Contents I. Introduction
Power-Law relationships are quite often used in dielectric modeling of composite systems [1]–[9]. These relationships model the effective permittivity of two-component systems using the volume fraction of each component according to equation (1). \epsilon _c^\beta = {\phi _1} \epsilon _1^\beta + \left( {1 - {\phi _1}} \right) \epsilon _3^\beta \eqno{\hbox{(1)}}
where and are the complex dielectric permittivity of the composite system, the filler and the matrix respectively, is the volume fraction of filler component of the composite system, and is a dimensionless parameter representing the shape and orientation of the filler particles within the bulk composite [1]. Common examples of this model are the linear mixtures model , the Birchak formula [2] and the Landau, Lifshitz, Looyenga formula [3]. The general two-component power-law model for complex permittivity has been used extensively for a wide range of material systems with varied success, including air-particulate composites [4]–[7], ceramic-ceramic composites [8] and polymer-ceramic composites [9]. More generally, for a composite comprised of n number of components, the power law mixtures model may be written as equation (2).
\epsilon _c^\beta = \sum\limits_{i = 1}^n {{\phi _i}} \epsilon _i^\beta \eqno{\hbox{(2)}}
where and are the complex dielectric permittivity of the composite system and any constituent component of the composite respectively, is the volume fraction of the constituent component, and is a dimensionless parameter representing the shape and orientation of the filler particles within the bulk composite. Although quite successfully used to model a wide range of composite systems, the power law mixtures model does not account for interactions between the components of the composite, which is a serious limitation.
