Conductivity of a periodic particle composite with spheroidal inclusions^*

Department of Physics, School of Physical Sciences, University of Surrey, Guildford, Surrey GU2 5XH, England

Corresponding author: n.harfield@surrey.ac.uk

Received: 20 July 1998
Revised: 13 October 1998
Accepted: 29 October 1998
Published online: 15 April 1999

Abstract

The effective electrical conductivity of a two-phase material consisting of a lattice of identical spheroidal inclusions in a continuous matrix is determined analytically. The inclusions are located at the node points of a simple-cubic lattice and the axis of rotation of each spheroid coincides with one of the lattice vectors, such that the spheroids are aligned with each other and with the lattice. With an electric field applied in the direction of the rotation axes of the spheroids, the electric potential is found by solving Laplace's equation. The solution is found by analytically continuing the interstitial field into the particle domain and replacing the particles with singular multipole source distributions. This yields an expression for the potential in the interstitial domain as a multipole expansion. Using Green's theorem, it can be shown that only the first coefficient in this expansion is required to determine the effective conductivity of the composite. The coefficients are determined by applying continuity conditions at the particle-matrix boundary and, in a novel approach, this is achieved by transforming the multipole expansion into an expansion in spheroidal harmonics. Results for spheres and prolate and oblate spheroids are compared with experimental data and previous theoretical work, and excellent agreement is observed.