📝 SummarXiv

Abstract

A long-standing problem has been a theoretical prediction of the densest packing fraction of random packings, $\Phi_{RCP}$, of same-size discs in $d=2$ and spheres in $3$. However, to minimize order, experiments and numerical simulations often use two-size discs. For practical purposes, then, a predictive theory for the packing fraction, $\Phi_{RCP}$, of the densest such bidisperse packings is more useful. A disorder-guaranteeing theory is formulated here to fill this gap, using an approach that led to an exact solution for monodisperse discs in $d=2$ [1]. $\Phi_{RCP}$ depends on the sizes ratio, $D$, and concentrations, $p$, of the disc types and the developed theory enables derivation of exact and rigorous upper and lower bounds on $\Phi_{RCP}(p,D)$, as well as an explicit prediction of it.