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 full 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 dynamic equations 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. Dendritic growth in solidification was a long-standing problem, and a theory was developed only recently that can describe it satisfactorily: 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. Hele-Shaw experiments in nematics exhibit dendritic fingers, and some speculated that here, effective shear thinning gives rise to dendrites. In the second part I present experimental results 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 fluid. This result shows that non-linear transport can also lead to dendritic fingers.

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