Fast generation of ultrastable computer glasses by minimization of an augmented potential energy

We present a model and protocol that enable the generation of extremely stable computer glasses at minimal computational cost. The protocol consists of an instantaneous quench in an augmented potential energy landscape, with particle radii as additional degrees of freedom. We demonstrate how our glasses' mechanical stability, which is readily tunable in our approach, is reflected in both microscopic and macroscopic observables. Our observations indicate that the stability of our computer glasses is at least comparable to that of computer glasses generated by the celebrated Swap Monte Carlo algorithm. Strikingly, some key properties support even qualitatively enhanced stability in our scheme: the density of quasilocalized excitations displays a gap in our most stable computer glasses, whose magnitude scales with the polydispersity of the particles. We explain this observation, which is consistent with the lack of plasticity we observe at small stress. It also suggests that these glasses are depleted from two-level systems, similarly to experimental vapor-deposited ultrastable glasses.

Figure 1

(a) Pairs of nearby particles in our model glass former interact via the pairwise potential $\phi (r_{ij},{\lambda }_i,{\lambda }_j)$ as given by Eq. (2), represented here by the thick blue line. We also plot the canonical Lennard-Jones potential (thin line) for comparison. (b) During the preparation of our glassy samples we allow the effective size DOF ${\lambda }_i$ to change; their fluctuations are governed by the potential $\mu ({\lambda }_i,{\lambda }_i^{\left(0\right)})$ given by Eq. (3), and plotted here for various values of the stiffness $k_{\lambda }$ as indicated by the legend.

Figure 2

Distributions $p\left(\lambda \right)$ of particles' effective size DOF, plotted against the dimensionless effective size $\lambda {\rho }^{1/3}$, for various values of $k_{\lambda }$ as indicated by the arrows. The dashed lines represent the distributions measured for particles whose target effective size during glass preparation was ${\lambda }^{\left(0\right)}=0.5$, and the solid lines represent those for which ${\lambda }^{\left(0\right)}=0.7$. Inset: the polydispersity $\mathrm{\Delta }$ is defined as the ratio of the effective sizes' standard deviation to their mean. We report $\mathrm{\Delta }$ for “small” and “large” particles, and the total polydispersity (calculated as the standard deviation to mean ratio, taken over all particles), vs the stiffnesses $k_{\lambda }$.

Figure 3

Pair correlations $g\left(r\right)$ between pairs of “large” particles (i.e., those with a target effective size ${\lambda }^{\left(0\right)}=0.70$ during glass preparation) measured in our glassy samples, plotted against the rescaled pairwise distances $r{\rho }^{1/3}$ for various values of the stiffness $k_{\lambda }$, increasing from thin to thick lines. We find no ordering upon decreasing $k_{\lambda }$. The inset shows how the density $\rho =N/V$ of our glassy samples increases as $k_{\lambda }$ is reduced. The horizontal dashed line marks the density of our $k_{\lambda }=\infty$ glasses.

Figure 4

Elastic properties of our glassy samples: (a) sample-to-sample standard deviation of the dimensionless shear stress $\delta \tilde{\sigma }$ (see text for definition and discussion) vs the stiffness of the size DOF $k_{\lambda }$. (b) $\delta \tilde{G}$ is the sample-to-sample standard deviation to mean ratio of the shear modulus, scaled by $\sqrt{N}$. (c) Sample-to-sample mean of the ratio of shear to bulk moduli. (d) Sample-to-sample mean Poisson's ratio of our glassy samples. The horizontal dashed lines represent the $k_{\lambda }=\infty$ values in all panels.

Figure 5

(a) Density of vibrational modes $D\left(\omega \right)$ vs rescaled frequency $\omega /{\omega }_0$ where ${\omega }_0\equiv c_s/{\rho }^{-1/3}$ and $c_s\equiv \sqrt{G/\rho }$ is the shear wave speed. (b) In the limit of large $k_{\lambda }$ a gap of order $1/\sqrt{k_{\lambda }}$ opens in $D\left(\omega \right)$ as the collapse indicates. Here we plot $D\left(\omega \right)$ for $k_{\lambda }={10}^5,3\times {10}^4$, and ${10}^4$. For smaller $k_{\lambda }$, the occurrence of the lowest-frequency phonons in $D\left(\omega \right)$ destroys the collapse. Inset: numerical validation of the frequency independence of the particlewise mean squared variation of the forces $f_i\equiv -\partial U/\partial {\lambda }_i$, induced by a quasilocalized vibrational mode of frequency $\omega$ in $k_{\lambda }=\infty$ glasses; see discussion in Sec. 5.

Figure 6

Particlewise distributions $p\left(v^2\right)$ of nonaffine displacements squared $v^2$, plotted against the dimensionless displacements squared $v^2{\rho }^{2/3}$, measured in our ensembles of glassy samples with various values of the size DOF $k_{\lambda }$, increasing from thin to thick lines.

Figure 7

Stress-strain curves for computer glasses made with $k_{\lambda }={10}^4,3\times {10}^3,{10}^3$, and ${10}^1$, from (a) to (d). Stresses are rescaled by their respective average steady-state values. The large insets show representative snapshots of the deformed solids; the color coding represents the magnitude of total nonaffine displacements $\delta x$ measured between $\gamma =0$ and $\gamma =0.13$, rescaled by the length ${\rho }^{-1/3}$. The small inset illustrates the geometry of the applied shear deformation.

Figure 8

Example stress-strain signals obtained in shear cycles, for $k_{\lambda }={10}^4$ (a), ${10}^3$ (b), and ${10}^2$ (c) and for maximal strains ${\gamma }_{\max }$ as indicated by the legend (growing from thin to thick). Curves are rescaled by the same steady-state stresses ${\sigma }_{\infty }$ as used in Fig. 7. For large $k_{\lambda }$ the dynamics is irreversible, even for small ${\gamma }_{\max }$. For intermediate $k_{\lambda }$ some plasticity sets in before the macroscopic shear-banding event; however, the system is able to nearly return to the zero-stress state upon reversing the strain. For our most stable glasses we observe almost no plasticity before the macroscopic shear band (see also Fig. 9 below).

Figure 9

Energy dissipated per unit volume (made dimensionless, see text) in a shear cycle of magnitude ${\gamma }_{\max }$, reported in linear (a) and logarithmic (b) scales.

Figure 10

(a) Stress-strain curves for computer glasses made with $k_{\lambda }={10}^2$, in which the two possible shear-banding geometries occur, as shown in panels (b) and (c). The color code of particles, which represents the magnitude of nonaffine displacements, is the same as in Fig. 7.