Are rare, long waiting times between rearrangement events responsible for the slowdown of the dynamics at the glass transition?

Non-Markovian Methods in Glass Transition

A model for the heterogeneity of local dynamics in polymer and other glass-forming materials is provided here. The fundamental characteristics of the glass transition phenomenology emerge when simulating a condensed matter open cluster that has a strong interaction with its heterogeneous environment. General glass transition features, such as non-exponential structural relaxations, the slowing down of relaxation times with temperature and specific off-equilibrium glassy dynamics can be reproduced by non-Markovian dynamics simulations with the minimum computer resources. Non-Markovian models are shown to be useful tools for obtaining insights into the complex dynamics involved in the glass transition phenomenon, including whether or not there is a need for a growing correlation length or the relationship between the non-exponentiality of structural relaxations and dynamic heterogeneity.

Continuous-time random-walk approach to supercooled liquids: Self-part of the van Hove function and related quantities

From equilibrium molecular dynamics (MD) simulations of a bead-spring model for short-chain glass-forming polymer melts we calculate several quantities characterizing the single-monomer dynamics near the (extrapolated) critical temperature [Formula: see text] of mode-coupling theory: the mean-square displacement g₀(t), the non-Gaussian parameter [Formula: see text] and the self-part of the van Hove function [Formula: see text] which measures the distribution of monomer displacements r in time t. We also determine these quantities from a continuous-time random walk (CTRW) approach. The CTRW is defined in terms of various probability distributions which we know from previous analysis. Utilizing these distributions the CTRW can be solved numerically and compared to the MD data with no adjustable parameter. The MD results reveal the heterogeneous and non-Gaussian single-particle dynamics of the supercooled melt near [Formula: see text]. In the time window of the early [Formula: see text] relaxation [Formula: see text] is large and [Formula: see text] is broad, reflecting the coexistence of monomer displacements that are much smaller ("slow particles") and much larger ("fast particles") than the average at time t, i.e. than [Formula: see text]. For large r the tail of [Formula: see text] is compatible with an exponential decay, as found for many glassy systems. The CTRW can reproduce the spatiotemporal dependence of [Formula: see text] at a qualitative to semiquantitative level. However, it is not quantitatively accurate in the studied temperature regime, although the agreement with the MD data improves upon cooling. In the early [Formula: see text] regime we also analyze the MD results for [Formula: see text] via the space-time factorization theorem predicted by ideal mode-coupling theory. While we find the factorization to be well satisfied for small r, both above and below [Formula: see text] , deviations occur for larger r comprising the tail of [Formula: see text]. The CTRW analysis suggests that single-particle "hops" are a contributing factor for these deviations.

Repetition and pair-interaction of string-like hopping motions in glassy polymers

The dynamics of many glassy systems are known to exhibit string-like hopping motions each consisting of a line of particles displacing one another. By using the molecular dynamics simulations of glassy polymers, we show that these motions become highly repetitive back-and-forth motions as temperature decreases and do not necessarily contribute to net displacements. Particle hops which constitute string-like motions are reversed with a high probability, reaching 73% and beyond at low temperature. The structural relaxation rate is then dictated not by a simple particle hopping rate but instead by the rate at which particles break away from hopping repetitions. We propose that disruption of string repetitions and hence also structural relaxations are brought about by pair-interactions between strings.

Spatial correlations of elementary relaxation events in glass-forming liquids

The dynamical facilitation scenario, by which localized relaxation events promote nearby relaxation events in an avalanche process, has been suggested as the key mechanism connecting the microscopic and the macroscopic dynamics of structural glasses. Here we investigate the statistical features of this process via numerical simulations of a model structural glass. First we show that the relaxation dynamics of the system occurs through particle jumps that are irreversible, and that cannot be decomposed in smaller irreversible events. Then we show that each jump does actually trigger an avalanche. The characteristics of this avalanche change upon cooling, suggesting that the relaxation dynamics crossovers from a noise dominated regime, where jumps do not trigger other relaxation events, to a regime dominated by the facilitation process, where a jump triggers more relaxation events.

Connecting short and long time dynamics in hard-sphere-like colloidal glasses

Glass-forming materials are characterized by an intermittent motion at the microscopic scale. Particles spend most of their time rattling within the cages formed by their neighbors, and seldom jump to a different cage. In molecular glass formers the temperature dependence of the jump features, such as the average caging time and jump length, characterizes the relaxation processes and allows for a short-time prediction of the diffusivity. Here we experimentally investigate the cage-jump motion of a two-dimensional hard-sphere-like colloidal suspension, where the volume fraction is the relevant parameter controlling the slowing down of the dynamics. We characterize the volume fraction dependence of the cage-jump features and show that, as in molecular systems, they allow for a short time prediction of the diffusivity.

Waiting time distribution for continuous stochastic systems

The waiting time distribution (WTD) is a common tool for analyzing discrete stochastic processes in classical and quantum systems. However, there are many physical examples where the dynamics is continuous and only approximately discrete, or where it is favourable to discuss the dynamics on a discretized and a continuous level in parallel. An example is the hindered motion of particles through potential landscapes with barriers. In the present paper we propose a consistent generalization of the WTD from the discrete case to situations where the particles perform continuous barrier crossing characterized by a finite duration. To this end, we introduce a recipe to calculate the WTD from the Fokker-Planck (Smoluchowski) equation. In contrast to the closely related first passage time distribution (FPTD), which is frequently used to describe continuous processes, the WTD contains information about the direction of motion. As an application, we consider the paradigmatic example of an overdamped particle diffusing through a washboard potential. To verify the approach and to elucidate its numerical implications, we compare the WTD defined via the Smoluchowski equation with data from direct simulation of the underlying Langevin equation and find full consistency provided that the jumps in the Langevin approach are defined properly. Moreover, for sufficiently large energy barriers, the WTD defined via the Smoluchowski equation becomes consistent with that resulting from the analytical solution of a (two-state) master equation model for the short-time dynamics developed previously by us [Phys. Rev. E 86, 061135 (2012)]. Thus, our approach "interpolates" between these two types of stochastic motion. We illustrate our approach for both symmetric systems and systems under constant force.

From cage-jump motion to macroscopic diffusion in supercooled liquids

The evaluation of the long term stability of a material requires the estimation of its long-time dynamics. For amorphous materials such as structural glasses, it has proven difficult to predict the long-time dynamics starting from static measurements. Here we consider how long one needs to monitor the dynamics of a structural glass to predict its long-time features. We present a detailed characterization of the statistical features of the single-particle intermittent motion, and show that single-particle jumps are the irreversible events leading to the relaxation of the system. This allows us to evaluate the diffusion constant on the time-scale of the jump duration, which is small and temperature independent, i.e. well before the system enters the diffusive regime. The prediction is obtained by analyzing the particle trajectories via a parameter-free algorithm.

Renewal events in glass-forming liquids

On cooling toward the glass transition temperature, glass-forming liquids display long periods of localized motion interrupted by fast "jumps" in the single-particle trajectories. Several theoretical models based on these single-particle jumps have been proposed, most prominently the continuous-time random walk (CTRW). The central assumption of the CTRW is that jumps are renewal events, i.e. that the internal clock of a particle can be reset upon a jump. In this paper, I present an easy-to-implement method to test whether jumps detected in a supercooled liquid or glass are renewal events or not. The test was applied to molecular dynamics simulations of a short-chain polymer melt, demonstrating that the jumps can in fact be treated as renewal events. The test further revealed that additional relaxation processes are present which are not accounted for in the CTRW picture, highlighting the limitations of this approach. The notion of renewal events in glass-forming systems could be a very important building block for the interpretation of aging and the glass transition. Furthermore, it could have practical implications for the study of non-equilibrium dynamics in glasses as well as mechanical rejuvenation.

Continuous-time random-walk approach to supercooled liquids. II. Mean-square displacements in polymer melts

The continuous-time random walk (CTRW) describes the single-particle dynamics as a series of jumps separated by random waiting times. This description is applied to analyze trajectories from molecular dynamics (MD) simulations of a supercooled polymer melt. Based on the algorithm presented by Helfferich et al. [Phys. Rev. E 89, 042603 (2014)], we detect jump events of the monomers. As a function of temperature and chain length, we examine key distributions of the CTRW: the jump-length distribution (JLD), the waiting-time distribution (WTD), and the persistence-time distribution (PTD), i.e., the distribution of waiting times for the first jump. For the equilibrium (polymer) liquid under consideration, we verify that the PTD is determined by the WTD. For the mean-square displacement (MSD) of a monomer, the results for the CTRW model are compared with the underlying MD data. The MD data exhibit two regimes of subdiffusive behavior, one for the early α process and another at later times due to chain connectivity. By contrast, the analytical solution of the CTRW yields diffusive behavior for the MSD at all times. Empirically, we can account for the effect of chain connectivity in Monte Carlo simulations of the CTRW. The results of these simulations are then in good agreement with the MD data in the connectivity-dominated regime, but not in the early α regime where they systematically underestimate the MSD from the MD.

Continuous-time random-walk approach to supercooled liquids. I. Different definitions of particle jumps and their consequences

Single-particle trajectories in supercooled liquids display long periods of localization interrupted by "fast moves." This observation suggests a modeling by a continuous-time random walk (CTRW). We perform molecular dynamics simulations of equilibrated short-chain polymer melts near the critical temperature of mode-coupling theory Tc and extract "moves" from the monomer trajectories. We show that not all moves comply with the conditions of a CTRW. Strong forward-backward correlations are found in the supercooled state. A refinement procedure is suggested to exclude these moves from the analysis. We discuss the repercussions of the refinement on the jump-length and waiting-time distributions as well as on characteristic time scales, such as the average waiting time ("exchange time") and the average time for the first move ("persistence time"). The refinement modifies the temperature (T) dependence of these time scales. For instance, the average waiting time changes from an Arrhenius-type to a Vogel-Fulcher-type T dependence. We discuss this observation in the context of the bifurcation of the α process and (Johari) β process found in many glass-forming materials to occur near Tc. Our analysis lays the foundation for a study of the jump-length and waiting-time distributions, their temperature and chain-length dependencies, and the modeling of the monomer dynamics by a CTRW approach in the companion paper [J. Helfferich et al., Phys. Rev. E 89, 042604 (2014)].