Mean-field theory, mode-coupling theory, and the onset temperature in supercooled liquids

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.

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.

Time scale for the onset of Fickian diffusion in supercooled liquids

We propose a quantitative measure of a time scale on which Fickian diffusion sets in for supercooled liquids, and we use Brownian dynamics computer simulations to determine the temperature dependence of this onset time in a Lennard-Jones binary mixture. The time for the onset of Fickian diffusion ranges between 6.5 and 31 times the relaxation time (the relaxation time is the characteristic relaxation time of the incoherent intermediate scattering function). The onset time increases faster with decreasing temperature than the relaxation time. Mean-squared displacement at the onset time increases with decreasing temperature.

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.

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.

Relaxation in a glassy binary mixture: comparison of the mode-coupling theory to a Brownian dynamics simulation

We solved the mode-coupling equations for the Kob-Andersen binary mixture using structure factors calculated from Brownian dynamics simulations of the same system. We found, as was previously observed, that the mode-coupling temperature T(c) inferred from simulations is about two times greater than that predicted by the theory. However, we find that many time-dependent quantities agree reasonably well with the predictions of the mode-coupling theory if they are compared at the same reduced temperature epsilon = (T - T(c))/T(c), and if epsilon is not too small. Specifically, the simulation results for the incoherent intermediate scattering function, the mean square displacement, the relaxation time, and the self-diffusion coefficient agree reasonably well with the predictions of the mode-coupling theory. We find that there are substantial differences for the non-Gaussian parameter. At small reduced temperatures the probabilities of the logarithm of single particle displacements demonstrate that there is hopping-like motion present in the simulations, and this motion is not predicted by the mode-coupling theory. The wave-vector-dependent relaxation time is shown to be qualitatively different from the predictions of the mode-coupling theory for temperatures where hopping-like motion is present.

Novel Computational Probes of Diffusive Motion

Self-diffusion constants, D, and the atomic-level processes that produce them have been investigated numerically for the binary-mixture Lennard-Jones (BMLJ) model and for liquid silica as described by the Van Beest-Kramer-Van Santen interaction model. The primary conceptual tool for this study is the joint probability distribution for single particles as a function of initial velocity and positional displacement at a given later instant. Self-diffusion constants can be expressed exactly in terms of this probability function. The numerical simulations for the BMLJ case reveal an unusual temperature effect; in contrast to the high-temperature behavior, particles with high initial velocities experience disproportionate retardation in forward displacement. In the silica modeling simulations, diffusive processes have been compared at constant-temperature "isodiffusive" pairs of states, demonstrating a significant role played by the amount of local tetrahedral order that is present in the medium.

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.

Molecular dynamics studies of heterogeneous dynamics and dynamic crossover in supercooled atomic liquids

Supercooled liquids near the glass transition exhibit the phenomenon of heterogeneous relaxation; at any specific time, a nominally homogeneous equilibrium fluid undergoes dynamic fluctuations in its structure on a molecular distance scale with rates that are very different in different regions of the sample. Several theoretical and simulation studies have suggested a change in the nature of the dynamics of fluids as they are supercooled, leading to the concept of a dynamic crossover that is often associated with mode coupling theory. Here, we will review the use of molecular dynamics computer simulation methods to investigate heterogeneous dynamics and dynamic crossovers in models of atomic liquids.

QUANTUM MODE-COUPLING THEORY: Formulation and Applications to Normal and Supercooled Quantum Liquids

▪ Abstract  We review our recent efforts to formulate and study a mode-coupling approach to real-time dynamic fluctuations in quantum liquids. Comparison is made between the theory and recent neutron scattering experiments performed on liquid ortho-deuterium and para-hydrogen. We discuss extensions of the theory to supercooled and glassy states where quantum fluctuations compete with thermal fluctuations. Experimental scenarios for quantum glassy liquids are briefly discussed.

Glassy behavior in an exactly solved spin system with a ferromagnetic transition

We show that applying simple dynamical rules to Baxter's eight-vertex model leads to a system which resembles a glass-forming liquid. There are analogies with liquid, supercooled liquid, glassy, and crystalline states. The disordered phases exhibit strong dynamical heterogeneity at low temperatures, which may be described in terms of an emergent mobility field. Their dynamics are well described by a simple model with trivial thermodynamics, but an emergent kinetic constraint. We show that the (second order) thermodynamic transition to the ordered phase may be interpreted in terms of confinement of the excitations in the mobility field. We also describe the aging of disordered states toward the ordered phase, in terms of simple rate equations.

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.

Renormalization group study of a kinetically constrained model for strong glasses

We derive a dynamic field theory for a kinetically constrained model, based on the Fredrickson-Andersen model, which we expect to describe the properties of an Arrhenius (strong) supercooled liquid at the coarse-grained level. We study this field theory using the renormalization group. For mesoscopic length and time scales, and for space dimension d>/=2 , the behavior of the model is governed by a zero-temperature dynamical critical point in the directed percolation universality class. We argue that in d=1 its behavior is that of compact directed percolation. We perform detailed numerical simulations of the corresponding Fredrickson-Andersen model on the lattice in various dimensions, and find reasonable quantitative agreement with the field theory predictions.

Quantum mode-coupling theory for binary mixtures

We extend the quantum mode-coupling theory of neat liquids to the case of binary mixtures, in order to study supercooled liquids where quantum fluctuations may compete with thermal fluctuations. We apply the theory to a generic model of a binary mixture of Lennard-Jones particles. Our treatment may be used to study quantum aging and exotic glass melting scenarios in structural supercooled quantum liquids.