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Predicting random close packing of binary hard-disk mixtures via third-virial-based parameters

Andrés Santos, Mariano López de Haro

TL;DR

This work tackles predicting the random close packing fraction $φ_{\text{mixt}}$ for binary hard-disk mixtures, where prior models show limited universality across size ratio $q$ and compositions. It introduces a third-virial-based parameter $μ$ with $μ = \frac{b_3-1-(\bar B_3-1)m_2}{b_3-3}$ and the relation $φ_{\text{mixt}} = φ_{\text{mono}} + μ(1-φ_{\text{mono}})$, together with $λ ≡ 1/(1-μ)$ and $\frac{φ_{\text{mixt}}}{1-φ_{\text{mixt}}} = \frac{λ}{1-φ_{\text{mono}}}-1$, to yield near-linear dependences. The approach achieves substantially better data collapse across $q$ than Brouwers' model and compares favorably with Zaccone's scheme. It naturally extends to polydisperse size distributions by evaluating the generalized moments and the reduced third virial coefficient, offering a practical universal framework for $RCP$ in disordered hard-disk systems. Overall, the method provides a simple, robust predictor for RCP with potential applications to jamming and granular matter in polydisperse settings.

Abstract

We propose a simple and accurate approach to estimate the random close packing (RCP) fraction of binary hard-disk mixtures. By introducing a parameter based on the reduced third virial coefficient of the mixture, we show that the RCP fraction depends nearly linearly on this parameter, leading to a universal collapse of simulation data across a wide range of size ratios and compositions. Comparisons with previous models by Brouwers and Zaccone demonstrate that our approach provides the most consistent and accurate predictions. The method can be naturally extended to polydisperse mixtures with continuous size distributions, offering a robust framework for understanding the universality of RCP in hard-disk systems.

Predicting random close packing of binary hard-disk mixtures via third-virial-based parameters

TL;DR

This work tackles predicting the random close packing fraction for binary hard-disk mixtures, where prior models show limited universality across size ratio and compositions. It introduces a third-virial-based parameter with and the relation , together with and , to yield near-linear dependences. The approach achieves substantially better data collapse across than Brouwers' model and compares favorably with Zaccone's scheme. It naturally extends to polydisperse size distributions by evaluating the generalized moments and the reduced third virial coefficient, offering a practical universal framework for in disordered hard-disk systems. Overall, the method provides a simple, robust predictor for RCP with potential applications to jamming and granular matter in polydisperse settings.

Abstract

We propose a simple and accurate approach to estimate the random close packing (RCP) fraction of binary hard-disk mixtures. By introducing a parameter based on the reduced third virial coefficient of the mixture, we show that the RCP fraction depends nearly linearly on this parameter, leading to a universal collapse of simulation data across a wide range of size ratios and compositions. Comparisons with previous models by Brouwers and Zaccone demonstrate that our approach provides the most consistent and accurate predictions. The method can be naturally extended to polydisperse mixtures with continuous size distributions, offering a robust framework for understanding the universality of RCP in hard-disk systems.
Paper Structure (4 sections, 12 equations, 4 figures)

This paper contains 4 sections, 12 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our proposal
  3. Extension to polydisperse mixtures
  4. Conclusions

Figures (4)

  • Figure 1: (a) $1/\phi_{\text{mixt}}$ versus the parameter $\mu_B$ [Eq. \ref{['1']}] and (b) $\phi_{\text{mixt}}/(1-\phi_{\text{mixt}})$ versus $\lambda_B$ [Eq. \ref{['2']}] for binary HD mixtures with size ratios $q=1.4$, $1.7$, $2$, and $3$. Symbols are simulation data note_12_2025, while the dashed and dashed-dotted lines correspond to Eqs. \ref{['1']} and \ref{['2']} with $\phi_{\text{mono}}=0.844$ and $\phi_{\text{mono}}=0.8425$, respectively.
  • Figure 2: (a) $\phi_{\text{mixt}}$ versus the parameter $\mu$ [Eq. \ref{['1bis']}] and (b) $\phi_{\text{mixt}}/(1-\phi_{\text{mixt}})$ versus $\lambda$ [Eq. \ref{['2bis']}] for binary HD mixtures with size ratios $q=1.4$, $1.7$, $2$, and $3$. Symbols are simulation data note_12_2025, and the dashed lines correspond to our proposal [Eqs. \ref{['26_2D']}] with $\phi_{\text{mono}}=0.844$.
  • Figure 3: $\phi_{\text{mixt}}$ as a function of the area fraction of large disks, $c_{\text{L}}$, for binary HD mixtures with size ratios $q=1.4$, $1.7$, $2$, and $3$. Symbols are simulation data note_12_2025. Solid lines show our predictions [Eq. \ref{['1bis']}] with $\phi_{\text{mono}}=0.844$, while the dashed-dotted and dashed lines correspond to Eq. \ref{['1']} with $\phi_{\text{mono}}=0.8425$ and Eq. \ref{['8']} with $\phi_{\text{mono}}=0.844$, respectively.
  • Figure 4: Plot of the parameter $\mu$ as a function of the dispersity parameter $s$ for a polydisperse mixture with a log-normal size distribution.