The effective ellipsoid: a method for calculating the permittivity of composites with multilayer ellipsoids

In order to calculate the effective permittivity of a mixture with multilayer ellipsoids, this paper presents a self-consistent approximation (SCA) on the basis of the Bruggeman’s analytical model. The effective permittivity of a mixture with aligned multilayer ellipsoids is derived directly from the linear system of equations, which are built using the boundary condition of the electric field on the confocal ellipsoidal interface in the ellipsoidal coordinate system. Furthermore, for a mixture with multilayer ellipsoids oriented randomly, an effective ellipsoid is introduced to substitute the original multilayer ellipsoid, and the permittivity of the effective ellipsoid is derived by jointly solving the two linear systems of equations for the situation of the original multilayer ellipsoid and that of the effective ellipsoid, then the effective permittivity of the mixture can be calculated by the existing Maxwell-Garnett formula. After comparisons, it is revealed that there is a good agreement between this SCA method and existing theories.