Dynamic Heterogeneity in a Glass Forming Fluid: Susceptibility, Structure Factor, and Correlation Length

We investigate the growth of dynamic heterogeneity in a glassy hard-sphere mixture for volume fractions up to and including the mode-coupling transition. We use an 80 000 particle system to test a new procedure to evaluate a dynamic correlation length $\xi (t)$: we determine the ensemble independent dynamic susceptibility ${\chi }_4(t)$ and use it to facilitate evaluation of $\xi (t)$ from the small wave vector behavior of the four-point structure factor. We analyze relations between the $\alpha$ relaxation time ${\tau }_{\alpha }$, ${\chi }_4({\tau }_{\alpha })$, and $\xi ({\tau }_{\alpha })$. We find that mode-coupling-like power laws provide a reasonable description of the data over a restricted range of volume fractions, but the power laws’ exponents differ from those predicted by the inhomogeneous mode-coupling theory. We find $\xi ({\tau }_{\alpha })\sim \ln ({\tau }_{\alpha })$ over the full range of volume fractions studied, which is consistent with an Adam-Gibbs-type relation.

Figure 1

(a) The first two terms contributing to ${\chi }_4({\tau }_{\alpha })$ given in Eq. (4). Also shown are ${\chi }_4({\tau }_{\alpha })$ obtained from Ornstein-Zernicke (OZ) extrapolation of $S_4(q;{\tau }_{\alpha })$ to $q=0$ and the sum of all the terms in Eq. (4). (b) The ratio ${\chi }_4({\tau }_{\alpha })/{\chi }_4({\tau }_{\alpha }){|}_{\varphi ,c}$.

Figure 2

(a) $\xi ({\tau }_{\alpha })$ versus $\varphi$ determined from the different fitting procedures discussed in the text; the labels denote the fitting procedure. The points are statistically the same, and the large error bars for $\xi ({\tau }_{\alpha })$ are due to the fits to $\ln [S_4(q;{\tau }_{\alpha })]$. (b) Scaling plot of $S_4(q;{\tau }_{\alpha })/S_4(q=0;{\tau }_{\alpha })$ versus $q\xi ({\tau }_{\alpha })$. The solid line is the Ornstein-Zernicke function used for the fits.

Figure 3

(a) The dynamics susceptibility ${\chi }_4({\tau }_{\alpha })$ versus the correlation length $\xi ({\tau }_{\alpha })$. (b) The correlation length $\xi ({\tau }_{\alpha })$ versus ${\tau }_{\alpha }$. The straight lines are power law fits over the range corresponding to $0.55\le \varphi \le 0.58$. The dashed line is a fit to $\xi \sim \ln ({\tau }_{\alpha })$ over the whole range of $\varphi$. The circles and the squares are the results from the 80 000 and the 10 000 particle simulations, respectively.