Density-Temperature-Softness Scaling of the Dynamics of Glass-Forming Soft-Sphere Liquids

We employ the principle of dynamic equivalence between soft-sphere and hard-sphere fluids [Phys. Rev. E 68, 011405 (2003)] to describe the interplay of the effects of varying the density $n$, the temperature $T$, and the softness (characterized by a softness parameter ${\nu }^{-1}$) on the dynamics of glass-forming soft-sphere liquids in terms of simple scaling rules. The main prediction is the existence of a dynamic universality class associated with the hard-sphere fluid, constituted by the soft-sphere systems whose dynamic parameters depend on $n$, $T$, and $\nu$ only through the reduced density $n^{*}\equiv n{\sigma }_{\mathrm{HS}}(T^{*},\nu )$. A number of scaling properties observed in recent experiments and simulations involving glass-forming fluids with repulsive short-range interactions are found to be a direct manifestation of this general dynamic equivalence principle.

Figure 1

Simulated data (solid symbols) and SCGLE theoretical predictions (solid lines) of the normalized long-time self-diffusion coefficient $D^{*}(n,T,\nu )$, as a function of $\varphi$ (inset) and as a function of ${\varphi }_{\mathrm{HS}}(n,T,\nu )$ (main figure), for the repulsive Yukawa fluid (squares), the truncated 6-12 Lennard-Jones fluid (triangles), and the hard-sphere fluid (circles, Ref. 11). The empty diamonds and the asterisks represent, respectively, the experimental master curve from Fig. 4 of Ref. 13 for the reduced viscosity (proportional to $1/D^{*}$) of microgel solutions and the data for $D_L(\varphi )$ of third- and fourth-generation dendrimer solutions in Fig. 3 of Ref. 18, scaled to collapse among themselves and with the HS master curve at low densities.

Figure 2

State space of the truncated 6-12 Lennard-Jones fluid, showing four isodiffusivity lines, including the freezing transition (dash-dotted line) and the ideal glass transition (solid line). In the inset we plot similar isodiffusivity lines for the soft-sphere system with harmonic repulsive potential of Ref. 5, together with the simulation results for the iso-${\tau }_{\alpha }$ lines labeled ${\tau }_{\alpha }={10}^1$, ${10}^2$, ${10}^3$, and ${10}^4$ in Fig. 1 of Ref. [5].

Figure 3

Theoretical (solid line) and simulated (symbols) hard-sphere master curve for $D_{\mathrm{HS}}^{*}({\varphi }_{\mathrm{HS}})$ from Fig. 1, but now plotted as a function of the combination $[{\varphi }_{\mathrm{HS}}{Z}_{\mathrm{CS}}({\varphi }_{\mathrm{HS}})]={\lambda }_{\nu }^3(T^{*})p^{*}/T^{*}$ (main panel) and $T^{*}/[p^{*}{\lambda }_{\nu }^3(T^{*})]$ (inset).

Figure 4

Molecular dynamics simulations reported in Figs. 1(a) and 1(b) of Xu et al. [4] for ${\tau }_{\alpha }$ of soft-sphere liquids, plotted here as the master curve for ${\tau }^{*}(T^{*},p^{*})$ as a function of the combination $[{\varphi }_{\mathrm{HS}}(\varphi ,T^{*},\nu )Z_{\mathrm{CS}}\mathbf{(}{\varphi }_{\mathrm{HS}}(\varphi ,T^{*},\nu )\mathbf{)}{]}^{-1}=[T^{*}/{\lambda }_{\nu }^3(T^{*})p^{*}]$. In the inset we show the simulation data, obtained by Xu et al., for the $p^{*}/T^{*}$ ratio (squares), compared with Eq. (5) (solid line).