Equilibrium Configurations of Hexagonal Columnar Liquid Crystals with Applications to Materials Science and Biology

Abstract

Using the mathematical theory of liquid crystals, we propose models for equilibrium configurations of the hexagonal columnar phase of lyotropic chromonic liquid crystals (LCLCs). These models are applied in a collection of settings in which the underlying phases are naturally observed, including LCLC toroidal superstructures, liquid crystal confined to a thin capillary, double-stranded DNA packed in bacteriophage viruses, and DNA toroidal clusters that form in the presence of condensing agents. Although the length scale of these superstructures ranges from nanometers to microns, they show the ability to sustain pressures of up to 60 atmospheres. In each setting, we investigate equilibrium configurations and discuss well-posedness of our model. The mathematical work focuses in the analysis of constrained free boundary problems for combined liquid crystal and elastic energies, with a main focus on the very rich structure of defect cores in lyotropic systems. We conclude with potential modifications to the model that could be used to incorporate information such as ionic concentrations present in the medium surrounding this liquid crystal phase.