How do shape anisotropy and spatial orientation of the constituents affect the permittivity of dielectric heterostructures?

The study of dielectric heterostructures has been advancing at a rapid pace. Much of the interest in these materials stems from the fact that their physical properties can be systematically tuned by variation of the size and shape of the constituents. Here we report on extensive computer simulations of the effective permittivity of dielectric periodic (deterministic) heterostructures, having monosized hard core inclusions of anisotropic shape (rod, ellipsoid) embedded in an otherwise homogeneous and isotropic matrix. The real and imaginary parts of the permittivity, in the quasistatic limit, are rigorously evaluated with the use of the PHI3D field calculation package and the resolution of boundary integral equations. In this article, we show that the effective permittivity has critical properties near a conduction threshold. The conduction threshold concentration can be significantly modified by the size, shape, and spatial arrangement of the constituents. More specifically, it obeys a square law dependence as a function of the aspect ratio, i.e., the ratio of the smaller dimension to the larger dimension in both the rodlike and ellipsoidal inclusions. The data exhibit a scaling behavior and can all be collapsed onto a single master curve, indicative of a remarkable universality in the conductivity property. The critical exponents which determine how the real and imaginary parts of the effective permittivity scale with the distance from the conduction threshold are determined. Our results are compared with the scaling prediction of the standard percolation theory for infinite three-dimensional random lattices of insulator-normal metal composite systems. We also observed that the conduction transition is shifted towards higher concentrations as the angle between the symmetry axis and the direction of the applied electric field increases. Increasing the contrast ratio, between the permittivity and the conductivity of the background medium and the inclusions, results in dramatic changes of the complex effective permittivity, depending on the geometry of the inclusions. The scale-dependent properties and the mechanism which govern criticality are related to the actual area of contacts between the inclusions. (C) 2000 American Institute of Physics. [S0021-8979(00)07723-9].