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

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.