Honors College Thesis
 

Twist in a Torus: Minimization of Nematic Configurations with Discontinuous Director Fields in Toroidal Coordinates

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https://ir.library.oregonstate.edu/concern/honors_college_theses/cz30q1551

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  • The ring disclination is a topological defect that may be suitable for light polarization inside of a nematic liquid crystal. Due to its stability and chirality, the ring disclination could also allow for theoretical applications to quantum and classical field theories as a model for fundamental particles. In order to model this defect within a nematic liquid crystal, we simulate a director field outside of a torus boundary. Equations are derived for the energy density of this configuration. Toroidal coordinates are implemented as a mesh on which the configuration is defined so that calculations remain stable for small tori. Deterministic iterative methods are developed for calculating the minimum energy of the system. Proof of the vector Laplacian as the correct minimization operator for generalized orthogonal coordinates is demonstrated using differential forms and Lagrange multipliers. The vector Laplacian is derived in toroidal coordinates and implemented in a hybrid regime with minimization of regions far from the torus surface occurring in Cartesian coordinates. This is implemented with mixed results. The minimized energy scales linearly with torus size as expected by dimensional analysis, but challenges remain for energy minimization at small sizes of torus. Possible solutions for extending the domain are discussed.
  • Key Words: Topological Defect, Nematic Liquid Crystal, Toroidal Coordinates
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