| Due to their anisotropic shape, liquid crystal molecules exert elastic forces that attempt to align themselves with their neighbors. Perfect alignment may be attainable for liquid crystals in a trivial geometry with harmonious boundary conditions, but antagonistic anchoring conditions or complex geometries introduce geometric frustration. When a system is geometrically frustrated, the preferred ordering of the liquid crystals is incompatible with the space in which the molecules are confined. This frustration results in distortions of the liquid crystal elastic field, and in soft systems, deformations of the enclosing boundary.;On surfaces patterned with chemical and/or topographical features, liquid crystals are often unable to satisfy their own alignment preference while meeting boundary conditions imposed by the pattern. These patterns can allow for control over liquid crystal orientation, as well as defect occurrence and location, all of which are important considerations in liquid crystal display and sensing applications. When the container housing the liquid crystal is non-rigid, the complexity of the problem increases. These systems exhibit geometric frustration, but allow shape deformation to alleviate some or all of this frustration at the expense of increasing surface tension. Understanding this interplay between shape and order provides insight into biological systems that exhibit shape deformation and guides the design of certain self-assembled systems.;In this thesis, we study shape deformation problems in the context of liquid crystal systems. We begin by examining patterned surfaces of flat circles and micro-posts in square arrays to develop an understanding of liquid crystal behavior in complex geometries. Next we develop a locally refined moving mesh finite element simulation model capable of solving shape deformation problems with evolving domains. This model is used to simulate nematic tactoids and the effect of isotropic inclusions in these tactoid systems. Then we determine the properties of nanoparticle shells that self assemble in a quenched nematic background, and look at the effect of shell morphology on their alignment angle. Finally, we consider thin films of smectic liquid crystals in an attempt to see the layer structure given a variety of anchoring conditions. |
