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# Relationship between local structure and relaxation in out-of-equilibrium glassy systems

Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved November 30, 2016 (received for review June 23, 2016)

## Significance

A key stumbling block in the study of out-of-equilibrium systems is that their properties depend not only on their current state but also on the manner in which the systems were prepared. Therefore, to understand generic systems out of equilibrium one must take into account the entire history of their preparation. Here we show that the history dependence of a widely studied model of a glass is captured by a quantity that depends only on the positions of the constituent particles. This quantity, called softness, is designed to be a structural order parameter for dynamics in glassy liquids. Our results suggest that a large class of out-of-equilibrium glasses can be understood by augmenting the usual state variables by the instantaneous softness distribution.

## Abstract

The dynamical glass transition is typically taken to be the temperature at which a glassy liquid is no longer able to equilibrate on experimental timescales. Consequently, the physical properties of these systems just above or below the dynamical glass transition, such as viscosity, can change by many orders of magnitude over long periods of time following external perturbation. During this progress toward equilibrium, glassy systems exhibit a history dependence that has complicated their study. In previous work, we bridged the gap between structure and dynamics in glassy liquids above their dynamical glass transition temperatures by introducing a scalar field called “softness,” a quantity obtained using machine-learning methods. Softness is designed to capture the hidden patterns in relative particle positions that correlate strongly with dynamical rearrangements of particle positions. Here we show that the out-of-equilibrium behavior of a model glass-forming system can be understood in terms of softness. To do this we first demonstrate that the evolution of behavior following a temperature quench is a primarily structural phenomenon: The structure changes considerably, but the relationship between structure and dynamics remains invariant. We then show that the relaxation time can be robustly computed from structure as quantified by softness, with the same relation holding both in equilibrium and as the system ages. Together, these results show that the history dependence of the relaxation time in glasses requires knowledge only of the softness in addition to the usual state variables.

In liquids cooled quickly enough so that crystallization is avoided, the dynamics become increasingly sluggish (1⇓–3) until

In recent papers we introduced a machine-learning approach to construct a “softness” field,

Here we show that a framework built on softness provides a coherent description of the out-of-equilibrium behavior of glassy liquids both above and below **1**) with the identical prefactors that we identified in the supercooled liquid. Thus, the characteristic multiplicity

Tanaka et al. (14) and Kawasaki and Tanaka (15) have also studied the connection of structure to dynamics in glassy systems in and out of equilibrium. They identified a structural length scale across a variety of systems from measures of bond-orientational order and related the relaxation time to this length scale. However, bond orientational order parameters alone are poor predictors of microscopic dynamics compared with softness (16). By contrast, softness correlates strongly with dynamics. It would be interesting in future work to consider the relationship between softness and the length scale proposed by Tanaka et al. (14) and Kawasaki and Tanaka (15).

## Background

We first summarize our method for computing the softness; for a more detailed description, see refs. 11⇓–13. We characterize the local structural environment of a central particle

Next we review the connection between softness and dynamics in equilibrium, introduced in ref. 12. We calculated the probability for a particle of softness *A*). In particular, we demonstrated (12) that

Both **1**, imply that dynamics will be independent of softness—and hence structure—when *A*, where all of the Arrhenius fits intersect at the same temperature,

To study the equilibration process and aging, we follow the procedure outlined by Kob and Barrat (7) using molecular dynamics. We first equilibrate an 80:20 binary Lennard-Jones mixture of

## History Dependence

In Fig. 1 we show inherent structures of the system at three different values of

For a quantitative analysis of evolution during aging, we investigate the connection between the changing structure and the increasingly sluggish dynamics of the aging system. The slow dynamics in the glass can be quantified in terms of the relaxation time

We plot *A* for

At later times the increase of relaxation time is well-described by a power law,

In Fig. 2*B* we plot the mean softness of particles,

## The Relationship Between Structure and Relaxation

To connect structure (softness) with dynamics, we consider the softness-dependent probability of rearrangement, *A* we plot *A* also shows the Arrhenius fits that we obtained in ref. 12 for the temperature range *B* we plot

The results of Figs. 2*B* and 3*A* imply that the description of the aging process is simplified considerably when viewed through the lens of softness. As a glass ages, it has long been recognized that the average energy barrier increases as the system becomes trapped in deeper and deeper minima (8, 9). Our results show that for particles of a given softness the energy barrier is unchanged. The average energy barrier increases with age simply because the distribution of softness shifts to lower values. Thus, the increasing relaxation time of glasses and supercooled liquids during aging is primarily structural in origin. Our results also imply that the history-dependent behavior of glasses can be understood in terms of local structure as quantified by the softness field.

We now consider the relationship between the relaxation time

To test this prediction we plot, in Fig. 4*A*, the relaxation time **3**. For each final temperature *A*) and denote the slope by *A*, as implied by Eq. **3**, that each of these fits intersects at *B* **3**. We find excellent agreement at all temperatures with **3** is a robust descriptor of relaxation in glassy systems both in and out of equilibrium.

## Discussion and Conclusion

Our expression for the relaxation time in Eq. **3** can be compared with previous models of glassy relaxation. In particular, we consider a parabolic form (19, 21), the VFT form, and the Bässler law (22) given, respectively, by**4** is not a free parameter; it is the onset temperature, which arises naturally in our framework. Each of these laws has been used to fit a large set of experimental relaxation time data for glassy liquids over many decades of relaxation time. To make the comparison we consider the temperature dependence of **3** with each of the models Eqs. **4**–**6**. We then compare the implied **3** is inconsistent with the VFT equation and the Bässler law for relaxation. This argument does not preclude modified forms of these laws that are asymptotically the same at lower temperatures with no pole at **3** on

We plot in Fig. 4*C* a comparison of **3** and the connection to the parabolic form by considering the implied average softness at the onset temperature. In particular, we see that when *A* the identical value, *C* implies that **3** and of the parabolic form, using two distinct sets of data.

As noted above the average softness, *B* should not decrease indefinitely even at the lowest temperatures. From Eq. **3**, a finite value of **3** breaks down at lower temperatures that we are unable to access or there is no divergence in

Finally, we investigate the question of whether or not the age of a glass may be inferred from its structure alone. To this end, we use the observation that the average softness of a system seems to follow the same function of *A*). We test the accuracy of this fit by predicting *B*). We suggest that this approach can be used to date disordered materials of unknown age, as long as a model can be fit to the age of another system of the same material as a function of its average softness or other sufficiently descriptive structural quantities that can be measured at different ages.

Our results show that the concept of softness is useful even for systems out of equilibrium at temperatures below the dynamical glass transition. Indeed, it would seem that history-dependent behavior in glasses can be understood in terms of local structure as quantified by the softness field, and that the connection between softness and the relaxation time is remarkably simple and independent of age. A common critique of numerical results such as the ones presented here in glass transition studies is that the timescales accessible in simulation are short compared with those observable in experiments. As a result, studies restricted to the equilibrium behavior of glassy liquids necessarily probe only properties at relatively high temperatures. It is encouraging that we observe exactly the same functional form in the equilibrium liquid and well inside the aging glass state for (*i*) the relation of softness to the probability of a rearrangement and (*ii*) the relation of the relaxation time to average softness (Eq. **3**). This agreement suggests that our results for these relations may not be hampered by limitations of computational modeling. These results, together with our demonstration that aging is structural, provide evidence that history dependence in glasses can be quantified using softness.

## Acknowledgments

This work was supported by US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering Award DE-FG02-05ER46199 (to S.S.S. and A.J.L.), a Harvard IACS Student Scholarship (to E.D.C.), and a Simons Investigator award from the Simons Foundation (to A.J.L.).

## Footnotes

↵

^{1}S.S.S. and E.D.C. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: ajliu{at}sas.upenn.edu.

Author contributions: S.S.S., E.D.C., and A.J.L. designed research; S.S.S. and E.D.C. performed research; S.S.S. and E.D.C. contributed new reagents/analytic tools; S.S.S. and E.D.C. analyzed data; and S.S.S., E.D.C., E.K., and A.J.L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1610204114/-/DCSupplemental.

Freely available online through the PNAS open access option.

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