From Subaging to Hyperaging in Structural Glasses

We demonstrate nonequilibrium scaling laws for the aging and equilibration dynamics in glass formers that emerge from combining a relaxation equation for the static structure with the equilibrium scaling laws of glassy dynamics. Different scaling regimes are predicted for the evolution of the structural relaxation time $\tau$ with age (waiting time $t_w$), depending on the depth of the quench from the liquid into the glass: “simple” aging ($\tau \sim t_w$) applies for quenches close to the critical point of mode-coupling theory (MCT) and implies “subaging” ($\tau \approx t_w^{\delta }$ with $\delta <1$) as a broad equilibration crossover for quenches to nearly arrested equilibrium states; “hyperaging” (or superaging, $\tau \sim t_w^{{\delta }^{\prime }}$ with ${\delta }^{\prime }>1$) emerges for quenches deep into the glass. The latter is cut off by non-mean-field fluctuations that we account for within a recent extension of MCT, the stochastic $\beta$-relaxation theory (SBR). We exemplify the scaling laws with a schematic model that quantitatively fits simulation data.

Figure 1

Structural-relaxation time $\tau$ as a function of waiting time $t_w$ after an instantaneous quench. Solid lines: schematic model, quenches from ${ϵ}_i=-0.5$ to ${ϵ}_f$. Inset: values in the Supplemental Material [52]. A dashed line indicates simple aging $\tau \sim t_w$, a dotted line hyperaging $\tau \sim t_w^{{\delta }^{\prime }}$ with ${\delta }^{\prime }=1.684$, and a dash-dotted line subaging $\tau \approx t_w^{\delta }$ with $\delta =0.9$. Thick dashed lines: SBR for ${ϵ}_f=0.01$ and 0.02. Symbols: simulation results for quasihard spheres from Ref. [29], quenched to various final packing fractions ${\phi }_f$, translated to schematic-model units ($\tau \mapsto 2\tau$, $t_w\mapsto 100t_w$).

Figure 2

Structural-relaxation time $\tau$ at various fixed waiting times $t_w$ as a function of the final point of the quench ${ϵ}_f$ (solid lines and symbols: theory and simulation as in Fig. 1; thick dashed lines from SBR). Thin dashed line, equilibrium divergence at the ideal glass transition point (${ϵ}_f=0$), ${\tau }_{\mathrm{eq}}\sim |{ϵ}_f{|}^{-\gamma }$ ($\gamma =2.46214$ corresponding to $\lambda =0.735$); dash-dotted lines, nonequilibrium asymptotes $\tau \sim B(t_w){ϵ}_f^{{\delta }^{\prime }}$.

Figure 3

Ratio $\tau /t_w$ as a function of waiting time $t_w$. Simulation data of Ref. [29] (filled symbols, data divided by 50; ${\phi }_c\approx 0.585$) and exemplary SBR results for ${ϵ}_f=-0.1$, 0, and 0.1 (solid lines, bottom to top). Dashed lines indicate the corresponding asymptotes of the ideal glass MCT.