The dielectric response of a lattice of dielectric spheres: convergence of multipole expansion methods

Abstract

The response to an external applied static electric field of a large spherical sample of a lattice of dielectric spheres embedded in a dielectric medium is analysed. The lattice cells are simple cubic and contain one dielectric sphere per cell. The response gives a dielectric constant for the lattice array. The electric field problem is written as an integral equation for an effective charge density on the surface of the dielectric spheres. The interactions of the responses of the spheres with one another are represented via lattice sums of external spherical harmonic potentials. The numerical implementation of the problem is carried out by truncating a spherical harmonic multipole expansion for this charge density at a set of increasing angular momentum numbers. The convergence of the solution as the truncation number increases is examined numerically. Some implications for the theoretical basis of simulations of the dielectric constant and viscosity of suspensions are discussed.