Equilibrium Sampling of Hard Spheres up to the Jamming Density and Beyond

We implement and optimize a particle-swap Monte Carlo algorithm that allows us to thermalize a polydisperse system of hard spheres up to unprecedentedly large volume fractions, where previous algorithms and experiments fail to equilibrate. We show that no glass singularity intervenes before the jamming density, which we independently determine through two distinct nonequilibrium protocols. We demonstrate that equilibrium fluid and nonequilibrium jammed states can have the same density, showing that the jamming transition cannot be the end point of the fluid branch.

Figure 1

Equilibrium relaxation time scale of polydisperse ($\mathrm{\Delta }=23\%$) hard spheres from conventional Monte Carlo simulations. The power law fit provides an estimate of the location of the avoided mode-coupling transition, ${\phi }_{\text{mct}}\approx 0.598$; activated relaxation fits, $\exp [B/({\phi }_0-\phi {)}^{\delta }]$, yield possible locations for the extrapolation of a diverging time scale, ${\phi }_0^{(1)}\approx 0.648$, when $\delta =1$ and ${\phi }_0^{(2)}\approx 0.672$ for $\delta =2$. The range of jamming densities $0.6487<{\phi }_J<0.6534$ using nonequilibrium protocols (the shaded area) overlaps with the range of the extrapolated dynamical divergences.

Figure 2

(a) Reduced pressure $Z$ during compressions at finite rates $\dot{P}$ with (right curves) and without (left curves) swap. The swap Monte Carlo method falls out of equilibrium at much larger volume fractions than conventional sampling. (b) Equilibrium measurements of the equation of state (the symbols) for different system sizes. Our equilibrium data go beyond the shaded area, which represents the determined range of jamming densities. In both panels, the dashed line is the Carnahan-Stirling empirical expression [33, 34].

Figure 3

(a) Memory effect in hard spheres. Time evolution of the volume fraction starting at $t=0$ from a jammed configuration in constant pressure swap Monte Carlo simulations. A nonmonotonic evolution of the density is observed, showing that jammed and fluid states are structurally distinct. At long times (note the break in the time axis), steady state fluctuations are observed both below and above ${\phi }_J$ around the value (dashed lines) obtained from the equation of state. (b) Pair distribution functions of rescaled distances $r_{ij}/[({\sigma }_i+{\sigma }_j)/2]$ are distinct for fluid and jammed states near $\phi =0.6534$.