The maximum packing fraction of disordered bidisperse discs: theory and predictions

Predicting the highest packing fraction that disordered packing of bidisperse discs can achieve, $ϕ_{max}$, is an important unresolved problem. Such a predictive theory is hindered by the density dependence on the packing protocol and by ensuring disorder. In bidisperse packings, this is more complicated because $ϕ_{max}$ is not a number but a function of the disc species size ratio, $D$, and one of the disc species concentration, $p$. These difficulties have been overcome in monodisperse disc packings by using the cell order distribution (COD)~\cite{Bl21}. Focusing on size ratios that prohibit rattlers, a COD-based theory is constructed here. It introduces bidisperse statistical closure equations, which combine an explicit disorder criterion with consistent cell and edge statistics, and yields $ϕ_{max}(p,D)$ and structural predictions. The a-priori disorder-guaranteeing criterion proposed here ensures fast exponential a decay of the occurrence probability of ordered clusters with size. Using a constrained maximum-entropy analysis to ensure maximum disorder, the occurrence probabilities of all the possible cell and edge configurations are derived in terms of $p$ and $D$ and they are found to be binomial. All the results are assembled to yield an exact prediction of $ϕ_{max}(p,D)$.