Exact solutions for transport properties of arrays of spheres

Abstract

The problems which involve calculation of transport properties (for example: dielectric constant, optical constants, thermal and electrical conductivity, resistance to fluid flow and Lame constants) of an array of spheres of one material embedded in another are mathematically analogous. The first order solution was given by J. C. Maxwell (1873) and the second order solution for spheres arranged in the simple cubic lattice was given by Lord Rayleigh (1892). The method of Lord Rayleigh is shown to be mathematically rigorous and capable of extension to give exact values of the transport property for each of the three cubic lattices. When applied to the conductivity problem, the solutions have the correct behavior in that, for perfectly conducting spheres, the array conductivity becomes infinite when the spheres touch. The solutions have been verified by comparison with experiment for the simple cubic and body centered cubic lattices. The exact solutions enable the limits of validity of the widely used Maxwell solution, also known as the Maxwell‐Garnett solution, to be obtained. We have used the exact solution for the regular lattice to establish the useful approximations for the transport properties of a disordered lattice.