Topological defects in confined nematic liquid crystal systems and the transitions between them

Abstract

The major theme of this thesis is the treatment of defect cores in uniaxial nematic liquid crystals. For simplicity, we prefer the Oseen-Frank formalism, where the orientational order of the uniaxial nematics is represented by unit vectors with head-tail symmetry. However, the defect core in this formalism is a tiny region where the unit vectors are not defined. This implies that when we evaluate the Oseen-Frank free-energy functional and solve the corresponding Euler-Lagrange equation, we should not admit differentiation and integration cross the defect core. In fact, we should either treat the defect core as a boundary or put it at the coordinate singularity of a special coordinate system. The first treatment is used in our numerical study of the defect transitions in the nematic bridges. The finite-difference method (with the use of the successive over-relaxation method) enables us to select the ground state after exhausting many possible defect structures. Our results confirm the existence of different types of equilibrium defect structures in the cylindrical bridge. Our results further imply that some different shapes of the lateral surfaces preserve the qualitative features of the defect structure diagram yet they can change the positions of the transition lines. However, the above-mentioned two treatments impede a general analytical theory of defects in nematics since they usually require exhaustive search or special geometries. Therefore, a better treatment may be to create the defect core during the calculation process. To test its feasibility, we conduct a numerical experiment by designing a special multigrid method for the study of equilibrium defect structures in the cylindrical bridge, where the crudest information of the defect core is expected to be contained on the coarsest grid and better information of the defect core is expected to be contained on the finer grid. Then, for the analytical study, we first experiment on the one-dimensional analog, where the solution is represented by Fourier series and the defect core is the jump discontinuity. We observe that the correct energy function can be obtained by properly eliminating an infinitely large part, and the resulting regularized energy function is equally effective with a finite number of its Fourier modes for the purpose of determining the equilibrium state. Based on that, some calculations are performed for two-dimensional nematics. We speculate that a finite number of Fourier modes of the regularized energy function may be enough to determine the equilibrium defect structures in nematics.