Maximally Random Jamming of Two-Dimensional One-Component and Binary Hard Disc Fluids

TL;DR

This paper develops a theoretical framework to predict the density at which two-dimensional hard disc fluids reach a maximally random jammed state, aligning well with experimental and simulation results.

Contribution

It introduces a bifurcation-based method to determine the transition point to random jamming in both one-component and binary hard disc fluids.

Findings

Predicted jamming density for one-component fluid: 0.84.

Binary fluid jamming density range: 0.84 to 0.87.

Results agree with experimental and simulation data.

Abstract

We report calculations of the density of maximally random jamming (aka random close packing) of one-component and binary hard disc fluids. The theoretical structure used provides a common framework for description of the hard disc liquid to hexatic, the liquid to hexagonal crystal and the liquid-to-maximally random jammed state transitions. Our analysis is based on locating a particular bifurcation of the solutions of the integral equation for the inhomogeneous single particle density at the transition between different spatial structures. The bifurcation of solutions we study is initiated from the dense metastable fluid, and we associate it with the limit of stability of the fluid, which we identify with the transition from the metastable fluid to a maximally random jammed state. For the one-component hard disc fluid the predicted packing fraction at which the metastable fluid to maximally random jammed state transition occurs is 0.84, in excellent agreement with the experimental value 0.84 \pm 0.02. The corresponding analysis of the limit of stability of a binary hard disc fluid with specified disc diameter ratio and disc composition requires extra approximations in the representations of the direct correlation function, the equation of state, and the number of order parameters accounted for. Keeping only the order parameter identified with the largest peak in the structure factor of the highest density regular lattice with the same disc diameter ratio and disc composition as the binary fluid, the predicted density of maximally random jamming is found to be 0.84 to 0.87, depending on the equation of state used, and very weakly dependent on the ratio of disc diameters and the fluid composition, in agreement with both experimental data and computer simulation data.