Dynamic Heterogeneities in a Simple Liquid over Different Time Scales

The heterogeneous features of the supercooled state over different time regimes are explored in a self-consistent mode-coupling mode. The exponent $a$ for the mean-square displacement $⟨r^2(t)⟩\sim t^a$, of a tagged particle is computed. The non-Gaussian parameter ${\alpha }_2(t)$ shows a peak in the short time regime in addition to a second peak over longer times. The position of the short-time peak in ${\alpha }_2(t)$ hardly shifts, while that of the other grows with density.

Figure 1

Non-Gaussian parameter ${\alpha }_2(t)$ vs $t^{*}=t/t_E$ for the simple model at $\eta =0.500$ (dashed), 0.510 (dot-dashed), and 0.540 (solid). The inset shows $t_{p2}$ vs $\eta$, and the solid line is the power-law fit showing divergence at ${\eta }_c=0.525$.

Figure 2

The long time peak in ${\alpha }_2$ vs $t^{*}$ at $\eta =0.550$ (dashed), $\eta =0.565$ (solid), and $\eta =0.580$ (dot-dashed) by using the extended MCT model. In the inset, the variation of $t_{p2}$ vs $\eta$ is shown for this model.

Figure 3

The normalized ${\Gamma }_s(q=0,t)$ vs $t^{*}$ at $\eta =0.565$. The dot-dashed line shows the stretched exponential fit, and the arrows indicate the $t^{*}$ position where the two peaks in ${\alpha }_2(t)$ occur.

Figure 4

$⟨r^2(t)⟩$ (in units of ${\sigma }^2$) vs $t^{*}$ with the extended model (solid line) and the simple model (dashed line) at $\eta =0.565$. The inset shows the decrease in the power-law exponent $a$ with $\eta$.

Figure 5

$G_s(r,t_{p2})$ vs $r^{*}=r/\sigma$ at $\eta =0.565$. The dot-dashed curve is the corresponding Gaussian distribution function $G_s^o(r,t_{p2})$. The inset shows the variation of $f_M$ (see text) vs $\eta$.

Figure 6

$[G_s(r,t_{p2})-G_s^o(r,t_{p2})]/G_s^o(r,t_{p2})$ vs $r^{*}$ for $\eta =0.550$ (dashed), $\eta =0.565$ (solid), and $\eta =0.580$ (dot-dashed). The inset shows $[G_s(r,t_{p2})-G_s^o(r,t_{p2})]$ at the same $\eta$ values.