Abstract
Topological defects in liquid crystals provide a versatile platform for studying how symmetry, confinement, and external fields generate singular and non-singular structures. In particular, apolar nematics and ferroelectric nematics exhibit distinct defect physics because polarity changes both the admissible topology and the role of electrostatics. However, a computational framework that simultaneously captures thermal fluctuations, boundary-frustrated fractional skyrmions, and depolarization-controlled polar textures remains lacking. This dissertation investigates the formation, transformation, and electrostatic selection of topological defects in confined liquid crystals through Fourier-space Monte Carlo sampling, Landau-de Gennes simulations, and reduced polar-electrostatic modeling. It first develops a Fourier-space Monte Carlo method for two-dimensional nematics, which reproduces the equipartition-governed elastic spectrum while describing defect annihilation and disk-confined defect dynamics within a collective continuum sampling scheme. It then introduces longitudinally split half-skyrmions and half-bimerons—distinct from conventional merons—and demonstrates their realization in thick achiral nematics via frustrated boundaries that stabilize non-singular strings. Furthermore, colloidal elastic dipole interactions with these strings are reported for the first time, revealing how dipoles decode local skyrmion topology through orientation and height selection. Emergent monopoles and antimonopoles are also shown to mediate reversible string transformations driven by inter-segment free-energy differences. Finally, it establishes a continuum model for ferroelectric nematics in which depolarization fields induce a chiral helical ground state and reproduce the oscillation, stripe, lattice, and homeotropic textures selected under ac driving. These findings establish how collective fluctuations, boundary frustration, and polar electrostatics govern topological structure selection in liquid crystals, and provide a practical framework for engineering responsive defects and textures in soft photonic and electro-optic systems.