The maximum packing fraction of disordered bidisperse discs: theory and predictions
Abstract
Predicting the highest packing fraction that disordered packing of bidisperse discs can achieve, ϕmax, 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 ϕmax 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 ϕmax(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 ϕmax(p,D).
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BibTeX
@misc{blumenfeld2026maximum,
title = {{The maximum packing fraction of disordered bidisperse discs: theory and predictions}},
author = {Raphael Blumenfeld},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02805},
eprint = {2611.02805},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02805},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The maximum packing fraction of disordered bidisperse discs: theory and predictions
AU - Raphael Blumenfeld
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02805
UR - https://arxiv.org/abs/2611.02805
AB - 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).
ER -