Theoretical Issues Relating to Thermally Reversible Gelation by

Jun 1, 2009 - ... the hierarchical bundling of fibers into “superfibers”, fiber branching, gelation through fiber impingement and the associated p...
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Theoretical Issues Relating to Thermally Reversible Gelation by Supermolecular Fiber Formation† Jack F. Douglas* Polymers Division, NIST, Gaithersburg, Maryland 20899 Received May 6, 2009 Existing models of the thermodynamics and dynamics of self-assembly are summarized to provide a context for discussing the difficulties that arise in modeling supermolecular fiber assembly and the formation of thermally reversible gels through fiber growth and branching. Challenging problems in this field, such as the physical origin of fibers of uniform diameter and fiber twisting, the kinetics of fiber growth, the hierarchical bundling of fibers into “superfibers”, fiber branching, gelation through fiber impingement and the associated phenomenon of fractal fiber network and spherulite formation, and the origin and control of structural polymorphism in the fiber and superfiber geometry, are discussed from a personal perspective. Suggestions are made for integrating current research efforts into a more coherent multiscale description of fiber formation and gelation on molecular, mesoscopic, and macroscopic scales.

Introduction The theoretical description of supermolecular self-assembly is limited to rather simple models such as linear and branched polymers or model compact structures such as spherical micelles and equilibrium vesicles. There has also been recent interest in modeling viral capsid and other protein shells such as clathrin in which proteins organize at equilibrium through the association and dissociation of monomers into “polymeric” cages. Wormlike micelles have been successfully modeled as a kind of equilibrium polymerization,20 but some evidence indicates that a sequential or nucleated assembly can be involved (see below). Specifically, Douglas et al.1 have shown theoretically that this type of activated assembly can alter the “cooperativity” of the self-assembly transition (quantifying the extent to which the self-assembly thermodynamic transition approaches a true phase transition) from simple equilibrium polymerization models that simply assume the unconstrained reversible association of molecules or particles into polymer chains. This type of a sequential or “activated” assembly is common in the formation of complex biological structures such as clathrin protein cages involved in endocytosis and the capsid shells of viruses. This phenenomenon has also been noted in small molecules that exhibit supermolecular chain assembly,2 and much effort has recently been made to develop the theory of activated or chemically initiated self-assembly; a Supporting Information file provides an extended list of references related to this and other topics indirectly related to fiber assembly. Many supermolecular polymers and fibers form chiral structures, and an equilibrium polymerization model has been developed that allows for a thermodynamic transition of the assembled chains from an angularly uncorrelated to a helical state. Whereas these self-assembly models provide insight into the formation of fibers by self-assembly and their thermally reversible gelation, fiber assembly involves the confrontation of many additional effects, and the next section describes a personal perspective on the recent modeling of particular aspects of fiber self-assembly. †

Part of the Molecular and Polymer Gels; Materials with Self-Assembled Fibrillar Networks special issue. (1) Douglas, J. F.; Dudowicz, J.; Freed, K. F. J. Chem. Phys. 2008, 128, 224901. (2) Jonkheijm, P.; van der Schoot, P; Schenning, A. P. H. J.; Meijer, E. W. Science 2006, 313, 80.

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Special Features of Fiber Self-Assembly Fiber formation3,4 is evidently a kind molecular self-assembly, but the assembly of fibers exhibits distinct features from the equilibrium polymerization of linear and branched polymer chains. The molecules within the fiber cross-section are normally highly ordered locally, as in a crystal, but along the fiber axis there can be considerable fluctuations and branching as in synthetic and equilibrium polymers. Fiber growth thus seems to be a hybrid process that is somehow intermediate between equilibrium polymerization and the formation of ordinary crystallized structures where 3D long-range molecular order exists over large distances. Gelation in assembled fiber systems is certainly a different physical process than the formation of macroscopic branched polymers through reversibly associating monomer units. This type of gel is often brittle and will break like a rigid solid rather than deform like a flexible rubbery material, or if they do not break, these stiff fiber networks strain stiffen rather than strain soften as in flexible polymer networks. Treatment of this type of self-assembly and the consequent gels formed from them requires the consideration of a whole series of basic theoretical questions. Why and How Fibers Form. First, how and why do fibers form in the first place? Then there is the question of what factors limit the diameter of the fibers, one of the more conspicuous features of this type of growth process. There have been several recent efforts to address these problems. The tendency of molecules and particles to form 1D polymeric structures is a natural consequence of having directional intermolecular potentials, as in fluids of magnetic nanoparticles and in nanoparticles such as CdSe quantum dots, which commonly have large electrical dipole moments. Dipolar interactions (or other highly directional interactions such a directional hydrogen bonding and π-π stacking interactions5,6) are often involved, often in concert and in combination with van der Waals and many-body “hydrophobic” (3) Terech, P.; Weiss, R. G. Chem. Rev. 1997, 97, 3133. (4) Molecular Gels: Materials with Self-Assembled Fibrillar Networks; Weiss, G., Terech, P., Eds.; Springer:The Netherlands, 2006. (5) Brunsveld, L.; Folmer, B. J. B.; Meijer, E. W.; Sijbesma, R. P. Chem. Rev. 2001, 101, 4071. (6) Sijbesma, R. P.; Beijer, F. H.; Brunsveld, L.; Folmer, B. J. B.; Ky Hirschberg, J. H. K.; Lange, R. F. M.; Lowe, J. K. L.; Meijer, E. W. Science 1997, 278, 1601.

Published on Web 06/01/2009

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interactions, in the organization of supermolecular polymers and fiber structures in low-molecular-mass gelators,7,8 and an important role for dipolar interactions also seems to be implicated in the formation of amyloid fibers from proteins. Proteins and synthetic polypeptides also normally exhibit rather large macrodipole interactions.9 Specifically, proteins typically have macrodipole moments in the range of about 50 D for small proteins to 2000 D for large proteins, which are frequently observed to exhibit fiber self-assembly (1 D (D) = 3.336  10-30 C m), and the dipole moment of synthetic helical polypeptides of interest in molecular electronics self-assembly applications increases remarkably linearly with the number of amino acid groups n with a slope of about 3.5 for n e 24.10 It is suggested that this basic property of proteins explains the generic tendency of proteins, and even short polypeptides, to form amyloid (“starchlike”) fibers.11-15 Van Workum and Douglas16 previously argued, and supported their hypothesis by simulation, that the correspondingly large multipole interactions of proteins could also explain the generic tendency of proteins to form nanotubes, closed-shell, and “open” self-assembled 2D sheetlike polymers by self-assembly. The common tendency of the amyloid fibers and nanotubes to form closed rings is highly suggestive of prevalent dipolar interactions because ring formation is characteristic of dipolar particle assembly in both measurements and simulations of dipolar fluids. Given the common interactions and morphological growth patterns found in amyloid fiber formation and other modes of biological fiber formation, we group protein fiber assembly together with synthetic low-molecular-mass gelator systems in the discussion below. Abdallah and Weiss17,18 have emphasized that low-molecularmass gelators do not always involve overtly anisotropic molecular interactions (e..g., alkane chains were found to exhibit this phenomenon), so dipolar or another highly directional direction having a similar symmetry cannot be the universal origin for this type of 1D growth form. Their observations then raise basic question about what molecular and particle systems exhibit this type of organization. Some insight into the findings of Abdallah and Weiss can be obtained from the recent observations of Akcora et al.,19 who found that spherical nanoparticles with grafted polymer layers could assemble into fibers (and in some cases membranes) when these particles were dispersed within a polymer matrix. This effect, which is also found in simulations of chain-grafted stabilized nanoparticles (fullerenes) in aqueous solutions,20,21 arises as a consequence of the direct interaction between the nanoparticles (7) Brotin, T.; R.; Utermohlen, R.; Fages, F.; Bouas-Larent, H.; Desverge, J. P. J. Chem. Soc., Chem. Commun. 1991, 416. (8) Sakamoto, A.; Ogata, D.; Shikata, T.; Urakawa, O.; Hanabusa, K. Polymer 2006, 47, 956. (9) Wada, A. Adv. Biophys. 1976, 9, 1. (10) Kimura, S. Department of Material Chemistry, Kyoto University; Personal communication, July 2, 2005. (11) Ramirez-Alvarado, M.; Merkel, J. S.; Regan, L. A. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 8979. (12) Chiti, F.; Calamai, M.; Taddei, N.; Stefani, M.; Ramponi, G.; Dobson, C. M. Proc. Nat. Acad. Sci. U.S.A. 2002, 96, 3590. (13) Podesta, A.; Tiana, G.; Milani, P.; Manno, M. Biophys. J. 2006, 90, 589. (14) Bucciantini, M.; Giannoni, E.; Chiti, F.; Baroni, F.; Formigli, L.; Zurdo, J. S.; Taddei, N; Ramponi, G.; Dobson, C. M.; Stefani, M. Nature 2002, 416, 507. (15) Kono, T.; Murata, K.; Nagayama, K. FEBS Lett. 1999, 454, 122. (16) (a) Van Workum, K.; Douglas, J. F. Phys. Rev. E 2005, 71, 031502. (b) Douglas, J. F.; Van Workum, K. J. Mater. Res. 2007, 22, 19. (17) Abdallah, D. J.; Weiss, R. G. Langmuir 2000, 16, 352. (18) Abdallah, D. J.; Weiss, R. G. Adv. Mater. 2000, 12, 1237. (19) Akcora, P.; Liu, H; Kumar, S. K.; Moll, J.; Li, Y.; Benicewicz, B. C.; Schadler, L. S.; Acehan, D.; Panagiotopoulos, A. Z.; Pryamitsyn, V.; Ganesan, V.; Ilavsky, J.; Thiyagarajan, P; Colby, R. H.; Douglas, J. F. Nat. Mater. 2009, 8, 354. (20) Bedrov, D.; Smith, G. D.; Li, L. W. Langmuir 2005, 21, 5251. (21) Hooper, J. B.; Bedrov, D.; Smith, G. B. Langmuir 2007, 23, 12032.

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(which in isolation have symmetric potentials). This interaction distorts or “polarizes” the segmental layer on the surface of the particles introduced to disperse the immiscible inorganic particles better. This directional interaction is an emergent property that is induced by a conformational change in the brush layer, rather than being an intrinsic property of the isolated particles. Autosteric interaction,22 the change in the energetic interactions for self-assembling species through intermolecular interaction, is a common regulatory mechanism for self-assembly in biological systems, and we can also expect such effects to arise commonly in molecular gelator systems. (Autosteric interaction is the analog of “allosteric” or “cooperative” interactions that arise in the mutual association23 of distinct chemical species, such as the classic case of the binding of oxygen to hemoglobin.) In simple “structureless” polymers without any overt amphiphilic character, chain folding is the most obvious mechanism by which this spontaneous breaking of the symmetry properties of the intermolecular interaction can occur. Chain folding is indeed sufficient for alkanes, along with other structural uniform biological molecules such as DNA, to self-assemble into fibers having a remarkably similar geometrical structure to that of synthetic gelator materials,24 and alkanes and even many high-molecularmass synthetic polymers, where many chain folds are required, exhibit fiber formation and gelation and other phenomenological characteristics of low-molecular-mass gelators (see discussion below). There are such striking similarities between polymer crystallization and gelator assembly morphologies that it is not clear whether these are really distinct phenomena. Directional interactions can also arise from associative interactions between species, such as ion association of a protein that triggers a protein conformational change or the formation of an amphiphilic complex of two molecules by mutual association that subsequently self-assembles. Doing the “Twist”. Next, we consider the common tendencies of assembled fibers to twist and for the fiber diameter to be constant under fixed thermodynamic conditions. One prevalent idea about this basic phenomenon is that the fiber diameter and their tendency to form twisted fibers are both derived from an interplay between the elastic distortion energy associated with the twisting of the assembled molecular threads composing the fiber and the lateral attractive interactions between the fibers.25 Other researchers have emphasized the role of molecular chirality in this fiber-twisting phenomenon.26 An interplay between long- and short-range intermolecular interactions or dipolar and quadrupolar interactions has reasonably been suggested to account for the formation of chiral fibers, such as tubulin, where competition between these interactions gives rise to a scale that determines the fiber diameter, the chirality, and even the topological characteristics.16,27,28 There are still other interesting arguments for the emergence of chirality in assembling fiber systems that emphasize molecular packing effects in the origin of chiral fibers.29,30 Theoretical discourse on this basic problem is ongoing.

(22) Caspar, D. L. D. Biophys. J. 1980, 32, 103. (23) Dudowicz, J.; Freed, K. F.; Douglas, J. F. J. Phys. Chem. B 2008, 112, 16193. (24) Ungar, G.; Stejny, J.; Keller, A.; Bidd, I.; Whiting, M. C. Science 1985, 386, 386. (25) Turner, M. S.; Briehl, R. W.; Ferrone, F. A.; Josephs, R. Phys. Rev. Lett. 2003, 90, 128103. (26) Grason, G. M.; Bruinsma, R. F. Phys. Rev. Lett. 2007, 99, 098101. (27) Fejer, S. N.; Wales, D. J. Phys. Rev. Lett. 2007, 99, 086106. (28) Miller, A.; Wales, D. J. J. Phys. Chem. B 2005, 109, 23109. (29) (a) Pauling, L. Disc. Farad. Soc. 1953, 170, 13. (b) Pauling, L. Science 1949, 110, 543. (30) Schnur, J. Science 1993, 262, 1669.

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How Fibers Grow in Length. There are also basic unanswered questions about how the fibers grow and even about what factors are qualitatively relevant to understanding their rate of growth. Some recent simulations have indicated a growth process in which polymer chains first self-assemble, as in the formation of equilibrium polymers, and these polymeric chains then subsequently bundle into fibers driven by their transverse intermolecular interactions.31 Observations on growing amyloid fibers, which seem to share many features of synthetic gelator molecule assembly, indicate that the fibers can grow with a welldefined velocity.32 Does this mean the fibers grow at a constant rate from the ends of the fibers, amounting to a kind of 1D crystallization process? The rate of fiber growth and gelation in fiber systems is also rather commonly observed to increase with cooling, provided the ordering occurs upon cooling. (Aqueous gelators, especially those of biological origin, can form fibers and gel upon heating, and thus heating can initiate fiber growth and accelerate fiber gelation in these systems. Self-assembly upon heating is not specific to fiber self-assembly; this phenomenon is also prevalent in viral capsid assembly, micelle formation, etc.) Although there is even greater uncertainty about the dynamics of the fiber assembly, recent fundamental calculations for the assembly kinetics of equilibrium polymer chains provide some insights.33 In the simplified model of supermolecular chain assembly kinetics developed by Sciortino et al.,33 the mass distribution is predicted to remain exponential throughout the polymerization process following a temperature jump into a regime below the concentration dependent polymerization line Tp(c), where c is the concentration of associating species. This transition line normally exhibits a nearly universal Arrhenius1 (alternatively, van’t Hoff or Eldrige-Ferry) variation in selfassembling systems, including low-molecular-mass gelators, micelle formation, thermally reversible associating polymer gels, living polymers, magnetic particle assembly, and so forth. This general phenomenology is extremely encouraging for the development of a general theory, despite the many complexities involved. In particular, Tp(c) defines the phase boundary of the self-assembly transition, and its determination is often an essential part of characterizing the entropic and enthalpic free-energy parameters governing self-assembly. (Strictly speaking, selfassembly transitions are frequently “rounded” or “infinite-order” thermodynamic transitions,34 so they are not phase transitions proper; phase boundary is thus a loose term introduced by analogy to phase boundaries used to characterize the thermodynamic parameters governing phase separation.) Sciortino et al.’s model of chain self-assembly kinetics33 also predicts that the rate of ordering becomes more rapid as the undercooling, the absolute value of the difference between Tp(c) and the temperature of polymerization, becomes larger.16 Thus, the rate of ordering increases with the magnitude of the thermodynamic driving force for ordering as in most thermodynamic transitions (provided that slowing down of the kinetics due to glass formation is avoided). The physical picture of the self-assembly kinetics provided by this analytic statistical mechanical model of chain assembly is remarkably simple.33 The system evolves from the high-temperature disordered state to the low-temperature equilibrium assem(31) Huisman, B. A. H.; Bolhuis, P. G.; Fasolino, A. Phys. Rev. Lett. 2008, 100, 188301. (32) Goldsbury, C.; Kistler, J.; Aebi, U.; Arvinte, T.; Cooper, G. J. S. J. Mol. Biol. 1999, 285, 33. (33) Sciortino, F.; DeMichele, C.; Douglas, J. F J. Phys.: Condens. Matter 2008, 20, 155101. (34) Dudowicz, J.; Freed, K. F.; Douglas, J. F. J. Chem. Phys. 2000, 113, 434.

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bled state by passing through a succession of equilibrium states having progressively lower temperature so that equilibrium theory can be used to calculate the properties of the system as a function of time where the effective temperature in this “aging” process is related to the waiting time tw after the initial temperature jump. Direct observation of the growth of fibers formed from the self-assembly of preassembled DNA tiles into nanotube fibers by atomic force microscopy has confirmed this mode of fiber growth.35 In particular, the size distribution was found to remain exponential throughout the growth process, the average length L of the fibers increased progressively in a sigmoidal fashion to its long-time-limit equilibrium value corresponding to the polymerization temperature, and the average long-time size exhibited an Arrhenius temperature dependence, all properties predicted by the chain assembly theory.7 Recently, a fundamental theory of stress relaxation in solutions of equilibrium polymers has also been developed,36 and this allows, in conjunction with the kinetic assembly theory just described, a description of the evolution (“aging”) of the viscoelastic properties (e.g., shear relaxation as a function of time and the evolving structural relaxation time) of these solutions as a function of the waiting time tw after jumping into the instable region for chain growth. This correspondence implies a specific relationship between temperature T in the equilibrium theory and tw. The kinetic assembly theory of polymer chain assembly has recently extended to branched equilibrium polymerization growth. These developments provide a solid foundation for developing a fundamental description of the fiber growth kinetics, but the theory will require further developments before it can be applied reliably to fiber growth. The time-resolved electron and atomic force microscopy study of intermediate filament formation by Portet et al.37 described in this issue reminds us of the limitations of the simple isodesmic or the simple model of the equilibrium association of monomers into polymeric structures without regard to constraints acting on the assembly process, such as thermal activation of the polymerization process, multifunctional associative interactions, assembly equilibria associated with the formation of subunits (e.g., the DNA tiles), topological constraints on the assembly structure (e.g., ring formation), geometrical confinement, and so forth, that can modify the cooperativity of the self-assembly process.1 The common case of thermal activation of the associating species (which leads to effects similar to the preassembly of the associating units into monomers for the subsequent assembly) leads to a qualitatively different mass distribution of the polymers. In this case, the mass distribution characteristically exhibits a peak at intermediate times as the system evolves toward the final equilibrium state where the mass distribution assumes its characteristic exponential form. Basically, the progressively growing equilibrium polymers have a tendency to “eat their young”,33 resulting in a transient depletion of the monomeric form of the assembling species. This is exactly the type of kinetic evolution observed by Portet et al.37 in their in vitro observations of the assembly of vimentin proteins into mature intermediate filaments. It is suggested that the assembly process of the “unit-length filaments” of vimentin, composed of about eight vimentin tetramers, plays a role in creating this cooperativity effect on the assembly dynamics.22 This type of preassembly is a rather common phenomenon in fiber assembly processes and has its analog in the formation of spherical micelle “monomers” in wormlike (35) Ekani-Nkado, A.; Kumar, A.; Fygenson, D. K. Phys. Rev. Lett. 2004, 93, 268301. (36) Stukalin, E.; Douglas, J. F.; Freed, K. F. J. Chem. Phys. 2008, 129, 094901. (37) Porte, P.; Mucke, N.; Kirmse, R; Langowski, J.; Beil, M.; Herrmann, H. Langmuir 2009, 25, 10.1021/la900509r (to appear in this issue).

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micelle formation and in the DNA tiles’ formation of EkaniNkado et al.35 in connection with the DNA nanotube formation. This complex evolution in the assembly dynamics of the intermediate filaments seen by Portet et al.37 is thus probably more of a norm than an exception, and more theoretical work is required in modeling the dynamics of self-assembly (and the corresponding problem of stress relaxation) subject to cooperativity-modifying constraints. Hierarchical Self-Assembly. Once the basic “protofibrils”, composed of some fixed number of molecular threads in amyloid fibers, have formed, then these structures often form hierarchical structures involving twisted fibrils of fibrils and twisted fibrils of the former fibrils. We are then led to a consideration of hierarchical self-assembly in which some basic units organize into a structure that is similar in form and interaction to the original species, which in turn organizes into larger such structures recursively. Dudowicz et al.38 have formulated a minimal model of hierarchical self-assembly in which n-gons assemble into n-gons having the same shape in a hierarchical fashion and have found that such systems (which can be realized experimentally) have an infinite hierarchy of nested self-assembly thermodynamic transitions in their model where there are distinct thermodynamic transition lines for each level of this hierarchical organization process. Moreover, this model reveals the coexistence between structures of different organizational complexity and thermodynamic transitions between levels of the organization as the temperature or the concentration of the associating species is varied. Nyrkova et al. and others following them39-41 have formulated a specific, albeit rather computationally complex, model of the hierarchical organization of gelator fibrils where the protofibrils form tapelike twisted structures reflecting the chirality of the gelator molecules, the intermolecular gelator molecule interactions, and the elastic deformation energy of the fibrils. The more complicated theory of Nyrkova et al.39,40 shares many qualitative features of the idealized hierarchical self-assembly model of Dudowicz et al.,38 and this model appears to be an excellent starting point for future theoretical efforts aimed at the quantitative modeling of hierarchical fibril self-assembly thermodynamics and dynamics. Fiber Branching and Spherulites. Another generic aspect of fibril assembly, and the associated gelation process, that makes this type of self-assembly process so practically useful, is a general tendency for the fibrils to branch upon varying temperature, gelator concentration and other thermodynamic parameters.42-44 This behavior invites comparison to polycrystalline growth in crystallization processes under far-from-equilibrium conditions where one observes a generic development of needlelike growth and the branching of these structures to form spherulites (polycrystalline growth forms having a spherical average form due to the random branching of the crystals forming these structures). Spherulites seem to be rather general, if not universal, characteristics of both synthetic and amyloid gelator systems. It is also normal in both of these classes of gelators to see a progressive change in gel morphology from locally aligned fibers forming a (38) Dudowicz, J.; Freed, K. F.; Douglas, J. F. “An Exactly Solvable Model of Hierarchical Self-Assembly”, J. Chem. Phys. (to appear). (39) Nyrkova, I. A.; Semenov, A. N.; Aggeli, A.; Bell, M.; Boden, N.; McLeish, T. C. B. Eur. Phys. J. B 2000, 17, 499. (40) Aggeli, A.; Nyrkova, I. A.; Bell, M.; Harding, R.; Carrick, L.; McLeish, T. C. B.; Semenov, A. N.; Boden, Proc. Nat. Acad. Sci. U.S.A. 2001, 98, 11857. (41) van Gestel, J.; Leeuw, S. W. Biophys. J. 2006, 90, 3134. (42) Galkin, O.; Vekilov, P. G. J. Mol. Biol. 2004, 336, 43. (43) Samuel, R. E.; Salmon, E. D.; Briehl, R. W. Nature 1990, 345, 833. (44) Forgacs, G.; Newman, S. A.; Hinner, B.; Maier, C. W.; Sackmann, E. Biophys. J. 2003, 84, 1272.

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nematic gel97 to fractal fiber and spherulitic geometries at high branching densities where the gels form through the impingement of the growing fiber network and where the branching rate of the fibers is relatively low and high, respectively. (See the schematic illustration of this type of gelation in Figure 1 of Raghavan.45) Increasing the thermodynamic driving force for ordering by varying the temperature or concentration evidently controls the extent of branching in these gels, and through the control of the thermodynamic conditions of assembly significant control of the properties of these systems can be obtained.46,47 Granasy et al.48 have recently made significant progress in modeling the origin of this type of polycrystalline growth based on phase field theory in the context of understanding spherulite growth in ordinary crystallization. In this coarse-grained field theory of spherulitic growth, the branching of the network is caused by the presence of overt structural heterogeneity (“dirt”) or dynamic heterogeneity (molecular clustering) associated with the intrinsic fluctuations of local ordering in self-assembling systems and glass-forming liquids that are predicted by the theory of Granasy et al. to give rise to a similar disordering effect on fibril organization. In particular, the dynamic heterogeneity engenders a breakdown of the Stokes-Einstein relation in glass-forming49,50 and selfassembling fluids51,52 that makes the rotational mobility relatively small in comparison with the mobility for translational molecular motions, which in turn stabilizes grain misorientations at the crystal that give rise to grains having generally a different orientation from that of the parent crystal. Thus, intrinsic or extrinsic sources of structural order in the fluid cause the growing ordered regions to “spray out” into fractal networks or spacefilling spherulites. This secondary nucleation mode of polycrystalline growth is recognized phenomenologically in the formation of spherultic structures in sickle cell hemoglobin and other amyloid proteins. An important implication of this model of the origin of polycrystalline growth is that molecular and nanoparticle additives that modify the fragility of glass formation can be expected to alter the branching frequency of the gelator network, thereby changing the properties rather considerably. Specifically, it is noted that glycerol is a known agent for reducing the extent of dynamic heterogeneity in aqueous fluid and sugar media and this additive and polyols are often added as stabilizing agents to enhance the degree of ordering of proteins.53 Salts can also be active agents in this regard, especially higher-valent salts, which could explain the impressively large range of properties exhibited by chitin-based fibrillar materials composing the exoskeleton of insects54 and the correlation of these property changes with salt concentration. At any rate, it should be interesting to observe the extent to which this class of additives alters the polycrystalline structure and mechanical properties of gelator systems. A shortcoming of the coarse-grained phase-field theory of Granasy et al.48 is that it does not really address the origin of the nearly universal tendency of crystals to form extended fiber structures at high undercooling, regardless of the symmetries of

(45) Raghavan, S. R. Langmuir 2009, 25, 10.1021/la901513w (to appear in this issue). (46) Huang, H.; Raghavan, S. R.; Terech, P.; Weiss, R. G. J. Am. Chem. Soc. 2006, 128, 15341. (47) Wang, R.; Liu, X.-Y.; Xiong, J.; Li, J. J. Phys. Chem. B 2006, 110, 7275. (48) Granasy, L.; Pusztai, T.; Tegze, G.; Warren, J. A.; Douglas, J. F. Phys. Rev. E 2005, 72, 011605. (49) Ediger, M. D. Annu. Rev. Phys. Chem. 2000, 51, 99. (50) Chang, I.; Silescu, H. J. Phys. Chem. B 1997, 101, 8974. (51) Bedrov, D.; Smith, G.; Douglas, J. F. Polymer 2004, 45, 3961. (52) Guo, L.; Luijten, E. J. Polym. Sci. B 2005, 43, 959. (53) Sousa, R. Acta Crystallogr. 1995, D51, 271. (54) Vincent, J. F. V.; Wegst, U. G. K. Anthropod Struct. Develop 2004, 33, 187.

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the crystal lattice formed at equilibrium. The initial fiber geometry is a basic input into this coarse-grained model. Novel Kind of Gelation. The problem of understanding the nature of solidification in these gels is also a matter demanding attention through simulation and theoretical development. The formation of a solid through the impingement of branched fiber structures of varying degrees of development through the secondary nucleation of fibers evidently has rather little to do with classical geometrical percolation theory, which considers the growth of geometrical connectivity in randomly placed bodies in space. It is true that the geometrical connection of the growing clusters is important for the formation of a gel with a nonvanishing shear modulus G, but these structures clearly have correlated internal structures. Goldbart and Goldenfeld55 have formulated a general theory of solidification in amorphous materials that provides a framework for considering this type of solidification, including the existence and interpretation of power law scaling in the frequency dependence of the real and imaginary components of the frequency-dependent shear modulus near the solidification transition that is often observed in the solidification of gelators and interpreted heuristically in terms of percolation theory. Gelator fibers also tend to be highly ordered locally, exhibiting well-defined crystallographic symmetries, and this tends to make the fibers themselves brittle or relatively stiff if they do not break readily, a point emphasized by Raghavan in his Perspective.45 Biological gel fibers are also characteristically highly ordered locally, but the many internal degrees of freedom of the complex molecules forming these fibers and their variable diameter with thermodynamic conditions allow these fibers to sustain more appreciable deformation before breaking, giving such fibers high structural integrity under mechanical deformation and a rigidity (shear modulus) that can be tuned over many orders of magnitude.56 The existence of fiber networks composed of this kind of stiff, unbreakable fiber leads to new elastic effects associated with the intrinsic nonlinear elasticity of these materials. In particular, biological fiber gels characteristically exhibit strain stiffening57 and negative first normal stresses,58 a behavior quite unlike ordinary rubbery materials composed of flexible polymer chains or fibers.59,60 The nonlinear elastic properties of stiff polymer networks has been argued to be essential to cell function, and similar elastic properties are found in polymer nanocomposites filled with carbon nanotubes.61,62 Following brittle network fracture or strain stiffening followed by abrupt strain softening63,64 in protein gelators or other stiff, tough fiber networks, the network must regenerate its form starting from a very different structure than the one existing after its initial growth. However, the fragments arising from the rupture of the gel provide nuclei for the reinitiation of growth where slow nucleation processes are not required to restart growth (55) (a) Goldenfeld, N.; Golbart, P. Phys. Rev. A 1987, 45, R-5343. (b) Golbart, P.; Goldenfeld, N. Phys. Rev. Lett. 1987, 58, 2676. (56) Knowles, T. P.; Fitzpatrick, A. W.; Meehan, S.; Mott, H. R. Science 2007, 318, 1900. (57) Storm, C.; Pastore, J. J.; MacKintosh, F. C.; Lubensky, T. M.; Jamney, P. A. Nature 2005, 435, 191. (58) Jamney, P. A.; McCormick, M. E.; Rammensee, S; Leight, J. L. Nat. Mater. 2007, 6, 48. (59) Treloar, L. R. G. The Physics of Rubber Elasticity; Oxford University Press: New York, 1949. (60) Han, W. H.; Horkay, F.; McKenna, G. B. Math. Mech. Solids 1999, 4, 139. (61) Kharchenko, S.; Douglas, J. F.; Obrzut, J.; Grulke, E. A.; Migler, K. B. Nat. Mater. 2004, 3, 564. (62) Cantournet, S.; Boyce, M. C.; Tsou, A. H. J. Mech. Phys. Solids 2007, 55, 1321. (63) Trepat, X.; Deng, L.; Ann, S. S.; Navajas, D.; Tschumperlin, D. J.; Gerthoffer, W. T.; Butler, J. P.; Fredberg, J. J. Nature 2007, 447, 592. (64) Chaudhuri, O.; Parekh, S. H.; Fletcher, D. A. Nature 2007, 445, 295.

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after structural breakdown, an effect of great significance in neural degenerative amyloid diseases (e.g., mad cow disease) where these “seeds” can be a source of disease through their initiation of fibril growth.65 Fiber regrowth requires the redissolution of relatively low energy fiber structures, a phenomenon that can be extremely sluggish so that the fiber network may tend to remain in a fractured state for some time without recovery. (See Raghavan.45) The aging and material history dependence of the self-assembling systems can be expected to complex strongly, and further modeling is evidently needed. The intermolecular interactions giving rise to fiber formation can be rather strong, leading to a common tendency for selfassembled fibers to resist dissolution once formed. This effect has obvious implications on the thermal stability of these fibers in applications of self-assembly66 and the pathological effect of insoluble amyloid fibers in many diseases.67,68 Of course, in some contexts the highly nonequilibrium nature of the self-assembled fiber material can be essential to biological function (e.g., cellulose fibers in trees, exoskeletons on insects and crabs, cartilage of animals), engendering a robust stability of the material once formed under varying thermodynamic conditions. Understanding and Controlling Polymorphism. The selfassembly of fibrils is often observed to be polymorphic, where distinct assembly morphologies seem to coexist in the ordering process. This polymorphism arises in the number of molecular chains within the basic protofilament fiber and in the number and configuration of such basic filaments in the protofilament fiber bundles. These structures apparently become fixed after a relatively short time, and the growth form becomes persistent thereafter. This phenomenon raises the difficult theoretical question of how these structures become “nucleated” and raises practical questions about how the form of these structures can be regulated for applications of self-assembled materials. This type of polymorphism is normally highly regulated in biological systems, so how do living systems do this? Van Workum and Douglas16 observed that model particles with multipole (quadrupole and hexapole corresponding to a triangular configuration of dipoles in a planar head-to-tail configuration) interactions, in conjunction with a short-range van der Waals interactions modeled by a Lennard-Jones potential, tend to form well-defined low-energy clusters having different symmetries and that the emergent symmetries of the seed structures encode the symmetry properties of the ultimate self-assembled structure. Classical nucleation theory, which assumes the formation of a spherical nucleus clearly, simply does not describe this situation. ten Wolde et al.69 have examined the simpler situation of the dipolar interaction version of this model (the Stockmayer fluid) and have found that the condensation of a gas of such particles into a droplet occurred first through the formation of polymer chains of dipoles that subsequently collapsed into droplets of a highly correlated fluid. Liu et al.70 have recently examined protein crystallization based on a model having directional interactions (multiple sticky spots to model the anisotropic interactions of real proteins) and found clear evidence that particle self-assembly can select out, within limits, the resulting protein crystal symmetry by seeding the crystallization process. By implication, we expect the (65) Padrick, S. B.; Miranker, A. D. Biochemistry 2002, 41, 4694. (66) Zhang, S.; Holmes, T.; Lockshin, C.; Rich, A. Proc. Natl. Acad. Sci. U.S.A. 1993, 90, 3334. (67) Koo, E. H.; Lansbury, P. T.; Kelly, J. W. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 9989. (68) Lomas, D.; Evans, D. L.; Finch, J. T.; Carrell, R. W. Nature 1992, 357, 605. (69) ten Wolde, P. R.; Oxtoby, D. W.; Frenkel, D. Phys. Rev. Lett. 2004, 92, 045502. (70) Liu, H.; Kumar, S. K.; Douglas, J. F. Phys. Rev. Lett., in press.

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existence of polymorphic seeds from the assembly process, and we should naturally expect the assembly structures to be polymorphic as observed in simulation by Van Workum and Douglas, who also showed that the polymorphism, as well as the fluctuations in the assembly time, can be controlled through the introduction of seeds that regulate both the assembly form and kinetics.16 These observations suggest a practical strategy for controlling the assembly morphology through a selective seeding process, an effect recognized to exist in biological selfassembly.27 The energy landscape approach to self-assembly71 involves a systematic determination, or at least a sampling, of the physically accessible particle cluster states, which should allow the development of a generalized nucleation theory describing the kinetics of formation of the various low-energy cluster states involved in the early stages of the self-assembly process. Because the formation of these clusters is a relatively rare event, transitionstate Monte Carlo sampling should be an appropriate computational tool for studying the emergence of polymorphic ordering in conjunction with the energy landscape method,71 at least for simpler molecular models where this type of computation should be possible.

Current Challenges for Modeling Fiber Growth and Fiber Gelation Despite progress at many levels, there are still many basic gaps in our theoretical understanding of fiber growth and gelation. First, there is a basic need to understand how this form of growth initiates. Fiber growth and spherulite formation is a common growth mode in crystallization under far-from-equilibrium conditions, and the reasons for this general tendency (essentially independent of the equilibrium crystal lattice cell geometry) are unclear, although recent work has suggested that a relatively rapid diffusionless growth mode under low temperatures or high supersaturation conditions is somehow associated with this phenomenon.72 This basic scientific problem points to the need for an improved theory of nucleation that is adapted to describe the physics of fiber growth. Molecular dynamics can play an important role in developing physical intuition about the physical factors that govern the initial stages of fiber growth, which could then serve as the basis for the development of such a theory. One reason that that theory is so undeveloped is that the early stage of fiber assembly is normally inaccessible experimentally. Simulation can also provide valuable information into what physical interactions give rise to a propensity to form into organize twisted filaments having a specific width. Once this issue is definitively resolved, then modeling efforts will be in better position to make progress in modeling hierarchical fiber assembly. Resolving the essential qualitative driving forces for this type of assembly should also provide a basic step in exerting phenom(71) Wales, D. J. Energy Landscapes; Cambridge University Press: New York, 2003. (72) Sun, Y.; Xi, H.; Chen, S.; Ediger, M. D.; Yu, L. J. Phys. Chem. B 2008, 112, 5594.

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enological control over fiber formation and gelation so that efforts on this problem are urgent. Once the nature of the fiber assembly at a molecular scale becomes better resolved, it will then be necessary to make advances in modeling the large-scale growth process leading to fiber branching and spherulite formation. The time scales involved are prohibitive for molecular dynamics, so it will then be necessary to incorporate information about the nucleation and form of the fibers from molecular simulation into the phase field or coarse-grained field models (lattice Boltzmann) of large-scale fiber growth. At present, phase-field simulations of spherulite growth48 are limited to two spatial dimensions, and a generalization to three dimensions and to sparse fiber networks will be required to describe the basic aspects of real 3D fiber networks, even in a highly coarse-grained sense. There is also a need for parallel studies aimed at better understanding how molecular additives alter the dynamic heterogeneity of crystallizing fluids on the molecular scale and the impact of these changes on the ultimate crystalline structure of the fiber network. Measurements are needed to complement these efforts and to validate effects found by simulation. Finally, efforts must be made in modeling the large changes in the viscosity of suspensions of polymerizing fibers and the resulting gel formation that arise during fiber growth and fiber impingement, respectively. Powerful finite element codes exist for this type of property calculation, but these computational methods need to be integrated with the phase-field modeling of the fiber growth process to make predictions about the evolution of gelator systems in the course of their self-organization. The successful modeling of gel formation in gelator systems then requires a multiscale, multidisciplinary approach that addresses the specific interactions, thermodynamics, and processes for fibril formation processes through a combination of molecular modeling and molecular dynamics simulation along with measurements validating these molecular models. Furthermore, large-scale simulation methods and complementary measurements are required to address the fiber branching processes that play a crucial role in regulating rigidity and other properties of this class of gels. Unconventional computations of this kind are extremely challenging in three dimensions, and a serious multidisciplinary effort will be required to obtain a satisfactory description of this phenomenon. However, many thermodynamic aspects of this type of self-assembly are shared rather generically by self-assembling systems so that a thermodynamic framework already exists for measuring parameters that serve to characterize the thermodynamic stability and interactions of these complex systems. Improving the theoretical understanding and characterization methods for simpler self-assembling systems will also help move the broad field of molecular assembly incrementally forward. Supporting Information Available: An extended list of references categorized according to section and topic of the main text. This material is available free of charge via the Internet at http://pubs.acs.org.

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