Lifetime of dynamical heterogeneity in a highly supercooled liquid

We numerically examine dynamical heterogeneity in a highly supercooled three-dimensional liquid via molecular-dynamics simulations. To define the local dynamics, we consider two time intervals: ${\tau }_{\alpha }$ and ${\tau }_{\text{ngp}}$. ${\tau }_{\alpha }$ is the $\alpha$ relaxation time, and ${\tau }_{\text{ngp}}$ is the time at which non-Gaussian parameter of the Van Hove self-correlation function is maximized. We determine the lifetimes of the heterogeneous dynamics in these two different time intervals, ${\tau }_{\text{hetero}}\left({\tau }_{\alpha }\right)$ and ${\tau }_{\text{hetero}}\left({\tau }_{\text{ngp}}\right)$, by calculating the time correlation function of the particle dynamics, i.e., the four-point correlation function. We find that the difference between ${\tau }_{\text{hetero}}\left({\tau }_{\alpha }\right)$ and ${\tau }_{\text{hetero}}\left({\tau }_{\text{ngp}}\right)$ increases with decreasing temperature. At low temperatures, ${\tau }_{\text{hetero}}\left({\tau }_{\alpha }\right)$ is considerably larger than ${\tau }_{\alpha }$, while ${\tau }_{\text{hetero}}\left({\tau }_{\text{ngp}}\right)$ remains comparable to ${\tau }_{\alpha }$. Thus, the lifetime of the heterogeneous dynamics depends strongly on the time interval.

Figure 1

(Color) Schematic illustration of two time intervals and their time separation.

Figure 2

${\tau }_{\alpha }$, ${\tau }_{\text{ngp}}$ versus the inverse temperature $1/T$. We use these two time intervals to define the local dynamics.

Figure 3

(Color) Visualization of the heterogeneous dynamics of particle species 1. The temperature is 0.253. The time intervals are $\left[t_0,t_0+{\tau }_{\alpha }\right]$ in (a) and $\left[t_0,t_0+{\tau }_{\text{ngp}}\right]$ in (b). The radii of the spheres are ${\left|\Delta {\mathit{r}}_j\left(t_0,t\right)\right|}^2/⟨{\left[\Delta {\mathit{r}}_j\left(t_0,t\right)\right]}^2⟩$, and the centers are at $\frac{1}{2}\left[{\mathit{r}}_j\left(t_0\right)+{\mathit{r}}_j\left(t_0+t\right)\right]$. The red and blue spheres represent ${\left|\Delta {\mathit{r}}_j\left(t_0,t\right)\right|}^2/⟨{\left[\Delta {\mathit{r}}_j\left(t_0,t\right)\right]}^2⟩\ge 1$ and ${\left|\Delta {\mathit{r}}_j\left(t_0,t\right)\right|}^2/⟨{\left[\Delta {\mathit{r}}_j\left(t_0,t\right)\right]}^2⟩<1$, respectively.

Figure 4

The time decay of $S_{\mathcal{D}}\left(q,t_s,t\right)$ at $t={\tau }_{\alpha }$, $q=0.38$ for $T=0.772–0.253$. $q=0.38$ is the smallest wave number in our simulation. Temperature decreases going right.

Figure 5

The wave number dependence of ${\tau }_h\left(q,t\right)$ at $t={\tau }_{\alpha }$ for $T=0.772–0.253$. Temperature decreases going up. The error bars represent the standard deviations of averaging data over initial times. The dotted line is ${\tau }_h\left(q,{\tau }_{\alpha }\right)\sim q^{-2}$.

Figure 6

The lifetime ${\tau }_{\text{hetero}}\left(t\right)$ for $t={\tau }_{\alpha }$, ${\tau }_{\text{ngp}}$ versus ${\tau }_{\alpha }$. The error bars represent the standard deviations of averaging data over initial times. The line ${\tau }_{\text{hetero}}\sim {\tau }_{\alpha }^{1.08\pm 0.02}$ is fitted for $t={\tau }_{\alpha }$, while the line ${\tau }_{\text{hetero}}\sim {\tau }_{\alpha }^{0.91\pm 0.03}$ is fitted for $t={\tau }_{\text{ngp}}$.