Multiple time scales hidden in heterogeneous dynamics of glass-forming liquids

A multitime probing of density fluctuations is introduced to investigate hidden time scales of heterogeneous dynamics in glass-forming liquids. Molecular-dynamics simulations for simple glass-forming liquids are performed and a three-time correlation function is numerically calculated for general time intervals. It is demonstrated that the three-time correlation function is sensitive to the heterogeneous dynamics and that it reveals couplings of correlated motions over a wide range of time scales. Furthermore, the time scale of the heterogeneous dynamics ${\tau }_{\text{hetero}}$ is determined by the change in the second time interval in the three-time correlation function. The present results show that the time scale of the heterogeneous dynamics ${\tau }_{\text{hetero}}$ becomes larger than the $\alpha$-relaxation time at low temperatures and large wavelengths. We also find a dynamical scaling relation between the time scale ${\tau }_{\text{hetero}}$ and the length scale $\xi$ of dynamical heterogeneity as ${\tau }_{\text{hetero}}\sim {\xi }^z$ with $z=3$.

Figure 1

(Color) Two-dimensional plots of the three-time correlation function $F_4\left(k,t_1,t_2,t_3\right)$ of component 1 particles at waiting time $t_2=0$ for various temperatures $T=0.473$ (a), 0.352 (b), 0.306 (c), and 0.289 (d). For each temperature, ${\tau }_{\alpha }$ is indicated by the arrow. The wave vector $k$ is chosen as $k=2\pi$.

Figure 2

(Color) Two-dimensional plots of the three-time correlation function $F_4\left(k,t_1,t_2,t_3\right)$ of component 1 particles for $T=0.289$ at several waiting time $t_2=0$ (a), $0.5{\tau }_{\alpha }$ (b), ${\tau }_{\alpha }$ (c), and $2{\tau }_{\alpha }$ (d). ${\tau }_{\alpha }$ is indicated by the arrow. The wave vector $k$ is chosen as $k=2\pi$.

Figure 3

Diagonal part of the three-time correlation function $F_4\left(k,t,t_2,t\right)$ of component 1 particles for $T=0.289$ at various waiting times $t_2=0$, $0.5{\tau }_{\alpha }$, ${\tau }_{\alpha }$, $2{\tau }_{\alpha }$, $3{\tau }_{\alpha }$, $4{\tau }_{\alpha }$, $6{\tau }_{\alpha }$, and $10{\tau }_{\alpha }$ from top to bottom. The wave vector $k$ is chosen as $k=2\pi$. The $\alpha$-relaxation time and the times at which the non-Gaussian parameter ${\alpha }_2\left(t\right)$ and the dynamic susceptibility ${\chi }_4\left(k=2\pi ,t\right)$ reach their peak values are indicated as ${\tau }_{\alpha }$, ${\tau }_{\text{NGP}}$, and ${\tau }_{{\chi }_4}$, respectively.

Figure 4

(Color online) Waiting time $t_2$ dependence of the integrated three-time correlation function ${\Delta }_{\text{hetero}}\left(k,t_2\right)/{\Delta }_{\text{hetero}}\left(k,0\right)$ for various temperatures. The waiting times are normalized by the $\alpha$ relaxation time ${\tau }_{\alpha }$ for each temperature. The wave vector $k$ is chosen as $k=2\pi$. The solid curve is determined by fitting with the stretched-exponential form for each temperature.

Figure 5

(Color online) Lifetime of heterogeneous dynamics ${\tau }_{\text{hetero}}$ normalized by the $\alpha$ relaxation ${\tau }_{\alpha }$ versus temperature $T$ for $k=\pi$ (circles), $2\pi$ (squares), $3\pi$ (diamonds), and $4\pi$ (triangles). Inset: relation between ${\tau }_{\text{hetero}}$ and ${\tau }_{\alpha }$ for $k=2\pi$. The straight line with the slope 1.5 is a viewing guide.