Theoretical studies of smectic C liquid crystals confined in a wedge. Stability considerations and Frederiks transitions

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Abstract
A general deformation of a smectic C liquid crystal is composed of five different distortions, each of which can be made independently. Here we show that to each of these distortions we can assign a simple vector operator. Use of these five basis operators enables us to write down the elastic free energy density as a quadratic form consisting of nine terms. We also discuss how the nine elastic constants defined by the elastic energy expansion must fulfil certain restrictions in terms of inequalities and a specific tilt angle dependence. Assuming the smectic layers to be incompressible, we examine how certain arrangements of the smectic layers can be stable due to an interplay between the incompressibility condition and the boundary conditions which we impose on the director. One such stable configuration is the wedge, where the smectic layers form parts of concentric cylinders with the common axis coinciding with the centre of the wedge. For such a system we discuss the different director configurations which can be achieved and their stability. We also discuss the possibility of inducing Frederiks transitions for some of these configurations and calculate the corresponding thresholds, thereby demonstrating the design of an experiment which would make it possible to measure those elastic constants which are related to the deformations of the smectic layers, constants which are normally difficult to determine experimentally.