Equilibrium Liquid Crystal Configurations: Energetics, Singularities and Applications

The research concerns the mathematical description of liquid crystals, an intermediate state of matter between solids and liquids. Liquid crystals are of technological importance because they are the working material of many electronic displays. A uniaxial nematic liquid crystal consists of rod-like molecules which at each point tend to align in a particular direction, like matches in a matchbox. This direction can be influenced by electric fields. The direction is determined by solving the governing partial differential equations that describe the mechanics of the liquid crystal. These equations depend in particular on the form that is assumed for the energy density of the liquid crystal. The proposal has various aims. The first is to examine when an arrow can be consistently assigned to the preferred direction. Such an arrow is not physical, since the molecules can be inverted end-to-end without changing their mechanical properties. However the existence of such an arrow is assumed in some widely-used theories, and we will study how the predictions of such theories compare to other theories in which no such arrow is assigned. The second aim is to investigate whether forms of the energy density different from those usually used could help to describe the behaviour of liquid crystals at defects, that is at points where the alignment direction is not uniquely defined. Related to this is the question of how the energy density can be determined experimentally. The third aim is to study when the equations have equilibrium solutions, in the context of theories which do not assign an unphysical arrow to the preferred direction, and whether these solutions have defects. Finally we will apply our work to the study of prototype new display devices, such as the Post Aligned Bistable Nematic (PABN) Device being developed at Hewlett-Packard laboratories.