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Freedericksz transitions in circular toroidal layers of smectic C liquid crystals

Kidd, J.E. and Constanda, Christian and Stewart, Iain W. (2001) Freedericksz transitions in circular toroidal layers of smectic C liquid crystals. IMA Journal of Applied Mathematics, 66 (4). pp. 387-409. ISSN 0272-4960

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The aim of this paper is to consider theoretically a Freedericksz transition for concentric toroidal layers of smectic C liquid crystal arising from a simple geometric setup, thereby extending the results of Atkin and Stewart [Q. Jl Mech. Appl. Math., 47, 1994] who considered spherical layers of smectic C in the usual cone and plate geometry. Application of smectic continuum theory leads, after suitable approximations are made, to a linear governing equilibrium equation which is satisfied by both the trivial solution and a variable solution involving Bessel functions. We are able to determine the critical magnitude cH of the magnetic field H at which this variable solution exists, and a standard energy comparison reveals that the variable solution is expected to be more energetically favourable than the zero solution provided H > 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 indication of possible future work.