Calculation of the Voronoi boundary for lens-shaped particles and spherocylinders

TL;DR

This paper develops analytical methods to calculate the Voronoi boundary for lens-shaped particles and spherocylinders, enabling improved estimation of packing fractions in non-spherical particle systems.

Contribution

It introduces a novel analytical approach to compute the Voronoi boundary for complex shapes like lens-shaped particles and spherocylinders, which was previously intractable.

Findings

Analytical expressions for Voronoi boundaries of lens-shaped particles.

Analytical expressions for Voronoi boundaries of spherocylinders.

Facilitates better mean-field theories for non-spherical particle packings.

Abstract

We have recently developed a mean-field theory to estimate the packing fraction of non-spherical particles [A. Baule et al., Nature Commun. (2013)]. The central quantity in this framework is the Voronoi excluded volume, which generalizes the standard hard-core excluded volume appearing in Onsager's theory. The Voronoi excluded volume is defined from an exclusion condition for the Voronoi boundary between two particles, which is usually not tractable analytically. Here, we show how the technical difficulties in calculating the Voronoi boundary can be overcome for lens-shaped particles and spherocylinders, two standard prolate and oblate shapes with rotational symmetry. By decomposing these shapes into unions and intersections of spheres analytical expressions can be obtained.