Analogy between dielectric relaxation and dielectric mixtures: Application of the spectral density representation

The spectral representation is an effecient tool to explore electrical properties of material mixtures. It separates the contributions of geometrical topology and intrinsic properties of the constituents in the system. The aim of this paper is to derive an expression for the spectral density representation, which favors dielectric relaxation phenomenon. This unfamiliar form is distinct in a way that the existing dielectric relaxation models and data analysis tools can be employed for extracting the spectral density function of a given system. Electrical properties of material mixtures have attracted researchers in academia and industry to seek a relation between overall composite properties and intrinsic properties of the constituents and their spatial arrangement inside the mixture [1]. Bergman [2] has proposed a mathematical way for representing the effective dielectric permittivity εe of a binary mixture as a function of permittivities of its constituents, εm and εi, and an integral equation, which includes the geometrical contributions. This theory is called the spectral density representation. Milton [3] gave the corrections, whereas, Golden and Papanicolaou [4] presented the rigorous derivation for the spectral representation. The permittivity εe of a mixture is expressed as follows in the spectral representation [5]