Equivalence of Glass Transition and Colloidal Glass Transition in the Hard-Sphere Limit

We show that the slowing of the dynamics in simulations of several model glass-forming liquids is equivalent to the hard-sphere glass transition in the low-pressure limit. In this limit, we find universal behavior of the relaxation time by collapsing molecular-dynamics data for all systems studied onto a single curve as a function of $T/p$, the ratio of the temperature to the pressure. At higher pressures, there are deviations from this universal behavior that depend on the interparticle potential, implying that additional physical processes must enter into the dynamics of glass formation.

Figure 1

Relaxation time versus different control parameters for a system of 1000 particles with harmonic repulsions. (a) Relaxation time $\tau$ versus temperature $T$ at different fixed pressures, $p$: $p=2\times {10}^{-7}$ (circles), $p=2\times {10}^{-6}$ (squares), $p=2\times {10}^{-5}$ (diamonds), and $p=2\times {10}^{-4}$ (upward triangles). (b) Relaxation time $\tau$ versus inverse pressure $1/p$ at different fixed temperatures: $T={10}^{-8}$ (downward triangles), $T={10}^{-7}$ (pluses), $T={10}^{-6}$ (crosses), and $T={10}^{-5}$ (stars).

Figure 2

Collapse of all the relaxation-time data shown in Fig. 1. (a) Scaled relaxation time, $\tau /\sqrt{m/p\sigma }$, versus scaled temperature $T/p{\sigma }^3$. (b) Scaled relaxation time $\tau /\sqrt{m/p\sigma }$ versus scaled pressure $p{\sigma }^3/ϵ$, for $T/p{\sigma }^3=0.08$ (circles), $T/p{\sigma }^3=0.1$ (squares) and $T/p{\sigma }^3=0.2$ (diamonds). Horizontal lines show the limiting values of $\tau \sqrt{p\sigma /m}$ as $p{\sigma }^3/ϵ\to 0$.

Figure 3

Scaled relaxation time, $\tau /\sqrt{m/p\sigma }$, versus scaled temperature $T/p{\sigma }^3$ for all the data for the harmonic potential (black) in Fig. 1 as well as for the Hertzian potential, $\alpha =5/2$ [red (medium gray)], the hard-sphere potential [magenta (light gray)], and the Weeks-Chandler-Andersen potential [blue (dark gray)]. Black solid curve is the Vogel-Fulcher fit: $y=0.59\exp [0.18/(x-0.045)]$, where $y=\tau /\sqrt{m/p\sigma }$ and $x=T/p{\sigma }^3$. Blue-dashed curve is a fit to the Elmatad-Chandler-Garrahan form: $y=3.1\exp [0.064(x^{-1}-4.72{)}^2]$.

Figure 4

Equation of state for three-dimensional systems at low $p{\sigma }^3/ϵ$ for hard-sphere systems (solid magenta circles) and systems with harmonic repulsions measured along both constant temperature and constant pressure trajectories. The dashed line is a fit to free-volume theory, $p{\sigma }^3/T=0.98\varphi /[1-(\varphi /{\varphi }_c{)}^{1/3}]$, with ${\varphi }_c=0.66$. The solid line is an empirical fit given by $p{\sigma }^3/T=\varphi /[1-(\varphi /1.13){]}^{4.35}$.