Ergodicity breaking of an inorganic glass aging near $T_g$ probed by elasticity relaxation

We performed a series of aging experiments with an inorganic glass (${\mathrm{As}}_2{\mathrm{Se}}_3$) at a temperature $T_2$ near the glass transition point $T_g$ by first relaxing it at $T_1$. The relaxations of Young's modulus were monitored, which were (almost if not ideally) exponential with $T_1$-dependent relaxation time τ, corroborating the Kovacs’ paradox in an inorganic glass. Associated with the divergence of τ, the quasiequilibrated Young's modulus $E_{\infty }$ does not converge either. An elastic model of relaxation time and a Mori-Tanaka analysis of $E_{\infty }$ lead to a similar estimate of the persistent memory of the history, illuminating ergodicity breaking within the accessible experimental time, as described in the Gardner transition theory. Experiments with different $T_2$ exhibit a critical temperature $T_p\sim T_g$, i.e., when $T_2>T_p$, both τ and $E_{\infty }$ converge. The results unveil a long-expected phenomenon that structural glass transition could be a zero-to-nonzero transition, manifested by a nonvanishing structural memory in aging when the temperature is below $T_p$ in the glass transition range. This demonstrates the existence of the ergodicity breaking deep in the glass state and $T_p$ could be the Gardner transition point of the structural glass.

Figure 1

The specimen, fixture, furnace, and the obtained sound signal in an IET experiment.

Figure 2

Two-step aging results of ${\mathrm{As}}_2{\mathrm{Se}}_3$ glass with $T_2=175{}^{\circ }\mathrm{C}$ and $T_1=T_2\pm 5$, 10, and $15{}^{\circ }\mathrm{C}$. Main plot: relaxation and quasiequilibrated Young's modulus at $T_2$, showing the divergence of Young's modulus relaxation time process, illustrating a $T_1$ dependence. Above-left inset: a temperature profile from 160 to $175{}^{\circ }\mathrm{C}$; above-right inset: plot of quasiequilibrium Young's modulus $E_{\infty }$ against $T_1$.

Figure 3

(a) Plots of the relative change of Young's modulus ${\delta }_E\left(t\right)$ with fitting results using Eq. (1), and plots of (b) relaxation time τ and (c) exponent β against $T_1$, indicating exponential relaxations.

Figure 4

Analysis of $T_1$ dependence of (a) relaxation time τ and (b) quasiequilibrium Young's modulus $E_{\infty }$ based on the elastic model [42] and Mori-Tanaka approach, respectively, with an inset in (b) illustrating the simplification of the $T_2$-aged glass to be a composite.

Figure 5

Temperature dependence of Young's modulus of ${\mathrm{As}}_2{\mathrm{Se}}_3$ glass. The Debye-Grüneisen is estimated by a linear fit.

Figure 6

Plot of the measure of memory persistence $V_f$ against the aging temperature $T_2$, indicating a clear transition at a critical temperature $T_p\in (177,180{}^{\circ }\mathrm{C})$ with bottom-left and upright insets showing the aging curves $E$($t$) at $T_2=180$ and $170{}^{\circ }\mathrm{C}$, respectively, after equilibrated at $T_1=T_2\pm 10{}^{\circ }\mathrm{C}$.

Figure 7

Variation of the free energy landscape and macrostate distribution in different protocols: when $T_2>T_p$, the barriers disappear at $T_2$ that the macroscopic properties of protocol 1 and protocol 2 converges due to the recovery of the ergodicity, even though $T_1<T_p$; when $T_2<T_p$, the barriers and fractal metabasins remains and the ergodicity breaks, so that different protocols results in discrepant macrostate distribution as the comprision of protocol 3 and protocol 4, which is the protocol dependence as the Gardner transition displays, even though $T_1>T_p$.