Mathematical analysis of composite systems of liquid crystal and polymer

This thesis deals with mathematical modeling of the composite material consisting of liquid crystal and polymer. Such systems are important in the applications. The mathematical work consists in minimizing the total free energy of a system in a geometrically complex domain. The total energy consists of the bulk Oseen-Frank free energy of the liquid crystal plus the surface contribution. The latter arises as a result of the contact interaction between the liquid crystal and the polymeric component. The first part of the work presents a two-dimensional prototype model. The effective equations for the composite configurations are obtained by means of homogenization limits, and subsequently analyzed and computed by numerical methods.;One of the main mathematical difficulties of such analyses is the presence of defects in the model. This becomes an important issue in the study of physically realistic three-dimensional configurations. In order to overcome such a difficulty, the original model is replaced by a relaxed one: while the unit vector constraint is omitted, a penalty term is included in the energy. The idea is borrowed from the Ginzburg-Landau modeling of superconductivity and the unit director condition will be satisfied approximately while allowing for defects (i.e., points or lines where the length of director is zero). The effective equations are studied, and the relaxation of the length of the directors is eventually removed through the limit of the penalty term. The external magnetic field is added and the effect on the composite system is discussed.;The last part of the thesis deals with systems of liquid crystal droplets in a polymeric isotropic matrix. The droplet configurations are of bipolar type, i.e., spherical-type shapes with two defects as opposite poles. The dipole line provides a new tool to construct anisotropic systems. A mathematical model for bipolar droplet is constructed and analyzed for both fixed and free boundary problems. The thesis concludes with a mean-field theory approach to model such droplet systems.