Explicit Analytical Solution for Random Close Packing in d=2 and d=3

TL;DR

This paper derives analytical formulas for the random close packing volume fractions in two and three dimensions, providing values consistent with experimental and simulation data, and discusses the implications for the nature of RCP and mechanical rigidity.

Contribution

It presents the first analytical derivation of RCP volume fractions in 2D and 3D using modified neighbor statistics, linking RCP to rigidity and order.

Findings

Derived $ ext{RCP}$ volume fractions: 0.659 in 3D and 0.886 in 2D.

Values align with experimental and numerical results.

RCP marks the onset of mechanical rigidity and maximal randomness.

Abstract

We present an analytical derivation of the volume fractions for random close packing (RCP) in both $d=3$ and $d=2$, based on the same methodology. Using suitably modified nearest neigbhour statistics for hard spheres, we obtain $\phi_{\mathrm{RCP}}=0.65896$ in $d=3$ and $\phi_{\mathrm{RCP}}=0.88648$ in $d=2$. These values are well within the interval of values reported in the literature using different methods (experiments and numerical simulations) and protocols. This order-agnostic derivation suggests some considerations related to the nature of RCP: (i) RCP corresponds to the onset of mechanical rigidity where the finite shear modulus emerges, (ii) the onset of mechanical rigidity marks the maximally random jammmed state and dictates $\phi_{\mathrm{RCP}}$ via the coordination number $z$, (iii) disordered packings with $\phi>\phi_{\mathrm{RCP}}$ are possible at the expense of creating some order, and $z=12$ at the FCC limit acts as a boundary condition.