Equilibration of concentrated hard-sphere fluids

We report a systematic molecular dynamics study of the isochoric equilibration of hard-sphere fluids in their metastable regime close to the glass transition. The thermalization process starts with the system prepared in a nonequilibrium state with the desired final volume fraction $\varphi$ for which we can obtain a well-defined nonequilibrium static structure factor $S_0(k;\varphi )$. The evolution of the $\alpha$-relaxation time ${\tau }_{\alpha }(k)$ and long-time self-diffusion coefficient $D_L$ as a function of the evolution time $t_w$ is then monitored for an array of volume fractions. For a given waiting time the plot of ${\tau }_{\alpha }(k;\varphi ,t_w)$ as a function of $\varphi$ exhibits two regimes corresponding to samples that have fully equilibrated within this waiting time [$\varphi ⩽\varphi {}^{(c)}(t_w)$] and to samples for which equilibration is not yet complete [$\varphi ⩾\varphi {}^{(c)}(t_w)$]. The crossover volume fraction $\varphi {}^{(c)}(t_w)$ increases with $t_w$ but seems to saturate to a value $\varphi {}^{(a)}\equiv \varphi {}^{(c)}(t_w\to \infty )\approx 0.582$. We also find that the waiting time $t_w^{\mathrm{eq}}(\varphi )$ required to equilibrate a system grows faster than the corresponding equilibrium relaxation time, $t_w^{\mathrm{eq}}(\varphi )\approx 0.27[{\tau }_{\alpha }^{\mathrm{eq}}(k;\varphi )]{}^{1.43}$, and that both characteristic times increase strongly as $\varphi$ approaches $\varphi {}^{(a)}$, thus suggesting that the measurement of equilibrium properties at and above $\varphi {}^{(a)}$ is experimentally impossible.

Figure 1

(a) Self-intermediate scattering function $F_S(k,\tau ;t_w)$ of a polydisperse hard-sphere system ($s=0.0866$) evaluated at $k=7.1$ at volume fraction $\varphi =0.575$ as a function of the correlation time $\tau$ for waiting times $t_w={10}^0,{10}^1,\dots ,{10}^5$. The inset in (a) shows the corresponding $S_0(k;\varphi )$ (solid line) and $S^{\mathrm{eq}}(k;\varphi )$ (dashed line). (b) Simulation data of the $\alpha$-relaxation time ${\tau }_{\alpha }(k;\varphi ,t_w)$ as a function of $t_w$ at fixed volume fraction. The asterisks highlight the points ($t_w^{\mathrm{eq}}(\varphi ),{\tau }_{\alpha }^{\mathrm{eq}}(k;\varphi )$).

Figure 2

The same simulation data of the $\alpha$-relaxation time ${\tau }_{\alpha }(k;\varphi ,t_w)$ as in Fig. 1, but now displayed as a function of volume fraction $\varphi$ at fixed waiting time $t_w$. The asterisks indicate, in this case, the crossover volume fraction $\varphi {}^{(c)}(t_w)$ at the various waiting times. The circles in the inset display the evolution of $\varphi {}^{(c)}(t_w)$ with waiting time, and the squares are the corresponding results in Fig. 5(a) of Ref. [14].

Figure 3

Volume fraction dependence of the scaled $\alpha$-relaxation time $\tau {}^{*}(k;\varphi ,t_w)\equiv k^2{D}^0{\tau }_{\alpha }(k;\varphi ,t_w)$. The solid (empty) squares denote simulation data of fully equilibrated (insufficiently equilibrated) systems. The solid line represents the predictions of the SCGLE theory. The dashed line is the fit with ${\tau }_{\alpha }(\varphi )={\tau }_{\infty }\exp [A({\varphi }_0-\varphi ){}^{-\delta }]$. The solid circles correspond to the experimental results of Fig. 13 of Ref. [10]. Inset (a) plots $\ln [{\tau }_{\alpha }(k;\varphi ,t_w)]$ as a function of $A({\varphi }_0-\varphi ){}^{-\delta }$ for $t_w={10}^5$ (squares), $\hspace{0.5em}{10}^4$ (diamonds), and ${10}^3$ (triangles), with the arrows pointing at the corresponding crossover volume fraction $\varphi {}^{(c)}(t_w)$. Inset (b) compares our simulation results (empty squares with line) for $\tau {}^{*}(k;\varphi ,t_w)$ vs $t_w^{*}\equiv k^2{D}^0{t}_w$ for $\varphi =0.58$ with the experimental data of Fig. 6 of Ref. [10] (empty circles) at $\varphi {}^{\text{exp}}=0.5876$.