Effective medium theory for dispersions of dielectric ellipsoids

A material composed of a mixture of distinct homogeneous media can be considered as a homogeneous one at a sufficiently large observation scale. In literature, a widely dealt topic is that of calculating the overall permittivity of a dispersion of dielectric spheres in a homogeneous matrix: the well-known Maxwell formula solves the problem in the case of very diluted suspensions. Moreover, this relationship has been adapted, to yield correct results even if the dispersion in not strongly diluted, by means of the so-called Bruggemanʹs procedure. In this paper, we apply this technique to perform a complete study on the equivalent permittivity of a dispersion of ellipsoids. The obtained solutions allow us to evaluate the effects of the shape of the inclusions on the overall electric behaviour of the mixture. In particular we find explicit expressions, which give the equivalent permittivity of the medium in terms of the eccentricities of the embedded ellipsoids and of some stoichiometric parameters. The treatment is carried out both for aligned ellipsoidal inclusions and for randomly oriented ellipsoids. In particular, new explicit relationships have been derived for dispersions of generally shaped randomly oriented ellipsoids.