Dynamical heterogeneity in a highly supercooled liquid: Consistent calculations of correlation length, intensity, and lifetime

We have investigated dynamical heterogeneity in a highly supercooled liquid using molecular-dynamics simulations in three dimensions. Dynamical heterogeneity can be characterized by three quantities: correlation length ${\xi }_4$, intensity ${\chi }_4$, and lifetime ${\tau }_{\mathrm{hetero}}$. We evaluated all three quantities consistently from a single order parameter. In a previous study [H. Mizuno and R. Yamamoto, Phys. Rev. E 82, 030501(R) (2010)], we examined the lifetime ${\tau }_{\mathrm{hetero}}\left(t\right)$ in two time intervals $t={\tau }_{\alpha }$ and ${\tau }_{\mathrm{ngp}}$, where ${\tau }_{\alpha }$ is the $\alpha$-relaxation time and ${\tau }_{\mathrm{ngp}}$ is the time at which the non-Gaussian parameter of the Van Hove self-correlation function is maximized. In the present study, in addition to the lifetime ${\tau }_{\mathrm{hetero}}\left(t\right)$, we evaluated the correlation length ${\xi }_4\left(t\right)$ and the intensity ${\chi }_4\left(t\right)$ from the same order parameter used for the lifetime ${\tau }_{\mathrm{hetero}}\left(t\right)$. We found that as the temperature decreases, the lifetime ${\tau }_{\mathrm{hetero}}\left(t\right)$ grows dramatically, whereas the correlation length ${\xi }_4\left(t\right)$ and the intensity ${\chi }_4\left(t\right)$ increase slowly compared to ${\tau }_{\mathrm{hetero}}\left(t\right)$ or plateaus. Furthermore, we investigated the lifetime ${\tau }_{\mathrm{hetero}}\left(t\right)$ in more detail. We examined the time-interval dependence of the lifetime ${\tau }_{\mathrm{hetero}}\left(t\right)$ and found that as the time interval $t$ increases, ${\tau }_{\mathrm{hetero}}\left(t\right)$ monotonically becomes longer and plateaus at the relaxation time of the two-point density correlation function. At the large time intervals for which ${\tau }_{\mathrm{hetero}}\left(t\right)$ plateaus, the heterogeneous dynamics migrate in space with a diffusion mechanism, such as the particle density.

Figure 1

Schematic illustration of the time configuration of the correlation functions of the particle dynamics: (a) the spatial correlation function of the particle dynamics in the time interval $[0,t]$ and (b) the time correlation function of the particle dynamics between the two time intervals $[0,t]$ and $[t_s+t,t_s+2t]$.

Figure 2

The wave-number dependence of (a) ${\tau }_{sa}\left(k\right)$ and (b) ${\tau }_{ca}\left(k\right)$ for particle species $a=1$ and 2. Temperatures are 0.772–0.253 from the lowest curve to the highest. The black line is ${\tau }_{sa}\left(k\right)=D_{sa}^{-1}{k}^{-2}$, where $D_{sa}$ is the diffusion constant of the single-particle motion calculated from the mean-square displacement.

Figure 3

The wave-number dependence of (a) $S_{\rho \rho }\left(k\right)$ and (b) ${\tau }_{\rho \rho }\left(k\right)$. The temperatures are 0.772–0.253 from the lowest curve to the highest.

Figure 4

Time intervals ${\tau }_{\alpha }$ and ${\tau }_{\mathrm{ngp}}$ vs the inverse temperature $1/T$. We use these two time intervals to define the local dynamics.

Figure 5

The distribution of the particle mobility $a_{1j}^2(t_0,t)$ for particle species 1. The time intervals are (a) $[t_0,t_0+{\tau }_{\alpha }](t={\tau }_{\alpha })$ and (b) $[t_0,t_0+{\tau }_{\mathrm{ngp}}](t={\tau }_{\mathrm{ngp}})$. The temperature is $0.253$. The radii of the spheres are $a_{1j}^2(t_0,t)$, and the centers are at ${\mathit{R}}_{1j}(t_0,t)$. The red and blue spheres represent $a_{1j}^2(t_0,t)⩾1$ (mobile particles) and $a_{1j}^2(t_0,t)<1$ (immobile particles), respectively.

Figure 6

The wave-number dependence of $S_4(q,t)$ for particle species 1 at $T=0.772$–0.253. The time interval $t$ is ${\tau }_{\alpha }$ in (a) and ${\tau }_{\mathrm{ngp}}$ in (b). Note that $S_4(q,t)$ was calculated with a larger system, $N={10}^5$.

Figure 7

(a) The correlation length ${\xi }_4\left(t\right)$ vs ${\tau }_{\alpha }$. (b) The intensity ${\chi }_4\left(t\right)$ vs the correlation length ${\xi }_4\left(t\right)$. The time intervals are $t={\tau }_{\alpha }$ and ${\tau }_{\mathrm{ngp}}$. The straight lines are power-law fits: ${\xi }_4\left({\tau }_{\alpha }\right)\sim {\tau }_{\alpha }^{0.1\pm 0.01}$ and ${\chi }_4\left({\tau }_{\alpha }\right)\sim {\xi }_4{\left({\tau }_{\alpha }\right)}^{3.2\pm 0.1}$. The dashed curve is a fit to ${\xi }_4\left({\tau }_{\alpha }\right)\sim {\left(\ln {\tau }_{\alpha }\right)}^{0.77\pm 0.12}$.

Figure 8

The correlation length ${\xi }_4\left(t\right)$ and the intensity ${\chi }_4\left(t\right)$ calculated by using $\delta {\widehat{\mathcal{D}}}_{\log 1}(\mathit{r},t_0,t)$, i.e., the distribution of the logarithm of the square particle displacements. (a) ${\xi }_4\left(t\right)$ vs ${\tau }_{\alpha }$. (b) ${\chi }_4\left(t\right)$ vs ${\xi }_4\left(t\right)$. The time intervals are $t={\tau }_{\alpha }$ and ${\tau }_{\mathrm{ngp}}$. The straight lines are power-law fits: ${\xi }_4\left({\tau }_{\alpha }\right)\sim {\tau }_{\alpha }^{0.1\pm 0.01}$ and ${\chi }_4\left({\tau }_{\alpha }\right)\sim {\xi }_4{\left({\tau }_{\alpha }\right)}^{3.5\pm 0.2}$. The dashed curve is a fit to ${\xi }_4\left({\tau }_{\alpha }\right)\sim {\left(\ln {\tau }_{\alpha }\right)}^{0.91\pm 0.12}$.

Figure 9

The time-interval dependence of the distribution of the particle mobility $a_{1j}^2(t_0,t)$ for particle species 1. The time intervals are (a) $t=0.001{\tau }_{\alpha }$, (b) $t=0.1{\tau }_{\alpha }$, (c) $t={\tau }_{\alpha }$, and (d) $t=100{\tau }_{\alpha }$. The temperature is $0.267$. The radii of the spheres are $a_{1j}^2(t_0,t)$, and the centers are at ${\mathit{R}}_{1j}(t_0,t)$. See also the caption for Fig. 5.

Figure 10

The wave-number dependence of $S_4(q,t)$ at $T=0.306$. The time intervals are (a) $0.05{\tau }_{\alpha }$, $0.1{\tau }_{\alpha }$, $0.5{\tau }_{\alpha }$, and ${\tau }_{\alpha }$ from the lowest curve to the highest, and (b) ${\tau }_{\alpha }$, $5{\tau }_{\alpha }$, $10{\tau }_{\alpha }$, $50{\tau }_{\alpha }$, $100{\tau }_{\alpha }$, and $300{\tau }_{\alpha }$ from the highest curve to the lowest. $S_4(q,t)$ is maximized at $t={\tau }_{\alpha }$. The dashed curve represents the static structure factor $S_{11}\left(q\right)$.

Figure 11

The wave-number dependence of ${\tau }_h(q,t)$ at $T=0.306$. The time intervals are $0.05{\tau }_{\alpha }$, $0.1{\tau }_{\alpha }$, $0.5{\tau }_{\alpha }$, ${\tau }_{\alpha }$, $5{\tau }_{\alpha }$, $10{\tau }_{\alpha }$, $30{\tau }_{\alpha }$, $50{\tau }_{\alpha }$, $70{\tau }_{\alpha }$, $100{\tau }_{\alpha }$, and $300{\tau }_{\alpha }$ from the lowest curve to the highest. The dashed curve is the relaxation time ${\tau }_{c1}\left(q\right)$ of the two-point density correlation function.

Figure 12

The wave-number dependence of ${\tau }_{hs}(q,t)$ at $T=0.306$. The time intervals are $0.05{\tau }_{\alpha }$, $0.1{\tau }_{\alpha }$, $0.5{\tau }_{\alpha }$, ${\tau }_{\alpha }$, $5{\tau }_{\alpha }$, $10{\tau }_{\alpha }$, $30{\tau }_{\alpha }$, $50{\tau }_{\alpha }$, $70{\tau }_{\alpha }$, $100{\tau }_{\alpha }$, and $300{\tau }_{\alpha }$ from the lowest curve to the highest. The dashed curve is the relaxation time ${\tau }_{s1}\left(q\right)$ of the self-part of the two-point density correlation function.

Figure 13

The time-interval dependence of ${\tau }_{\mathrm{hetero}}\left(t\right)$. Temperatures are 0.473 in (a), 0.306 in (b), 0.267 in (c), and 0.253 in (d). The time interval $t$ and the lifetime ${\tau }_{\mathrm{hetero}}\left(t\right)$ are normalized by ${\tau }_{\alpha }$. The dotted line indicates the value of ${\tau }_{c1}\left(q\right)$ at $q=0.38$, where ${\tau }_{c1}\left(q\right)$ is the relaxation time of the two-point density correlation function.