Theoretical and computational analysis of a Ginzburg-Landau type energy model for smectic-C* liquid crystals with defects
Abstract
This dissertation investigates properties of a smectic C* liquid crystal film containing defects that give rise to distinctive spiral patterns in the film's texture. A Ginzburg-Landau type model describes this phenomena and the investigation provides a detailed analysis of minimal energy configurations for the film's director field. The study demonstrates the existence of a limiting location for the defects (vortices) so as to minimize a reduced energy. It is proved that if the degree of the boundary data is positive then the vortices each have degree +1 and that they are located away from the boundary. A renormalized energy function is constructed that depends on the vortices, the boundary values, and the field's pattern inside the domain. It is proved that the limit of the energy functional minus the sum of the energy around the vortices, as the G-L parameter ε tends to zero, is equal to this renormalized energy. This dissertation also presents a spectral-Galerkin method for the Euler-Lagrange equations of the minimizers to the generalized Ginzburg-Landau energy in polar geometries. This utilizes previous work done with the classical Ginzburg-Landau energy functional. This method is based on a variational formulation that incorporates essential pole condition(s) at the origin due to the Fourier expansion of the solution. A stability analysis shows that conditions on the stability term ensures the scheme is stable and error estimates are provided. The computational complexity of using the Chebyshev-Galerkin method on the radial component is quasi-optimal.
Degree
Ph.D.
Advisors
Phillips, Purdue University.
Subject Area
Applied Mathematics|Mathematics
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