Effect of instantaneous and continuous quenches on the density of vibrational modes in model glasses

Computational studies of supercooled liquids often focus on various analyses of their “underlying inherent states”—the glassy configurations at zero temperature obtained by an infinitely fast (instantaneous) quench from equilibrium supercooled states. Similar protocols are also regularly employed in investigations of the unjamming transition at which the rigidity of decompressed soft-sphere packings is lost. Here we investigate the statistics and localization properties of low-frequency vibrational modes of glassy configurations obtained by such instantaneous quenches. We show that the density of vibrational modes grows as ${\omega }^{\beta }$ with $\beta$ depending on the parent temperature $T_0$ from which the glassy configurations were instantaneously quenched. For quenches from high temperature liquid states we find $\beta \approx 3$, whereas $\beta$ appears to approach the previously observed value $\beta =4$ as $T_0$ approaches the glass transition temperature. We discuss the consistency of our findings with the theoretical framework of the soft potential model, and contrast them with similar measurements performed on configurations obtained by continuous quenches at finite cooling rates. Our results suggest that any physical quench at rates sufficiently slower than the inverse vibrational time scale—including all physically realistic quenching rates of molecular or atomistic glasses—would result in a glass whose density of vibrational modes is universally characterized by $\beta =4$.

Figure 1

Stress autocorrelation function (see text for definition) measured in equilibrium simulation runs at temperatures $T=2.00,1.20,0.85,0.70,0.60,0.56,0.54,0.53,$ and 0.52, decreasing from left to right. Inset: the relaxation times ${\tau }_{\alpha }$ vs $1/T$, determined by $c\left({\tau }_{\alpha }\right)=1$, as indicated by the dashed horizontal line of the main panel.

Figure 2

Density of vibrational modes $D\left(\omega \right)$ measured in the ensembles of glassy samples quenched from the parent temperatures $T_0=2.00,0.70,0.60,0.56,0.54,0.53,$ and 0.52, decreasing from top to bottom. The frequency axis is scaled by ${\omega }_0=3.0$ for visualization purposes. The dash-dotted line fitted to the $T_0=0.60$ data set corresponds to $D\left(\omega \right)\sim {\omega }^{3.4}$.

Figure 3

Participation ratio $e$ of vibrational modes vs frequency $\omega$, for glassy samples instantaneously quenched from a parent temperature $T_0$ as indicated in the figure. The shaded gray areas cover the second through ninth deciles of data, and the circles represent the mean participation ratio binned over frequency. The frequency axes are scaled by ${\omega }_0=3.0$ for visualization purposes. Stronger localization is observed as $T_0\to T_g$.

Figure 4

Density of vibrational modes $D\left(\omega \right)$ measured in emsembles of glassy samples quenched continuously at rates as indicated by the legend, starting from equilibrium configurations at $T=1.00$. Both continuous lines correspond to the ${\omega }^4$ law. The frequency axis is scaled by ${\omega }_0=3.0$ for visualization purposes.

Figure 5

Potential energy per particle of glassy samples averaged over (a) ensembles created by instantaneous quenches from the parent temperature $T_0$, and (b) ensembles created by continuous quenches at quench rates $\dot{T}$. The dashed horizontal line shows that the $T_0=0.60$ ensemble and the $\dot{T}={10}^{-3}$ ensemble have very similar energies per particle (see text for further discussion).