Permittivity of a multiphase and isotropic lattice of spheres at low frequency

A solution methodology is presented in this article to compute the effective permittivity for a multiphase lattice of dielectric and/or conducting spheres at low frequencies. It is assumed that the lattice is effectively isotropic. This methodology relies on two central developments. The first is a T-matrix solution for a multiphase lattice of spheres immersed in a uniform electric field. This solution is presented in a succinct matrix-vector notation and is valid for any lattice type. The second development is a simple and accurate equation for the effective permittivity that incorporates all mutual coupling between the spheres. Results are shown in this article for three situations. The first is a two-phase system of conducting spheres (used for verification) and the second is a dielectric-conductor (cermet composite) lattice of spheres. The third and final result is from a lattice containing a cluster of conducting spheres. It is suggested that this last material type displays a behavior in between that of random materials and two-phase lattices due to “permittivity enhancement” at low volume fraction. It is also shown that the Maxwell Garnett formula is not nearly as accurate for this cluster lattice, also because of this enhancement effect.