Analytical solution for the polydisperse random close packing problem in 2D

Abstract

An analytical theory for the random close packing density, $\ddot{I}{•}_{\text{RCP}}$, of polydisperse hard disks is provided using an equilibrium model of crowding [A. Zaccone, Phys. Rev. Lett. 128, 028002 (2022)] which has been justified on the basis of extensive numerical analysis of the maximally random jammed (MRJ) line in the phase diagram of hard spheres [Anzivino et al., J. Chem. Phys. 158, 044901 (2023)]. The solution relies on the equations of state for the hard disk fluid and provides predictions for $\ddot{I}{•}_{\text{RCP}}$ as a function of the ratio, $s$, of the standard deviation of the distribution of disk diameters to its mean. For a power-law size distribution with $s=0.246$, the theory yields $\ddot{I}{•}_{\text{RCP}}=0.892$, which compares well with the most recent numerical estimate $\ddot{I}{•}_{\text{RCP}}=0.905$ based on the Monte-Carlo swap algorithms [Ghimenti, Berthier, van Wijland, Phys. Rev. Lett. 133, 028202 (2024)].