Inclusion's distribution pattern's influence on mixture's effective permittivity in two dimension utilizing finite difference method (FDM)

Abstract In this paper, a numerical technique based on two dimensional finite difference method (FDM) is utilized to study the effective permittivity of mixtures with various patterns of inclusions’ distributions. The inclusions are assumed to be periodically distributed in the host medium. The influence of the inclusions’ shape, permittivity, volume fraction and randomness are investigated in the simulation. Three categories of inclusions’ types are considered, which are solid, hollow and randomly distributed inclusion. The shapes of solid inclusion type include square, cylinder, diamond and plus. The shapes of hollow inclusion type include square, cylinder, diamond and grid. The shapes of randomly distributed inclusion type include fine-grained inclusions and coarse-grained inclusions. The results of FDM methods are also compared with that of the analysis. Several analysis formulas are considered, such as Maxwell-Garnett, Bruggeman, Birchak, Looyenga and Lichtenecker. For all shapes of solid patterns except the plus, the simulated effective dielectric properties of mixtures are in good agreement with that obtained by Maxwell-Garnett prediction formula. When the contrast of permittivities between the inclusions and the environment is high, the numerical results for the square solid pattern match well with the analytical results provided by the Maxwell-Garnett formula. For other patterns, no analytical formulas can be implemented appropriately. The FDM results of hollow patterns deviate from the theoretical results significantly. In addition, for mixtures with randomly distributed inclusions, the effective permittivity achieved by the numerical technique varies almost linearly with volume fraction, which does not agree well with any analysis formula.