VOLUME85, NUMBER 16 PHYSICAL REVIEW LETTERS 16OCTOBER2000Soap Froths and Crystal StructuresP. Ziherl* and Randall D. KamienDepartment of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396(Received 17 July 2000)We propose a physical mechanism to explain the crystal symmetries found in macromolecular andsupramolecular micellar materials. We argue that the packing entropy of the hard micellar cores isfrustrated by the entropic interaction of their brushlike coronas. The latter interaction is treated as asurface effect between neighboring Voronoi cells. The observed crystal structures correspond to theKelvin and Weaire-Phelan minimal foams. We show that these structures are stable for reasonable arealentropy densities.PACS numbers: 83.70.Hq, 61.50.Ah, 82.70. –yDendritic polymers [1–3], hyperbranched star polymers[4,5], and diblock copolymers [6,7] represent a new classof molecular assemblies all of which form a variety ofcrystalline lattices, many of which are not close packed.These assemblies are all characterized by compact coresand brushlike, soft coronas. These systems might be mod-eled by treating the micelles as sterically interacting hardspheres and it would follow that their crystalline phasesshould be stackings of hexagonal-close-packed (hcp) lay-ers. Recently [8] it has been shown that the face-centered-cubic (fcc) lattice maximizes the total entropy and sohard-sphere crystals should form fcc structures. Note thatthe entropic difference between the various hcp latticesis a global issue: the local arrangement of spheres is thesame for all close-packed variants and thus the latticecannot be predicted from nearest-neighbor interactions.In order to understand the richness of crystal symmetriesin the micellar systems, we propose an additional globalconsideration: we add an interaction proportional to theinterfacial surface area between the cages which containeach micelle (Voronoi cells). Though approaches basedon self-consistent field theory and two-body interactionscan yield non-close-packed lattices [9,10], we proposea universal explanation for a host of new structures andpresent a new paradigm for the rational design and controlof macromolecular assemblies [11].The interfacial interaction arises through the entropy ofthe brushlike coronas of the micelles. Because of con-straints on their conformations, the brushes suffer an en-tropic penalty proportional to the interfacial area betweenthe Voronoi cells surrounding each sphere. Thus they favorarea-minimizing structures, precisely the type of structuresthat dry foams might make. Over a century ago, LordKelvin proposed that a body-centered-cubic (bcc) foamstructure had the smallest surface-to-volume ratio [12], butin 1994 Weaire and Phelan found that a structure basedon the A15 lattice [13] was more efficient. We note thatneither the bcc nor A15 structures are close packed andthus there is a fundamental frustration between the hard-core volume interaction and the surface interaction due tooverlapping soft coronas.For concreteness, in this paper we focus on structuresobserved in a family of dendrimer compounds consistingof a compact poly(benzyl ether) core segment and a dif-fuse dodecyl corona [1,2]. These conical dendrimers self-assemble in spherical micelles which subsequently arrangeinto the A15 lattice (Fig. 1). The interaction between themicelles is primarily steric, i.e., repulsive and short range.The micellar architecture suggests that the potential is char-acterized by three regimes. At large distances, the micellesdo not overlap and the interaction vanishes. As the coro-nas begin to overlap, the entropy of the brushlike coronasdecreases, which gives rise to a soft repulsion between themicelles. Finally, at small separations the coronas begin topenetrate the compact cores: this is very unfavorable andgives rise to hard-core repulsion. This energy landscapeis in qualitative agreement with recent, detailed moleculardynamics simulations [14].Although both originate in steric interaction, the two re-pulsive regimes are characterized by very different func-tional behaviors. The hard part of the potential results in aFIG. 1. Various lattices: (a) Face-centered cubic, (b) body-centered cubic, (c) A15 lattice, and (d) columnar representationof A15 lattice. In the A15 lattice, columnar and interstitial sitesare drawn in grey and black, respectively.3528 0031-9007兾00兾85(16)兾3528(4)$15.00 © 2000 The American Physical SocietyVOLUME85, NUMBER 16 PHYSICAL REVIEW LETTERS 16OCTOBER2000restricted positional entropy of the micelles which dependson the free volume, the difference between the actual andthe hard-core volumes. The soft part comes from the de-creased orientational entropy of the chains within the over-lapping coronas. The matrix of overlapping coronas can bethought of as a compressed bilayer and thus the free vol-ume may be written as a product of the interfacial area Aand the average spacing between the hard cores d so thatat any given densityAd 苷 const . (1)Though this approximation ignores the curvature of brush-like coronas, the dendrimers are relatively close and weexpect this constraint to hold in this system. Since the re-pulsion decreases monotonically with distance, the systemwill favor a maximum thickness d and will thus tend tominimize the interfacial area, hence our proposed interfa-cial interaction, which is incompatible with the bulk freeenergy minimized by a close-packed arrangement of mi-celles. In the following, we compare the free energies offcc, bcc, and A15 lattices and estimate the strength of theinterfacial interaction such that the structure of the micel-lar crystal is dictated by the minimal-area principle.The calculation of the bulk free energies of condensedsystems is fairly complicated even for hard-sphere systemsand the best theoretical results are obtained numerically. Itis interesting to note that elaborate analytic models, such asthe high-density analog of the virial expansion [15] and theweighted-density-functional approximation [16], are onlyslightly better than the simple cellular free-volume theory[16,17]. The free-volume theory is a high-density approxi-mation where each micelle is contained in a cell formed byits neighbors, and the communal entropy associated withthe correlated motion of micelles is neglected.Within this theory, the positional entropy of a micelle isdetermined by the configurational space of its Voronoi orWigner-Seitz
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