Nonequilibrium dynamics of four-point correlations of collective density fluctuations in a supercooled liquid

In this paper we study the four-point correlation function ${\chi }_4$ of collective density fluctuations in a nonequilibrium liquid. The equilibration is controlled by a modified stretched exponential behavior $\{exp[-{(t_w/\tau )}^{\beta }]\}$ having the relaxation time $\tau$ dependent on the aging time $t_w$. Similar aging behavior has been seen experimentally in supercooled liquids. The basic equations of fluctuating nonlinear hydrodynamics are solved here numerically to obtain ${\chi }_4$ for equilibrium and non equilibrium states. We also identify a dynamic length scale $\xi$ from the equilibrated function. $\xi \left(T\right)$ grows with fall of temperature $T$. From a broader perspective, we demonstrate here that the characteristic signatures of dynamical heterogeneities in a supercooled liquid, observed previously in computer simulations of the dynamics of a small number of particles, are also present in the coarse grained equations of generalized hydrodynamics.

Figure 1

The normalized four-point functions ${\chi }_4\left(t\right)$ at $q=q_m$ vs time $t$ at $n_0^{*}=1.10$ and $T^{*}=1.0$ (circles), $0.8$ (squares), $0.7$ (triangles), $0.6$ (diamonds), $0.5$ (stars). Solid lines are the best fit curves of Lorentzian form.

Figure 2

The normalized two-point function $C\left(t\right)$ at $n_0^{*}=1.10$ and $T^{*}=0.8$ (solid), $0.7$ (dashed), and $0.6$ (dashed-dotted). In the inset the solid and dashed curves indicate the respective power law fits predicted in MCT for $T^{*}=0.8$.

Figure 3

The peak ${\chi }^P$ of ${\chi }_4\left(t\right)$ at $t=t_p$ appear to diverge around $T_o=0.2$ with exponent $\alpha =1.21$. Inset shows ${\chi }^P\sim t_p^{\mu }$ behavior with the exponent $\mu =0.47$.

Figure 4

The $\alpha$-relaxation time ${\tau }_{\alpha }$ vs $T$. The fit shows a power law divergence around $T_c=0.4$.

Figure 5

The normalized ${\chi }^P\left(q\right)/{\chi }^P\left(0\right)$ for density $n_0^{*}=1.10$ and different temperatures $T^{*}$ = 1.0 (circles), 0.8 (squares), 0.7 (diamonds), 0.6 (triangles), and 0.5 (stars) plotted with corresponding $q\xi \left(T\right)$. Dashed line is the best fit to an Ornstein-Zernike form (see text). Inset (a) divergence $\xi$ around the $T_o=0.2$ and exponent $1.4;$ (b) ${\chi }^P(q=0)\equiv {\chi }_0\sim {\xi }^{(2-\eta )}$ with $2-\eta =2.1$.

Figure 6

The nonequilibrium ${\chi }_4(t,t_w)$ vs $t$ for different values of the waiting time $t_w$= 200 (circles), 400 (diamonds), 600 (triangles), 800 (squares), and 1000 (stars) corresponding to $T^{*}=0.4$ and $n_0^{*}=1.10$. Inset: peak time $t_p$ vs $t_w$.

Figure 7

At $T^{*}=0.6$ and $n_0^{*}=1.10$, parametric plot of normalized ${\chi }_4(t,t_w)/{\chi }^P\left(t_w\right)$ vs $1-\tilde{C}(t,t_w)$ (see text). Equilibration with waiting time $t_w$: Inset (a) the peak value ${\chi }^P$; (b) $f\left(t_w\right)$ (defined in text).

Figure 8

Data collapse (solid line) on the scaling function $g\left(t_w\right)$ (defined in the text) vs $t_w$ corresponding to four different frequencies $\omega {\tau }_0$ = 0.0001 (circle), 0.0005 (diamond), 0.001 (triangle), and 0.01 (star) at $T^{*}=0.6$ and $n_0^{*}=1.10$. Scaling function corresponding to two-point functions (dashed line). The $t_w$ dependence of relaxation times of MKWW scaling functions: Inset (a) ${\tau }_4\left(t_w\right)$ for four-point functions; (b) ${\tau }_2\left(t_w\right)$ for two-point functions.