Random packing fraction of binary hyperspheres with small or large size difference: a geometric approach

TL;DR

This paper develops a geometric model to analyze the random packing fraction of binary hyperspheres across different dimensions, comparing predictions with computational results and deriving asymptotic behaviors for large size differences.

Contribution

It introduces a unified geometric approach to predict binary hypersphere packing fractions, extending classical theories to higher dimensions and large size ratios, with validation against computational data.

Findings

Good agreement between model predictions and computational results in various dimensions.

Derived asymptotic approximation for large size ratios showing proportionality to u^-1.

Presented a phase diagram for binary packing fractions in high-dimensional spaces.

Abstract

The random packing fraction of binary particles in D-dimensional Euclidean space R^D is studied using a geometric approach. First, the binary packing fraction of assemblies with small size difference are studied, using the excluded volume model by Onsager for particles in three-dimensional space (D = 3). According to this model the packing increase by bidispersity is proportional to (1 - f)(u^D - 1)^2, with f as monosized packing fraction, u as size ratio and D as space dimension. The model predictions are compared with computational results for disks in two dimensions (D = 2) and hyperspheres in the large-dimension limit (D to infinity), yielding good agreement. Subsequently, the packing of hyperspheres with large size difference is modeled, employing the classic theory of Furnas. This theory, developed for three dimensions, starts from an infinite size ratio of larger and smaller particles (u to infinity). Here, the pertaining equations are applied to hyperspheres, and successfully compared with computational results for hyperspheres in the large-dimension limit. Furthermore, an asymptotic approximation of the binary packing fraction for large size ratio is derived, which shows that the first order variation of the Furnas packing fraction (u^-1 = 0) is proportional to (2 - f)u^-1. Finally, a scaled D-dimensional binary packing graph is presented, governing a simplified phase diagram that borders the binary random packing fraction of amorphous assemblies. To summarize, basic space-filling and geometric (athermal) theories on simple hard spheres appear to be a valuable tool for the study of hyperspheres packing and amorphization.