Modelling nematic liquid crystal in fractal dimensions

Abstract

In this paper we present a fractal Berreman's model which account for azimuthal surface anchoring of nematic liquid crystals which play a leading role nowadays in biomedical field and pharmaceutical controlled release dosage forms. The model is based on the concept of “product-like fractal measure” in anisotropic media which permits to describe liquid crystals formation in terms of fractal dimensions. It was observed that fractal dimensions affect both the polar azimuthal angle and the Franck excess elastic energy density of the distortion. For fractal dimensions close to unity, our analysis reveals the emergence of fluctuations and large deformations in the dynamical variables of the theory which suggest the presence of non-linear elastic instabilities or emergence of defect structures in liquid crystals. These fluctuations fade away for fractal dimensions much less than unity. We have evaluated the elastic energy per unit of area which is found to depend on the square of the fractal dimension. This leads to a decrease of the saturation voltage which is necessary to reduce the elastic energy of liquid crystals films in order to minimize the elastic energy. This reduction may describe nematic liquid crystals free from deformations, singularities or weak anchoring surfaces.