Using the $s$ ensemble to probe glasses formed by cooling and aging

From length scale distributions characterizing frozen amorphous domains, we relate the $s$ ensemble method with standard cooling and aging protocols for forming glass. We show that in a class of models where space-time scaling is in harmony with that of experiment, the spatial distributions of excitations obtained with the $s$ ensemble are identical to those obtained through cooling or aging, but the computational effort for applying the $s$ ensemble is generally many orders of magnitude smaller than that of straightforward numerical simulation of cooling or aging. We find that in contrast to the equilibrium ergodic state, a nonequilibrium length scale characterizes the anticorrelation between excitations and encodes the preparation history of glass states.

Figure 1

Soft spots and space-time bubbles. (a) Burnished excitations for the $d=2$ supercooled WCA liquid mixture [10], with $\mathrm{\Delta }{\overline{r}}_i=\left[\right|{\overline{\mathbf{r}}}_i\left(\mathrm{\Delta }t\right)-{\overline{\mathbf{r}}}_i\left(0\right){\left|\right]}_{\mathrm{iso}}$. (b) Excitation lines for a subsection of the system shown in (a). (c) The functions $Z\left(r\right)$ and $F\left(r\right)$ demonstrating correlation holes for inactive subsystems of a supercooled $d=3$ WCA liquid mixture [23].

Figure 2

Trajectories of excitations in the East model. (a) Equilibrium dynamics for $T=0.45$ over a time scale spanning about 50 structural relaxation times at that temperature. (b) Aging dynamics after a quench from $T=1$ to $T=0.25$. (c) Cooling at a rate $\nu ={10}^{-5}$. (d) Trajectory from the $s$ ensemble at $T=0.72$ and $s>s^{*}\approx {10}^{-2}$, trajectories running for about 1/2 a structural relaxation time at that temperature.

Figure 3

First-order dynamical phase transition in the $s$ ensemble with activity measured in terms of enduring kinks. (a) Probability of observing a trajectory with intensive activity $A$, for several $t_{\mathrm{obs}}$ (arrow indicates the direction of increasing $t_{\mathrm{obs}})$ for the $d=1$ East model at $T=0.72$ and $N=64$. (b) $A$ as a function of $s$ for the same systems sampled in (a). The value of $s^{*}$ is plotted as a function of $t_{\mathrm{obs}}$ in the inset for different system sizes $N$. (c) Susceptibility $\chi \left(s\right)$ as a function of $s$. The inset shows the peak of the susceptibility $\chi \left(s^{*}\right)$ as a function of $t_{\mathrm{obs}}$ for different $N$.

Figure 4

Distributions of East-model glasses. (a) The relative concentration of excitations a distance $\ell$ from one of the least active subregions in the equilibrium model at temperature $T=0.5$ (b) Distributions of domain lengths $P\left(\ell \right)$ for systems aged, cooled, and driven with $s$, all designed to yield ${\ell }_{\mathrm{ne}}\approx 10$. Here, $\ell$ represents the distance between frozen excitations (called “superspins” [9]). Superspins were identified by rapidly quenching final configurations, thus removing fleeting short-length-scale fluctuations and revealing the underlying domain structure. Aging was done at $T=0.25$ for $t_{\mathrm{age}}=2.5\times {10}^5$ after quenching from $T=0.4$. Cooling to $T=0$ was done with a cooling rate of $\nu ={10}^{-6}$. $s$ ensemble trajectories were carried out at $T=0.72$ for a time duration of $t_{\mathrm{obs}}=320$. (c)–(e) Growth of ${\ell }_{\mathrm{ne}}$ as a function of relevant time variables. Dashed lines in (c) refer to ${\ell }_{\mathrm{ne}}={(t_{\mathrm{age}}/{\tau }_{\mathrm{o}})}^{1/\tilde{\beta }\gamma }$; dashed line in (d) refers to ${\ell }_{\mathrm{eq}}\left(T_{\mathrm{g}}\right)$; dashed lines in (e) refer to ${\ell }_{\mathrm{ne}}={(t_{\mathrm{obs}}/{\tau }_{\mathrm{o}})}^{1/\tilde{\beta }\gamma }$.

Figure 5

(a) A typical configuration of the 2D system contains many hexagonal clusters (colored). The clusters are locally stable. (b), (c) Probability of sixfold clusters (3D WCA) and hexagonal clusters (2D WCA) at different times. The cluster distributions do not exhibit net growth with time at the state points considered in the main text. The last configuration is separated in time from the first configuration by more than ${10}^2$ structural relaxation times.