Spiraling Out Of Disorder: Investigating The Behavior Of Nematic Liquid Crystalline Systems And Soft Vortex Matter Under Geometrical Confinement
Self-assembly of ordered phases can be exploited to design new functional materials. Such structures can be heavily influenced by the confining geometry, which may be incommensurate with the ordering of the phase. In this thesis, I investigate two soft matter systems in which the interplay between the preferred ordering and confining geometry leads to emergent behaviors. I demonstrate that loxodromes, spirals that form a fixed angle relative to the principal directions on a surface of revolution, are key to understanding the behavior of both systems. In the first project, I theoretically investigate the twisting behavior of a highly ordered spindle-shaped polymer system that has nematic orientational ordering. Using geometric arguments, I develop a model that describes the twisting structure as loxodromes with fixed length. I show that the model captures the observed relationship between the spindle aspect ratio and surface twist angle as well as the observed transition from achiral to chiral. Next, I investigate the energetics of twisted spindles and show that loxodromes minimize the free energy on a spindle surface if a length constraint is included. I then show that in the bulk the twisting solution is well approximated by loxodromes. Extending ideas from the geometric model to the bulk, I show that including the length constraint makes twisted structures favorable over a larger parameter range than the bulk Frank free energy alone. In the second project, I use molecular dynamics simulations to determine the lattice ground states of vortices in a type-II superconductor confined to the surface of a conical frustum. I demonstrate that the confining geometry is incommensurate with an ideal lattice, resulting in structural lattice transitions. I show that the topological defect patterns at the transitions depend on properties of the two adjacent lattice structures. Using properties of the ground states, I demonstrate that inducing a density gradient in the lattice allows the creation of defect-free structures. I show that the idealized defect-free state is a conformal crystal, the construction of which involves loxodromes. This construction therefore provides a natural example of a configuration in which loxodromes appear.