Random close packing fraction of bidisperse discs: theoretical derivation

Predicting theoretically the highest density, which a disordered packing of discs can achieve, has been a long-standing unresolved problem. Such predictions are hindered by two difficulties - the dependence of the density on the packing procedure and ensuring disorder. A theory that overcomes these difficulties has been developed recently for mono-disperse disc packing~\cite{Bl21}. However, to minimise order, experiments and numerical simulations often use two-size discs and a prediction of the highest possible packing fraction, $\phi_{RCP}$, for these packings is arguably more useful. This problem is more complex because in such packings, $\phi_{RCP}$ is not a number but a function of the sizes ratio, $D$, and concentrations, $p$, of the disc types. A disorder-guaranteeing theory is formulated here to derive $\phi_{RCP}(p,D)$ under some assumptions, using the concept of the cell order distribution. Exact upper and lower bounds on the densest disordered packing fraction are also derived.