Atomistic mechanism of physical ageing in glassy materials
Using molecular simulations, we identify microscopic relaxation events of individual particles in ageing structural glasses, and determine the full distribution of relaxation times. We find that the memory of the waiting time elapsed since the quench extends only up to the first relaxation event, while the distribution of all subsequent relaxation times (persistence times) follows a power law completely independent of history. Our results are in remarkable agreement with the well known phenomenological trap model of ageing. A continuous time random walk (CTRW) parametrized with the atomistic distributions captures the entire bulk diffusion behavior and explains the apparent scaling of the relaxation dynamics with during ageing, as well as observed deviations from perfect scaling.
Glasses (including metallic glasses) Theory and modeling of the glass transition Fluctuation phenomena, random processes, noise, and Brownian motion
The dynamics of glass forming materials is characterized by slow, spatially heterogeneous and temporally intermittent processes [1, 2, 3, 4, 5, 6]. Long periods of caged particle vibrations are interrupted by rapid, localized structural relaxations. Dynamical heterogeneity near the glass transition leads to stretched exponential relaxation of correlation and response functions, as well as subdiffusive, non-Gaussian transport. In the glassy state, configurational degrees of freedom continue to evolve slowly. This phenomenon is called physical ageing  and is a defining and universal feature of nonequilibrium glassy dynamics . Structural relaxations become increasingly sluggish with the waiting time elapsed since vitrification, and almost all material properties change as a result.
Ageing often manifests itself as a simple rescaling of the slow, configurational part of the global correlation functions , where is the ageing exponent. This scaling behaviour has been observed in a wide range of disordered materials  and is particularly pronounced in polymer glasses , colloidal suspensions [9, 10], and physical gels . Recent simulations of structural glasses show that scaling also applies approximately to the distribution of local correlation functions  and the distribution of particle displacements (van Hove function) , which superimpose when plotted at times of equal mean. Scaling of local correlations can be proved to hold exactly for certain spin glass models whose dynamical action obeys time reparametrization invariance . Despite the apparently universal nature of this phenomenon , the physical mechanism behind it remains elusive.
In this Letter, we directly observe the atomistic dynamics underlying physical ageing on a single particle level through molecular dynamics simulations of two model systems for atomic and polymeric glasses. We consider both the well-known amorphous 80/20 binary Lennard-Jones mixture , as well as a bead-spring model for polymer glasses [17, 18], where particles interact via the Lennard-Jones potential and are coupled together by stiff springs to form chains of length 10. Twenty thousand particles each are simulated in a periodic simulation box. For the polymer (80/20) model, the initial ensemble is equilibrated at (4.0), and then quenched rapidly to (0.33) to form a glass. The system is aged at constant pressure for various waiting times and then the dynamics are followed as a function of the measurement time . All results are quoted in Lennard-Jones units.
In both models, the effect of ageing can be immediately observed through the one-dimensional mean squared displacement shown in Fig. 1, which exhibits three regimes: a short time vibrational regime precedes a long plateau characterized by “caged” motion, followed by a cage escape or -relaxation regime. The cage escape time increases with increasing , but -curves for different waiting times can be superimposed in the cage escape regime by rescaling time with . At long times , scaling of the mean squared displacement fails for both models, and curves for the different ages merge. Experimental creep compliance curves show the same behavior, which is a well-known consequence of subageing () .
We proceed to analyze the intermittent dynamics in these systems by decomposing single particle trajectories into discrete hopping events. To investigate the hopping dynamics, a subset of five thousand particle trajectories are recorded. A running average of the atom position and its standard deviation is found for each atom over a small time window of 400 (40 individual snapshots of the particle positions) . A hop is identified when the standard deviation is greater than a threshold of . This method differs from the usual technique of identifying hops via a threshold in the hop distance [5, 21, 22], and effectively captures hops at all length scales through their increased activity during the relaxation event. While the correlation factor between hops is nearly zero, approximately of hops are either forward or backward correlated. Removing these hops does not affect the results.
Fig. 2 shows a typical atomic trajectory with the hops identified using this criterion. The initial hop times , the times between all subsequent hops , also called the persistence time, and the displacements were binned to form distribution functions , , and . The initial hop time distribution is shown in Fig. 3(a) for the Lennard-Jones mixture and in (d) for the polymer. For all waiting times, it appears to take the form of two power laws. The exponents are waiting time independent, but the crossover time between the two regimes shifts towards longer times with increasing .
Figures 3(b) and (e) show that the distributions also decay as a power law: (Lennard-Jones mixture) and (polymer). By contrast, is independent of and shows no ageing effects. The one-dimensional hop displacement distribution (Fig. 3(c) and (f)) is sharply peaked at zero displacement, and is also independent of waiting time. This is perhaps not surprising, as it has been shown that the density and the dynamical correlation lengths vary little during ageing . has a purely exponential form for the mixture, whereas in the polymer model it decays more rapidly in the tails due to chain connectivity effects. Both the first hop time distribution, and the displacement distribution are in qualitative agreement with results obtained by analyzing hops between adjacent inherent structures [23, 24, 25].
The measured distributions have much in common with the well-known phenomenological trap model of ageing . In the trap model, as in our results, is constant and continuously evolves towards longer times in the ageing regime. In fact, even the form of our measured distributions is similar. The distributions and decay as weak power laws at long times with an exponent , where is the noise temperature which governs activated hopping between states . In this formulation, is either a constant (exponential density of states), or more realistically [23, 24, 25], a slowly varying function of time (Gaussian density of states) . We do not observe any curvature in , but cannot rule out its existence at longer timescales. While our results clearly show subageing behavior, the ageing exponent in the trap model is strictly unity . However, this holds true only in the limit . We have observed that depending on the quench conditions, there can be a long lived transient regime where correlation functions approximately scale with an ageing exponent .
To confirm that our analysis captures all important physics, we recompute the ageing dynamics through stochastic realizations of a continuous time random walk (CTRW) using the measured hop distributions , and as input. For the polymer model, a pure random walk is insufficient as the chain connectivity induces subdiffusive Rouse dynamics. The mean squared displacement of the atoms as a function of the number of hops is found to be ( for the binary mixture). In the CTRW, the Rouse behaviour is accounted for by choosing a new particle position from a Gaussian distribution widening as at every step. This procedure creates the proper displacement statistics. In both systems, the initial displacement is chosen such that the mean squared displacement coincides with the MD data at the shortest measured relaxation time. This is slightly higher than the actual vibrational amplitude, but accounts for relaxations which occur at times down to the vibrational timescale, but could not be captured otherwise.
The full distribution of displacements (van Hove function)
as well the as the mean squared displacements are compared with MD data in Fig. 4 for the polymer model. As time progresses, a Gaussian caged peak at small displacements loses particles to widening exponential tails. The origin of the exponential tails has recently generated intense interest, as they appear to be a universal feature of structural glasses [28, 22, 29, 30, 31]. In our calculation, they arise naturally from the wide distribution of persistence times. The CTRW reproduces the true ageing dynamics exceedingly well, with no fit parameters. The same conclusion holds for the binary mixture. As in the trap model, the power law form of is sufficient to generate ageing. Distributions evolved via the CTRW from the shortest waiting time alone closely track measured distributions. Ageing emerges here as a self-generating, dynamical phenomenon.
Our results clearly show that there are two relevant timescales in the relaxation dynamics of the glass: the first hop time which experiences ageing, and the subsequent hop times which are independent. This would seem to refute the hypothesis that the dynamics are universally reparametrized with waiting time as has been suggested for spin glasses , and may serve to explain some deviations from local dynamical scaling that have been observed. In ref. , the authors note that the distribution of local correlation functions for a binary Lennard-Jones glass superimposes to first order at times where the global correlation function is equal. However, good collapse was found only for large and small global correlation with significant deviations for and .
We further investigate the local dynamical scaling by using the CTRW model to calculate the van Hove distribution for three waiting times and three different values of the mean in Fig. 5. Note that here is generated by evolving the system directly in the ageing CTRW. The superposition of the van Hove functions displays the same behavior reported in ref. . At short times (large ), the van Hove functions closely superimpose at times of equal mean. In this regime, few atoms have hopped and their mean squared displacement and van Hove function are dominated by the distribution , which scales with . For the mid-range of the correlation functions, the overlap of the van Hove functions becomes weaker. The long waiting time distributions have a sharper trapped peak and wider tails than the short function. This is due to the fact that the width of the tails is controlled by , whereas the height of the caged peak is controlled by . At very long times (small ), the mean number of hops per atom is much greater than one, and the waiting time independent behavior begins to dominate the dynamics. In this case, the mean squared displacement curves merge and there is trivial superposition of the van Hove functions.
To summarize, we have presented a complete characterization of particle diffusion in ageing structural glasses. We find that on a microscopic level, time is not simply reparameterized during ageing. Only the first hop time depends on the waiting time, and the particle subsequently “forgets” its age. The evolution of the first hop time distributions is remarkably close to that assumed in the trap model with annealed disorder. Although this model is a rather abstract picture of glassy dynamics, our results indicate that it may be closer to physical reality on a single particle level than previously thought. An explicit mapping of the MD data onto a CTRW with measured distributions of hop times and displacements is able to completely reproduce all dynamical features of the relaxing glass, and provides an explanation for the scaling with and small deviations thereof.
We thank the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canadian Foundation for Innovation (CFI) for financial support.
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