Soft Report - Dipartimento di Fisica - Sapienza

Multi-Scale Coarse-Graining of Diblock Copolymer Self-Assembly: From Monomers to Ordered MicellesFig. 1: (a) Single configurationof 5000 dumbbells atdensity beyond the cmc.Red and blue indicate theideal and self avoiding strandsof the copolymersrespectively. (b) Position ofcentre of mass of the micellesfor the same system. (c)Ordered phase at slightlyhigher density.Block copolymers in solution show a remarkable tendencyto self-assemble into a bewildering range ofdisordered (liquid-like) and ordered structures, dependingon the macromolecular composition, therelative sizes of the blocks, solvent selectivity, polymerconcentration and temperature [1]. A frequent,partial scenario sees the copolymers aggregate intopolydisperse spherical micelles at low polymer concentration;upon lowering the temperature or increasingthe concentration, the micelles undergo adisorder-order transition onto a cubic lattice. Atheoretical understanding of copolymer phase behaviourgenerally relies on self-consistent field theorysimilar to that applied earlier to copolymer melts[2]. A more molecular approach is ob-viously verydifficult, in view of the wide range of length scalesinvolved, from themonomer level to themesoscopic scales characterisingordered micellarstructures.We have recently implementeda two-stagecoarse graining strategyto investigate the micellizationand disorder-ordertransition of a simplemicroscopic model ofdiblock copolymers in aselective solvent [3]. Westart considering AB copolymersmade up oftwo blocks of equalnumbers M A=M B=M ofmonomers on a cubiclattice. The solvent selectivitywith respect toA and B monomers isrepresented by modellingthe A blocks as idealchains (I) and the Bblocks as self and mutuallyavoiding walks (S).Moreover A and B blocksof the same or differentcopolymers are assumedto be mutually avoiding(S). The model is clearlyathermal and the polymerdensity is the onlythermodynamic variable.With typical polymersizes M~10 3 , and expectedmicelle aggregationnumbers n~10 2 ,very large system sizeswould be needed to generatethe tens of micellesrequired to studytheir ordered and disorderedstructures. At thefirst stage of our coarsegrainingprocedure, ABcopo-lymers are mappedonto “ultrasoft” dumbbells with effective interactionsbetween the centres of mass (CM) of A and B blockson different copolymers, and an intramolecular“tethering” potential between the CM's of the twoblocks of the same copolymer. These effective interactions,of purely entropic origin, are obtained byinverting the A-A, A-B, and B-B CM-CM intermolecularand intramolecular pair distri-bution functions inthe low density limit [4]. The obtained effective potentialsare roughly gaussian in shape with an amplitudeof a few k BT, and a range of the order of thegyration radius of the copolymer. Earlier experiencewith homopolymers shows that assuming the transferabilityof zero density potentials to finite density isnot an unreasonable approximation well into thesemi-dilute regime. By Monte Carlo simulation of theeffective dumbbell model, we have observed that micellizationsets in beyond a critical micellar concentrationwith the ideal A-blocks forming the densecore and the B-blocks forming the corona. With aperiodic sample of 5000 AB copolymers beyond thecmc (fig. 1) we observe about 50 moderatelypolydisperse micelles. A further increase of concentrationproduces a disordered-ordered transition,from a liquid-like structure into a micellar cubicphase with defects, in qualitative agreement withexperimental observation. At the second level ofcoarse graining we consider the structure of thesystem of micelles regarded as spherical particles.Within the HNC theoretical framework [5], an effectivemicelle-micelle pair potential can be extractedfrom the pair distribution function between micellecentres of mass. The resulting effective pair potentialexhibits a relatively soft repulsion that culminates ata value of roughly 12 K BT at the origin, followed by ashallow attractive well. This kind of potentials cangive rise to a freezing transition at sufficiently highdensities as we have indeed observed. Extension ofthe above coarse-graining strategy to more generalmodels is being pursed in order to reproduce the richexperimental phase diagram of dilute diblock copolymerssolutions in selective solvent.References[1] T.Lodge et al, Faraday Discussions 128, 1 (2005)[2] L. Leibler, Macromolecules 13, 1602 (1980);G.H. Frederickson and E. Helfand, J. Chem. Phys.87, 697 (1987); M.W. Matsen and M. Schick, Phys.Rev. Lett. 72, 2660 (1994).[3] C. Pierleoni, C.I. Addison, J.-P. Hansen and V.Krakoviack, submitted to Phys. Rev. Letts., January2006; cond-mat/0601417.[4] C.I. Addison, J.P. Hansen, V. Krakoviack and A.A.Louis, Molec. Phys. 103, 3045 (2005).[5] J.-P. Hansen and I.R. McDonald, “The Theory ofSimple Liquids”.AuthorsCarlo Pierleoni (a) , Chris Addison (b) , Jean-Pierre Hansen(b) and Vincent Krakoviack (c)(a) INFM CRS-SOFT, and Department of Physics, U. ofL'Aquila, I-67010 L'Aquila, Italy,(b) Dept. ofChemistry, University of Cambridge, Cambridge CB21EW, United Kingdom(c) Laboratoire de Chimie, Ecole Normale Supérieurede Lyon, 69364 Lyon Cedex 07, France.95SOFT Scientific Report 2004-06