Contact function, uniform-thickness shell volume, and convexity measure for 3D star-shaped random particles

Abstract

Using a spherical harmonic series, the three-dimensional shape of star-shaped particles (e.g. most powders, sand, gravel) can be represented mathematically as readily as can a sphere, cube, or ellipsoid. In principle, any particle parameter, such as volume, surface area, moment of inertia tensor, or integrated mean curvature, can be easily computed. In this paper, we extend this list by developing three important algorithms, which have been found useful for regular particles, and adapt them to the case of random star-shaped particles using spherical harmonic series. These three algorithms are: a two-particle contact function, how to add a uniform-thickness shell to a single particle, and estimation of the convexity of a single particle using an algorithm for computing the convex hulls of non-convex particles. A derivation and numerical examples are shown for each algorithm. The two-particle contact function may be used in random particle placement programs, so that random-shaped particles may be used in suspension models as readily as spheres or ellipsoids. New results are shown for how the volume of a uniform-thickness shell, normalized by the surface area of the original particle, depends on the shell thickness t. For a sphere, the exact expression is cubic in t, with the linear coefficient unity and the quadratic coefficient equal to the reciprocal of the sphere radius. For star-shaped particles, the coefficient of the linear term is also unity. For all star-shaped particles considered, the coefficient of the quadratic term in t is approximately equal to the reciprocal of the radius of the volume-equivalent sphere, to within a few percent. The convexity measure, using convex hulls, adds to the list of useful shape parameters for random star-shaped particles and is shown to distinguish between different particle sets.