Why is Random Close Packing Reproducible?

We link the thermodynamics of colloidal suspensions to the statistics of regular and random packings. Random close packing has defied a rigorous definition yet, in three dimensions, there is near universal agreement on the volume fraction at which it occurs. We conjecture that the common value of ${\varphi }_{\mathrm{rcp}}\approx 0.64$ arises from a divergence in the rate at which accessible states disappear. We relate this rate to the equation of state for a hard-sphere fluid on a metastable, noncrystalline branch.

Figure 1

Measured equation of state $p$ (in units of $\rho T$) versus volume fraction $\varphi$ for a hard-sphere fluid from experiments by Rutgers et al. [17] (circles) and simulations by Rintoul and Torquato [18] (diamonds) and Speedy [19] (squares). The dashed curve is the Carnahan-Starling equation of state. In (a) we show two fits to free volume theory, corresponding to ${\varphi }_{\max }=0.635$ (solid), which is the best fit to the experimental data for the equilibrium liquid branch, and ${\varphi }_{\max }=0.6465$ (dotted), which is the best fit to the numerical data for the metastable branch. In (b) we show free volume theory for ${\varphi }_{\max }=0.6465$ (dotted) and ${\varphi }_{\max }=0.640$ (solid). Part (a) of the figure shows the quality of the fit to the equilibrium liquid branch while part (b) shows the quality of the fit to the metastable branch. Inset to (a): the variation in the mean-square error $\Delta$ between free volume theory and the experimental data as a function of ${\varphi }_{\max }$.