Energy Considerations for Parabolic Cyclides in SmA Liquid Crystals

Abstract

ABSTRACT For Dupin cyclides in the SmA liquid crystal phase, Kleman and Lavrentovich [1] found that increasing the saddle-splay constant K¯ caused the minimum of the total elastic energy to occur at a decreased value of the eccentricity e, close to unity. We extend this work to the parabolic cyclides and present an analytical expression for the total elastic energy that is finite over a suitable range of values for the latus rectum, conventionally denoted in this context by −4ℓ (>0). The length of the latus rectum characterises the parabolic focal conic structure analogously to the eccentricity of the Dupin cyclide focal conics. We demonstrate that the total energy is minimised at a particular value of ℓ. It is further observed that the usual saddle-splay elastic term acts independently of ℓ and that varying the value of K¯ does not affect the actual value of ℓ at which the minimum of the elastic energy occurs.