Electrostatic interactions of arbitrarily dispersed spheres with spherically transversely isotropic constituents

We propose a theoretical framework for evaluation of electrostatic potentials in an unbounded isotropic matrix containing many arbitrarily dispersed spherical inclusions subjected to a remotely prescribed potential field. The inclusions could be isotropic or spherically transversely isotropic, and may have different sizes with different conductivities. The approach is based on a multipole expansion formalism, together with a construction of consistency conditions and translation operators. This procedure generalizes the approach of the classic work of Lord Rayleigh [L. Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of a medium, Philos. Mag. 34 (1892) 481-502] for a periodic array to an arbitrary dispersion of spheres. In the formulation, we expand the potential field versus various local coordinates with origins positioned at each inclusion's center. We show that the coefficients of field expansions can be written in the form of an infinite set of linear algebraic equations. Numerical results are presented for a few different configurations. For the case of an infinite space containing two spheres subjected to a uniform intensity, we have verified our solutions with those obtained from the bispherical coordinate transformation. The derived field solutions can be used to assess the effective conductivity of a random heterogeneous medium.