Freedericksz transitions in circular toroidal layers of smectic C liquid crystals

The aim of this paper is to consider theoretically a Freedericksztransition for concentric toroidal layers of smectic C liquidcrystal arising from a simple geometric setup, thereby extendingthe results of Atkin & Stewart [Q. Jl Mech. Appl. Math.,47, 1994] who considered spherical layers of smectic C in theusual cone and plate geometry. Application of smectic continuumtheory leads, after suitable approximations are made, to a lineargoverning equilibrium equation which is satisfied by both thetrivial solution and a variable solution involving Bessel functions.We are able to determine the critical magnitude _cH of the magneticfield H at which this variable solution exists, and a standardenergy comparison reveals that the variable solution is expectedto be more energetically favourable than the zero solution providedH > _cH. Numerical examples of critical thresholds are given,which are comparable to those in the literature for nematics.The paper ends with a discussion section and some indicationof possible future work.