Pattern formation in liquid crystals: The Saffman -Taylor instability and the dynamics of phase separation

This dissertation focuses on two topics relevant to the field of pattern formation in liquid crystals. The first is thermally induced phase separation of mixtures of nematic liquid crystals and isotropic fluids. This topic is investigated theoretically and numerically. This work develops a theory, based on mean field theory, that describes phase separation in nematic liquid crystals. The theory is dynamic and fully non-local. In order to fully capture the physical features of the system, the nematic liquid crystal is described with a three-dimensional tensorial order parameter, which retains biaxiality at interfaces and the first order nematic-isotropic transition. The theory addresses the issue that including all symmetry allowed gradient terms to second order in the free energy density leads to an ill-posed minimization problem for the free energy functional. This was pointed out by Oldano and Barbero and is colloquially known as the "K13 problem." The theory is used to derive fully non-local generalized forces of an n-ary mixture of particles interacting via London forces. Steric interactions are included by means of the pair-distribution function. The dissertation also develops a tensor-valued Langevin equation, used to apply thermal noise to the dynamic equations. The dynamic equations were implemented for three spatial dimensions and simulated on a cluster of four personal computers.;Understanding dendritic growth in solidification was a long-standing problem, and a theory that can describe it satisfactorily was developed only recently: microscopic solvability theory, which identifies surface tension anisotropy as a critical ingredient. Hele-Shaw experiments are a hydrodynamic analog to the solidification problem and were once instrumental in verifying the critical role of surface tension anisotropy. Nematics in Hele-Shaw experiments exhibit dendritic fingers, and it was speculated that here, effective shear thinning gives rise to dendrites. In the secand part of the dissertation experimental results are presented of radial Hele-Shaw experiments where a shear-thinning fluid, a dilute solution of xanthan in water, is driven by air through a narrow gap. The experiments clearly show dendritic fingers, possibly for the first time in an isotropic system. This result shows that non-linear transport can also give rise to stable dendritic fingers.