Entropic barriers, activated hopping, and the glass transition in colloidal suspensions

Multistep relaxation in equilibrium polymer solutions: a minimal model of relaxation in "complex" fluids

We examine the rheological and dielectric properties of solutions of equilibrium self-assembling particles and molecules that form polydisperse chains whose average length depends on temperature and concentration (free association model). Relaxation of the self-assembling clusters proceeds by motions associated either with cluster rotations, with diffusive internal chain dynamics, or with interchain entanglement interactions. A hierarchy of models is used to emphasize different physical effects: Unentangled rodlike clusters, unentangled flexible polymers, and entangled chains. All models yield a multistep relaxation for low polymer scission rates ("persistent polymers"). The short time relaxation is nearly exponential and is dominated by the monomeric species and solvent, and the long time relaxation is approximately a stretched exponential, exp[-(t/tau)(beta)], a behavior that arises from an averaging over the equilibrium chain length distribution and the internal relaxation modes of the assembled structures. Relaxation functions indicate a bifurcation of the relaxation function into fast and slow contributions upon passing through the polymerization transition. The apparent activation energy for the long time relaxation becomes temperature dependent, while the fast monomeric relaxation process remains Arrhenius. The effective exponent beta(T), describing the long time relaxation process, varies monotonically from near unity above the polymerization temperature to a low temperature limit, beta approximately 13, when the self-assembly process is complete. The variation in the relaxation function with temperature is represented as a function of molecular parameters, such as the average chain length, friction coefficient, solvent viscosity, and the reaction rates for particle association and dissociation.

Cooperative activated dynamics in dense mixtures of hard and sticky spheres

The coupled activated dynamics in dense mixtures of repulsive and sticky hard spheres is studied using stochastic nonlinear Langevin equation theory. The effective free energy surface, barriers, saddle point trajectories, and mean first passage times depend in a rich manner on mixture composition, (high) total volume fraction, and attractive interaction strength. In general, there are three types of saddle point trajectories or relaxation pathways: a pure sticky or pure repulsive particle displacement keeping the other species localized, and a cooperative motion involving repulsive and attractive particle displacements. The barrier for activated hopping usually increases with the ratio of sticky to repulsive particle displacement. However, at intermediate values of the displacement ratio it can attain a broad plateau value, and can even exhibit a local maximum, and hence nonmonotonic behavior, at high sticky particle mixture compositions if the attraction strength is modest. The mean first passage, or hopping, times are computed using multidimensional Kramers theory. In most cases the hopping time trends reflect the behavior of the barrier height, especially as the sticky particle attraction strengths become large. However, there are dramatic exceptions associated with cooperative repulsive and attractive particle trajectories where the barriers are high but a greatly enhanced number of such trajectories exist near the saddle point.

Effect of the number and placement of polymer tethers on the structure of concentrated solutions and melts of hybrid nanoparticles

We have generalized and applied the microscopic polymer reference interaction site model theory to study intermolecular pair correlation functions and collective structure factors of concentrated solutions and melts of spherical nanoparticles carrying one, two, or four tethered polymer chains. A complex interplay of entropy (translational, conformational, and packing) and enthalpy (particle-particle attraction) leads to different structural arrangements with distinctive small- and wide-angle scattering signatures. Strong concentration fluctuations indicative of aggregate formation and/or a tendency for microphase separation occur as the total packing fraction and/or particle-particle attraction strength increase. In analogy with block copolymers, the microphase spinodal curve is estimated by extrapolation of the inverse of the amplitude of the small-angle scattering peak. As the number of tethered chains on nanoparticles increases, the microphase separation boundary spinodal occurs at higher particle-particle attraction strength or lower temperature. For nanoparticles with two tethers, increasing the angle between the attached chains shifts the microphase spinodal to lower temperatures. For nanoparticles with four tethers, the structural correlations are insensitive to various symmetric placements. The tendency for microphase transition is enhanced upon asymmetrically placing all four tethers on one side of the particle due to the high anisotropy of steric hindrance.

Connections of activated hopping processes with the breakdown of the Stokes-Einstein relation and with aspects of dynamical heterogeneities

We develop an extended version of the mode-coupling theory (MCT) for glass transition, which incorporates activated hopping processes via the dynamical theory originally formulated to describe diffusion-jump processes in crystals. The dynamical-theory approach adapted here to glass-forming liquids treats hopping as arising from vibrational fluctuations in the quasiarrested state where particles are trapped inside their cages, and the hopping rate is formulated in terms of the Debye-Waller factors characterizing the structure of the quasiarrested state. The resulting expression for the hopping rate takes an activated form, and the barrier height for the hopping is "self-generated" in the sense that it is present only in those states where the dynamics exhibits a well defined plateau. It is discussed how such a hopping rate can be incorporated into MCT so that the sharp nonergodic transition predicted by the idealized version of the theory is replaced by a rapid but smooth crossover. We then show that the developed theory accounts for the breakdown of the Stokes-Einstein relation observed in a variety of fragile glass formers. It is also demonstrated that characteristic features of dynamical heterogeneities revealed by recent computer simulations are reproduced by the theory. More specifically, a substantial increase of the non-Gaussian parameter, double-peak structure in the probability distribution of particle displacements, and the presence of a growing dynamic length scale are predicted by the extended MCT developed here, which the idealized version of the theory failed to reproduce. These results of the theory are demonstrated for a model of the Lennard-Jones system, and are compared with related computer-simulation results and experimental data.

Exploring the potential energy landscape of glass-forming systems: from inherent structures via metabasins to macroscopic transport

In this review a systematic analysis of the potential energy landscape (PEL) of glass-forming systems is presented. Starting from the thermodynamics, the route towards the dynamics is elucidated. A key step in this endeavor is the concept of metabasins. The relevant energy scales of the PEL can be characterized. Based on the simulation results for some glass-forming systems one can formulate a relevant model system (ideal Gaussian glass-former) which can be treated analytically. The macroscopic transport can be related to the microscopic hopping processes, using either the strong relation between energy (thermodynamics) and waiting times (dynamics) or, alternatively, the concepts of the continuous-time random walk. The relation to the geometric properties of the PEL is stressed. The emergence of length scales within the PEL approach as well as the nature of finite-size effects is discussed. Furthermore, the PEL view is compared to other approaches describing the glass transition.

Theory of physical aging in polymer glasses

A statistical segment scale theory for the physical aging of polymer glasses is proposed and applied. The approach is based on a nonlinear stochastic Langevin equation of motion and the concept of an effective free energy which quantifies temporary localization, collective barriers, and the activated segment hopping process. The key collective structural variable that plays the role of the dynamic order parameter for aging is the experimentally measurable nanometer and longer wavelength amplitude of density fluctuations, S0 . The degree of local cooperativity, and the bare activation energy of the high-temperature Arrhenius process, are determined in the molten state by utilizing experimental measurements of the glass temperature and dynamic crossover time, respectively. A first-order kinetic equation with a time varying rate is proposed for the temporal evolution of S0 which is self-consistently and nonlinearly coupled with the mean segmental relaxation time. The theory has been applied to study physical aging of the alpha relaxation time, shear relaxation modulus, amplitude of density fluctuations, cohesive energy, absolute yield stress, and fictive temperature of polymethylmethacrylate and other glasses over a range of temperatures. Temperature-dependent logarithmic and effective power-law aging is predicted at intermediate times. Time-aging time superposition is found for the mechanical relaxation function. A strongly asymmetric aging response is predicted for up and down temperature jump experiments. Comparison of the approach with the classic phenomenological model is presented.

A metastable van der waals gel: transitioning from weak to strong attractions

Here we describe a method to create gels where the gel point is decoupled from gel elastic properties. Working with charge stabilized polystyrene latex particles with diameters, D, of 508-625 nm at ionic strengths of 0.1-1 M, the gel volume fraction is varied from 0.10-0.35 through the addition of less than monolayer coverage of hexaethylene glycol monododecyl ether (C6E12). At each surfactant concentration, the gel volume fraction depends on the background ionic strength. The changes in gel point with surfactant concentration suggest the strength of interparticle attraction decreases with increasing surfactant concentration. These changes are not reflected in the gel moduli, which are independent of surfactant concentration and ionic strength. We propose a model to describe this behavior based on gelation due to localization in a shallow truncated van der Waals minimum produced by the surfactant acting as a steric stabilizing layer. The surfactant remains mobile on the surface. Below the gel volume fraction, the time particles spend in the truncated well are not sufficient for the surfactant to be displaced such that the particles can only sample the shallow well. Above the gel volume fraction, particles are localized in the truncated van der Waals minima for sufficient periods of time to displace the surfactant layers with the result being that the particles fall into a primary van der Waals minimum. The result is gel points sensitive to surfactant concentration but moduli that are independent of the gel volume fraction.

Glassy dynamics and kinetic vitrification of isotropic suspensions of hard rods

A nonlinear Langevin equation (NLE) theory for the translational center-of-mass dynamics of hard nonspherical objects has been applied to isotropic fluids of rigid rods. The ideal kinetic glass transition volume fraction is predicted to be a monotonically decreasing function beyond an aspect ratio of two. The functional form of the decrease is weaker than the inverse aspect ratio. Vitrification occurs at lower volume fractions for corrugated tangent bead rods compared to their smooth spherocylinder analogs. The ideal glass transition signals a crossover to activated dynamics, which is estimated to be observable before the nematic phase boundary is encountered if the aspect ratio is less than roughly 25. Calculations of the glassy elastic shear modulus and absolute yield stress reveal a roughly exponential growth with volume fraction. The dependence of entropic barriers and mean barrier hopping times on concentration for rods of variable aspect ratios can be collapsed quite well based on a difference volume fraction variable that quantifies the distance from the ideal glass boundary. Full numerical solution of the NLE theory via stochastic trajectory simulation was performed for tangent bead rods, and the results were compared to their hard sphere analogs. With increasing shape anisotropy the characteristic length scales of the nonequilibrium free energy increase and the magnitude of the localization well and entropic barrier curvatures decreases. These changes result in a significant aspect ratio dependence of dynamical properties and time correlation functions including weaker intermediate time subdiffusive transport, stronger two-step decay of the incoherent dynamic structure factor, longer mean alpha relaxation time, and stronger wavevector-dependent decoupling of relaxation times and the self-diffusion constant. The theoretical results are potentially testable via computer simulation, confocal microscopy, and dynamic light scattering.

Theory of gelation, vitrification, and activated barrier hopping in mixtures of hard and sticky spheres

Naive mode coupling theory (NMCT) and the nonlinear stochastic Langevin equation theory of activated dynamics have been generalized to mixtures of spherical particles. Two types of ideal nonergodicity transitions are predicted corresponding to localization of both, or only one, species. The NMCT transition signals a dynamical crossover to activated barrier hopping dynamics. For binary mixtures of equal diameter hard and attractive spheres, a mixture composition sensitive "glass-melting" type of phenomenon is predicted at high total packing fractions and weak attractions. As the total packing fraction decreases, a transition to partial localization occurs corresponding to the coexistence of a tightly localized sticky species in a gel-like state with a fluid of hard spheres. Complex behavior of the localization lengths and shear moduli exist because of the competition between excluded volume caging forces and attraction-induced physical bond formation between sticky particles. Beyond the NMCT transition, a two-dimensional nonequilibrium free energy surface emerges, which quantifies cooperative activated motions. The barrier locations and heights are sensitive to the relative amplitude of the cooperative displacements of the different species.

Large-amplitude jumps and non-Gaussian dynamics in highly concentrated hard sphere fluids

Our microscopic stochastic nonlinear Langevin equation theory of activated dynamics has been employed to study the real-space van Hove function of dense hard sphere fluids and suspensions. At very short times, the van Hove function is a narrow Gaussian. At sufficiently high volume fractions, such that the entropic barrier to relaxation is greater than the thermal energy, its functional form evolves with time to include a rapidly decaying component at small displacements and a long-range exponential tail. The "jump" or decay length scale associated with the tail increases with time (or particle root-mean-square displacement) at fixed volume fraction, and with volume fraction at the mean alpha relaxation time. The jump length at the alpha relaxation time is predicted to be proportional to a measure of the decoupling of self-diffusion and structural relaxation. At long times corresponding to mean displacements of order a particle diameter, the volume fraction dependence of the decay length disappears. A good superposition of the exponential tail feature based on the jump length as a scaling variable is predicted at high volume fractions. Overall, the theoretical results are in good accord with recent simulations and experiments. The basic aspects of the theory are also compared with a classic jump model and a dynamically facilitated continuous time random-walk model. Decoupling of the time scales of different parts of the relaxation process predicted by the theory is qualitatively similar to facilitated dynamics models based on the concept of persistence and exchange times if the elementary event is assumed to be associated with transport on a length scale significantly smaller than the particle size.

Collisions, caging, thermodynamics, and jamming in the barrier hopping theory of glassy hard sphere fluids

An ultralocal limit of the microscopic single particle barrier hopping theory of glassy dynamics is proposed which allows explicit analytic expressions for the characteristic length scales, energy scales, and nonequilibrium free energy to be derived. All properties are shown to be controlled by a single coupling constant determined by the fluid density and contact value of the radial distribution function. This parameter quantifies an effective mean square force exerted on a tagged particle due to collisions with its surroundings. The analysis suggests a conceptual basis for previous surprising findings of multiple inter-relationships between characteristics of the transient localized state, the early stages of cage escape, non-Gaussian or dynamic heterogeneity effects, and the barrier hopping process that defines the alpha relaxation event. The underlying physical picture is also relevant to fluids of nonspherical molecules and sticky colloidal suspensions. The possibility of a unified view of liquid dynamics is suggested spanning the range from dense gases to the zero mobility jammed state.

Rheology modification in mixed shape colloidal dispersions. Part II: mixtures

We report the results of a comprehensive study of the rheological properties of a series of mixed colloid systems where the shape of one of the components has been varied systematically. Specifically we have measured the oscillatory, transient (creep) and continuous steady shear flow behaviour of a 2.5 wt% dispersion in water of a well-characterised hectorite clay modified by the addition of a series of aluminasol colloidal particles whose shape varies systematically from rod (boehmite) to platelet (gibbsite) to sphere (alumina-coated silica), all having essentially the same smallest dimension, which is similar to that of the hectorite. The particle characterisation and rheological properties of the pure components have recently been reported in Part I of this series (Soft Matter, 2007, 3, 1145). The mixtures show the same general behaviour as the pure systems, displaying a complex 'yield space' transition from an elastoviscous gel at low applied stresses to a viscous, weakly elastic, shear-thinning liquid at high stresses. The unifying theme of this work is that the addition of 0.25 wt% of the minor component in all cases results in dramatic enhancements to the dispersion rheological properties. At the same time the magnitude of this effect depends on the shape of the particles. Shear moduli, low stress viscosities and effective yield stresses all increase in the additive order rods < platelets < spheres, with enhancements for the latter being up to a factor of 500 and typically 20. At the same time the critical failure strains for the gels decreased in the same order - the strongest gels are also the most fragile in this sense. The physicochemical factors underlying this behaviour are discussed and a simple qualitative model described. While no complete explanation or model can be proposed at this stage, the study provides a quantitative model-system baseline for mixed colloidal dispersions already used for industrial applications (e.g. oilwell-drilling fluids) and suggests ways in which such fluids may be optimised and controlled.

Intermolecular forces and the glass transition

Random first-order transition theory is used to determine the role of attractive and repulsive interactions in the dynamics of supercooled liquids. Self-consistent phonon theory, an approximate mean field treatment consistent with random first-order transition theory, is used to treat individual glassy configurations, whereas the liquid phase is treated using common liquid-state approximations. Free energies are calculated using liquid-state perturbation theory. The transition temperature, T*A, the temperature where the onset of activated behavior is predicted by mean field theory; the lower crossover temperature, T*C, where barrierless motions actually occur through fractal or stringy motions (corresponding to the phenomenological mode coupling transition temperature); and T*K, the Kauzmann temperature (corresponding to an extrapolated entropy crisis), are calculated in addition to T*g, the glass transition temperature that corresponds to laboratory cooling rates. Relationships between these quantities agree well with existing experimental and simulation data on van der Waals liquids. Both the isobaric and isochoric behavior in the supercooled regime are studied, providing results for DeltaCV and DeltaCp that can be used to calculate the fragility as a function of density and pressure, respectively. The predicted variations in the alpha-relaxation time with temperature and density conform to the empirical density-temperature scaling relations found by Casalini and Roland. We thereby demonstrate the microscopic origin of their observations. Finally, the relationship first suggested by Sastry between the spinodal temperature and the Kauzmann temperatures, as a function of density, is examined. The present microscopic calculations support the existence of an intersection of these two temperatures at sufficiently low temperatures.

Ideal vitrification, barrier hopping, and jamming in fluids of modestly anisotropic hard objects

Our recent theory for the glassy dynamics of fluids and suspensions of hard nonspherical objects is applied to several modestly anisotropic shapes. The role of bond length and aspect ratio is studied for diatomics, triatomics, and spherocylinders. As spherical symmetry is broken the ideal kinetic glass transition volume fraction of all objects increases linearly with aspect ratio with the same slope, in surprising agreement with the jamming phase diagram of hard granular ellipsoids. The ideal glass boundary of all shapes is a nonmonotonic function of aspect ratio which is also in qualitative accord with the jamming behavior of spherocylinders and ellipsoids. The maximum glass volume fraction shifts to higher values, and larger aspect ratios, as the object becomes smoother. Suggestions for why the nonequilibrium jamming and kinetic ideal glass formation (dynamical crossover) boundaries are similar are advanced. Beyond the ideal glass volume fraction the nonequilibrium free energy acquires a localization well and entropic barrier. Although its form is highly nonuniversal, if different shapes are compared at constant barrier height then a good collapse is found. Collapse of the volume fraction dependence of the barrier height for different shapes is also predicted for modest shape anisotropy, but increasingly fails as the aspect ratio exceeds 2. For a given volume fraction the mean barrier hopping times are nonmonotonic functions of aspect ratio. The functional form of this dependence, and order of magnitude variation with aspect ratio, is distinct for each object.

Solidity of viscous liquids. V. Long-wavelength dominance of the dynamics

This paper is the fifth in a series exploring the physical consequences of the solidity of highly viscous liquids. Paper IV proposed a model where the density field is described by a time-dependent Ginzburg-Landau equation of the nonconserved type with rates in k space of the form Gamma0+Dk2. If a is the average intermolecular distance, the model assumes that DGamma0a2. This inequality expresses a long-wavelength dominance of the dynamics, which implies that the Hamiltonian (free energy) to a good approximation may be taken to be ultralocal, i.e., with the property that equal-time field fluctuations are uncorrelated in space. Paper IV also briefly discussed how to generalize the model by including the molecular orientational fields, the stress tensor fields, and the potential energy density field. In the present paper it is argued that this is the simplest model consistent with the following three experimental facts: (1) Viscous liquids approaching the glass transition do not develop long-range order; (2) the glass has lower compressibility than the liquid; (3) the alpha process involves several decades of relaxation times shorter than the mean relaxation time. The paper proceeds to list six further experimental facts of viscous liquid dynamics and shows that these follow naturally from the model.

Dynamics of interacting Brownian particles: A diagrammatic formulation

We present a diagrammatic formulation of a theory for the time dependence of density fluctuations in equilibrium systems of interacting Brownian particles. To facilitate derivation of the diagrammatic expansion, we introduce a basis that consists of orthogonalized many-particle density fluctuations. We obtain an exact hierarchy of equations of motion for time-dependent correlations of orthogonalized density fluctuations. To simplify this hierarchy we neglect contributions to the vertices from higher-order cluster expansion terms. An iterative solution of the resulting equations can be represented by diagrams with three- and four-leg vertices. We analyze the structure of the diagrammatic series for the time-dependent density correlation function and obtain a diagrammatic interpretation of reducible and irreducible memory functions. The one-loop self-consistent approximation for the latter function coincides with mode-coupling approximation for Brownian systems that was derived previously using a projection operator approach.

Effective separation of forces in a mode coupling theory of self-diffusion

A mode coupling theory (MCT) expression for the self-diffusion coefficient follows simply when the soft fluctuating intermolecular forces are projected along a collective densitylike variable. The projected forces separate into two parts: from the gradient of the direct correlation function (dcf), and from the short range forces. The time correlation function of the dcf-derived forces is related to the excess entropy, as shown by Ali [J. Chem. Phys. 124, 144504 (2006)], and this relationship is evaluated for two variations of MCT. As for hard spheres, the derivation of an analogous MCT is beset by a number of singularities that kinetic theory could not remove. A justifiable MCT for hard sphere fluids may not exist.

Crossover behavior and multistep relaxation in a schematic model of the cut-off glass transition

We study a schematic mode-coupling model in which the ideal glass transition is cut off by a decay of the quadratic coupling constant in the memory function. (Such a decay, on a time scale tau I , has been suggested as the likely consequence of activated processes.) If this decay is complete, so that only a linear coupling remains at late times, then the alpha relaxation shows a temporal crossover from a relaxation typical of the unmodified schematic model to a final strongly slower-than-exponential relaxation. This crossover, which differs somewhat in form from previous schematic models of the cutoff glass transition, resembles light-scattering experiments on colloidal systems, and can exhibit a "slower-than- alpha " relaxation feature hinted at there. We also consider what happens when a similar but incomplete decay occurs, so that a significant level of quadratic coupling remains for t>>tau I . In this case the correlator acquires a third, weaker relaxation mode at intermediate times. This empirically resembles the beta process seen in many molecular glass formers. It disappears when the initial as well as the final quadratic coupling lies on the liquid side of the glass transition, but remains present even when the final coupling is only just inside the liquid (so that the alpha relaxation time is finite, but too long to measure). Our results are suggestive of how, in a cutoff glass, the underlying "ideal" glass transition predicted by mode-coupling theory can remain detectable through qualitative features in dynamics.

Molecular theory of physical aging in polymer glasses

A molecular level theory for the physical aging of polymer glasses is proposed. The nonequilibrium time evolution of the amplitude of long wavelength density fluctuations, and its influence on activated barrier hopping, plays an essential role. The theory predicts temperature-dependent apparent power-law aging of the segmental relaxation time and logarithmic aging of thermodynamiclike properties, in good accord with experiments. A physical origin for the quantitative nonuniversal aspects based on the amplitude of quenched density fluctuations is suggested.

Dynamics in inhomogeneous liquids and glasses via the test particle limit

We show that one may view the self-part and the distinct-part of the van Hove dynamic correlation function of a simple fluid as the one-body density distributions of a binary mixture that evolve in time according to dynamical density functional theory. For a test case of soft-core Brownian particles the theory yields results for the van Hove function that agree quantitatively with those of our Brownian dynamics computer simulations. At sufficiently high densities the free energy landscape underlying the dynamics exhibits a barrier as a function of the mean particle displacement, shedding new light on the nature of glass formation. For hard spheres confined between parallel planar walls the barrier height oscillates in phase with the local density, implying that the mobility is maximal between layers, which should be experimentally observable in confined colloidal dispersions.

Theory of relaxation and elasticity in polymer glasses

The recently developed activated barrier hopping theory of deeply supercooled polymer melts [K. S. Schweizer and E. J. Saltzman, J. Chem. Phys. 121, 1984 (2004)] is extended to the nonequilibrium glass state. Below the kinetic glass temperature T(g), the exact statistical mechanical relation between the dimensionless amplitude of long wavelength density fluctuations, S(0), and the thermodynamic compressibility breaks down. Proper extension of the theory requires knowledge of the nonequilibrium S(0) which x-ray scattering experiments find to consist of a material specific and temperature-independent quenched disorder contribution plus a vibrational contribution which varies roughly linearly with temperature. Motivated by these experiments and general landscape concepts, a simple model is proposed for S(0)(T). Deep in the glass state the form of the temperature dependence of the segmental relaxation time is found to depend sensitively on the magnitude of frozen in density fluctuations. At the (modest) sub-T(g) temperatures typically probed in experiment, an effective Arrhenius behavior is generically predicted which is of nonequilibrium origin. The change in apparent activation energy across the glass transition is determined by the amplitude of frozen density fluctuations. For values of the latter consistent with experiment, the theory predicts a ratio of effective activation energies in the range of 3-6, in agreement with multiple measurements. Calculations of the shear modulus for atactic polymethylmethacrylate above and below the glass transition temperature have also been performed. The present work provides a foundation for the formulation of predictive theories of physical aging, the influence of deformation on the alpha relaxation process, and rate-dependent nonlinear mechanical properties of thermoplastics.

Ideal glass transitions, shear modulus, activated dynamics, and yielding in fluids of nonspherical objects

An extension of naive ideal mode coupling theory (MCT) and its generalization to treat activated barrier hopping and glassy dynamics in fluids and suspensions composed of nonspherical hard core objects is proposed. An effective center-of-mass description is adopted. It corresponds to a specific type of pre-averaging of the dynamical consequences of orientational degrees of freedom. The simplest case of particles composed of symmetry-equivalent interaction sites is considered. The theory is implemented for a homonuclear diatomic shape of variable bond length. The naive MCT glass transition boundary is predicted to be a nonmonotonic function of the length-to-width or aspect ratio and occurs at a nearly unique value of the dimensionless compressibility. The latter quantifies the amplitude of long wavelength thermal density fluctuations, thereby (empirically) suggesting a tight connection between the onset of localization and thermodynamics. Localization lengths and elastic shear moduli for different aspect ratio and volume fraction systems approximately collapse onto master curves based on a reduced volume fraction variable that quantifies the distance from the ideal glass transition. Calculations of the entropic barrier height and hopping time, maximum restoring force, and absolute yield stress and strain as a function of diatomic aspect ratio and volume fraction have been performed. Strong correlations of these properties with the dimensionless compressibility are also found, and nearly universal dependences have been numerically identified based on property-specific nondimensionalizations. Generalization of the approach to rigid rods, disks, and variable shaped molecules is possible, including oriented liquid crystalline phases.

Role of solvation forces in the gelation of fumed silica–alcohol suspensions

Aggregation and gelation kinetics of fumed silica were investigated by altering the solvent-surface interactions. Native and surface-modified (hydrophobic) fumed silica particles were dispersed in short-chain linear alcohols. Based on the kinetics of aggregation and gelation, we show that the solvent-surface interactions have a tremendous impact on the bulk suspension properties. The gelation kinetics were qualitatively similar in all of the fumed silica-alcohol samples, and the gel times for all the alcohols were captured on a master curve requiring two parameters. The two parameters, the stability ratio and critical volume fraction, describe the two regimes of gelation. At low concentrations, gelation occurs due to aggregation of the particles diffusing over a potential barrier (15-25 kT). The rate of aggregation and time to gelation then scales with the stability ratio. At high particle loadings, gelation occurs at a critical volume fraction due to localization in a secondary minimum with a depth of 3-4 kT. These observations are supported by evidence of hydrogen bonding between the solvent and the particle, creating oscillatory solvation forces that govern the magnitude of these two parameters.

Non-Gaussian effects, space-time decoupling, and mobility bifurcation in glassy hard-sphere fluids and suspensions

Brownian trajectory simulation methods are employed to fully establish the non-Gaussian fluctuation effects predicted by our nonlinear Langevin equation theory of single particle activated dynamics in glassy hard-sphere fluids. The consequences of stochastic mobility fluctuations associated with the space-time complexities of the transient localization and barrier hopping processes have been determined. The incoherent dynamic structure factor was computed for a range of wave vectors and becomes of an increasingly non-Gaussian form for volume fractions beyond the (naive) ideal mode coupling theory (MCT) transition. The non-Gaussian parameter (NGP) amplitude increases markedly with volume fraction and is well described by a power law in the maximum restoring force of the nonequilibrium free energy profile. The time scale associated with the NGP peak becomes much smaller than the alpha relaxation time for systems characterized by significant entropic barriers. An alternate non-Gaussian parameter that probes the long time alpha relaxation process displays a different shape, peak intensity, and time scale of its maximum. However, a strong correspondence between the classic and alternate NGP amplitudes is predicted which suggests a deep connection between the early and final stages of cage escape. Strong space-time decoupling emerges at high volume fractions as indicated by a nondiffusive wave vector dependence of the relaxation time and growth of the translation-relaxation decoupling parameter. Displacement distributions exhibit non-Gaussian behavior at intermediate times, evolving into a strongly bimodal form with slow and fast subpopulations at high volume fractions. Qualitative and semiquantitative comparisons of the theoretical results with colloid experiments, ideal MCT, and multiple simulation studies are presented.

Microstructure and rheology of thermoreversible nanoparticle gels

Naïve mode coupling theory is applied to particles interacting with short-range Yukawa attractions. Model results for the location of the gel line and the modulus of the resulting gels are reduced to algebraic equations capturing the effects of the range and strength of attraction. This model is then applied to thermo reversible gels composed of octadecyl silica particles suspended in decalin. The application of the model to the experimental system requires linking the experimental variable controlling strength of attraction, temperature, to the model strength of attraction. With this link, the model predicts temperature and volume fraction dependencies of gelation and modulus with five parameters: particle size, particle volume fraction, overlap volume of surface hairs, and theta temperature. In comparing model predictions with experimental results, we first observe that in these thermal gels there is no evidence of clustering as has been reported in depletion gels. One consequence of this observation is that there are no additional adjustable parameters required to make quantitative comparisons between experimental results and model predictions. Our results indicate that the naïve mode coupling approach taken here in conjunction with a model linking temperature to strength of attraction provides a robust approach for making quantitative predictions of gel mechanical properties. Extension of model predictions to additional experimental systems requires linking experimental variables to the Yukawa strength and range of attraction.

Activated hopping and dynamical fluctuation effects in hard sphere suspensions and fluids

Single particle Brownian dynamics simulation methods are employed to establish the full trajectory level predictions of our nonlinear stochastic Langevin equation theory of activated hopping dynamics in glassy hard sphere suspensions and fluids. The consequences of thermal noise driven mobility fluctuations associated with the barrier hopping process are determined for various ensemble-averaged properties and their distributions. The predicted mean square displacements show classic signatures of transient trapping and anomalous diffusion on intermediate time and length scales. A crossover to a stronger volume fraction dependence of the apparent nondiffusive exponent occurs when the entropic barrier is of order the thermal energy. The volume fraction dependences of various mean relaxation times and rates can be fitted by empirical critical power laws with parameters consistent with ideal mode-coupling theory. However, the results of our divergence-free theory are largely a consequence of activated dynamics. The experimentally measurable alpha relaxation time is found to be very similar to the theoretically defined mean reaction time for escape from the barrier-dominated regime. Various measures of decoupling have been studied. For fluid states with small or nonexistent barriers, relaxation times obey a simple log-normal distribution, while for high volume fractions the relaxation time distributions become Poissonian. The product of the self-diffusion constant and mean alpha relaxation time increases roughly as a logarithmic function of the alpha relaxation time. The cage scale incoherent dynamic structure factor exhibits nonexponential decay with a modest degree of stretching. A nearly universal collapse of the different volume fraction results occurs if time is scaled by the mean alpha relaxation time. Hence, time-volume fraction superposition holds quite well, despite the presence of stretching and volume fraction dependent decoupling associated with the stochastic barrier hopping process. The relevance of other origins of dynamic heterogeneity (e.g., mesoscopic domains), and comparison of our results with experiments, simulations, and alternative theories, is discussed.

Nonlinear elasticity and yielding of nanoparticle glasses

We employ experiment and theory to explore the nonlinear elasticity and yielding of concentrated suspensions of nanoparticles which interact via purely repulsive forces. These glassy suspensions are found to exhibit high exponent power law or simple exponential dependences of the shear elastic modulus and perturbative yield stress on nanoparticle volume fraction, as well as a monotonic decrease of the perturbative yield strain with increasing concentration. Our experimental observations are in good agreement with the predictions of a recently developed microscopic statistical mechanical theory, which describes glassy dynamics based on a nonequilibrium free energy that incorporates local cage correlations and activated barrier hopping processes [(1) Schweizer, K. S.; Saltzman, E. J. J. Chem. Phys. 2003, 119, 1181. (2) Saltzman, E. J.; Schweizer, K. S. J. Chem. Phys. 2003, 119, 1197. (3) Kobelev, V.; Schweizer, K. S. Phy. Rev. E 2005, 71, 021401].

Entropy theory of polymer glass formation revisited. I. General formulation

A generalized entropy theory of glass formation is developed by merging the lattice cluster theory for the thermodynamics of semiflexible polymer melts at constant pressure with the Adam-Gibbs relation between the structural relaxation time and the configurational entropy. Since experimental studies have suggested that the relative rigidity of the chain backbone and the side groups is an essential parameter governing the nature of glass formation in polymers, we incorporate this rigidity disparity parameter, along with monomer structure, into our new theoretical description of the polymer fluid thermodynamics. Our entropy theory is compared with alternative theories that describe the rate of structural relaxation in glass-forming liquids in terms of an activated rate process.

Derivation of a microscopic theory of barriers and activated hopping transport in glassy liquids and suspensions

A recently proposed microscopic activated barrier hopping theory [K. S. Schweizer and E. J. Saltzman, J. Chem. Phys. 119, 1181 (2003)] of slow single-particle dynamics in glassy liquids, suspensions, and gels is derived using nonequilibrium statistical mechanics. Fundamental elements underlying the stochastic nonlinear Langevin equation description include an inhomogeneous liquid or locally solid-state perspective, dynamic density-functional theory (DDFT), a local equilibrium closure, and a coarse-grained free-energy functional. A dynamic Gaussian approximation is not adopted which is the key for avoiding a kinetic ideal glass transition. The relevant excess free energy is of a nonequilibrium origin and is related to dynamic force correlations in the fluid. The simplicity of the approach allows external perturbations to be rather easily incorporated. Dynamic heterogeneity enters naturally via mobility fluctuations associated with the stochastic barrier-hopping process. The derivation both identifies the limitations of the theory and suggests new avenues for its systematic improvement. Comparisons with ideal mode-coupling theory, alternative DDFT approaches and a field theoretic path-integral formulation are presented.

Theory of glassy dynamics in conformationally anisotropic polymer systems

A mode coupling theory for the ideal glass transition temperature, or crossover temperature to highly activated dynamics in the deeply supercooled regime, T(c), has been developed for anisotropic polymer liquids. A generalization of a simplified mode coupling approach at the coarse-grained segment level is employed which utilizes structural and thermodynamic information from the anisotropic polymer reference interaction site model theory. Conformational alignment or/and coil deformation modifies equilibrium properties and constraining interchain forces thereby inducing anisotropic segmental dynamics. For liquid-crystalline polymers a small suppression of T(c) with increasing nematic or discotic orientational order is predicted. The underlying mechanism is reduction of the degree of coil interpenetration and intermolecular repulsive contacts due to segmental alignment. For rubber networks chain deformation results in an enhanced bulk modulus and a modest elevation of T(c) is predicted. The theory can also be qualitatively applied to systems that undergo nonuniversal local deformation and alignment, such as polymer thin films and grafted brush layers, and large elevations or depressions of T(c) are possible. Extension to treat directionally dependent collective barrier formation and activated hopping is possible.

Clustering and mechanics in dense depletion and thermal gels

We report on the microstructure and mechanical properties (elastic modulus) of concentrated depletion and thermal gels of octadecyl-coated silica particles for different values of the strength of interaction--polymer concentration for depletion gels and temperature for thermal gels. The depletion gels are composed of dense clusters and voids, while the thermal gels are devoid of clusters. Shear breaks up clusters in depletion gels while it induces clustering in the thermal gels. In both of these gels, the microstructure recovers to the presheared state upon cessation of shear. The recovery of the elastic modulus mimics the microstructure in the sense that the elastic modulus recovers to the presheared sheared state after shearing is stopped. Calculations of the gel boundary by modeling the interactions with an effective one-component square-well model reveals that suspensions with similar ranges of attraction gel at the same volume fraction at a fixed strength of attraction. Calculations of the elastic modulus using the naïve mode coupling theory for depletion gels are in good agreement with experimental measurements provided clustering is taken into account and have the same magnitude as the elastic moduli of thermal gels with similar strengths of attraction. These calculations, in addition to the experimental observations reinforce the point that the microscopic parameter determining the elastic modulus of dense gels and its recovery is the localization length which is only a fraction of the particle diameter and not the structure on the length scale of the particle diameter and larger.

Nonlinear elasticity and yielding of depletion gels

A microscopic activated barrier hopping theory of the viscoelasticity of colloidal glasses and gels has been generalized to treat the nonlinear rheological behavior of particle-polymer suspensions. The quiescent cage constraints and depletion bond strength are quantified using the polymer reference interaction site model theory of structure. External deformation (strain or stress) distorts the confining nonequilibrium free energy and reduces the barrier. The theory is specialized to study a limiting mechanical description of yielding and modulus softening in the absence of thermally induced barrier hopping. The yield stress and strain show a rich functional dependence on colloid volume fraction, polymer concentration, and polymer-colloid size asymmetry ratio. The yield stress collapses onto a master curve as a function of the polymer concentration scaled by its ideal mode-coupling gel boundary value, and sufficiently deep in the gel is of an effective power-law form with a universal exponent. A similar functional and scaling dependence of the yield stress on the volume fraction is found, but the apparent power-law exponent is nonuniversal and linearly correlated with the critical gel volume fraction. Stronger gels are generally, but not always, predicted to be more brittle in the strain mode of deformation. The theoretical calculations appear to be in accord with a broad range of observations.

Dynamic yielding, shear thinning, and stress rheology of polymer-particle suspensions and gels

The nonlinear rheological version of our barrier hopping theory for particle-polymer suspensions and gels has been employed to study the effect of steady shear and constant stress on the alpha relaxation time, yielding process, viscosity, and non-Newtonian flow curves. The role of particle volume fraction, polymer-particle size asymmetry ratio, and polymer concentration have been systematically explored. The dynamic yield stress decreases in a polymer-concentration- and volume-fraction-dependent manner that can be described as apparent power laws with effective exponents that monotonically increase with observation time. Stress- or shear-induced thinning of the viscosity becomes more abrupt with increasing magnitude of the quiescent viscosity. Flow curves show an intermediate shear rate dependence of an effective power-law form, becoming more solidlike with increasing depletion attraction. The influence of polymer concentration, particle volume fraction, and polymer-particle size asymmetry ratio on all properties is controlled to a first approximation by how far the system is from the gelation boundary of ideal mode-coupling theory (MCT). This emphasizes the importance of the MCT nonergodicity transition despite its ultimate destruction by activated barrier hopping processes. Comparison of the theoretical results with limited experimental studies is encouraging.

Caging and mosaic length scales in plaquette spin models of glasses

We consider two systems of Ising spins with plaquette interactions. They are simple models of glasses which have dual representations as kinetically constrained systems. These models allow an explicit analysis using the mosaic, or entropic droplet, approach of the random first-order transition theory of the glass transition. We show that the low-temperature states of these systems resemble glassy mosaic states, despite the fact that excitations are localized and that there are no static singularities. By means of finite-size thermodynamics we study a generalized caging effect whereby the system is frozen on short length scales, but free at larger length scales. We find that the freezing length scales obtained from statics coincide with those relevant to dynamic correlations, as expected in the mosaic view. The simple nucleation arguments of the mosaic approach, however, do not give the correct relation between freezing lengths and relaxation times, as they do not capture the transition states for relaxation. We discuss how these results make a connection between the mosaic and the dynamic facilitation views of glass formers.

Fragility of Glass-Forming Polymer Liquids

The fragility of polymeric glass-forming liquids is calculated as a function of molecular structural parameters from a generalized entropy theory of polymer glass-formation that combines the Adam-Gibbs (AG) model for the rate of structural relaxation with the lattice cluster theory (LCT) for polymer melt thermodynamics. Our generalized entropy theory predicts the existence of distinct high and low temperature regimes of glass-formation that are separated by a thermodynamically well-defined crossover temperature T(I) at which the product of the configurational entropy and the temperature has an inflection point. Since the predicted temperature dependence of the configurational entropy and structural relaxation time are quite different in these temperature regimes, we introduce separate definitions of fragility for each regime. Experimentally established trends in the fragility of polymer melts with respect to variations in polymer microstructure and pressure are interpreted within our theory in terms of the accompanying changes in the chain packing efficiency.

Direct computation of characteristic temperatures and relaxation times for glass-forming polymer liquids

Characteristic temperatures and structural relaxation times for different classes of glass-forming polymer liquids are computed using a revised entropy theory of glass formation that permits the chain backbone and the side groups to have different rigidities. The theory is applied to glass formation at constant pressure or constant temperature. Our calculations provide new insights into physical factors influencing the breadth of the glass transition and the associated growth of relaxation times.

Relaxation in a glassy binary mixture: mode-coupling-like power laws, dynamic heterogeneity, and a new non-Gaussian parameter

We examine the relaxation of the Kob-Andersen Lennard-Jones binary mixture using Brownian dynamics computer simulations. We find that in accordance with mode-coupling theory the self-diffusion coefficient and the relaxation time show power-law dependence on temperature. However, different mode-coupling temperatures and power laws can be obtained from the simulation data depending on the range of temperatures chosen for the power-law fits. The temperature that is commonly reported as this system's mode-coupling transition temperature, in addition to being obtained from a power law fit, is a crossover temperature at which there is a change in the dynamics from the high-temperature homogeneous, diffusive relaxation to a heterogeneous, hopping-like motion. The hopping-like motion is evident in the probability distributions of the logarithm of single-particle displacements: approaching the commonly reported mode-coupling temperature these distributions start exhibiting two peaks. Notably, the temperature at which the hopping-like motion appears for the smaller particles is slightly higher than that at which the hopping-like motion appears for the larger ones. We define and calculate a new non-Gaussian parameter whose maximum occurs approximately at the time at which the two peaks in the probability distribution of the logarithm of displacements are most evident.

Barrier hopping, viscous flow, and kinetic gelation in particle-polymer suspensions

The naive mode coupling-polymer reference interaction site model (MCT-PRISM) theory of gelation and elasticity of suspensions of hard sphere colloids or nanoparticles mixed with nonadsorbing polymers has been extended to treat the emergence of barriers, activated transport, and viscous flow. The barrier makes the dominant contribution to the single particle relaxation time and shear viscosity, and is a rich function of the depletion attraction strength via the polymer concentration, polymer-particle size asymmetry ratio, and particle volume fraction. The dependences of the barrier on these three system parameters can be accurately collapsed onto a single scaling variable, and the resultant master curve is well described by a power law. Nearly universal master curves are also constructed for the hopping or alpha relaxation time for system conditions not too close to the ideal MCT transition. Based on the calculated barrier hopping time, a theory for kinetic gel boundaries is proposed. The form and dependence on system parameters of the kinetic gel lines are qualitatively the same as obtained from prior ideal MCT-PRISM studies. The possible relevance of our results to the phenomenon of gravity-driven gel collapse is studied. The general approach can be extended to treat nonlinear viscoelasticity and rheology of polymer-colloid suspensions and gels.

Strain softening, yielding, and shear thinning in glassy colloidal suspensions

A microscopic theory for the dependence on external strain, stress, and shear rate of the transient localization length, elastic modulus, alpha relaxation time, shear viscosity, and other dynamic properties of glassy colloidal suspensions is formulated and numerically applied. The approach is built on entropic barrier hopping as the elementary physical process. The concept of an ideal glass transition plays no role, and dynamical slowing down is a continuous, albeit precipitous, process with increasing colloid volume fraction. The relative roles of mechanically driven motion versus thermally activated barrier hopping and transport have been studied. Various scaling behaviors are found for the relaxation time and shear viscosity in both the controlled stress and shear rate mode of rheological experiments. Apparent power law and/or exponential dependences of the elastic modulus and perturbative and absolute yield stresses on colloid volume fraction are predicted. A nonmonotonic dependence of the absolute yield strain on volume fraction is also found. Qualitative and quantitative comparisons of calculations with experiments on high volume fraction glassy colloidal suspensions show encouraging agreement, and multiple testable predictions are made. The theory is generalizable to treat nonlinear rheological phenomena in other soft glassy complex fluids including depletion gels.