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Nematic liquid crystals (LCs) have been an area of intense investigation. In this thesis, we employ the numerical calculations based on the Landau--de Gennes theory to model nematic LC systems under the effect of chirality, external fields, and anchoring conditions. In chapters two and three, we investigate the rich morphological behaviors in confined cholesteric liquid crystals (ChLCs), arising from the balance between elasticity, chirality, and surface energy. More specifically, a systematic study of micrometer-sized ChLC droplets and shells is presented. In ChLC droplets, with increasing chirality, a continuous transition is observed from a twisted bipolar structure to a radial spherical structure all within a narrow range of chirality. During such a transition, a bent structure is predicted by simulations and confirmed by experimental observations. The influence of chirality and surface interactions are discussed in the context of the potential use of ChLC droplets as stimuli-responsive materials for reporting molecular adsorbates. To broaden the spectrum of defect textures, the third chapter examines more complex geometries of boundaries, shells. We propose a dimensionless parameter to represent the chirality in shell geometry and characterize phase boundaries in the chirality-thickness phase diagram. Importantly, we observe that, in uniform shells, the structural transition, in response to both chirality and shell geometry, is accompanied by an abrupt change of defect positions. Its response to chirality is more sensitive compared to that of ChLC droplet with same size, implying an enhanced performance of the potential use as sensors. Moreover, we demonstrate that non-chiral and chiral nematic shells exhibit distinct equilibrium positions of inner drop in shells, which are governed by shell chirality. In the fourth chapter, we study the effect of strong uniform magnetic field on the reorientations of the director fields and defects on tetravalent namatic liquid crystal shells. Two different cases, in terms of thickness gradient, are studied: i) homogeneous shells with four s=1/2 defects in tetrahedral arrangement, and ii) inhomogeneous shells with four 1/2 defects localized in the thinner part of the shells. Even though the defects finally move to the poles determined by the magnetic field in all cases, the dynamics of defect motions we observe are extremely rich and compelling. We investigated the underlying mechanism and demonstrate that the disclination walls, which depend on the direction of magnetic fields with respect to the defect orientations, are essential in determining the defect motions. Moreover, we report a hybrid-splay-bend disclination wall for the first time, to our knowledge. Our corresponding experimental observations show a good agreement with simulation results, hence validate the detailed structures of disclination walls and their evolution mechanisms. The last chapter examines the evolution of defect configurations, including recently reported elastic hexadecapoles and other multipoles, by tuning the preferred tilt angle of degenerate conic anchoring. In order to gain a systematic understanding of hexadecapoles and other related structures, a new continuum model for anchoring is introduced here at the level of a Landau-de Gennes free energy functional. New types of angular and radial dependencies for colloidal interactions are predicted and confirmed experimentally as a function of preferred tilt angle. In particular, the new model predicts a new type of elastic dipole whose stability decreases as preferred tilt angle increases, as well as the dipole-hexadecapole transformation, which is confirmed by our experimental observations. Taken together, the results of simulations and experiments presented here suggest that new and previously unanticipated avenues may exist for design of self-assembled crystal lattice structures via control of tilt angle.


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