Predicting the dielectric nonlinearity of anisotropic composite materials via tensorialanalysis

Abstract

The discovery of new materials with peculiar optical properties as well as the prediction oftheir behaviour given the microstructure is a matter of remarkable interest in thecommunity of material scientists. A complete theory allowing such a prediction is not yetavailable. We have formulated a theory able to analytically predict the effective second- andthird-order nonlinear electrical behaviour of a dilute dispersion of randomly orientedanisotropic nonlinear spheres in a linear host. The inclusion medium has non-vanishingsecond- and third-order nonlinear hypersusceptibilities. As a result, the overall compositematerial is nonlinear but isotropic because of the random orientation of the inclusions. Wederive the expressions for the equivalent permittivity and for the Kerr equivalenthypersusceptibility in terms of the characteristic electric tensors describing theelectrical behaviour of the spheres. The complete averaging over inclusion positionsand orientations led to general results in the dilute limit. We show that theseresults are consistent with earlier theories and that they provide null second-orderhypersusceptibility as expected in a macroscopically isotropic medium. This theorygeneralizes the well-known Maxwell-Garnett formula and it can be easily specialized toany of the 32 crystallographic symmetry classes. Despite this study assumingstatic conditions, it can be generalized to the sinusoidal regime, pointing at aninteresting way to engineer optically active materials with desired behaviour.