Effective-medium theories for two-phase dielectric media

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Abstract
Two different effective-medium theories for two-phase dielectric composites are considered. They are the effective medium approximation (EMA) and the differential effective medium approximation (DEM). Both theories correspond to realizable microgeometries in which the composite is built up incrementally through a process of homogenization. The grains are assumed to be similar ellipsoids randomly oriented, for which the microgeometry of EMA is symmetric. The microgeometry of DEM is always unsymmetric in that one phase acts as a backbone. It is shown that both EMA and DEM give effective dielectric constants that satisfy the Hashin–Shtrikman bounds. A new realization of the Hashin–Shtrikman bounds is presented in terms of DEM. The general solution to the DEM equation is obtained and the percolation properties of both theories are considered. EMA always has a percolation threshold, unless the grains are needle shaped. In contrast, DEM with the conductor as backbone always percolates. However, the threshold in EMA can be avoided by allowing the grain shape to vary with volume fraction. The grains must become needlelike as the conducting phase vanishes in order to maintain a finite conductivity. Specifically, the grain-shape history for which EMA reproduces DEM is found. The grain shapes are oblate for low-volume fractions of insulator. As the volume fraction increases, the shape does not vary much, until at some critical volume fraction there is a discontinuous transition in grain shape from oblate to prolate. In general, it is not always possible to map DEM onto an equivalent EMA, and even when it is, the mapping is not preserved under the interchange of the two phases. This is because DEM is inherently unsymmetric between the two phases.