Microstructures and topological defects induced by a spherical particle in a nematic liquid crystal

In this dissertation the Landau-de Gennes expansion for the free energy density of a distorted, nematic liquid crystal is combined with Doi's and Marrucci-Greco molecular theories to study the orientation field surrounding a spherical particle with homeotropic surface anchoring and uniform far-field alignment. The model includes the effects of elastic anisotropy, i.e., unequal Frank elastic constants. The molecular orientation field, described by the order parameter tensor, is determined by numerical minimization of the global free energy via a relaxation-type algorithm. The formation of two distinct microstructures was observed, namely, a symmetric Saturn Ring (SR) pattern with equatorially-positioned line defect, and an asymmetric configuration with a Hyperbolic Hedgehog point defect (PD) located at one of the poles. For large enough particles, the global energy of the PD pattern is smaller than that of the SR, but both structures may be observable, as the transition from one state to the other requires overcoming a finite activation energy. Below a critical particle size, all trial microstructures are unstable to small perturbations, except for the SR pattern towards which all simulations converge. Non-equatorial rings also emerged occasionally, but they were found to be intermediate, higher energy states that slowly evolved to either the SR or PD configurations. A detailed analysis of the structure and topology of defects reveals that, for our range of parameters, the PD is not a small SR, but is endowed with a much richer structure. In order to minimize the number of simulations required to track the SR-PD energy crossover, scaling relations for the dependence of the global energy of each pattern on the dimensionless Landau coefficients (i.e., particle size) were derived. These functional relations are in satisfactory agreement with the simulations results. Boundary conditions are also discussed, and an analytical description of the interfacial boundary layer is carried out.