Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution

Abstract

Analytic approximations for percolation points in two–dimensional and three–dimensional particulate arrays have been reported for only a very few, simple particle geometries. Here, an analytical approach is presented to determine the percolative properties (i.e. statistical cluster properties) of permeable ellipsoids of revolution. We generalize a series expansion technique, previously used by other authors to study arrays of spheres and cubes. Our analytic solutions are compared with Monte Carlo simulation results, and show good agreement at low particle aspect ratio. At higher aspect ratios, the analytic approximation becomes even more computationally intensive than direct simulation of a number of realizations. Additional simulation results, and simplified, closed–form bounding expressions for percolation thresholds are also presented. Limitations and applications of the asymptotic expressions are discussed in the context of materials design and design of sensor arrays.