Emergent facilitation behavior in a distinguishable-particle lattice model of glass

We propose an interacting lattice gas model of structural glass characterized by particle distinguishability and site-particle-dependent random nearest-neighboring particle interactions. This incorporates disorder quenched in the configuration space rather than in the physical space. The model exhibits nontrivial energetics while still admitting exact equilibrium states directly constructible at arbitrary temperature and density. The dynamics is defined by activated hopping following standard kinetic Monte Carlo approach without explicit facilitation rule. Kinetic simulations show emergent dynamic facilitation behaviors in the glassy phase in which motions of individual voids are significant only when accelerated by other voids nearby. This provides a microscopic justification for the dynamic facilitation picture of structural glass.

Figure 1

(a) Schematic diagram of a region with distinguishable particles randomly colored. The arrows indicate a possible sequence of hops by four particles arranged in a line. The dynamics is equivalently described by four hops of a single void in the reversed direction. (b) The particle displacements alter the nearest neighbor pairings and hence the pair interactions (indicated by black lines) along the whole path.

Figure 2

(a) Particle mean square displacement (MSD) against $t$ in log-log scale for $T=$ 0.170, 0.190, 0.216, 0.250, 0.296, 0.363, 0.470, 0.666, 1.142, 4.000, and void density ${\varphi }_v=0.01$ with the highest $T$ at the top. (b) Arrhenius plot of $D$ for ${\varphi }_v=$ 0.005, 0.008, 0.013, 0.021, 0.035, 0.056, 0.092, 0.149, 0.242, 0.392, with the highest ${\varphi }_v$ at the top.

Figure 3

(a) Decays of self-intermediate scattering function $F_s(q,t)$ in linear-log scale, for the same values of $T$ used in Fig. 2, with $T$ decreasing from left to right. Wave number $q=(2\pi /L)q^{\prime }=\pi /5$ and ${\varphi }_v=0.01$ are used here. (b) Same data as in (a) in log-log-versus-log scale. Data corresponding to $F_s(q,t)<{10}^{-3}$ are noisy and are omitted. The slope of the linear region at large $t$ with ${10}^{-3}\le F_s(q,t)\le 0.9$ gives the stretching exponent $\mathbf{\beta }$.

Figure 4

(a) Stretching exponent $\mathbf{\beta }$ plotted against $1/T$ for ${\varphi }_v=0.01$. (b) Violation of the Stokes-Einstein relation, $D{\tau }_{\alpha }=\text{constant}$ where ${\tau }_{\alpha }$ is a relaxation time.

Figure 5

${\chi }_4\left(t\right)$ for ${\varphi }_v=0.01$ and the same values of $T$ used in Fig. 2. $T$ decreases from left to right.

Figure 6

(a) Particle diffusion coefficient $D$ against void density ${\varphi }_v$ in log-log scale for values of $T$ used in Fig. 2 with the highest $T$ at the top. (b) Scaling exponent $\alpha$ against $1/T$ obtained from linear fits to data in (a) with ${\varphi }_v\le 0.05$.

Figure 7

(a) A snapshot from a small-scale simulation on a $40\times 40$ lattice with 1592 particles and 8 voids, i.e., ${\varphi }_v=0.005$. It shows the final positions of voids (black squares) after a short simulation duration of $\mathrm{\Delta }\tau ={10}^{-3}$ at $T=0.5$. Particles with net displacements 0, 1, and $>1$ during the period are shaded in white, pink, and red, respectively. Each thin line shows the trajectory of a void and is colored randomly. (b) Similar to (a) with $T=0.16$ and $\mathrm{\Delta }\tau =5\times {10}^4$. In both (a) and (b), the particle MSD during the period is about 0.5. Particle dynamics are shown in videos in the Supplemental Material [28].

Figure 8

Probabilities $P_{\mathrm{ret}}$ and $P_2$ for returning and nonreturning second hops against $1/T$ for ${\varphi }_v=0.01$.

Figure 9

(a) Plot of energy per particle $E/N$ against time $t$ from four independent runs adopting elementary (solid curves) and direct (dashed curves) initialization algorithms. The black dotted line shows $〈E〉/N$ from Eq. (A36). (b) A semi-log plot of the probability distribution $p_t\left(V_{ij{s}_i{s}_j}\right)$ of realized interaction $V_{ij{s}_i{s}_j}$ at time $t$ from simulations adopting the elementary initialization algorithm. $p_t\left(V_{ij{s}_i{s}_j}\right)$ for $t\ge 256000$ has converged to $p_{eq}$ from Eq. (A33) indicated by the black dashed line. For both (a) and (b), $T=0.170$ and ${\varphi }_v=0.01$.