Cholesteric Liquid Crystals: the Blue Phase and the Analogy with HELIUM-3 - a.

Part I of this thesis presents a unified view of some of the theoretical progress on the blue phase of cholesteric liquid crystals. We find that the elastic and bulk cholesteric energies are separately minimized by different forms of the order parameter, and view the problem of minimizing the full free energy as a competition between these incompatible forms. Near the transition to the high temperature disordered phase, the bulk and elastic contributions may be comparable, and complicated, non-uniform structures can appear as compromise solutions to the minimization problem. We interpret these structures as the blue phases. The two predominant models for the blue phase, which use either a director or second rank tensor order parameter, are seen in this picture as forms applicable in two limits, when the transition from the disordered phase is strongly and weakly first order, respectively. Both methods of constructing the order parameters, and the relation between them, is presented. In addition, we consider the application of geometric theories of frustration to cholesterics, and construct the exact solution to the minimization problem on a sphere in four dimensions. Implications for the flat space system are discussed. Part II of this thesis concerns the problem of describing long wavelength distortions in cholesterics. We consider a proposed analogy between cholesterics and superfluid ('3)He-A, with the uniform twist in cholesterics corresponding to stable superflow in helium. The difficulty with such a description lies in attempting to define variables which are well-defined in the distorted system and which vary slowly enough to justify a gradient expansion of the distortion energy. We show that by requiring the variables in such a theory to be slowly varying, as in the superfluid case, we can only describe director configurations which are not stationary points of the free energy. We propose a solution which includes rapid variations of the director and corrects this description in such a way to satisfy the stationary conditions.