Lattice model of living polymerization. III. Evidence for particle clustering from phase separation properties and “rounding” of the dynamical clustering transition

Approach to hyperuniformity in a metallic glass-forming material exhibiting a fragile to strong glass transition

We investigate a metallic glass-forming (GF) material (Al₉₀Sm₁₀) exhibiting a fragile-strong (FS) glass-formation by molecular dynamics simulation to better understand this highly distinctive pattern of glass-formation in which many of the usual phenomenological relations describing relaxation times and diffusion of ordinary GF liquids no longer apply, and where instead genuine thermodynamic features are observed in response functions and little thermodynamic signature is exhibited at the glass transition temperature, T_g. Given the many unexpected similarities between the thermodynamics and dynamics of this metallic GF material with water, we first focus on the anomalous static scattering in this liquid, following recent studies on water, silicon and other FS GF liquids. We quantify the "hyperuniformity index" H of our liquid, which provides a quantitative measure of molecular "jamming". To gain insight into the T-dependence and magnitude of H, we also estimate another more familiar measure of particle localization, the Debye-Waller parameter 〈u²〉 describing the mean-square particle displacement on a timescale on the order of the fast relaxation time, and we also calculate H and 〈u²〉 for heated crystalline Cu. This comparative analysis between H and 〈u²〉 for crystalline and metallic glass materials allows us to understand the critical value of H on the order of 10^-3 as being analogous to the Lindemann criterion for both the melting of crystals and the "softening" of glasses. We further interpret the emergence of FS GF and liquid-liquid phase separation in this class of liquids to arise from a cooperative self-assembly process in the GF liquid.

Dynamic heterogeneity, cooperative motion, and Johari-Goldstein [Formula: see text]-relaxation in a metallic glass-forming material exhibiting a fragile-to-strong transition

We investigate the Johari-Goldstein (JG) [Formula: see text]-relaxation process in a model metallic glass-forming (GF) material ([Formula: see text]), previously studied extensively by both frequency-dependent mechanical measurements and simulation studies devoted to equilibrium properties, by molecular dynamics simulations based on validated and optimized interatomic potentials with the primary aim of better understanding the nature of this universal relaxation process from a dynamic heterogeneity (DH) perspective. The present relatively low temperature and long-time simulations reveal a direct correspondence between the JG [Formula: see text]-relaxation time [Formula: see text] and the lifetime of the mobile particle clusters [Formula: see text], defined as in previous DH studies, a relationship dual to the corresponding previously observed relationship between the [Formula: see text]-relaxation time [Formula: see text] and the lifetime of immobile particle clusters [Formula: see text]. Moreover, we find that the average diffusion coefficient D nearly coincides with [Formula: see text] of the smaller atomic species (Al) and that the 'hopping time' associated with D coincides with [Formula: see text] to within numerical uncertainty, both trends being in accord with experimental studies. This indicates that the JG [Formula: see text]-relaxation is dominated by the smaller atomic species and the observation of a direct relation between this relaxation process and rate of molecular diffusion in GF materials at low temperatures where the JG [Formula: see text]-relaxation becomes the prevalent mode of structural relaxation. As an unanticipated aspect of our study, we find that [Formula: see text] exhibits fragile-to-strong (FS) glass formation, as found in many other metallic GF liquids, but this fact does not greatly alter the geometrical nature of DH in this material and the relation of DH to dynamical properties. On the other hand, the temperature dependence of the DH and dynamical properties, such as the structural relaxation time, can be significantly altered from 'ordinary' GF liquids.

Predictive relation for the α-relaxation time of a coarse-grained polymer melt under steady shear

We examine the influence of steady shear on structural relaxation in a simulated coarse-grained unentangled polymer melt over a wide range of temperature and shear rates. Shear is found to progressively suppress the α-relaxation process observed in the intermediate scattering function, leading ultimately to a purely inertially dominated β-relaxation at high shear rates, a trend similar to increasing temperature. On the basis of a scaling argument emphasizing dynamic heterogeneity in cooled liquids and its alteration under material deformation, we deduce and validate a parameter-free scaling relation for both the structural relaxation time τα from the intermediate scattering function and the "stretching exponent" β quantifying the extent of dynamic heterogeneity over the entire range of temperatures and shear rates that we can simulate.

Lattice theory of competitive binding: Influence of van der Waals interactions on molecular binding and adsorption to a solid substrate from binary liquid mixtures

The reversible binding of molecules to surfaces is one of the most fundamental processes in condensed fluids, with obvious applications in the molecular separation of materials, chromatographic characterization, and material processing. Motivated in particular by the ubiquitous occurrence of binding processes in molecular biology and self-assembly, we have developed a lattice type theory of competitive molecular binding to solid substrates from binary mixtures of two small molecule liquids that interact between themselves by van der Waals forces in addition to exhibiting binding interactions with the solid surface. The derived theory, in contrast to previously existing theoretical frameworks, enables us to investigate the influence of van der Waals interactions on interfacial binding and selective molecular adsorption. For reference, the classic Langmuir theory of adsorption is recovered when all van der Waals interaction energies between the molecules in the bulk liquid phase and those on the surface are formally set to zero. Illustrative calculations are performed for the binding of molecules to a solid surface from pure liquids and from their binary mixtures. The properties analyzed include the surface coverage θ, the binding transition temperature T_bind, the individual surface coverages, θ_A and θ_C, and the relative surface coverages, σ_AC≡θ_A/θ_C or σ_CA≡θ_C/θ_A. The latter two quantities coincide with the degrees of adsorption directly determined from experimental adsorption measurements. The Langmuir theory is shown to apply formally under a wide range of conditions where the original enthalpies (Δh or Δh_A and Δh_C) and entropies (Δs or Δs_A and Δs_C) of the binding reactions are simply replaced by their respective "effective" counterparts (Δh_eff or Δh_A^eff and Δh_C^eff and Δs_eff or Δs_A^eff and Δs_C^eff), whose values depend on the strength of der Waals interactions and of the "bare" free energy parameters (Δh or Δh_A and Δh_C, and Δs or Δs_A and Δs_C). Numerous instances of entropy-enthalpy compensation between these effective free energy parameters follow from our calculations, confirming previous reports on this phenomenon obtained from experimental studies of molecular binding processes in solution.

Valence, loop formation and universality in self-assembling patchy particles

Patchy particles have emerged as an attractive model to mimic phase separation and self-assembly of globular proteins solutions, colloidal patchy particles, and molecular fluids where directional interactions are operative. In our previous work, we extensively explored the coupling of directional and isotropic interactions on both the phase separation and self-assembly in a system of patchy particles with five spots. Here, we extend this work to consider different patch valences and isotropic interaction strengths with an emphasis on self-assembly. Although the location of self-assembly transition lines in the temperature-density plane depend on a number of parameters, we find universal behavior of cluster size that is dependent only on the probability of a spot being bound, the patch valence, and the density. Using these principles, we quantify both the mass distribution and the shape for all clusters, as well as clusters containing loops. Following the logical implications of these results, combined with a simplified version of a mean-field theory that incorporates Flory-Stockmayer theory, we find a universal curve for the temperature dependence of cluster mass and a universal curve for the fraction of clusters that contain loops. As the curves are dependent on the particle valence, such results provide a method for parameterizing patchy particle models using experimental data.

Coupling of isotropic and directional interactions and its effect on phase separation and self-assembly

The interactions of molecules and particles in solution often involve an interplay between isotropic and highly directional interactions that lead to a mutual coupling of phase separation and self-assembly. This situation arises, for example, in proteins interacting through hydrophobic and charged patch regions on their surface and in nanoparticles with grafted polymer chains, such as DNA. As a minimal model of complex fluids exhibiting this interaction coupling, we investigate spherical particles having an isotropic interaction and a constellation of five attractive patches on the particle's surface. Monte Carlo simulations and mean-field calculations of the phase boundaries of this model depend strongly on the relative strength of the isotropic and patch potentials, where we surprisingly find that analytic mean-field predictions become increasingly accurate as the directional interactions become increasingly predominant. We quantitatively account for this effect by noting that the effective interaction range increases with increasing relative directional to isotropic interaction strength. We also identify thermodynamic transition lines associated with self-assembly, extract the entropy and energy of association, and characterize the resulting cluster properties obtained from simulations using percolation scaling theory and Flory-Stockmayer mean-field theory. We find that the fractal dimension and cluster size distribution are consistent with those of lattice animals, i.e., randomly branched polymers swollen by excluded volume interactions. We also identify a universal functional form for the average molecular weight and a nearly universal functional form for a scaling parameter characterizing the cluster size distribution. Since the formation of branched clusters at equilibrium is a common phenomenon in nature, we detail how our analysis can be used in experimental characterization of such associating fluids.

String model for the dynamics of glass-forming liquids

We test the applicability of a living polymerization theory to describe cooperative string-like particle rearrangement clusters (strings) observed in simulations of a coarse-grained polymer melt. The theory quantitatively describes the interrelation between the average string length L, configurational entropy Sconf, and the order parameter for string assembly Φ without free parameters. Combining this theory with the Adam-Gibbs model allows us to predict the relaxation time τ in a lower temperature T range than accessible by current simulations. In particular, the combined theories suggest a return to Arrhenius behavior near Tg and a low T residual entropy, thus avoiding a Kauzmann "entropy crisis."

Cooperative polymerization of one-patch colloids

We numerically investigate cooperative polymerization in an off-lattice model based on a pairwise additive potential using particles with a single attractive patch that covers 30% of the colloid surface. Upon cooling, these particles self-assemble into small clusters which, below a density-dependent temperature, spontaneously reorganize into long straight tubes. We evaluate the partition functions of clusters of all sizes to provide an accurate description of the chemical reaction constants governing this process. Our calculations show that, for intermediate sizes, the partition functions retain contributions from two different structures, differing in both energy and entropy. We illustrate the microscopic mechanism behind the complex polymerization process in this system and provide a detailed evaluation of its thermodynamics.

The glassy and supercooled state of elemental sulfur: vibrational modes, structure metastability, and polymer content

We report a detailed investigation of vibrational modes, structure, and dynamics of elemental sulfur in the glassy and the supercooled state, using Raman scattering and ab initio calculations. Polarized Raman spectra are recorded--for sulfur quenched from 473 K--over a broad temperature range from 93 K to 273 K where the supercooled liquid crystallized. The temperature induced shifts of the majority of the vibrational modes are determined and compared with the corresponding ones of crystalline sulfur. Analysis of the reduced isotropic spectra showed that the structure of the quenched product is composed of eight member rings (S8) and polymeric chains (Sμ) with a relative fraction comparable to that of the parent liquid at 473 K. Low temperature spectra, where spectral line broadening due to thermal effects is limited, revealed that two different polymeric species are present in the glass with distinct vibrational frequencies. Their interpretation was assisted by ab initio calculations used to simulate the vibrational frequencies of polymeric chains S(8k) (k = 1, ..., 7). Theoretical results exhibit an increasing breathing mode frequency for sulfur chains up to k = 2, although it remains constant beyond the above value. The polymeric content is metastable; heating the glass above its glass transition temperature, T(g), destabilizes the chains and drives them back to the more thermodynamically stable rings. This bond interchange mechanism provides the structural origin of a secondary relaxation process in supercooled sulfur reported long ago, which has been also considered as a complication in the correct fragility estimation of this material. Finally, the Boson peak of the glass was found to exhibit strong temperature dependence even at temperatures below T(g).

Phase field method for nonequilibrium dynamics of reversible self-assembly systems

Phase field methods are extended to describe the nonequilibrium dynamics of reversible self-assembly systems, an extension that is complicated by the mutual coupling of many non-conserved order parameters into a set of highly nonlinear partial differential equations. Further complications arise because the sum of all non-conserved order parameters equals a conserved order parameter. The theory is developed for the simplest model of reversible self-assembly in which no additional constraints are imposed on the self-assembly process since the extension to treat more complex self-assembly models is straightforward. Specific calculations focus on the time evolution of the cluster size distribution for a free association system that is rapidly dropped from one ordered state to a more ordered state within the one-phase region. The dynamics proceed as expected, thereby providing validation of the theory which is also capable of treating systems with spatial inhomogeneities.

Lattice cluster theory of associating telechelic polymers. III. Order parameter and average degree of self-assembly, transition temperature, and specific heat

The lattice cluster theory of strongly interacting, structured polymer fluids is applied to determine the thermodynamic properties of solutions of telechelic polymers that may associate through bifunctional end groups. Hence, this model represents a significant albeit natural extension of a diverse array of prior popular equilibrium polymerization models in which structureless "bead" monomers associate into chain-like clusters under equilibrium conditions. In particular, the thermodynamic description of the self-assembly of linear telechelic chains in small molecule solvents (initiated in Paper II) is systematically extended through calculations of the order parameter Φ and average degree <N> of self-assembly, the self-assembly transition temperature T(p), and the specific heat C(V) of solutions of telechelic molecules. Special focus is placed on examining how molecular and thermodynamic parameters, such as the solution composition φ, temperature T, microscopic interaction energies (ε(s) and ε), and length M of individual telechelic chains, influence the computed thermodynamic quantities that are commonly used to characterize self-assembling systems.

Theoretical issues relating to thermally reversible gelation by supermolecular fiber formation

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.

Lattice model of equilibrium polymerization. VII. Understanding the role of "cooperativity" in self-assembly

Cooperativity is an emergent many-body phenomenon related to the degree to which elementary entities (particles, molecules, organisms) collectively interact to form larger scale structures. From the standpoint of a formal mean field description of chemical reactions, the cooperativity index m, describing the number of elements involved in this structural self-organization, is the order of the reaction. Thus, m for molecular self-assembly is the number of molecules in the final organized structure, e.g., spherical micelles. Although cooperativity is crucial for regulating the thermodynamics and dynamics of self-assembly, there is a limited understanding of this aspect of self-assembly. We analyze the cooperativity by calculating essential thermodynamic properties of the classical mth order reaction model of self-assembly (FAm model), including universal scaling functions describing the temperature and concentration dependence of the order parameter and average cluster size. The competition between self-assembly and phase separation is also described. We demonstrate that a sequential model of thermally activated equilibrium polymerization can quantitatively be related to the FAm model. Our analysis indicates that the essential requirement for "cooperative" self-assembly is the introduction of constraints (often nonlocal) acting on the individual assembly events to regulate the thermodynamic free energy landscape and, thus, the thermodynamic sharpness of the assembly transition. An effective value of m is defined for general self-assembly transitions, and we find a general tendency for self-assembly to become a true phase transition as m-->infinity. Finally, various quantitative measures of self-assembly cooperativity are discussed in order to identify experimental signatures of cooperativity in self-assembling systems and to provide a reliable metric for the degree of transition cooperativity.

Insensitivity to salt of assembly of a rigid biopolymer aggrecan

Many polyelectrolytes, ranging from sulfonated polystyrene to DNA, exhibit a strong sensitivity of their phase behavior to salt concentration, especially to higher valence salts, which often lead to phase separation. We show that the stiff polyelectrolyte aggrecan exhibits a qualitatively different behavior. Specifically, the scattering properties of aggrecan solutions are exceptionally insensitive to the addition of calcium salt, conferring on aggrecan the role of an ion reservoir mediating calcium metabolism in cartilage and bone, and also providing osmotic resilience to compressive load.

Gel-like behavior in aggrecan assemblies

Aggrecan, a large biological polyelectrolyte molecule with a bottlebrush shape, forms complexes with hyaluronic acid (HA) that provide compressive resistance in cartilage. In solutions of aggrecan alone, the concentration dependence of the osmotic pressure Pi is marked by self-assembly of the molecules into aggregates. When HA is added to the solution at low aggrecan concentration c, the osmotic pressure is reduced, but in the physiological concentration range this trend is reversed. The osmotic modulus c partial differentialPi partial differentialc, which determines load bearing resistance, is enhanced in the HA-containing solutions. Dynamic light scattering (DLS) measurements show that the aggregates behave like microgels and that they become denser as the aggrecan concentration increases. The degree of densification is greatest at large distance scales in the microgels, but decreases at short distance scales. Measurements at higher resolution, involving small angle neutron scattering and small angle x-ray scattering (SAXS), confirm that at length scales shorter than 1000 angstroms, the density is independent of the concentration and that the individual bottlebrushes in the microgels retain their identity. The absence of collective diffusion modes in the relaxation spectrum, measured by DLS and neutron spin echo, corroborates the lack of interpenetration among the aggrecan subunits in the microgel. Complexation with HA modifies the long-range spatial organization of the microgels. Comparison of the scattering pattern of the individual aggrecan molecules obtained from SAXS measurements with that of the complexes measured by DLS shows that the aggrecan-HA structure is denser and is more uniform than the random microgels. This enhanced space-filling property allows higher packing densities to be attained, thus, optimizing resistance to osmotic compression.

The polymerization of actin: extent of polymerization under pressure, volume change of polymerization, and relaxation after temperature jumps

The protein actin can polymerize from monomeric globular G-actin to polymeric filamentary F-actin, under the regulation of thermodynamic variables such as temperature, pressure, and compositions of G-actin and salts. We present here new measurements of the extent of polymerization (phi) of actin under pressure (P), for rabbit skeletal muscle actin in H2O buffer in the presence of adenosine triposphate and calcium ions and at low (5-15 mM) KCl concentrations. We measured phi using pyrene-labeled actin, as a function of time (t) and temperature (T), for samples of fixed concentrations of initial G-actin and KCl and at fixed pressure. The phi(T,P) measurements at equilibrium have the same form as reported previously at 1 atm: low levels of polymerization at low temperatures, representing dimerization of the actin; an increase in phi at the polymerization temperature (Tp); a maximum in phi(T) above Tp) with a decrease in phi(T) beyond the maximum, indicating a depolymerization at higher T. From phi(T,P) at temperatures below Tp, we estimate the change in volume for the dimerization of actin, DeltaVdim, to be -307+/-10 ml/mol at 279 K. The change of Tp with pressure dTp/dP=(0.3015+/-0.0009) K/MPa=(30.15+/-0.09) mK/atm. The phi(T,P) data at higher T indicate the change in volume on propagation, DeltaVprop, to be +401+/-48 ml/mol at 301 K. The phi(t) measurements yield initial relaxation times rp(T) that reflect the behavior of phi(T) and support the presence of a depolymerization temperature. We also measured the density of polymerizing actin with a vibrating tube density meter, the results of which confirm that the data from this instrument are affected by viscosity changes and can be erroneous.

Biological Soft Materials

With one or two exceptions, biological materials are "soft", meaning that they combine viscous and elastic elements. This mechanical behavior results from self-assembled supramolecular structures that are stabilized by noncovalent interactions. It is an ongoing and profound challenge to understand the self-organization of biological materials. In many cases, concepts can be imported from soft-matter physics and chemistry, which have traditionally focused on materials such as colloids, polymers, surfactants, and liquid crystals. Using these ideas, it is possible to gain a new perspective on phenomena as diverse as DNA condensation, protein and peptide fibrillization, lipid partitioning in rafts, vesicle fusion and budding, and others, as discussed in this selective review of recent highlights from the literature.

Self-assembly of patchy particles into polymer chains: A parameter-free comparison between Wertheim theory and Monte Carlo simulation

The authors numerically study a simple fluid composed of particles having a hard-core repulsion, complemented by two short-ranged attractive (sticky) spots at the particle poles, which provides a simple model for equilibrium polymerization of linear chains. The simplicity of the model allows for a close comparison, with no fitting parameters, between simulations and theoretical predictions based on the Wertheim perturbation theory. This comparison offers a unique framework for the analytic prediction of the properties of self-assembling particle systems in terms of molecular parameters and liquid state correlation functions. The Wertheim theory has not been previously subjected to stringent tests against simulation data for ordering across the polymerization transition. The authors numerically determine many of the thermodynamic properties governing this basic form of self-assembly (energy per particle, order parameter or average fraction of particles in the associated state, average chain length, chain length distribution, average end-to-end distance of the chains, and the static structure factor) and find that predictions of the Wertheim theory accord remarkably well with the simulation results.

The Par-3 NTD adopts a PB1-like structure required for Par-3 oligomerization and membrane localization

The evolutionarily conserved Par-3/Par-6/aPKC complex is essential for the establishment and maintenance of polarity of a wide range of cells. Both Par-3 and Par-6 are PDZ domain containing scaffold proteins capable of binding to polarity regulatory proteins. In addition to three PDZ domains, Par-3 also contains a conserved N-terminal oligomerization domain (NTD) that is essential for proper subapical membrane localization and consequently the functions of Par-3. The molecular basis of NTD-mediated Par-3 membrane localization is poorly understood. Here, we describe the structure of a monomeric form of the Par-3 NTD. Unexpectedly, the domain adopts a PB1-like fold with both type-I and type-II structural features. The Par-3 NTD oligomerizes into helical filaments via front-to-back interactions. We further demonstrate that the NTD-mediated membrane localization of Par-3 in MDCK cells is solely attributed to its oligomerization capacity. The data presented in this study suggest that the Par-3 NTD is likely to facilitate the assembly of higher-order Par-3/Par-6/aPKC complex with increased avidities in targeting the complex to the subapical membrane domain and in binding to other polarity-regulating proteins.

Does equilibrium polymerization describe the dynamic heterogeneity of glass-forming liquids?

A significant body of evidence indicates that particles with excessively high or low mobility relative to Brownian particles form in dynamic equilibrium in glass-forming liquids. We examine whether these "dynamic heterogeneities" can be identified with a kind of equilibrium polymerization. This correspondence is first checked by demonstrating the presence of a striking resemblance between the temperature dependences of the configurational entropy s(c) in both the theory of equilibrium polymerization and the generalized entropy theory of glass formation in polymer melts. Moreover, the multiple characteristic temperatures of glass formation are also shown to have analogs in the thermodynamics of equilibrium polymerization, supporting the contention that both processes are varieties of rounded thermodynamic transitions. We also find that the average cluster mass (or degree of polymerization) varies in nearly inverse proportionality to s(c). This inverse relation accords with the basic hypothesis of Adam-Gibbs that the number of particles in the cooperatively rearranging regions (CRR) of glass-forming liquids scales inversely to s(c) of the fluid. Our identification of the CRR with equilibrium polymers is further supported by simulations for a variety of glass-forming liquids that verify the existence of stringlike or polymeric clusters exhibiting collective particle motion. Moreover, these dynamical clusters have an exponential length distribution, and the average "string" length grows upon cooling according to the predictions of equilibrium polymerization theory. The observed scale of dynamic heterogeneity in glass-forming liquids is found to be consistent with this type of self-assembly process. Both experiments and simulations have revealed remarkable similarities between the dynamical properties of self-assembling and glass-forming liquids, suggesting that the development of a theory for the dynamics of self-assembling fluids will also enhance our understanding of relaxation in glass-forming liquids.

Actin polymerization under pressure: A theoretical study

An extended Flory-Huggins-type equilibrium polymerization theory for compressible systems is used to describe experimental data for the unusual pressure and temperature dependence of the equilibrium polymerization of G-actin to F-actin. The calculations provide rich insights into the reaction mechanism and the thermodynamics of actin polymerization at the molecular level. Volume changes associated with individual steps of the mechanism are calculated to be DeltaVactiv=(s1*-s1)upsilon0=+1553 mlmol for the activation reaction, DeltaVdim=(s2-s1*)upsilon0=-3810 mlmol for dimerization, and DeltaVprop=(sP-s1)upsilon0=+361 mlmol for the propagation reaction, where s1upsilon0, s1*upsilon0, s2upsilon0, and sPupsilon0 are the monomer volumes in the G-actin monomer, the activated G-action, the dimer, and higher polymers, respectively. Comparison with experimental measurements is made, and discrepancies are discussed.

Physical properties of sulfur near the polymerization transition

Acoustical measurements, electron spin resonance, and Raman spectroscopy have been employed to probe sulfur over the temperature range 80-180 degrees C, which includes the polymerization transition and the supercooled liquid state. Acoustical properties (sound velocity, absorption, and impedance) have been studied with both longitudinal and transverse waves at frequencies between 500 kHz and 22 MHz. The results confirm that polymeric sulfur is a solution of long chain molecules in monomeric solvent, and that the polymerization transition is not a second-order phase transition, as was proposed theoretically. Sulfur is a viscous liquid, but not viscoelastic, both below and above the polymerization transition temperature. It is shown that the classical Navier-Stokes theory is not applicable to the sound absorption in liquid sulfur in the highly viscous state.

On the extent of polymerization of liquid sulfur at very high temperatures

The extent of polymerization of liquid sulfur up to 731 K has been obtained using Raman scattering. The data reveal the absence of a maximum in the polymerization curve. The obtained results are discussed in the spirit of computer simulations that suggest the existence of a maximum which depends on the bond breaking energy.

Lattice model of equilibrium polymerization. V. Scattering properties and the width of the critical regime for phase separation

Dynamic clustering associated with self-assembly in many complex fluids can qualitatively alter the shape of phase boundaries and produce large changes in the scale of critical fluctuations that are difficult to comprehend within the existing framework of theories of critical phenomena for nonassociating fluids. In order to elucidate the scattering and critical properties of associating fluids, we consider several models of equilibrium polymerization that describe widely occurring types of associating fluids at equilibrium and that exhibit the well defined cluster geometry of linear polymer chains. Specifically, a Flory-Huggins-type lattice theory is used, in conjunction with the random phase approximation, to compute the correlation length amplitude xi(o) and the Ginzburg number Gi corresponding, respectively, to the scale of composition fluctuations and to a parameter characterizing the temperature range over which Ising critical behavior is exhibited. Our calculations indicate that upon increasing the interparticle association energy, the polymer chains become increasingly long in the vicinity of the critical point, leading naturally to a more asymmetric phase boundary. This increase in the average degree of polymerization implies, in turn, a larger xi(o) and a drastically reduced width of the critical region (as measured by Gi). We thus obtain insight into the common appearance of asymmetric phase boundaries in a wide range of "complex" fluids and into the observation of apparent mean field critical behavior even rather close to the critical point.

Symmetry, equivalence, and molecular self-assembly

Molecular self-assembly at equilibrium is fundamental to the fields of biological self-organization, the development of novel environmentally responsive polymeric materials, and nanofabrication. Our approach to understanding the principles governing this process is inspired by existing models and measurements for the self-assembly of actin, tubulin, and the ubiquitous icosahedral shell structures of viral capsids. We introduce a family of simple potentials that give rise to the self-assembly of linear polymeric, random surface ("membrane"), tubular ("nanotube"), and hollow icosahedral structures that are similar in many respects to their biological counterparts. The potentials involve equivalent particles and an interplay between directional (dipolar, multipolar) and short-range (van der Waals) interactions. Specifically, we find that the dipolar potential, having a continuous rotational symmetry about the dipolar axis, gives rise to chain formation, while particles with multipolar potentials, having discrete rotational symmetries (square quadrupole or triangular ring of dipoles or "hexapole"), lead to the self-assembly of open sheet, nanotube, and hollow icosahedral geometries. These changes in the geometry of self-assembly are accompanied by significant changes in the kinetics of the organization.

Compressible models of equilibrium polymerization

Flory-Huggins-type models of equilibrium polymerization are extended to describe compressible systems and, hence, the pressure dependence of thermodynamic properties. The theory is developed for three different mechanisms of equilibrium polymerization (the free association, monomer-activated polymerization, and chemically initiated polymerization models). In contrast to previous approaches for describing the pressure dependence, the theory delineates the thermodynamic consequences of the size disparities between solvent molecules, unpolymerized monomers, and the monomers within polymers. Basic thermodynamic properties (the extent of polymerization, density, heat capacities C(P) and C(V), etc.) are calculated analytically as functions of pressure, temperature, and composition of the associating species. Illustrative calculations refer to systems that polymerize upon cooling and demonstrate general agreement with numerous experimental trends. Comparisons with results from other theories are also discussed.

Equilibrium polymerization in the Stockmayer fluid as a model of supermolecular self-organization

A diverse range of molecular self-organization processes arises from a competition between directional and isotropic van der Waals intermolecular interactions. We conduct Monte Carlo simulations of the Stockmayer fluid (SF) with a large dipolar interaction as a minimal self-organization model and focus on basic thermodynamic properties that are needed to characterize the polymerization transition that occurs in this fluid. In particular, we determine the polymerization transition lines from the maximum in the specific heat, C(v), and the inflection point in the extent of polymerization, Phi. We also characterize the geometry (radius of gyration R(g), chain length L, chain topology) of the clusters that form in this associating fluid as a function of temperature, T, and concentration, rho . The pressure, P, and the second virial coefficient, B2, were determined, since these properties contain essential information about the strength of the isotropic (van der Waals) interactions. Our simulations indicate that the locations of the polymerization lines are quantitatively consistent with a model of equilibrium polymerization with the enthalpy of polymerization ("sticking energy") fixed by the minimum in the intermolecular potential. The polymerization transition in the SF is accompanied by a topological transition from predominantly linear to ring polymers upon cooling that is driven by the minimization of the dipolar energy of the clusters. We also find that the basic interaction parameters describing polymerization and phase separation in the SF can be estimated based on the existing theory of equilibrium polymerization, but the theory must be refined to account for ring formation in order to accurately describe the configurational properties of this model self-organizing fluid.

Flory-Huggins model of equilibrium polymerization and phase separation in the Stockmayer fluid

The competition between chain formation and phase separation in the Stockmayer fluid (SF) of dipolar particles is analyzed using a renormalized Flory-Huggins model of equilibrium polymerization. Calculated critical properties (T(c), phi(c), Z(c)) for the SF compare favorably with simulations over a wide range of the dimensionless dipolar (or "sticking") energy mu*. We find that the polymerization transition preempts phase separation for a large mu*, i.e., (mu*)2 >22.