer thickness slightly independent of the temperature. And for all alcohols there are 

slight tendencies towards a lower bilayer thickness as the temperature is increased. 

Perspectives 

There are several threads to pick up after this project. More SAXS experiments would 

either confirm or reject the idea of a crystalline sub-system. If it could be confirmed, one 

could look for an equilibrium in which a larger part or the whole system is a crystalline 

structure to see if it is an independent phase. Even in the case of a negative confirmation, 

it would be interesting to perform light scattering experiments on the system in order to 

find the aggregate size distribution. 

Neutron scattering (supported by simulations, e.g. Molecular dynamics) on the system 

using deuteriated alcohol might reveal the preferred location of the alcohol within the 

membrane. 

103

104

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A 

The course of events 

The work with this thesis started in early January 2006. Inspired by the work of Ulf 

Rørbæk Pedersen ([59]) we decided to work with long chain alcohols (5-7 carbon atoms) 

and their influence on phospholipids with a chainlength of 14-18. The prime experimental 

method should be Small-Angle X-ray Scattering (SAXS) as we had the Kratky camera 

at our disposal at the university. Over the following weeks we were taught how to do the 

sample preparation and did some pilot experiments with DMPC in order to reproduce 

some of the experiment in [59]. After a thorough review of a lot of the literature in 

this field of research and a number of discussions on possible subjects, we decided to 

investigate the alcohol induced interdigitated phase of DPPC. In the literature there were 

examples of pentanol and hexanol as inducers, whereas heptanol never had been reported 

as being an inducer. SAXS experiments could be used in the structural investigation 

of both the interdigitated phase and the liquid crystalline phase of DPPC. These two 

phases differ most significantly in the thickness of the bilayer, reflected in the electron 

density profile through the bilayer, which can be measured by SAXS. By choosing the 

three alcohols, pentanol, hexanol and heptanol, we had the possibility of investigating 

the chain-dependence of the bilayer thickness, rechecking the results obtained in the 

literature with pentanol and hexanol as inducers for the interdigitated phase as well as 

heptanol as non-inducer. 

The first pilot experiments with DPPC and pentanol, hexanol and heptanol, respectively, 

were conducted in mid-March 2006. We did a lot of pilot experiments with 

different lipid to alcohol ratios while reviewing the literature for references, which could 

serve as a basis for our experiments. Quite early in the working process we decided to 

use DSC in order to locate the phase transition temperature for the transition between 

the interdigitated phase and the liquid crystalline phase. In the beginning of April some 

introductory DSC scans were conducted with the help of Brian Igarashi (The ”Glass and 

Time” group at IMFUFA, University of Roskilde, Denmark) at the instrument belonging 

to the research centre ”Glass and Time”. 

In the beginning of May we began using the scal-1 microcalorimeter. This instrument 

is very sensitive and only very dilute (0.05 w%) solutions could be used. As the solutions 

were so dilute a correspondingly high lipid to alcohol ratio was used in order to make 

sure, that the lipids were interdigitated prior to performing the experiments. These 

experiments convinced us that the samples were stable with regards to temperature 

and that the phase transition temperature was unaffected by cooling and heating the 

sample. We ran a number of DSC scans on lipid solutions containing different amounts 

of alcohol. These scans showed that the phase transition temperature decreased with 

increasing alcohol concentration as predicted in the literature. The results from the 

scal-1 DSC supported our hypothesis. But the measured phase transition temperatures 

were found in a system 100 times more dilute than the solution used for the SAXS 

experiments. For this reason we turned to another, less sensitive DSC instrument, the 

DSC7 in mid-August. 

111

112 The course of events 

In June and July we calibrated the Kratky camera with regards to beam, q-vector and 

temperature. The temperature calibration (performed with a NTC-resistor) was done 

with both silicone oil and air in the cuvette. At first only the calibration with air was 

done, but as silicone oil after all has a more similar consistency to the lipid suspension 

the calibration was also done with this liquid. 

The DSC experiments performed on the DSC7 showed that the phase transition 

temperature of DPPC decreased when adding alcohol to the sample. The results did not 

differ notably from the results obtained with the DSC7. During the following months no 

SAXS experiments were performed without a preceding DSC scan. Besides the location of 

the phase transition temperature the shape of the peak also contributed to our conviction 

that the vesicles below the phase transition temperature were interdigitated. 

The underlying idea behind making use of ITC was that using lipids in the L β ′ and 

adding alcohol while measuring the reaction heat would reveal when the lipid membrane 

was saturated and hence interdigitated. By doing the experiment at a constant temperature 

below the phase transition temperature, the bilayers should turn from the gel 

phase, (L β ′), to the interdigitated phase as alcohol was added to the suspension. Different 

experimental procedures were tested before the solvent null was chosen. The first 

experiments were conducted with the lipid suspension in the reaction cell and alcohol 

dissolved in water was added from the syringe. This experimental method was skipped 

due to several reasons e.g. a too large amount of lipid was used for each experiment 

(more than 1 mL lipid suspension per experiment) and the large contribution to the total 

heat from the heat of dilution of the alcohol. In mid-December 2006 the solvent-null 

method was conducted with DPPC and hexanol, and subsequently the experiments with 

pentanol and heptanol were carried out. 

In the autumn of 2006 the SAXS experiments were done. During this period of 

time we ran into several problems with the samples and experimental equipment. For 

the samples especially the DPPC suspensions with heptanol were unstable in the sense 

that they turned gel-like after being kept at room temperature for only a few hours. 

Some of the samples even turned gel-like at 52 ◦ C. There were no systematism in these 

observations. But every time the one or more of the samples turned into a gel-like liquid 

we discarded the sample series and prepared a new quantity of ULV of DPPC. 

The SAXS spectra of all the different measurements (all three alcohols, low and high 

temperature) were reproduced after a number of experimental series, which mostly were 

discarded by computer-connection problems. When the computer lost the connection 

to the temperature-regulator, it froze and stopped counting the incoming photons, the 

sample was still exposed to radiation though. And as we assigned importance to treating 

the samples the same, we chose not to use spectra from samples, which had been exposed 

to radiation longer than the necessary 2 times 18 hours. 

Before reproducing the spectra we only used the models from [59] to fit to our data. 

We did not want to spend a lot of time working and developing the modelling procedure 

before we were sure the spectra could be reproduced. In the process of obtaining the 

SAXS spectra we repeatedly compared the spectra obtained in the low and high temperature 

region to see if they differed. We expected them to differ as we assumed the spectra 

to reflect two different lipid phases. As the spectra obtained in the low temperature and 

high temperature region indeed differed, we continued the measurements. 

After reproducing the spectra we developed and expanded the models (primarily 

adapted from [59]) to cover the interdigitated phase as well as the liquid fluid phase. 

These models did fit the main features of the raw data spectra but none of the models 

took the small, distinct peaks into account. At first the peaks were considered as noise

on the spectra. Not until all the SAXS spectra had been reproduced it became clear that 

it was in fact not noise but distinct reproducible peaks, which we subsequently tried to 

fit to a hexagonal lattice structure. 

113

114

B 

Theory 

The nonrelativistic quantum mechanical scattering experiment 

The scattering experiment has played an important role in the development of quantum 

theory and is perhaps the most important technique. Rutherford discovered the nucleus 

with α-particles (α+ 14 N → p+ 17 O) and Franck-Hertz discovered the existence of atomic 

energy levels by observation of electrons scattering off mercury vapor. 

The timedependent Schrödinger equation i d |Ψ〉 = H|Ψ〉 has the general solution |Ψ〉 = 

dt 

U(t) |ψ〉 = exp(−iHt) |ψ〉 where U(t) is the evolution operator and H is the Hamiltonian, 

which consists of the Hamiltonian for the free particle, H 0 = p 2 /2m, and timeindependent 

potential, V (x). U 0 (t) ≡ exp(−iH 0 t) 

A long time before and a long time after 1 the collision the particles can be considered 

as free particles. Accordingly, the actual wave function will tend to the free wave function 

asymptotes. 

U(t)|ψ〉 −−−−→ 

t→−∞ U0 (t)|ψ in 〉 (B.1) 

U(t)|ψ〉 −−−−→ 

t→+∞ U0 (t)|ψ out 〉 (B.2) 

Figure B.1 A figure showing a classical interpretation of the Møller operators (figure from 

Taylor [76, p. 30]). 

The actual wave function |ψ〉 is connected to the scattering asymptotes (|ψ in 〉 and 

|ψ out 〉) by the isometric Møller operators (Ω + and Ω − ) 

|ψ〉 = lim 

t→−∞ U(t)† U 0 (t)|ψ in 〉 ≡ Ω + |ψ in 〉 

|ψ〉 = lim 

t→+∞ U(t)† U 0 (t)|ψ out 〉 ≡ Ω − |ψ out 〉 

(B.3) 

(B.4) 

1 On the atomic scale, for the actual scattering process 10 −10 seconds are considered very slow. 

115

116 Theory 

If we define the scattering operator, S ≡ Ω † − Ω +, this gives the relation between the 

asymptotes, since |ψ〉 = Ω + |ψ in 〉 = Ω − |ψ out 〉 

|ψ out 〉 = Ω † − |ψ〉 = Ω† − Ω +|ψ in 〉 = S|ψ in 〉 (B.5) 

All scattering propabilities can be calculated from S, e.g. the propability of scattering 

from the initial (combined) state |ψ in 〉 = |i〉 to the final (combined) state |ψ out 〉 = |f〉 is 

proportional to |〈f|S|i〉| 2 and the differential cross section is, Als-Nielsen & McMorrow 

[3, p. 265] 

( ) 2 ∫ 

dσ V 

dΩ = 1 

2π 4 c 4 |〈f|S|i〉| 2 E f δ(E f − E i )dE f 

(B.6) 

where E f and E i are the energy of final and initial state, respectively. Thus S contains 

all the information of experimental interest - unfortunately, S is not easily calculated.

C 

Fourier transform 

To model the electron density profile of the lipid bilayer from the measured intensity it 

is necessary to Fourier transform the Gaussians representing the three different electron 

density profiles to find the form factors. Basicly the Fourier transform of a given function 

f(x) is defined by [28] 

F(q) = 

∫ ∞ 

and the inverse Fourier transform as 

−∞ 

f(x)exp(iqx)dx, 

(C.1) 

f(x) = 1 ∫ ∞ 

F(q)exp(−iqx)dq. 

2π −∞ 

(C.2) 

The Fourier transform is linear, since if f(x) and g(x) have Fourier transforms F(x) 

and G(x) respectively then 

∫ ∞ 

−∞ 

∫ ∞ 

[c 1 f(x) + c 2 g(x)] exp(iqx)dx = c 1 f(x)exp(iqx)dx 

−∞ 

∫ ∞ 

+c 2 g(x)exp(iqx)dx 

−∞ 

= c 1 F(q) + c 2 G(q). (C.3) 

This is useful as the three electron densities have different form factors which must 

be added. 

The Fourier transform of a shifted function is 

F shift (q) = 

Substituting u = x − x 0 gives 

∫ ∞ 

−∞ 

f(x − x 0 )exp(iqx)dx 

(C.4) 

F shift (q) = 

∫ ∞ 

−∞ 

= exp(iqx 0 ) 

f(u)exp(iq(u + x 0 ))du 

∫ ∞ 

−∞ 

f(u)exp(iqu)du 

(C.5) 

(C.6) 

= exp(iqx 0 )F(q) (C.7) 

i.e. the factor exp(iqx 0 ) is multiplied onto the Fourier transform of the original 

function f(x). This result is commonly known as the Shift theorem. 

The evaluation of equation C.1 is simplified as 

117

118 Fourier transform 

F(q) = 

= 

∫ ∞ 

−∞ 

∫ ∞ 

−∞ 

∫ ∞ 

f(x)cos(qx)dx + i f(x)sin(qx) 

−∞ 

f(x)cos(qx)dx 

(C.8) 

(C.9) 

as the integral of an antisymmetic function on a symmetric interval is zero, and the 

product of the symmetric Gaussian and the antisymmetric sine function is antisymmetric. 

The Gaussian 

g(x) = Aexp(− x2 

σ 2 ) 

(C.10) 

can be Fourier transformed analytically by use of C.1 and C.8, 

G(q) = 

∫ ∞ 

0 

= 2A 

= 2A 

= 2A 

) 

Aexp 

(− x2 

σ 2 cos(xq)dx 

∫ ∞ 

0 

∫ ∞ 

0 

∫ ∞ 

0 

= 2Aexp 

Re 

(exp(− x2 

Re 

(exp(− x2 

( 

Re 

( 

− qσ 2 

( 

= 2Aexp − qσ 2 

( 

= 2Aexp − qσ ) ( 2 

Re σ 

2 

σ 2 )exp(iqx) ) 

dx 

σ 2 + iqx) ) 

dx 

) 2 

) 

exp 

(− x2 

σ 2 − iqσ − ( qσ 2 e )2 dx 

) 2 

∫ ∞ 

) 

Re 

(exp(− x2 

0 σ 2 − iqσ 2 )2 dx 

) 2 

(∫ ∞ 

) 

Re σ exp(−z 2 ) dz 

0 

√ ) π 

2 

(C.11) 

(C.12) 

(C.13) 

(C.14) 

(C.15) 

(C.16) 

(C.17) 

= σA √ ( qσ 

π exp − 

2 

) 2 

, (C.18) 

which is also a Gaussian but with amplitude σA √ π and a width parameter of 2σ −1 . 

The substitution 

z = x σ − iqσ 2 , 

(C.19) 

was made, yielding dx = σdz and the square 

was completed. 

( 

− x2 x 

) 2 ( qσ 

) 2 

σ 2 + iqx = − σ − iqσ − (C.20) 

2 2

119 

Electron density to form factor 

The electron density is defined as a sum of Gaussians 

ρ(z) = ∑ A n 

√ exp 

(− (z − z n) 2 ) 

n 2πσ 

2 n 

2σn 

2 

(C.21) 

The form factor is the Fourier transform of the electron density, using the shift theorem 

equation C.5, 

and the fact that, [72], 

F{ρ(z)} = F{ρ(z − z n )} = F{ρ(z)} exp(iqz n ) 

F { exp(−bx 2 ) } = 1 2 

which in this case corresponds to 

√ ( ) π 

b exp − α2 

4b 

(C.22) 

(C.23) 

F 

{ } 

A 

√ 

2πσ 

2 exp(−(2σ2 ) −1 z 2 ) 

= 1 √ ( 

A π 

√ 

2 2πσ 

2 (2σ 2 ) −1 exp q 2 ) 

− 

4 · (2σ 2 ) −1 

= A 2 exp(−1 2 q2 σ 2 ) 

Yielding, with equation C.3, the form factor 

F{ρ(z)} = ∑ ( 

A n 

2 exp − 1 ) 

2 q2 σn 

2 exp(iqz n ) = F(q) 

n 

(C.24) 

(C.25)

120

D 

Experimental series 

When a new series of experiments was planned, a new quantity of lipid solution was 

prepared. Whenever a new quantity was made, it was numbered consecutively. The last 

three series were only made to reproduce results obtained with previous series. 

Series 7 DPPC with hexanol. It was intended to check whether there was a difference 

in measuring from 25 ◦ C to 45 ◦ C, the so-called up-scan, and from 45 ◦ C to 25 ◦ C, 

the so-called down-scan. After preparing the samples, different subsamples were 

made containing different amounts of hexanol, equivalent to a ratio of 1:2, 1:4, 

1:6 and 1:9, and they were kept in the Eppendorph Thermomixer at 30 ◦ C. The 

DSC scans showed that the phase transition temperature decreased with increasing 

hexanol concentration. The SAXS experiments were performed using the 1:2 

ratio DPPC:hexanol sample. Using a MTC-script made sure that the spectra were 

measured at 25 ◦ , 30 ◦ , 40 ◦ and 45 ◦ , respectively. The selected temperatures were 

chosen given the DSC results. Concentration: 1.62±0.061 w%. 

Series 8 DPPC with pentanol. After preparing the samples, different subsamples were 

made containing different amounts of pentanol, equivalent to a ratio of 1:2, 1:4, 

and they were kept in the Eppendorph Thermomixer at 30 ◦ C. The DSC scans 

showed that the phase transition temperature decreased with increasing hexanol 

concentration. The SAXS experiments were performed using the 1:2 lipid:pentanol 

sample. An MTC-script was made, making sure the spectra was measured at 25 ◦ , 

30 ◦ , 40 ◦ and 45 ◦ , respectively. These temperatures were chosen given the DSC 

results. Concentration: 3.07±0.026 w%. 

Series 9 DPPC with pentanol and hexanol. After preparing the samples, different subsamples 

were made containing different amounts of pentanol and hexanol, equivalent 

to a ratio of 1:2, and they were kept in the Eppendorph Thermomixer at 52 ◦ C. 

The DSC scans showed that the phase transition temperature decreased drastically 

in the samples containing hexanol. The pentanol samples also decreased the 

phase transition temperature. The SAXS experiments were performed on the 1:2 

DPPC:pentanol sample first. An MTC-script was made, making sure the spectra 

was measured at 25 ◦ , 30 ◦ , 40 ◦ and 45 ◦ , respectively. These temperatures were 

chosen given the DSC results. The SAXS experiments were performed on the 1:2 

DPPC:hexanol was measured at 20 ◦ , 25 ◦ , 35 ◦ and 40 ◦ , respectively. These temperatures 

were chosen given the DSC results. Concentration: 4.32±0.063 w%. 

Series 10 DPPC with heptanol. After preparing the samples heptanol was added, equivalent 

to a ratio of 1:2, and they were kept in the Eppendorph Thermomixer at 

52 ◦ C. The DSC scans showed that the phase transition temperature was around 

35 ◦ C. The SAXS experiments were performed at 25 ◦ C, 30 ◦ C, 40 ◦ C and 45 ◦ C. 

The heptanol sample was gellike after the last SAXS experiment. Concentration: 

3.56±0.021 w%. 

Series 11 Pure DPPC. After preparing the multilamellar vesicles several DSC scans were 

performed, and a phase transition temperature of 41 ◦ C was found. SAXS exper- 

121

122 Experimental series 

iments were performed, identifying the repeating structure of the lipid bilayers 

inside the MLV. Concentration: ∼5.12 w% 

Series 12 DPPC with pentanol, hexanol and heptanol. After preparing the samples 

pentanol, hexanol or heptanol was added, equivalent to a ratio of 1:2 (1:3 for 

heptanol), and the samples were kept in the Eppendorph Thermomixer until used. 

The SAXS experiments were only performed on the pentanol and hexanol samples 

as the heptanol sample turned gellike and was partly evaporated, probably due to 

a too high alcohol concentration. The pentanol SAXS experiments were performed 

at 25 ◦ C, 30 ◦ C, 40 ◦ C and 45 ◦ C. The hexanol SAXS experiments were performed at 

25 ◦ C and 35 ◦ C. Concentration: 2.1±0.187 w%. 

Series 13 DPPC with pentanol, hexanol and heptanol. After preparing the samples pentanol, 

hexanol or heptanol was added, equivalent to a ratio of 1:2, and the samples 

were kept in the Eppendorph Thermomixer until used. The SAXS experiments 

were only performed on the pentanol and hexanol samples as the heptanol sample 

turned gellike and was partly evaporated. The pentanol SAXS experiments were 

performed at 25 ◦ C and 45 ◦ C. The hexanol SAXS experiments were performed at 

25 ◦ C and 35 ◦ C. Concentration: 4.59±0.046 w%. 

Series 13b Concentration: 2.22±0.040 w%. Used for ITC. 


pentanol, hexanol or heptanol was added, equivalent to a ratio of 1:2, and the 

samples were kept in the Eppendorph Thermomixer until used. The pentanol SAXS 

experiments were performed at 25 ◦ C and 45 ◦ C. The hexanol SAXS experiments 

were performed at 25 ◦ C and 45 ◦ C. The heptanol SAXS experiments were performed 

at 25 ◦ C and 45 ◦ C. Concentration: 4.29±0.039 w%. 

Series 14b Concentration: 4.77±0.034 w%. Used for ITC. 


pentanol, hexanol or heptanol was added, equivalent to a ratio of 1:2, and the 

samples were kept in the Eppendorph Thermomixer until used. The pentanol SAXS 

experiments were performed at 25 ◦ C and 45 ◦ C. The hexanol SAXS experiments 

were performed at 25 ◦ C and 45 ◦ C. The heptanol SAXS experiments were performed 

at 25 ◦ C and 45 ◦ C. Concentration: 3.90±0.024 w%. 

Series 15b Concentration: 4.52±0.054 w%. Used for ITC introductory experiments. 

Series 16 Concentration: 4.43±0.017 w%. Used for ITC introductory experiments. 

Series 17b Concentration: 3.93±0.038 w%. Kept at 25 ◦ C until alcohol was added. To 

this lipid suspension pentanol and hexanol, respectively was added. The lipidalcohol 

suspension was kept shaken at 52 ◦ C until used for ITC experiments. 


this lipid suspension pentanol and hexanol, respectively was added. The lipidalcohol 



this lipid suspension hexanol and heptanol, respectively was added. The lipidalcohol 


Series 20 Pure DPPC. Made for SAXS experiments on ULV vesicles. Concentration: 

∼4.8 w%.

E 

ITC 

The partition coefficient is defined as: 

K x = X a,b 

X a,w 

(E.1) 

where 

n alc,lipid 

X a,b = 

n alc,lipid + n DPPC 

n alc,free 

X a,w = 

n alc,free + n water 

The used n-alcohols, lipid and the free alcohol concentration calculated from the linear 

n-Alcohol Lipid Free alcohol conc. K x 

1 Pentanol (28.7 mM) DPPC (53.5 mM) 22.5 mM 261 

2 Pentanol (66.6 mM) DPPC (45.5 mM) 53.9 mM 225 

3 Hexanol (10.5 mM) DPPC (45.5 mM) 7.9 mM 367 

4 Hexanol (15.1 mM) DPPC (54.3 mM) 3.7 mM* 2553* 


6 Hexanol (22.6 mM) DPPC (54.3 mM) 11.8 mM /3.9 mM* 749/ 3546* 


8 Heptanol (7.3 mM) DPPC (45.5 mM) 2.5 mM/ 2.9 mM* 2166/1713* 

Table E.1 The first column represents the used alcohol and the related concentration in the 

syringe. The second column represents the lipid concentration, the third column the calculated 

free alcohol concentration and the fourth column the calculated partition coefficient. The 

figures marked by stars are explained in section 4.1. 

correlation in figures E.1 and E.2 and the partition coefficients are shown i table E.1. All 

the experimental series have been conducted using the MSC-ITC instrument, but hexanol 

no. 3, which has been conducted on the VP-ITC instrument. For every plot a measure 

for the standard deviation is made by using the definition for standard deviation: 

S = √ 1 ∑i=n 

(y − ỹ) 

n − 1 

2 (E.2) 

where y is the measured heat, ỹ is the fitted value for the heat and n is the number of 

measurements. 

i=1 

123

124 ITC 

80 

28.7 mM pentanol in 54.3 mM lipid (test 4) 

250 

66.6 mM pentanol in 45.5 mM lipid 

60 

200 


40 

20 

0 

−20 


150 

100 

50 

0 

−50 

−40 

−100 

−60 

18 20 22 24 26 28 30 


−150 

50 52 54 56 58 60 62 


(a) The free alcohol concentration is found 

to be 22.5 mM. y=9.1·x-203.24, S=9.65. 

(b) The free alcohol concentration is found 

to be 53.9 mM. y=30.6·1.6·10 3 , S=13.9. 

2000 

31.3 mM hexanol in 53.5 mM lipid 

150 


1500 

100 


1000 

500 

0 

−500 


50 

0 

−50 

−100 

−1000 

−150 

−1500 

10 12 14 16 18 20 


−200 

5 6 7 8 9 10 


(c) The free alcohol concentration is found to 

be 13.8 mM. y=316.2·x-4.4·10 3 , S=31.3. 

(d) The free alcohol concentration is found 

to be 7.9 mM. y=57.5 ·x-456.6, S=4.7. 

Figure E.1 Results from two pentanol experimental series and two hexanol experimental series. 

The graphs represent the integrated heat contributions as functions of the concentration of 

alcohol in the reaction cell. The free alcohol concentration is equivalent to the intersection of 

the graph with the axis of abscisses. Each point is equivalent to an experimental series.

125 

1000 


300 

21.6 mM hexanol in 54.3 mM lipid (test 7) 

800 

200 

600 


400 

200 

0 

−200 


100 

0 

−100 

−400 

−200 

−600 

0 2 4 6 8 10 


−300 

10 11 12 13 14 15 16 17 


(a) The free alcohol concentration is found 

to be 3.7 mM. y=174.2·x-544.9, S=84.4. 

(b) The free alcohol concentration is found 

to be 13.5 mM. y=75.9·x-1.0·10 3 , S=24.3. 

2000 


400 

7.3 mM heptanol in 45.5 mM lipid (test 1) 

1500 

300 


1000 

500 

0 

−500 


200 

100 

0 

−100 

−200 

−1000 

−300 

−1500 

0 5 10 15 20 


−400 

1 1.5 2 2.5 3 3.5 4 


(c) The free alc. conc: 11.6 mM(*) and 

3.9 mM (o). y(*)=123.7·x-1.5·10 3 , S=257.1. 

y(o)=262.5·x-1.0·10 3 , S=68.4. 

(d) The free alc. conc: 2.5 mM(*) and 

2.9 mM(o). y(*)=230.8 ·x-566.1, S=26.7. 

y(o)=214.8·x-614.5, S=19.1. 

Figure E.2 Results from three hexanol experiment series and one heptanol series. The graphs 

represent the integrated heat contributions as functions of the concentration of alcohol in the 

reaction cell. The free alcohol concentration is equivalent to the intersection of the graph with 

the axis of abscisses. Each point is equivalent to an experimental series.

126 ITC 

Thermodynamical reflections on the ITC results 

Through the use of the partition coefficient several thermodynamical parameters can be 

calculated, since the free Gibbs energy is defined as 

− ∆G 0 transfer = R · T · ln(K x) 

∆G 0 transfer = ∆H 0 transfer − T∆S 0 transfer 

(E.3) 

(E.4) 

Here R is the universal gas constant, T is the temperature in Kelvin, ∆H 0 transfer is 

calculated for the transfer of alcohol into the vesicle. The ∆Htransfer 0 was calculated from 

the slope of the solvent-null curve, see figure 4.1, and the amount of alcohol transferred 

to the lipid-vesicle was obtained from equation 4.2 1 . ∆S 0 transfer is the entropy for the 

transfer. The thermodynamic parameters are listed in table E.2. 

n-Alcohol K x ∆G 0 transfer ∆H 0 transfer ∆S 0 transfer 

(kJ/mol alc.) (kJ/mol alc.) (J/(mol · K) alc.) 

1 Pentanol 261 -13.8 0.32 47.5 

2 Pentanol 225 -13.4 0.46 46.6 

3 Hexanol 367 -14.7 4.3 63.7 

4 Hexanol 2553* -19.5* 3.0* 75.2* 

5 Hexanol 524 -15.5 2.1 59.3 

6 Hexanol 749/ 3546* -16.4/-20.27* 2.7/3.25* 64.2/78.9* 

7 Hexanol 986 -17.1 3.44 68.9 

8 Heptanol 2166/1713* -19.0/-18.5* 9.1/9.28* 94.47/93.0* 

Table E.2 Partition coefficients and thermodynamic parameters calculated from solvent-null 

experiments. 

As an alcohol is mixed with a lipid membrane the configurational entropy increases 

resulting in a stabilization with regard to an energetic standpoint. This is due to an additional 

entropic term, the configurational entropy, which is competing with the thermal 

entropy in changing the free energy of the lipid membrane [29]. Neither the configurational 

nor the thermal entropy can be measured directly. The former is deduced from 

the partitioning, the latter from the enthalpy change of the equilibrium. A membrane 

equilibrium sensitive to partitioning of alcohols exhibits a large change in the configurational 

entropy and a smaller change in the enthalpy, which is exactly the case in this 

work. 

As the alcohol partitions differently into the the L βI , the P β ′ and the L α phase, the 

free energy, ∆G, will vary accordingly [29]. This is also seen in the results from this work, 

table E.2. Here the free energy for transferring alcohol into the interdigitated phase is 

higher (less negative) than the free energy for transferring alcohol into the ripple phase. 

Another contribution to the increase in entropy is entropy gain of the water from 

the hydration layer and of the acyl chains, as well as the head group disordering due 

to the partitioning of the alcohol molecules [73]. This gain in entropy will continue 

until the vesicles are saturated. This can explain the larger entropy associated with the 

1 ∆H 0 transfer is calculated as ∆H= α·4.186·C lipid·V sample 

n alc,lipid 

. Here α is the slope from the solvent-null curve, 

and 4.186 is the universal gas constant.

hexanol and pentanol samples, which were kept at room temperature for 24 hours prior 

to the experiments. The vesicles in these samples were not saturated when the main 

experiments were conducted. 

Classical hydrophobic interactions are also characterised by positive entropic driven 

forces. And this indicates the importance of the water structure around the non-polar 

moieties involved in the partitioning process. As the acyl chains of the alcohol are moved 

from the aqueous to the lipid environment hydration water is released and this causes a 

growth in entropy. In addition the acyl chains of the alcohol most likely have a higher 

degree of mobility when in the membrane than in water [73]. 

127

128

F 

DSC 

Several scans using the scal-1 microcalorimeter with different scanrates resulted in figure 

F.1. 

Figure F.1 The baseline of the scal-1 microcalorimeter obtained with different scanrates. 

129

130 DSC 

18000 


16000 

scan 1 

scan 11 

14000 

mV 

12000 

10000 

8000 

6000 

10000 

36 38 40 42 44 46 48 50 


DPPC : heptanol 

8000 

scan 1 

scan 9 

6000 

mV 

4000 

2000 

0 

36 38 40 42 44 


Figure F.2 Both figures show 12 successive scans plotted on top of each other. There is 

practically no difference between the 12 scans and the location of the peak does not differ. A 

lipid to pentanol/heptanol ratio of 1:2, 0.05 weight % lipid and a scanrate of 1 K/min was used. 

There is an average standard deviation of 1082.4 on the pentanol/DPPC scan, and an average 

standard deviation of 382.5 on the heptanol/DPPC scan, both relative to the first scan. These 

scans have been obtained by using the scal-1 microcalorimeter.

131 

14 

Enthalpy of the phase transition 

heat flow [mW] 

13.5 

13 

12.5 

12 

11.5 

11 

10.5 

10 

9.5 

100 200 300 400 500 600 700 800 

time [sec] 

Figure F.3 The phase transition enthalpy of pentanol/DPPC, series 09. Lipid to alcohol ratio 

1:2. The blue line is the raw DSC data, the red line is the raw DSC data added the uncertainty 

calculated and represented in figure 4.2. For this particular scan the calculated enthalpy is 

64.5 ± 11 kJ/mol. The vertical lines represent the interval of the phase transition.

132

G 

SAXS 

Two different experimental series with DPPC and hexanol were conducted to show that 

there were no difference between starting the measurements in the low temperature area 

and the high temperature area. 


Serie 7 


8 

6 

4 

25 °C (downscan) 

25 °C (upscan) 



2 

0 

0 0.1 0.2 0.3 0.4 0.5 

q [Å -1 ] 

Figure G.1 Low temperature. 

133

134 SAXS 

8 


Serie 7 

6 






4 

2 

0 

0 0.1 0.2 0.3 0.4 0.5 0.6 

q [Å -1 ] 

Figure G.2 High temperature.

H 

Modelling results 

The electron density profile graphs have been scaled such that every graph has the same 

amplitude for the inner head group Gauss function. 

135

136 Modelling results 

H.1 Pure DPPC 


Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 

Figure H.1 Low temperature (25 ◦ C), Series 20

H.1 Pure DPPC 137 


Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 

Figure H.2 High temperature (45 ◦ C), Series 20


series 20 series 20 

sym 

sse 20.2 13 

A 0.88 ± 0.07 0.5 ± 1 

z h / Å 20.1 ± 0.9 10 ± 60 

σ h / Å 5.6 ± 0.5 7 ± 20 

σ t / Å 6 ± 2 9 ± 200 

asym 

sse 20.4 25.9 

A1 0.6 ± 4 0.5 ± 30 

A2 0.6 ± 4 0.5 ± 30 

z h / Å 17.5 ± 0.6 20 ± 40 

σ h / Å 7.4 ± 0.7 7 ± 10 

σ t / Å 14 ± 1 9 ± 100 

sym-mlv 

sse 10.2 4.78 

A 0.6 ± 0.01 0.589 ± 0.004 

z h / Å 17.9 ± 0.4 18.8 ± 0.2 

σ h / Å 7.7 ± 0.4 3.8 ± 0.2 

σ t / Å 17.6 ± 0.2 5.1 ± 0.3 

A b 0.019 ± 0.001 0.0156 ± 0.0005 

q 0 / Å −1 0.0928 ± 0.0006 0.0936 ± 0.0004 

3d 

sse 20.3 13 

A 0.9 ± 0.1 0.7 ± 10 

z h / Å 20.1 ± 0.9 10 ± 60 

σ h / Å 5.6 ± 0.5 7 ± 20 

σ t / Å 6 ± 2 9 ± 200 

3d-mlv 

sse 10.3 4.76 

A 1.14 ± 0.08 0.79 ± 0.03 

z h / Å 17.9 ± 0.6 18.8 ± 0.2 

σ h / Å 8.4 ± 0.6 3.8 ± 0.2 

σ t / Å 21.7 ± 0.5 5.1 ± 0.3 

A b 0.035 ± 0.001 0.0156 ± 0.0005 

q 0 / Å −1 0.0921 ± 0.0004 0.0936 ± 0.0004 

sym-4g 

sse 20.1 12.7 

A 1.8 ± 0.2 1.03 ± 0.01 

z h / Å 20 ± 1 0.2 ± 5000 

σ h / Å 5.5 ± 0.7 10 ± 70 

z t / Å 4 ± 3 5 ± 0.1 

σ t / Å 3 ± 7 6.1 ± 0.6

H.2 Pentanol 139 

H.2 Pentanol 


Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 




Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 

Figure H.4 Low temperature (25 ◦ C), Series 15. Please notice the sym is hidden behind sym-4g.



Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 

Figure H.5 High temperature (45 ◦ C), Series 13.



Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 

Figure H.6 High temperature (45 ◦ C), Series 15.


low temp. 

high temp. 

series 14 series 15 series 13 series 15 

sym 

sse 13 35.1 15 19.5 

A 0.79 ± 0.07 0.55 ± 0.04 0.5 ± 0.002 0.5 ± 0.1 

z h / Å 18 ± 1 13 ± 2 15 ± 6 15 ± 10 

σ h / Å 6.9 ± 0.6 9 ± 1 6 ± 2 6 ± 4 

σ t / Å 6 ± 2 13.7 ± 0.3 8 ± 20 9 ± 30 

asym 

sse 12.5 34.5 15.1 23.8 

A1 0.74 ± 0.04 0.51 ± 0.06 0.5 ± 100 0.5 ± 30 

A2 0.9 ± 0.06 0.5 ± 0.2 0.5 ± 100 0.5 ± 30 

z h / Å 18.6 ± 0.9 10 ± 20 15 ± 9 17 ± 2 

σ h / Å 6.4 ± 0.5 10 ± 6 6 ± 4 4.5 ± 0.8 

σ t / Å 5 ± 1 14 ± 4 8 ± 30 6 ± 3 

sym-mlv 

sse 12.4 35.2 10.4 15.3 

A 0.5 ± 1 0.8 ± 0.3 0.504 ± 0.003 0.532 ± 0.01 

z h / Å 8 ± 300 18 ± 5 17.3 ± 0.4 15.6 ± 0.3 

σ h / Å 10 ± 60 6 ± 2 3.7 ± 0.3 5.3 ± 0.5 

σ t / Å 10 ± 50 6 ± 6 4.8 ± 0.7 10 ± 2 

A b 0.004 ± 0.001 0.0011 ± 0.0007 0.0129 ± 0.0009 0.013 ± 0.001 

q 0 / Å −1 0.077 ± 0.002 0.122 ± 0.01 0.0854 ± 0.0007 0.085 ± 0.001 

3d 

sse 12.8 48.8 15 19.5 

A 0.67 ± 0.06 0.8 ± 0.4 0.7 ± 1 0.8 ± 2 

z h / Å 18 ± 1 13 ± 7 15 ± 6 15 ± 9 

σ h / Å 6.9 ± 0.6 10 ± 4 6 ± 2 6 ± 4 

σ t / Å 6 ± 2 17 ± 3 8 ± 20 9 ± 30 

3d-mlv 

sse 12.8 35.1 10.4 15.3 

A 0.67 ± 0.07 0.7 ± 0.3 0.66 ± 0.05 1.03 ± 0.08 

z h / Å 17 ± 3 18 ± 4 17.3 ± 0.4 15.6 ± 0.3 

σ h / Å 7 ± 1 6 ± 2 3.7 ± 0.3 5.3 ± 0.5 

σ t / Å 6 ± 3 6 ± 6 4.8 ± 0.7 10 ± 2 

A b 0.0008 ± 0.0005 0.0012 ± 0.0007 0.0129 ± 0.0009 0.013 ± 0.001 

q 0 / Å −1 0.137 ± 0.007 0.122 ± 0.009 0.0855 ± 0.0007 0.085 ± 0.001 

sym-4g 

sse 12.8 35.1 16.7 19.4 

A 1 ± 3 1.1 ± 0.2 1 ± 7e − 05 1.009 ± 0.008 

z h / Å 7 ± 500 13 ± 4 1 ± 50 0.2 ± 10000 

σ h / Å 10 ± 90 9 ± 3 10 ± 3 9 ± 300 

z t / Å 0.8 ± 3000 0.5 ± 30000 0.05 ± 2 4.7 ± 0.5 

σ t / Å 10 ± 100 10 ± 1000 10 ± 3 6 ± 0.7


H.3 Hexanol 


Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 


H.3 Hexanol 145 


Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 




Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 




Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 




sym 

sse 72.5 33.5 18.5 15.8 

A 0.77 ± 0.03 0.76 ± 0.02 0.501 ± 0.003 0.56 ± 0.02 

z h / Å 19.9 ± 0.5 19.2 ± 0.4 16 ± 2 18 ± 2 

σ h / Å 4.3 ± 0.4 4.4 ± 0.4 5 ± 0.9 5 ± 0.9 

σ t / Å 5 ± 1 4.7 ± 0.9 6 ± 3 5 ± 2 

asym 

sse 69.7 39 18.7 23.1 

A1 0.5 ± 200 0.7 ± 20 0.5 ± 30 0.5 ± 20 

A2 0.5 ± 200 0.7 ± 20 0.5 ± 30 0.5 ± 20 

z h / Å 16 ± 1 18.5 ± 0.8 16 ± 4 18.4 ± 0.5 

σ h / Å 8 ± 1 5.9 ± 0.5 5 ± 2 4.4 ± 0.4 

σ t / Å 15.1 ± 0.7 4.7 ± 1 7 ± 5 4.6 ± 0.6 

sym-mlv 

sse 39.6 25.1 13.9 14.8 

A 0.5 ± 0.01 0.51 ± 0.07 0.509 ± 0.003 0.573 ± 0.007 

z h / Å 16 ± 2 10 ± 30 17.6 ± 0.4 18.5 ± 0.5 

σ h / Å 10 ± 2 11 ± 9 3.7 ± 0.4 4.3 ± 0.4 

σ t / Å 19 ± 1 14 ± 8 4.8 ± 0.7 4.8 ± 0.8 

A b 0.016 ± 0.003 0.0032 ± 0.0007 0.0127 ± 0.001 0.0057 ± 0.0009 

q 0 / Å −1 0.089 ± 0.001 0.115 ± 0.003 0.084 ± 0.0009 0.084 ± 0.002 

3d 

sse 73.4 33.8 18.5 15.8 

A 0.9 ± 0.1 0.8 ± 0.09 0.6 ± 0.2 0.6 ± 0.1 

z h / Å 19.9 ± 0.5 19.2 ± 0.4 16 ± 2 18 ± 2 

σ h / Å 4.3 ± 0.4 4.4 ± 0.3 5 ± 0.9 5 ± 0.9 

σ t / Å 5 ± 1 4.7 ± 0.9 6 ± 3 5 ± 2 

3d-mlv 

sse 39.5 25.2 13.9 14.8 

A 1 ± 0.2 0.7 ± 1 0.65 ± 0.06 0.64 ± 0.06 

z h / Å 16 ± 2 10 ± 30 17.6 ± 0.4 18.5 ± 0.5 

σ h / Å 10 ± 2 11 ± 9 3.7 ± 0.4 4.3 ± 0.4 

σ t / Å 19 ± 1 15 ± 8 4.8 ± 0.7 4.8 ± 0.8 

A b 0.016 ± 0.003 0.0033 ± 0.0007 0.0126 ± 0.001 0.0057 ± 0.0009 

q 0 / Å −1 0.089 ± 0.001 0.114 ± 0.003 0.084 ± 0.0009 0.084 ± 0.002 

sym-4g 

sse 71.8 33.3 18.5 15.8 

A 1.57 ± 0.07 1.55 ± 0.07 1.003 ± 0.006 1.13 ± 0.03 

z h / Å 20.2 ± 0.6 19.6 ± 0.7 16 ± 2 18 ± 2 

σ h / Å 4 ± 0.6 4.2 ± 0.6 5 ± 1 5 ± 0.9 

z t / Å 3.5 ± 0.8 3.2 ± 1 0.1 ± 3000 0.09 ± 2000 

σ t / Å 2 ± 4 2 ± 6 6 ± 50 5 ± 40



Measured 

Sym model, sse=212 

Asym model, sse=217 


3d model, sse=212 


Sym-4g model, sse=211 

0.1 0.2 0.3 0.4 0.5 

q / Å -1 

-40 -20 0 20 40 


Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 

Figure H.11 Low temperature (25 ◦ C), Series 9.



Measured 

Sym model, sse=120 



3d model, sse=120 


Sym-4g model, sse=120 

0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 

Figure H.12 Low temperature (25 ◦ C), Series 13.


low temp. 

series 9 series 13 

sym 

sse 212 120 

A 0.65 ± 0.04 0.66 ± 0.06 

z h / Å 10 ± 4 13 ± 3 

σ h / Å 11 ± 2 9 ± 1 

σ t / Å 4.9 ± 0.7 4.9 ± 1 

asym 

sse 217 124 

A1 0.7 ± 20 0.6 ± 50 

A2 0.7 ± 20 0.6 ± 50 

z h / Å 13 ± 3 1 ± 1000 

σ h / Å 9 ± 2 10 ± 100 

σ t / Å 4.5 ± 0.9 7 ± 1 

sym-mlv 

sse 37.1 37.9 

A 0.86 ± 0.05 0.6 ± 0.2 

z h / Å 15.3 ± 0.4 15 ± 2 

σ h / Å 2.2 ± 0.5 4 ± 2 

σ t / Å 5 ± 1 7 ± 9 

A b 0.137 ± 0.004 0.063 ± 0.002 

q 0 / Å −1 0.1174 ± 0.0002 0.1193 ± 0.0003 

3d 

sse 212 120 

A 0.29 ± 0.03 0.37 ± 0.02 

z h / Å 10 ± 4 13 ± 3 

σ h / Å 11 ± 2 9 ± 1 

σ t / Å 4.9 ± 0.7 4.9 ± 1 

3d-mlv 

sse 37.1 38.3 

A 1.9 ± 0.1 1.6 ± 0.06 

z h / Å 15.3 ± 0.4 14.9 ± 0.1 

σ h / Å 2.2 ± 0.5 3.9 ± 0.2 

σ t / Å 5 ± 1 13 ± 0.4 

A b 0.137 ± 0.004 0.076 ± 0.003 

q 0 / Å −1 0.1174 ± 0.0002 0.1208 ± 0.0003 

sym-4g 

sse 211 120 

A 1.3 ± 0.2 1.4 ± 0.3 

z h / Å 11 ± 7 14 ± 6 

σ h / Å 11 ± 3 8 ± 2 

z t / Å 3 ± 2 3 ± 4 

σ t / Å 3 ± 6 3 ± 9


H.4 Heptanol 


Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 


H.4 Heptanol 153 


Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 




Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 


H.4 Heptanol 155 


Measured 







0.1 0.2 0.3 0.4 0.5 

q / Å -1 


-40 -20 0 20 40 

Sym model 

Asym model 

Sym-mlv model 

3d model 

3d-mlv model 

Sym-4g model 

z / Å 



low temp. 

high temp. 


sym 

sse 41.8 41 17.3 35.3 

A 0.6 ± 0.3 0.55 ± 0.01 0.5 ± 0.1 0.5 ± 2 

z h / Å 17 ± 9 17.8 ± 0.5 10 ± 30 10 ± 200 

σ h / Å 7 ± 3 7 ± 0.6 6 ± 10 7 ± 60 

σ t / Å 9 ± 30 15.2 ± 0.7 9 ± 80 9 ± 500 

asym 

sse 72.8 121 18.8 43.6 

A1 0.5 ± 40 0.5 ± 40 0.5 ± 100 0.5 ± 100 

A2 0.5 ± 40 0.5 ± 40 0.5 ± 100 0.5 ± 100 

z h / Å 20 ± 30 15 ± 2 18.6 ± 0.2 10 ± 20 

σ h / Å 7 ± 10 7 ± 3 4.6 ± 0.3 6 ± 6 

σ t / Å 9 ± 80 12 ± 4 3.9 ± 0.3 8 ± 50 

sym-mlv 

sse 9.58 23.3 8.61 18 

A 0.515 ± 0.008 0.50001 ± 1e − 04 0.52 ± 0.003 0.561 ± 0.006 

z h / Å 17.1 ± 0.5 2 ± 8 18 ± 0.3 17.9 ± 0.4 

σ h / Å 8 ± 0.5 13.6 ± 0.6 3.6 ± 0.3 3.7 ± 0.4 

σ t / Å 17.6 ± 0.2 13.7 ± 0.6 4.9 ± 0.5 4.8 ± 0.7 

A b 0.044 ± 0.002 0.033 ± 0.002 0.0169 ± 0.0007 0.026 ± 0.001 

q 0 / Å −1 0.0824 ± 0.0002 0.0796 ± 0.0005 0.088 ± 0.0005 0.0888 ± 0.0005 

3d 

sse 42 42 17.2 35.2 

A 0.8 ± 1 0.82 ± 0.09 0.7 ± 5 0.7 ± 30 

z h / Å 17 ± 9 20.3 ± 0.5 10 ± 30 10 ± 200 

σ h / Å 7 ± 3 4.4 ± 0.5 6 ± 10 7 ± 60 

σ t / Å 9 ± 20 5 ± 1 9 ± 80 9 ± 500 

3d-mlv 

sse 15 35.7 8.58 18 

A 0.91 ± 0.06 0.89 ± 0.09 0.71 ± 0.04 0.73 ± 0.07 

z h / Å 20.1 ± 0.2 20.3 ± 0.3 18 ± 0.3 17.9 ± 0.4 

σ h / Å 3.9 ± 0.2 4.1 ± 0.3 3.6 ± 0.3 3.7 ± 0.4 

σ t / Å 4.9 ± 0.5 5 ± 0.7 4.9 ± 0.5 4.8 ± 0.7 

A b 0.027 ± 0.001 0.011 ± 0.001 0.0169 ± 0.0007 0.026 ± 0.001 

q 0 / Å −1 0.0822 ± 0.0004 0.081 ± 0.001 0.088 ± 0.0005 0.0888 ± 0.0005 

sym-4g 

sse 41.8 41.6 16.9 34.3 

A 1 ± 100 1.39 ± 0.05 1.009 ± 0.004 1.029 ± 0.008 

z h / Å 10 ± 9000 20.6 ± 0.8 0.2 ± 8000 0.2 ± 6000 

σ h / Å 8 ± 1000 4.2 ± 0.8 10 ± 200 10 ± 90 

z t / Å 8 ± 200 4 ± 1 4.9 ± 0.1 4.8 ± 0.1 

σ t / Å 7 ± 1000 3 ± 4 5.6 ± 0.6 5.1 ± 0.6

I 

Matlab source code 

RUCSAXS.M 

100 % Process RUCSAXS data . The f uncti on loads the data f i l e s , s u b t r a c t s 

% background , converts from channel to q values , w r i t e s processed 

% f i l e s ( explained l a t e r ) , d i s p l a y s graphs and returns a [ q , c ] matrix 

% ( q ~ q−v e c t o r magnitude , c ~ photon count ) 

% In order to s u b t r a c t the background , the transmission f a c t o r has to 

105 % be c a l c u l a t e d . The sample and background should coincide at high q−v a l u e s . 

% 

% [ q , c ] = rucsaxs ( fn , sb =1, wf=1, sg =1) 

% 

% fn : filename , ( p a t t e r n s allowed , e . g . fn = ’07 hex2_1 . p ∗ ’) 

110 % i f more than one f i l e matches the pattern , the s p e c t r a are added 

% up to one spectrum , which i s processed . The measuring time i s 

% looked up in the INF−f i l e . I f no INF−f i l e i s found , the user i s 

% prompted f o r measuring time . 

% 

115 % sb : s u b t r a c t background ( the background f i l e i s s p e c i f i e d in another f i l e ) 

% (1=yes , 0=no , d e f a u l t =1) 

% 

% wf : write f i l e s (1=yes , 0=no , d e f a u l t =1), the w r i t t e n f i l e s are : 

% [ fn ] . dat : s p e c t r a in count pr . sec pr . channel 

120 % [ fn ] . norm . dat : normalized s p e c t r a 

% [ fn ] . i n f o . dat : ’ [ n tmt davg ch_interval bmt trans i n t e g r a l e ] ’ 

% Warning : the f i l e s are overwritten , i f they e x i s t ! 

% 

% sg : show graphs (1=yes , 0=no , d e f a u l t =1) 

125 % 

% Please note t h a t you have to s p e c i f y s e v e r a l v a r i a b l e s in the 

% " r u c s a x s _ s e t t i n g s .m’ f i l e , t h i s i n c l u d e s the q−c a l i b r a t i o n data and 

% r e f e r e n c e s to a background sprectrum . 

% 

130 % q_pr_c : q pr . channel 

% q0 : the q v e c t o r magnitude of the d i r e c t beam 

% bfn : background s p e c t r a name 

% 

% Example : [ q , c]= rucsaxs ( ’07 hex2_1 . p ∗ ’ ) ; 

135 % 

f u n c t i o n [ q , cmb ] = rucsaxs ( fn , v arargin ) 

%%%%%%%%%%%%%%%%%%%%%%% 

140 %%% Check i nput %%% 

%%%%%%%%%%%%%%%%%%%%%%% 

157

158 Matlab source code 

% Default return v a l u e s 

q = 0 ; 

145 cmbn = 0 ; 

% Default f uncti on parameters 

sb = 1 ; % yes , s u b t r a c t background 

wf = 1 ; % yes , write output f i l e s 

150 sg = 1 ; % yes , show graphs 

% Number of o p t i o n a l v a r i a b l e s 

nv = length ( v arargin ) ; 

i f nv >= 3 sg = cell2mat ( v arargin ( 3 ) ) ; end 

155 i f nv >= 2 wf = cell2mat ( v arargin ( 2 ) ) ; end 

i f nv >= 1 sb = cell2mat ( v arargin ( 1 ) ) ; end 

%%%%%%%%%%%%%%%%%%%%%%% 

%%% Find f i l e s %%% 

160 %%%%%%%%%%%%%%%%%%%%%%% 

165 

% Get f i l e parts from [ fn ] 

[ path , name , ext , versn ] = f i l e p a r t s ( fn ) ; 

i f isempty ( path ) path = ’ . ’ ; end 

pattern = [ path f i l e s e p name ext ] ; 

r u c s a x s _ f i l e s = d i r ( pattern ) ; 

f c = length ( r u c s a x s _ f i l e s ) ; 

i f ( f c == 0) error ( s p r i n t f ( ’ F i l e ␣ not ␣ found : ␣"%s " ’ , fn ) ) ; end 

170 d a t a f i l e s = [ ] ; 

tmt = 0 ; % Total measuring time 

% Turn on diary ( l o g ) 

i f wf 

175 t_name = name ; 

i f t_name == ’ ∗ ’ t_name = ’ a l l ’ ; end 

dfn = [ path f i l e s e p t_name ext ’ . log ’ ] ; 

i f f c > 1 dfn = [ path f i l e s e p t_name ’ . log ’ ] ; end 

d iary ( dfn ) ; 

180 end % i f 

d isp ( s p r i n t f ( ’%d␣ f i l e ( s ) ␣ found . ’ , f c ) ) ; 

%%%%%%%%%%%%%%%%%%%%%%% 

185 %%% Load s e t t i n g s %%% 

%%%%%%%%%%%%%%%%%%%%%%% 

% Please read the r u c s a x s _ s e t t i n g s .m f i l e f o r information 

% about each of the parameters 

190 [ q_pr_ch , q0 , bfn , bmt , ch_first , ch_last , sbx , dgpfn ] = rucsaxs_settings ; 

ch_interval=f l o o r ( ch_last−c h _ f i r s t +1); 

ch_pr_q=1/q_pr_ch ; 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

159 

195 %%% Read INF f i l e s %%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

f o r i =1: f c 

rucsaxs_fn = [ path f i l e s e p r u c s a x s _ f i l e s ( i ) . name ] ; 

200 d a t a f i l e . filename = rucsaxs_fn ; 

d a t a f i l e . no = 0 ; 

d a t a f i l e . abs_temperature = 0 ; 

d a t a f i l e . delay = 0 ; 

d a t a f i l e . measuring_time = 0 ; 

205 d a t a f i l e s = [ d a t a f i l e s ; d a t a f i l e ] ; 

end 

pattern = [ path f i l e s e p name ’ . INF ’ ] ; 

i n f o _ f i l e s = d i r ( pattern ) ; 

210 count = length ( i n f o _ f i l e s ) ; 

f o r i =1: count 

info_fn = [ path f i l e s e p i n f o _ f i l e s ( i ) . name ] ; 

[ i f p a t h ifname i f e x t i f v e r s n ] = f i l e p a r t s ( info_fn ) ; 

215 % Read the contents of the i nput INF f i l e 

contents = textread ( info_fn , ’%s ’ , ’ whitespace ’ , ’ \n ’ ) ; 

n = length ( contents ) ; 

% Find data 

220 ln = 0 ; % S c r i p t data l i n e 

f o r j =1:n 

s t r = cell2mat ( contents ( j ) ) ; 

% Determine channel count f o r p o s i t i o n and energy spectrum 

225 i f f i n d s t r ( str , ’ [ channels ] ’ ) > 0 & j+2 0 

[T,R] = s t r t o k ( str , ’ : ’ ) ; 

[ ll_energy , c , e , ni ] = s s c a n f ( s t r r e p (R, ’ : ␣ ’ , ’ ’ ) , ’%d ’ ) ; 

end % i f 

i f f i n d s t r ( str , ’PSD1␣Upper␣ Limit : ’ ) > 0 

240 [T,R] = s t r t o k ( str , ’ : ’ ) ; 

[ ul_energy , c , e , ni ] = s s c a n f ( s t r r e p (R, ’ : ␣ ’ , ’ ’ ) , ’%d ’ ) ; 

end % i f 

i f ( f i n d s t r ( str , ’No ’ ) > 0 & f i n d s t r ( str , ’Temp ’ ) > 0 & . . . 

245 f i n d s t r ( str , ’ Hold ’ ) > 0 & f i n d s t r ( str , ’ Real␣Time ’ ) > 0) 

ln = j +1; 

break ;


250 

end % i f 

end % f o r 

% I t e r a t e s c r i p t data l i n e s 

k = ln ; 

while k

161 

305 

[ t_path , t_name , t_ext , versn ] = f i l e p a r t s ( dfn ) ; 

s=load ( [ t_path f i l e s e p t_name s t r r e p ( t_ext , ’P ’ , ’E ’ ) ] ) ; 

e=e+s ( 2 : cc_energy +1); 

end % f o r 

t f c = sum( c ) ; % t o t a l f oton count 

d isp ( s p r i n t f ( ’ Total ␣ foton ␣ count :%14d␣ (%0.1 e ) ’ , tfc , t f c ) ) ; 

q=transpose ( ( 1 : length ( c ))/ ch_pr_q−q0 ) ; 

310 c_err=s q r t ( c ) ; 

r p i=c/tmt ; % Raw p o s i t i o n i n t e n s i t y 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% S u b t r a c t i n g background %%% 

315 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

load_bg=load ( bfn ) ; 

bg=rot90 ( z e r o s (1 , ch_interval ) , −1); 

bg_err=bg ; 

320 f o r i =1: ch_interval 

bg ( i )=load_bg ( i+c h _ f i r s t ) ; 

bg_err ( i )= s q r t ( bg ( i ) ) ; 

end 

bg=bg/bmt ; 

325 bg_err=bg_err /bmt ; 

% Calculate transmission 

trans_bp =752; %%% f i r s t channel to use 

trans_np=50; %%% number of p o i n t s 

330 trans_ep=trans_bp+trans_np ; %%% end point 

i n t=trans_bp : trans_ep ; 

trans=mean( c ( i n t )/ tmt)/mean( load_bg ( i n t )/bmt ) ; 

d isp ( s p r i n t f ( ’ Transmission : ␣␣%6.2 f ␣ ( channels ␣%3d␣ through ␣%3d) ’ , trans , . . . 

335 trans_bp , trans_ep ) ) ; 

s t r = s p r i n t f ( [ ’The␣ transmission ␣ i s ␣ the ␣ r a t i o ␣between ␣ the ␣ average␣ ’ . . . 

’ i n t e n s i t y ␣ pr . ␣ channel ␣ of ␣ the ␣ current ␣ data ␣ s e t ␣and␣ ’ . . . 

’ the ␣background ␣ data ␣ s e t . ’ ] ) ; 

%disp ( s t r ) ; 

340 

% c a l c u l a t i n g c minus background 

cmb=c ( c h _ f i r s t : ch_last )/ tmt−bg∗ trans ; 

cmb_err=(( c_err ( c h _ f i r s t : ch_last )/ tmt).^2+ bg_err . ^ 2 ) . ^ 0 . 5 ; 

345 % C a l c u l a t i n g the normalized spectrum with the trapez −f uncti on 

% i n t e g r a l e=sum(cmb )∗( q(2)−q (1)) 

i n t e g r a l e=trapz ( q ( c h _ f i r s t : ch_last ) ,cmb ) ; 

cmbn=cmb/ i n t e g r a l e ; 

cmbn_err=cmb_err/ i n t e g r a l e ; 

350 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% Save data f i l e s % 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


355 [ path2 name2 ext2 versn2 ] = f i l e p a r t s ( d a t a f i l e s ( 1 ) . filename ) ; 

base = [ path f i l e s e p name2 ext ] ; % base path & f i lename 

i f name == ’ ∗ ’ name = ’ a l l ’ ; end 

i f f c > 1 base = [ path f i l e s e p name ] ; end 

360 e_vek=transpose ( [ 1 : 1 : length ( e ) ] ) ; 

i f wf 

% Save ( q , c ) " raw " data to f i l e 

s a v e f i l e =[ base ’ . dat ’ ] ; 

365 out = [ q ( c h _ f i r s t : ch_last ) cmb cmb_err ] ; 

save ( s a v e f i l e , ’ out ’ , ’−ASCII ’ ) ; 

% Save the normalized spectrum 

s a v e f i l e =[ base ’ . norm . dat ’ ] ; 

370 out = [ q ( c h _ f i r s t : ch_last ) cmbn cmbn_err ] ; 


% Save div info , computer readable 

s a v e f i l e =[ base ’ . i n f o . dat ’ ] ; 

375 out=[n tmt ch_interval bmt trans i n t e g r a l e ] ; 


% Save info , counts pr second f o r p l o t counts pr sec /q 

s a v e f i l e =[ base ’ . inten . dat ’ ] ; 

380 out=[q ( c h _ f i r s t : ch_last ) r p i ( c h _ f i r s t : ch_last ) bg ] ; 


%Save info , energy spectrum 

s a v e f i l e =[ base ’ . energy . dat ’ ] ; 

385 out=[e_vek e ] ; 


% turn diary o f f diary 

390 end % i f 

395 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% P l o t t i n g graphs %%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

legend_str = s t r r e p (name2 , ’_ ’ , ’− ’ ) ; 

i f f c > 1 legend_str = s t r r e p (name , ’_’ , ’− ’ ) ; end 

% Raw data p l o t 

400 i f sg 

f i g u r e (105) 

hold on 

p l o t ( rpi , ’ bx ’ , ’ MarkerSize ’ , 2 . 0 ) 

405 x l a b e l ( ’ Channel ’ ) ; 

y l a b e l ( ’ I n t e n s i t y ␣/␣Counts ␣ pr . ␣ second ’ ) ;

163 

t i t l e ( ’Raw␣ data ␣ spectrum ’ ) ; 

legend ( [ legend_str ’ ␣raw␣ data ’ ] ) ; 

410 % Setup f i g u r e 

s e t ( gca , ’ XGrid ’ , ’ on ’ ) ; 

s e t ( gca , ’ Units ’ , ’ centimeters ’ ) ; 

s e t ( gca , ’ Position ’ , [ 3 , 2 , 1 1 , 8 ] ) ; 

v = a x i s ; 

415 a x i s ( [ 1 , cc_position , 0 , v ( 4 ) ] ) ; 

lower_limit_line = l i n e ( [ c h _ f i r s t c h _ f i r s t ] , [ 0 v ( 4 ) ] ) ; 

s e t ( lower_limit_line , ’ LineStyle ’ , ’− ’ ) ; 

s e t ( lower_limit_line , ’ Color ’ , ’ k ’ ) ; 

upper_limit_line = l i n e ( [ ch_last ch_last ] , [ 0 v ( 4 ) ] ) ; 

420 s e t ( upper_limit_line , ’ LineStyle ’ , ’− ’ ) ; 

s e t ( upper_limit_line , ’ Color ’ , ’ k ’ ) ; 

i f wf 

p r i n t ( ’−r300 ’ , ’−depsc ’ , [ base ’ . raw . c o l o r . eps ’ ] ) ; 

425 p r i n t ( ’−r300 ’ , ’−deps ’ , [ base ’ . raw . eps ’ ] ) ; 

end % i f 

430 

435 

hold 

end 

o f f 

% Plot r p i and bg without error bars 

i f sg 

f i g u r e ( 1 1 0 ) ; 

hold on 

p l o t ( q ( c h _ f i r s t : ch_last ) , bg , ’ k . ’ , ’ MarkerSize ’ , 4 . 0 ) ; 

%errorbar ( q , bg , bg_err ) 

p l o t ( q ( c h _ f i r s t : ch_last ) , r p i ( c h _ f i r s t : ch_last ) , ’ bx ’ , ’ MarkerSize ’ , 2 . 0 ) ; 

%errorbar ( q , c , c_err ) 

440 x l a b e l ( ’ q␣ [ ␣Å^{␣ −1}] ’ ) ; 

y l a b e l ( ’ I n t e n s i t y ␣ [ counts ␣ pr . ␣ second ] ’ ) ; 

t i t l e ( ’ Background ␣ spectrum␣and␣raw␣ data ␣ spectrum ’ ) ; 

legend ( [ legend_str ’ ␣ background ’ ] , [ legend_str ’ ␣raw␣ data ’ ] ) ; 

445 % Setup f i g u r e 




v = a x i s ; 

450 a x i s ( [ 0 , 0.55 , 0 , v ( 4 ) ] ) ; 

i f wf 

p r i n t ( ’−r300 ’ , ’−depsc ’ , [ base ’ . cb . c o l o r . eps ’ ] ) ; 

p r i n t ( ’−r300 ’ , ’−deps ’ , [ base ’ . cb . eps ’ ] ) ; 

455 end % i f 

hold 

end 

o f f


460 % Plot cmb without error bars 

i f sg 

f i g u r e (100) 

hold on 

465 p l o t ( q ( c h _ f i r s t : ch_last ) , cmb , ’ bx ’ , ’ MarkerSize ’ , 2 . 0 ) ; 

%errorbar ( q , cmb , cmb_err , ’ . ’ ) 

x l a b e l ( ’ q␣/␣Å^{␣−1} ’ ) ; 

y l a b e l ( ’ I n t e n s i t y ␣/␣ counts ␣ pr . ␣ second ’ ) ; 

t i t l e ( ’ Spectrum ␣ with ␣ subtracted␣background ’ ) ; 

470 

% Setup f i g u r e 




475 v = a x i s ; 

a x i s ( [ 0 , 0.55 , 0 , v ( 4 ) ] ) ; 

legend ( legend_str ) ; 

i f 

wf 

480 p r i n t ( ’−r300 ’ , ’−depsc ’ , [ base ’ . cmb . c o l o r . eps ’ ] ) ; 

p r i n t ( ’−r300 ’ , ’−deps ’ , [ base ’ . cmb . eps ’ ] ) ; 

i f i s u n i x 

cmd = s p r i n t f ( ’ gracebat ␣−settype ␣xydy␣%s ␣−param␣%s ␣−s a v e a l l ␣%s ’ , . . . 

[ base ’ . dat ’ ] , dgpfn , [ base ’ . cmb . agr ’ ] ) ; 

485 % [ status , r e s u l t ] = system (cmd ) ; 

end % i f 

end % i f 

490 end 

hold 

o f f 

% Plot energy spectrum 

i f sg 

f i g u r e (120) 

495 hold on 

p l o t ( e , ’ bx ’ , ’ MarkerSize ’ , 2 . 0 ) ; 

x l a b e l ( ’ Energy ␣ channel ’ ) ; 

y l a b e l ( ’ Total ␣ counts ’ ) ; 

500 t i t l e ( ’ Energy ␣ spectrum ’ ) ; 

% Setup f i g u r e 



505 s e t ( gca , ’ Position ’ , [ 3 , 2 , 1 1 , 8 ] ) ; 

v = a x i s ; 

a x i s ( [ 1 , cc_energy , 0 , v ( 4 ) ] ) ; 


510 lower_limit_line = l i n e ( [ ll_energy ll_energy ] , [ 0 v ( 4 ) ] ) ; 

s e t ( lower_limit_line , ’ LineStyle ’ , ’− ’ ) ; 

s e t ( lower_limit_line , ’ Color ’ , ’ k ’ ) ;

165 

upper_limit_line = l i n e ( [ ul_energy ul_energy ] , [ 0 v ( 4 ) ] ) ; 

s e t ( upper_limit_line , ’ LineStyle ’ , ’− ’ ) ; 

515 s e t ( upper_limit_line , ’ Color ’ , ’ k ’ ) ; 

i f wf 

p r i n t ( ’−r300 ’ , ’−depsc ’ , [ base ’ . e . c o l o r . eps ’ ] ) ; 

p r i n t ( ’−r300 ’ , ’−deps ’ , [ base ’ . e . eps ’ ] ) ; 

520 end % i f 

hold 

end 

o f f 

525 q = q ( c h _ f i r s t : ch_last ) ; 

RUCSAXS_SETTINGS.M 

f u n c t i o n 

[ q_pr_ch , q0 , bfn , bmt , ch_first , ch_last , sbx , dgpfn ] = rucsaxs_settings 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% Data processing parameters % 

529 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

% q−c a l i b r a t i o n constants 

q_pr_ch=8.0318 e −4; 

% d i r e c t beam 

534 q0 =0.1759; 

539 

% Background spectrum f i l e name 

bfn=’ /home/ grupper/ alkohol / rucsaxs/background /background . dat ’ ; 

bmt=64800; % background spectrum measuring time 

% Channels to use 

c h _ f i r s t = 250; 

ch_last = 900; 

544 % Beam p r o f i l e along x−a x i s 

% ( h o r i z o n t a l a x i s perpendicular to beam f l u x ) 

% f i t t e d to a gaussian f uncti on with a spread of 

sbx = 0 . 1 4 6 8 1 ; 

% centered around 0! 

549 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% Graph parameters % 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

554 % Default Grace parameters f i lename 

dgpfn = ’ /home/ grupper/ alkohol / s c r i p t s / d e f a u l t . grace . par ’ ; 

DSC7.M 

% Process data f i l e ( s ) from the DSC7 equipment in øSren Hvidts l a b o r a t o r y . 

% The s c r i p t reads the [ i f n ] . d7 f i l e , p l o t s the data and 

% ( o p t i o n a l l y ) w r i t e s an [ i f n ] . dat f i l e containing the (T, J ) data s e t . 

559 %


% f uncti on [T, J ] = dsc7 ( i f n , s t r = ’ ’ , sg =1, wf=1, pr=0) 

% i f n − i nput f i l e name , p a t t e r n s are allowed 

% s t r − t i t l e s t r i n g of the graph produced . ( d e f a u l t : ’ ’ ) 

% sg − show graphs (1=yes , 0=no , d e f a u l t : 1) 

564 % wf − write output f i l e , [ i f n ] . dat (1=yes , 0=no , d e f a u l t : 1) 

% containing (T, J) data s e t . 

% pr − pans reversed ( Sample and r e f e r e n c e pans reversed ) 

% 

% Example : 

569 % 

% dsc7 ( ’ hex2 ∗ ’ , ’DSC on ULVs (DPPC) pertubated by hexanol ’ , 1 , 0 ) ; 

% Shows a (T, J) graph with the t i t l e above , and does not write the output f i l e . 

574 

f u n c t i o n [T, J ] = dsc7 ( ifn , v arargin ) 

% Default f uncti on parameters 

J = 0 ; 

T = 0 ; 

t i t l e _ s t r = ’ ’ ; 

579 legend_str = ’ ’ ; 

sg = 1 ; % yes , show graphs 

wf = 1 ; % yes , write output f i l e s 

pr = 0 ; % no , pans are not reversed 

graph_y_max = 0 ; 

584 graph_y_min = 0 ; 

% Number of o p t i o n a l v a r i a b l e s 

nv = length ( v arargin ) ; 

i f nv >= 4 pr = cell2mat ( v arargin ( 4 ) ) ; end 

589 i f nv >= 3 wf = cell2mat ( v arargin ( 3 ) ) ; end 

i f nv >= 2 sg = cell2mat ( v arargin ( 2 ) ) ; end 

i f nv >= 1 t i t l e _ s t r = cell2mat ( v arargin ( 1 ) ) ; end 

% Get f i l e parts from i f n 

594 [ path , name , ext , versn ] = f i l e p a r t s ( i f n ) ; 

i f isempty ( path ) path = ’ . ’ ; end 

ext = ’ . d7 ’ ; 


f i l e s = d i r ( pattern ) ; 

599 count = length ( f i l e s ) ; 

i f count == 0 

ext = ’ . D7 ’ ; 


f i l e s = d i r ( pattern ) ; 

604 count = length ( f i l e s ) ; 

end % i f 


error ( s p r i n t f ( ’ F i l e ␣ not ␣ found␣ using ␣ pattern : ␣%s ’ , pattern ) ) ; 

609 end % i f 

f o r i =1: count 

T = 0 ;

167 

614 

J = 0 ; 

fn = [ path f i l e s e p f i l e s ( i ) . name ] ; 

d isp ( s p r i n t f ( ’ Current ␣ f i l e : ␣%s ’ , fn ) ) ; 

% Read i nput f i l e 

619 contents = textread ( fn , ’%s ’ , ’ whitespace ’ , ’ \n ’ ) ; 

measurement_count = str2num ( char ( contents ( 8 ) ) ) ; 

scan_rate = str2num ( char ( contents ( 9 ) ) ) ; 

T_start = str2num ( char ( contents ( 7 ) ) ) ; 

T_final = str2num ( char ( contents ( 1 3 ) ) ) ; 

624 Time_at_T_start = str2num ( char ( contents ( 1 5 ) ) ) ; 

J_axis_max = str2num ( char ( contents ( 1 4 ) ) ) ; 

metainfo = char ( contents ( 1 ) ) ; 

id = s t r c a t ( metainfo ( 2 : 3 0 ) ) ; 

J_factor = J_axis_max / 10000; 

629 i f pr 

J_factor = −1 ∗ J_factor ; 

end 

i f length ( t i t l e _ s t r ) == 0 

t i t l e _ s t r = id ; 

634 end 

d isp ( s p r i n t f ( ’ ID : ␣"%s "\ t \ tScanrate : ␣%3d␣ ’ , id , scan_rate ) ) ; 

% Process data 

time = ( T_final − T_start ) / scan_rate ; 

639 x = 58:57+ measurement_count ; 

T = T_start + (x−58) ∗ ( T_final − T_start ) / measurement_count ; 

f o r j =58:57+measurement_count 

J ( j −57) = ( str2num ( char ( contents ( j ) ) ) − J ( 1 ) ) ∗ J_factor ; 

end 

644 Tmin = min (T) ; Tmax = max(T) ; deltaT = Tmax−Tmin ; 

Jmin = min ( J ) ; Jmax = max( J ) ; deltaJ = Jmax−Jmin ; 

i f sg 

% Plot data 

649 f i g u r e ( 1 ) ; hold on ; p l o t (T, J , ’ k . ’ , ’ MarkerSize ’ , 4 . 0 ) ; hold o f f ; 


x l a b e l ( ’ Temperature␣/␣\ circC ’ ) ; 

y l a b e l ( ’ Heat␣ flow ␣/␣mW’ ) ; 

t i t l e ( t i t l e _ s t r ) ; 

654 s e t ( gca , ’ Units ’ , ’ centimeters ’ ) ; 


i f length ( legend_str ) == 0 

legend_str = char ( id ) ; 

e l s e 

659 legend_str = char ( legend_str , id ) ; 

end 

% Set a x i s 

v = a x i s ; 

664 i f Jmax > graph_y_max 

graph_y_max = ( f l o o r (Jmax /10) + 1) ∗ 10;


end 

i f Jmin < graph_y_min 

graph_y_min = f l o o r ( Jmin /10) ∗ 10; 

669 end 

a x i s ( [ v ( 1 ) , v ( 2 ) , graph_y_min , graph_y_max ] ) ; 


tx = deltaT /8 + Tmin ; 

674 ty = Jmax − 1∗ deltaJ /8; 

text ( tx , ty , s p r i n t f ( ’ Scanrate : ␣%d␣K␣/␣min . ’ , scan_rate ) ) ; 

679 

ty = Jmax − 2∗ deltaJ /8; 

text ( tx , ty , s p r i n t f ( ’ Delay ␣ at ␣T−s t a r t : ␣%d␣min . ’ , Time_at_T_start ) ) ; 


text ( tx , ty , s p r i n t f ( ’ ID : ␣%s ’ , id ) ) ; 


684 f n s t r = s t r r e p ( fn , ’_’ , ’− ’ ) ; 

text ( tx , ty , s p r i n t f ( ’ F i l e : ␣%s ’ , f n s t r ) ) ; 

end % i f 

end % i f 

689 % Write output f i l e 

i f wf 

data = [ transpose (T) , transpose ( J ) ] ; 

[ path , name , extension , versn ] = f i l e p a r t s ( fn ) ; 

694 % Set output f i lename 

ofn = [ path f i l e s e p name ’ . dat ’ ] ; 

save ( ofn , ’ data ’ , ’−ASCII ’ ) ; 

end % i f 

end % f o r 

699 

i f sg 


end % i f 

F_SYM.M 

% Returns the form f a c t o r from a 1−dimensional symmetric 

% e l e c t r o n d e n s i t y p r o f i l e f o r a b i l a y e r . 

704 % 

% f_sym ( q , [A, zh , sh , s t ] , [ ] ) 

% 

% q ~ s c a t t e r i n g v e c t o r 

% A ~ amplitude 

709 % zh ~ center of gauss peak 

% sh ~ spread of head gauss f uncti on 

% s t ~ spread of t a i l gauss f uncti on 

% 

714 f u n c t i o n f = f_sym (q , parameters , constants )

169 

A = parameters ( 1 ) ; 

zh = parameters ( 2 ) ; 

sh = parameters ( 3 ) ; 

719 s t = parameters ( 4 ) ; 

f = (A ∗ ( fourier_gauss (q , zh , sh ) + fourier_gauss (q,−zh , sh ) ) − fourier_gauss (q , 0 , s t ) ) . / q ; 

I_SYM.M 

% Returns the i n t e n s i t y from a point s c a t t e r e r f o r 

% a symmetric e l e c t r o n d e n s i t y p r o f i l e f o r a b i l a y e r . 

% 

724 % I_sym(q , [A, zh , sh , s t ] , [ ] ) 

% 



% zh ~ center of gauss peak 

729 % sh ~ spread of head gauss f uncti on 


% 

734 

f u n c t i o n I = I_sym(q , parameters , constants ) 

f = f_sym (q , parameters , constants ) ; 

I = f . ∗ conj ( f ) ; 

F_ASYM.M 

% Returns the form f a c t o r from a 1−dimensional asymmetric 


% 

739 % f_asym(q , [ A1, A2, zh , sh , s t ] , [ ] ) 

% 

% Inner Gauss f uncti on 

% A1 ~ amplitude f o r gauss f uncti on 



% 

% Outer Gauss f uncti on 


% 

749 % Tail 


f u n c t i o n f = f_asym (q , parameters , constants ) 

754 A1 = parameters ( 1 ) ; 

A2 = parameters ( 2 ) ; 



s t = parameters ( 5 ) ; 

759 

f = (A1∗ fourier_gauss (q,−zh , sh ) + A2∗ fourier_gauss (q , zh , sh ) − fourier_gauss (q , 0 , s t ) ) . / q ;


I_ASYM.M 


% an asymmetric e l e c t r o n d e n s i t y p r o f i l e f o r a b i l a y e r . 

% 

% I_asym( q , [ A1, A2, zh , sh , s t ] , [ ] ) 

764 % 





769 % 



% 

% Tail 

774 % s t ~ spread of t a i l gauss f uncti on 

f u n c t i o n I = I_asym(q , parameters , constants ) 

f = f_asym (q , parameters , constants ) ; 

779 I = f . ∗ conj ( f ) ; 

I_SYM_MLV.M 

779 % Returns the i n t e n s i t y from a point s c a t t e r e r f o r 

% a symmetric e l e c t r o n d e n s i t y p r o f i l e f o r a b i l a y e r . 

% 

% I_sym_mlv ( q , [A, zh , sh , st , Ab , q0 ] , [ phase ] ) 

% 

784 % A ~ amplitude f o r gauss f uncti on 




% 

789 % M u l t i l a m e l l a r c o r r e c t i o n 

% Ab ~ the r e l a t i v e amount of m u l t i l a m e l l a r v e s i c l e s ( mlv/ u l v ) 

% q0 ~ F i r s t bragg peak of mlv 

% 

% phase ~ 1 ( low temp ) or 2 ( high temp ) 

794 

f u n c t i o n I = I_sym_mlv(q , parameters , constants ) ; 



799 sh = parameters ( 3 ) ; 


Ab = parameters ( 5 ) ; 

q0 = parameters ( 6 ) ; 

phase = constants ( 1 ) ; 

804 

Isym = I_sym(q , [A, zh , sh , s t ] , [ ] ) ; 

I = Isym . ∗ (1 + Ab. ∗ S_mlv(q , [ q0 ] , [ phase ] ) ) ;

171 

I_ASYM_MLV.M 


% an asymmetric e l e c t r o n d e n s i t y p r o f i l e f o r a b i l a y e r . 

809 % 

% I_asym_mlv( q , [A, zh , sh , st , Ab , q0 ] , [ A_f , zh_f , sh_f , phase ] ) 

% 


% A ~ amplitude f o r gauss f uncti on 

814 % zh ~ center of gauss peak 


% 


% A_f ~ amplitude weighing f a c t o r 

819 % zh_f ~ peak weighing f a c t o r 

% sh_f ~ spread weighing f a c t o r 

% 

% Tail 


824 % 

% M u l t i l a m e l l a r c o r r e c t i o n 



% 

829 % phase ~ 1 ( low temp ) or 2 ( high temp ) 

f u n c t i o n I = I_asym_mlv(q , parameters , constants ) ; 


834 zh = parameters ( 2 ) ; 





839 A_f = constants ( 1 ) ; 

zh_f = constants ( 2 ) ; 

sh_f = constants ( 3 ) ; 


844 Iasym = I_asym(q , [A, zh , sh , s t ] , [ A_f , zh_f , sh_f ] ) ; 

I = Iasym . ∗ (1 + Ab. ∗ S_mlv(q , [ q0 ] , [ phase ] ) ) ; 

F_3D.M 



% ( from : Brzustowicz & Brunger ) 

849 % 

% f_3d_bb(q , [A, zh , sh , s t ] , [ r0 ] ) 

% 

% A ~ amplitude f o r gauss f uncti on 



% s t ~ spread of t a i l gauss f uncti on


% r0 ~ v e s i c l e radius 

% 

859 f u n c t i o n f = f_3d_bb (q , parameters , constants ) 




864 s t = parameters ( 4 ) ; 

r0 = constants ( 1 ) ; 

f = (A∗ fourier_spheric al _gauss (q , r0−zh , sh ) . . . 

+ A∗ fourier_spheric a l_gauss (q , r0+zh , sh ) . . . 

869 − fourier_spheric al _gauss (q , r0 , s t ) ) . / q ; 

I_3D.M 


% a 3d symmetric e l e c t r o n d e n s i t y p r o f i l e f o r a b i l a y e r . 

% 

% I_3d ( q , [A, zh , sh , s t ] , [ ] ) 

% 

874 % q ~ s c a t t e r i n g v e c t o r 





879 % 

f u n c t i o n I = I_3d(q , parameters , constants ) 

R0 = 600; % Å 

884 R_s = 120; % Å 

R_start = R0−R_s∗2; 

R_end = R0+R_s∗2; 

% checked with 8 , 16 , 32 , 64 , 128 , 256 

889 % above 64 , a l l graphs are i d e n t i c a l 

R_steps = 64; 

R_stepsize = (R_end − R_start )/ R_steps ; 

R = R_start : R_stepsize : R_end ; 

894 R_dist = gauss (R, R0 , R_s ) ; 

N = length ( q ) ; 

I = z e r o s (1 ,N) ; 

899 % I t e r a t e v e s i c l e radius 

f o r j =1:R_steps 

f = f_3d (q , parameters , [R( j ) ] ) ; 

I_R = f . ∗ conj ( f ) ; 

I = I + R_dist ( j ) ∗ I_R ; 

904 end ;

173 

I_3D_MLV.M 


% a 3d e l e c t r o n d e n s i t y p r o f i l e f o r a b i l a y e r . 

% 

% I_3d_mlv ( q , [A, zh , sh , st , Ab , q0 ] , [ phase ] ) 

% 

909 % A ~ amplitude f o r gauss f uncti on 




% 

914 % M u l t i l a m e l l a r c o r r e c t i o n 



% 


919 

f u n c t i o n I = I_3d_mlv(q , parameters , constants ) ; 








929 

I3d = I_3d(q , [A, zh , sh , s t ] , [ ] ) ; 

I = I3d . ∗ (1 + Ab. ∗ S_mlv(q , [ q0 ] , [ phase ] ) ) ; 

F_SYM_4G.M 


% e l e c t r o n d e n s i t y p r o f i l e f o r a b i l a y e r using 4 gauss f u n c t i o n s . 

934 % 

% f_sym_4g( q , [A, zh , sh , zt , s t ] , [ ] ) 

% 



939 % zh ~ center of gauss peak f o r the heads 


% z t ~ center of gauss peak f o r the t a i l s 


% 

944 

f u n c t i o n f = f_sym_4g(q , parameters , constants ) 




zt = parameters ( 4 ) ; 

s t = parameters ( 5 ) ;


f = (A ∗ ( fourier_gauss (q , zh , sh ) + fourier_gauss (q,−zh , sh ) ) . . . 

954 − fourier_gauss (q,−zt , s t ) − fourier_gauss (q , zt , s t ) ) . / q ; 

I_SYM_4G.M 

954 % Returns the form f a c t o r from a 1−dimensional symmetric 

% e l e c t r o n d e n s i t y p r o f i l e f o r a b i l a y e r using 4 gauss f u n c t i o n s . 

% 

% I_sym_4g ( q , [A, zh , sh , zt , s t ] , [ ] ) 

% 

959 % q ~ s c a t t e r i n g v e c t o r 


% zh ~ center of gauss peak f o r the heads 


% z t ~ center of gauss peak f o r the t a i l s 

964 % s t ~ spread of t a i l gauss f uncti on 

% 

f u n c t i o n I = I_sym_4g(q , parameters , constants ) 

969 f = f_sym_4g (q , parameters , constants ) ; 

I = f . ∗ conj ( f ) ; 

S_MLV.M 

% Returns the s t r u c t u r e f a c t o r f o r m u l t i l a m e l l a r v e s i c l e s . 

% 

% S_mlv(q , [ q0 ] , [ phase ] ) 

% 

974 % q0 ~ Bragg peak of mlv 

% sb ~ spred of Bragg peak 


979 

f u n c t i o n S = S_mlv(q , parameters , constants ) 



switch phase 

984 case {1} % low temperature 

sb = 0 . 0 1 2 8 ; % f i t t e t to ( q , I ) of MLV sample 

S = gauss (q , q0 , sb ) + 0.22∗ gauss (q ,2∗ q0 , sb ) + 0.022∗ gauss (q ,3∗ q0 , sb ) ; 

case {2} % high temperature 

sb = 0 . 0 1 5 9 ; % f i t t e t to ( q , I ) of MLV sample 

989 S = gauss (q , q0 , sb ) + 0.3∗ gauss (q ,2∗ q0 , sb ) ; 

otherwise 

error ( ’ Phase ␣ parameter ␣must␣be␣ e i t h e r ␣1␣ ( low␣temp) ␣ or ␣2␣ ( high ␣temp) ’ ) ; 

end ; 

SMEARING.M 

% Returns the smeared i n t e n s i t y 

% 

994 % smearing ( I_func , q , parameters , s t e p s )

175 

% 

% I_func ~ I n t e n s i t y f uncti on from a point s c a t t e r e r model 


% parameters ~ Parameters f o r the i n t e n s i t y f uncti on 

999 % s t e p s ~ The number of i n t e g r a t i o n s t e p s 

% 

f u n c t i o n I_smeared = smearing ( I_func , q , parameters , constants ) 

1004 %%%%%%%%%%%%%%%%%%%%%%% 

%%% Load s e t t i n g s %%% 

%%%%%%%%%%%%%%%%%%%%%%% 

% Please read the r u c s a x s _ s e t t i n g s .m f i l e f o r information 

1009 % about each of the parameters 

[ q_pr_ch , q0 , bfn , bmt , ch_first , ch_last , sbx , dgpfn ] = rucsaxs_settings ; 

% " s t e p s " found by t r y i n g 2 ,4 ,8 ,16 ,32 ,10000 

% There was no d i f f e r e n c e between 32 and 10000 above q =0.025 (beam stop ) . 

1014 s t e p s = 32; 

% I n t e r v a l along x−a x i s 

x = −2∗sbx :4∗ sbx / s t e p s :2∗ sbx ; 

1019 % Beam p r o f i l e along x−a x i s 

Px = transpose ( gauss (x , 0 , sbx ) ) ; 

N = length ( q ) ; 

f o r i =1:N 

1024 I_p = f e v a l ( I_func , s q r t (q ( i ) . ^ 2 + x . ^ 2 ) , parameters , constants ) ’ ; 

I_smeared ( i ) = trapz (x , Px . ∗ I_p ) ; 

end 

FIT.M 

% Fit m u l t i p l e models to m u l t i p l e data s e t s . 

% 

% f uncti on r e s u l t s = f i t ( saxs_data , max_iterations ) ; 

1029 % 

% saxs_data ~ 1xN s t r u c t array 

% data : [ 3 x651 double ] 

% a l c o h o l : n 

% phase : n 

1034 % s e r i e s : n 

% modeling_results : [ 1 x1 s t r u c t ] 

% 

% Each model has the f o l l o w i n g s t r u c t u r e 

% modeling_results . ( model ) 

1039 % parameters : [ Nx1 double ] 

% converged : 0 or 1 

% r e s i d u a l s : [561 x1 double ] 

% jacobian : [561xN double ] 

% sse : d 

1044 % i t e r a t i o n s : n


% time : d 

% constants : [Mx1 double ] 

% l a b e l s : {Nx1 c e l l } 

% l a t e x _ l a b e l s : {Nx1 c e l l } 

1049 % 

1054 

f u n c t i o n r e s u l t s = f i t ( saxs_data , max_iterations ) ; 

r e s u l t s = saxs_data ; 

% data s e t s 

N = length ( saxs_data ) ; 

% i t e r a t e data s e t s 

1059 f o r i =1:N 

% do nothing , i f there i s no data 

i f saxs_data ( i ) . data (1 ,1) == 0 

continue ; 

1064 end 

1069 

a l c o h o l = alcohol_str ( saxs_data ( i ) . a l c o h o l ) ; 

s e r i e s = saxs_data ( i ) . s e r i e s ; 

phase = phase_str ( saxs_data ( i ) . phase ) ; 

log_str = ’ \ nAlcohol : ␣%s , ␣Phase : ␣%s , ␣ S e r i e s :%3d ’ ; 

d isp ( s p r i n t f ( log_str , alcohol , phase , s e r i e s ) ) ; 

q = saxs_data ( i ) . data ( 1 , : ) ; 

1074 I = saxs_data ( i ) . data ( 2 , : ) ; 

model_names = fieldnames ( saxs_data ( i ) . modeling_results ) ; 

M = length ( model_names ) ; 

1079 % I t e r a t e models 

f o r j =1:M 

% current model 

model = saxs_data ( i ) . modeling_results . ( model_names { j } ) ; 

1084 % current model parameters 

parameters = model . parameters ; 

par_count = length ( parameters ) ; 

% do nothing i f there i s no parameters 

1089 i f parameters (1) == 0 

continue ; 

end 

% current model constants 

1094 constants = model . constants ; 

log_str = ’ F i t t i n g ␣ to ␣model : ␣%s ’ ; 

d isp ( s p r i n t f ( log_str , model_names { j } ) ) ;

177 

1099 % or i f the f i t t i n g already 

% has been done b e f o r e and has converged 

i f model . converged == 1 

log_str = ’ Modeling␣ has ␣ converged ␣ before , ␣ skipping ␣model . ’ ; 

d isp ( s p r i n t f ( log_str ) ) ; 

1104 

par_str = ’ ’ ; 

f o r k = 1 : par_count 

pattern = [ ’ ␣ ’ model . l a b e l s {k} ’=%.3g , ’ ] ; 

i f k == par_count 

1109 pattern = [ ’ ␣ ’ model . l a b e l s {k} ’=%.3g ’ ] ; 

end 

par_str = [ par_str s p r i n t f ( pattern , parameters ( k ) ) ] ; 

end 

d isp ( s p r i n t f ( ’SSE : ␣%.8g , ␣ Parameters : ␣[% s ] \ n ’ , model . sse , par_str ) ) ; 

1114 

continue ; 

end 

% parameter count 

1119 P = length ( parameters ) ; 

% g e t s t a r t time 

time_before = c l o c k ; 

1124 [ fitted_parameters , r e s i d u a l s , jacobian , i t e r a t i o n s , error_code ] = . . . 

n l i n f i t (q , I , model_names { j } , parameters , constants , max_iterations ) ; 

1129 

% g e t end time 

process_time = etime ( clock , time_before ) ; 

r e s u l t s ( i ) . modeling_results . ( model_names { j } ) . time = . . . 

r e s u l t s ( i ) . modeling_results . ( model_names{ j } ) . time + process_time ; 

r e s u l t s ( i ) . modeling_results . ( model_names { j } ) . r e s i d u a l s = r e s i d u a l s ; 

r e s u l t s ( i ) . modeling_results . ( model_names { j } ) . jacobian = jacobian ; 

1134 r e s u l t s ( i ) . modeling_results . ( model_names { j } ) . parameters = fitted_parameters ; 

r e s u l t s ( i ) . modeling_results . ( model_names { j } ) . s s e = transpose ( r e s i d u a l s )∗ r e s i d u a l s ; 

r e s u l t s ( i ) . modeling_results . ( model_names { j } ) . i t e r a t i o n s = . . . ; 

r e s u l t s ( i ) . modeling_results . ( model_names{ j } ) . i t e r a t i o n s + i t e r a t i o n s ; 

1139 i f error_code == 0 

r e s u l t s ( i ) . modeling_results . ( model_names{ j } ) . converged = 1 ; 

e l s e 

r e s u l t s ( i ) . modeling_results . ( model_names{ j } ) . converged = 0 ; 

end 

1144 end % models 

end % data 

NLINFIT.M 

% NLINFIT Nonlinear l e a s t −squares data f i t t i n g by the Gauss−Newton method . 

% 

% NLINFIT(X,Y, ’MODEL’ ,BETA0, maxiter ) f i n d s the c o e f f i c i e n t s of the nonlinear


% f uncti on d e s c r i b e d in MODEL. MODEL i s a user s u p p l i e d f uncti on having 

1149 % the form y = f ( beta , x ) . That i s MODEL returns the p r e d i c t e d values of y 

% given i n i t i a l parameter esti mates , beta , and the independent v a r i a b l e , X. 

% [BETA,R, J ] = NLINFIT(X,Y, ’MODEL’ ,BETA0) returns the f i t t e d c o e f f i c i e n t s 

% BETA the r e s i d u a l s , R, and the Jacobian , J , f o r use with NLINTOOL to 

% produce error esti mates on p r e d i c t i o n s . 

1154 % 

% B.A. Jones 12−06−94. 

% Copyright ( c ) 1993−98 by The MathWorks , Inc . 

% $Revision : 2.10 $ $Date : 1997/11/29 01:46:10 $ 

% 

1159 % Modifies by Ulf øæRrbk Pedersen , may 2005 

% Modificati ons by Thomas Hecksher , january 2007 

% 

f u n c t i o n [ beta , r , J , iterations_used , error_code ] = . . . 

1164 n l i n f i t (X, y , model , beta0 , constants , maxiter ) 

error_code = 0 ; % OK 

iterations_use d = 0 ; 

1169 n = length ( y ) ; 

i f min ( s i z e ( y ) ) ~= 1 

error ( ’ Requires ␣a␣ v ector ␣ second␣ input␣argument . ’ ) ; 

end 

y = y ( : ) ; 

1174 

p = length ( beta0 ) ; 

beta0 = beta0 ( : ) ; 

J = z e r o s (n , p ) ; 

1179 beta = beta0 ; 

betanew = beta + 1 ; 

i t e r = 0 ; 

b e t a t o l = 1.0E−4; 

r t o l = 1.0E−4; 

1184 s s e = 1 ; 

s s e o l d = s s e ; 

par_count = length ( beta ) ; 

while ( norm ( ( betanew−beta ) . / ( beta+s q r t ( eps ) ) ) > b e t a t o l | . . . 

1189 abs ( sseold −s s e )/( s s e+s q r t ( eps ) ) > r t o l ) & i t e r < maxiter 

i f i t e r > 0 , 

beta = betanew ; 

end 

1194 i t e r = i t e r + 1 ; 

y f i t = normalize (X, transpose ( f e v a l ( ’ smearing ’ , model ,X, beta , constants ) ) ) ; 

r = y − y f i t ; 

s s e o l d = r ’ ∗ r ; 

1199 par_str = ’ ’ ; 

f o r i = 1 : par_count

179 

pattern = ’ ␣%.3g , ’ ; 

i f i == par_count 

pattern = ’ ␣%.3g ’ ; 

1204 end 

par_str = [ par_str s p r i n t f ( pattern , beta ( i ) ) ] ; 

end 

d isp ( s p r i n t f ( ’ I t e r ␣%4d , ␣SSE␣%.8g , ␣Par␣[% s ] ’ , i t e r , sseold , par_str ) ) ; 

1209 f o r k = 1 : p , 

d e l t a = z e r o s ( s i z e ( beta ) ) ; 

d e l t a (k ) = s q r t ( eps )∗ beta ( k ) ; 

yplus = normalize (X, transpose ( f e v a l ( . . . 

’ smearing ’ , model ,X, beta+delta , constants ) ) ) ; 

1214 J ( : , k ) = ( yplus − y f i t )/( s q r t ( eps )∗ beta ( k ) ) ; 

end 

1219 

1224 

Jplus = [ J ; 1 . 0 E−2∗eye (p ) ] ; 

r p l u s = [ r ; z e r o s (p , 1 ) ] ; 

% Levenberg −Marquardt type adjustment 

% Gauss−Newton s t e p −> J\ r 

% LM s t e p −> inv (J ’ ∗ J+constant ∗ eye ( p ))∗ J ’ ∗ r 

step = Jplus \ r p l u s ; 

betanew = beta + step ; 

betanew = abs ( betanew ) ; % only accept p o s i t i v e parameter v a l u e s 

yfitnew = normalize (X, transpose ( f e v a l ( . . . 

1229 ’ smearing ’ , model ,X, betanew , constants ) ) ) ; 

rnew = y − yfitnew ; 

s s e = rnew ’ ∗ rnew ; 

i t e r 1 = 0 ; 

while s s e > s s e o l d & i t e r 1 < 12 

1234 step = step / s q r t ( 1 0 ) ; 

betanew = beta + step ; 

betanew = abs ( betanew ) ; % only accept p o s i t i v e parameter v a l u e s ; 

yfitnew = normalize (X, transpose ( f e v a l ( . . . 

’ smearing ’ , model ,X, betanew , constants ) ) ) ; 

1239 rnew = y − yfitnew ; 

s s e = rnew ’ ∗ rnew ; 

i t e r 1 = i t e r 1 + 1 ; 

end 

end 

1244 

iterations_used = i t e r ; 

i f 

1249 end 

i t e r == maxiter 

error_code = 1 ; % I n t e g r a l did not converge 

T_SET.M 

1249 f u n c t i o n out=T_set (T_abs ) ; 

% C a l c u l a t e s the temperature to be s e t from the d e s i r e d temperature .


% The c a l i b r a t i o n was done in August with a s i l i c o n e o i l i n s i d e the c u v e t t e . 

% 

% f uncti on T_set (T_abs ) 

1254 % T_abs i s the d e s i r e d temperature ( Celcius ) . 

% T_set return the s e t temperature ( Celcius ) . 

% 

% example : T = T_set ( 3 0 ) ; % returns 32.4254 

1259 out= ( ( 2 7 3 . 1 5 + T_abs − 60.459) / 0.79421) − 2 7 3 . 1 5 ; 

T_ABS.M 

1259 f u n c t i o n out = T_abs( T_set ) ; 

% C a l c u l a t e s the a b s o l u t e temperature from the s e t temperature . 

% The c a l i b r a t i o n was done in August with a s i l i c o n e o i l i n s i d e the c u v e t t e . 

% 

% f uncti on T_abs( T_set ) 

1264 % T_set i s the s e t temperature ( Celcius ) . 

% T_abs i s the returned a b s o l u t e temperature ( Celcius ) . 

% 

% example : T = T_abs ( 3 2 . 4 ) ; % returns ~30 

1269 out = 0.79421 ∗ ( T_set + 273.15) − 273.15 + 6 0 . 4 5 9 ; 

LIPID_WATER_MASS_RATIO.M 

1269 % out=lipid_water_mass_ratio ( numberOfDropTimeToUse ) 

% 

% pans . dat : weight of emty t r a y s 

% drops . dat : vap . data op drops 

% l i p i d . dat : end weight of t r a y s 

1274 % 

% Example : [ wp wp_err]= lipid_water_mass_ratio (3) 

% 

1279 

f u n c t i o n out=lipid_water_mass_ratio ( numberOfDropTimeToUse ) 

%%% Loading data 

pans=transpose ( load ( ’ pans . dat ’ ) ) ; 

drops=transpose ( load ( ’ drops . dat ’ ) ) ; 

l i p i d=transpose ( load ( ’ l i p i d . dat ’ ) ) ; 

1284 [ numberOfTrays , pansncol]= s i z e ( pans ) ; 

[ numberOfTrays , l i p i d n c o l ]= s i z e ( l i p i d ) ; 

[ numberOfTrays , numberOfDropTimes]= s i z e ( drops ) ; 

%%% Running through t r a y s 

1289 f o r i =1:numberOfTrays 

[ ’ ␣␣␣−−−␣ tray ␣number␣ ’ num2str ( i ) ’ ␣−−−’ ] 

%%% g e t t i n g tray weight 

pans_weight=mean( pans ( i , : ) ) ; 

1294 pans_weight_err=std ( pans ( i , : ) ) ; 

lipid_weight=mean( l i p i d ( i , : ) ) ; 

lipid_weight_err=std ( l i p i d ( i , : ) ) ,

181 

1299 

lw=lipid_weight −pans_weight ; 

lw_err=s q r t ( pans_weight_err^2+lipid_weight_err ^ 2 ) ; 

%%% g e t t i n g the weight from drops data 

c l e a r i n t ; i n t=numberOfDropTimes−numberOfDropTimeToUse+1:numberOfDropTimes ; 

x=i n t ; 

[ par , s s ]= p o l y f i t (x , drops ( i , i n t ) , 1 ) ; 

1304 drops_weight=par (2)− pans_weight ; 

1309 

%%% C a l c u l a t i n g wheight % 

wp( i )=lw/ drops_weight ; 

wp_err ( i )=lw_err/drops_weight ; 

%%% p l o t data 

f i g u r e ( 1 ) ; hold on ; p l o t ( pans ( i , : ) , ’ x−’ ) ; hold o f f 

f i g u r e ( 2 ) ; hold on ; p l o t ( l i p i d ( i , : ) , ’ x−’ ) ; hold o f f 

f i g u r e ( 3 ) ; hold on ; p l o t ( 1 : numberOfDropTimes , drops ( i , : ) , ’ o−’ , [ 0 x ] , . . . 

1314 p o l y v a l ( par , [ 0 x ] ) , ’ x : ’ ) ; hold o f f 

end 

%%% o v e r a l l weight % 

WP=mean(wp) 

1319 WP_err=std (wp)/ s q r t ( numberOfTrays) 

f i g u r e ( 4 ) ; 

hold on 

errorbar ( 1 : numberOfTrays , wp, wp_err ) ; 

p l o t ( [ 1 numberOfTrays ] ,WP∗[1 1 ] , ’−−’ ) ; 

1324 p l o t ( [ 1 numberOfTrays ] , (WP+WP_err ) ∗ [ 1 1 ] , ’ : ’ ) ; 

p l o t ( [ 1 numberOfTrays ] , (WP−WP_err ) ∗ [ 1 1 ] , ’ : ’ ) ; 

hold o f f 

1329 

out=[WP; WP_err ] ; 

%%% Saving data 

savedata =[WP WP_err ] ; 

s a v e f i l e =[ ’ kons . dat ’ ] ; 

d isp ( [ ’ Saving ␣ data ␣ to ␣ f i l e : ␣ ’ s a v e f i l e ] ) ; 

1334 save ( s a v e f i l e , ’ savedata ’ , ’−ASCII ’ ) ; 

ALCOHOL_LIPID_RATIO 

1334 f u n c t i o n out=a l c o h o l _ l i p i d _ r a t i o ( acl , ma, ms , lwr ) 

% C a l c u l a t e s the alcohol −l i p i d r a t i o 

% (DPPC i s i mplied and the output has the same unit as ms i nput ) . 

% 

% f uncti on a l c o h o l _ l i p i d _ r a t i o ( acl , ma, ms, lwr ) 

1339 % a c l − a l c o h o l chain l e n g t h 

% ma − mass a l c o h o l 

% ms − mass sample ( water + l i p i d ) 

% lwr − l i p i d water r a t i o 

1344 % Assignments 

M_lipid = 7 3 4 . 1 ; % g/mol (DPPC) 

M_alc = M_alcohol ( a c l ) ;


% C a l c u l a t i o n 

1349 out = ( M_lipid ∗ ma ∗ (1 + lwr ) ) / (M_alc ∗ lwr ∗ ms ) ; 

% Output 

%disp ( s p r i n t f ([ ’%1.3 e ’ ] , out ) ) ;

Index 

Adiabatic Scanning Calorimeter 36 

Anaesthetic potency 10 

Asymmetric 1d model 85 

Atomic Form Factor 26 

Backgammon technique 44 

Background 68 

Baseline 60 

Bilayer 3, 4, 5, 6, 9, 10, 11, 12, 14 

Bilayer normal 6 

Bilayer thickness 7, 8, 93 

Biphasic effect 11 

Block collimation 42 

Bragg reflections 90 

Bragg’s Law 28 

Brilliance 18 

Centrifuge 49 

Coherence 19 

Compensating DSC 36 

Compton scattering 24, 29 

Compton scattering length 29 

Cuvette 43 

De Broglie wavelength 17 

Detector 41, 44 

Di-palmityol-Phosphatidyl-Choline (DPPC) 

5 

Differential Scanning Calorimetry (DSC) 36, 

59 

DSC 7 38 

Electron density profile 14, 79 

Electron propability 26 

Extruder 49, 51 

Gravimetric determination 49 

H II 90 

Hexagonal structure 90 

Hydrophobic effect 12 

Hydrophobic force 4 

Incoherence 19 

Isothermal Titration Calorimetry (ITC) 12, 

33, 55 

Kratky camera 41 

L α 6, 7, 9, 10, 12, 14 

L c 9 

L β ′ 9 

L βI 9, 14 

Laue Condition 27 

Lipid 3 

Longitudinal Coherence length 19 

Lorentz factor 82 

Main phase transition temperature 11, 59 

Microfurnace 38 

Miller indices 27 

Mole fractions 12 

Monochromator 42 

MSC-ITC 33 

Multilamellar vesicle (MLV) 4 

P β ′ 9, 10 

Pan (DSC) 39 

Partition coefficient 10, 12, 14, 33 

Phase diagram 10 

Phase transition 5, 6, 11 

Phospholipid 3, 4, 5, 7, 8, 9, 10 

Photoelectric absorption 17 

Photon 17 

Polydispersity 52 

Power compensating DSC 38 

Pretransition 59, 62 

q-calibration 45 

Sample preparation 49 

Scal-1 microcalorimeter 36 

Scanrate 61 

Scattering cross section 21 

Simulation 12 

Smearing 78, 79 

Solvent-null method 12, 14, 33 

Spherically symmetric 3d model 87 

Structure factor 89 

183

184 Index 

Symmetric 1d model 81 

Symmetric 4g model 84 

Synchrotron 18 

Temperature calibration 47 

Transition temperature 63 

Thermostatic shield 37 

Thomson scattering 24 

Thomson scattering length 23, 29 

Threshold concentration 12, 59 

Transition Enthalpy 62 

Transverse Coherence length 19 

Unilamellar vesicle (ULV) 4, 8 

Unit Cell Structure Factor 27 

van der Waals force 4 

VP-ITC 33 

X-ray source 41

(DPPC) vesicles - dirac

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