Flexoelectric Induced Vanishing of the Cholesteric Helix

Abstract

In the cholesteric liquid crystal helix structure, the director, n, rotates in a plane about an axis, t 0 ⊥ n, with a constant twist, n curln = -q. The inverse helix pitch is defined by q = 2π/pitch. Here we show that in the limit of a small electric field, E, applied perpendicular to t 0, a solution to the minimizer of the elastic free energy, including a linear coupling between E and splay/bend deformations of n (the flexoelectric term), is one where the director develops a small periodic component parallel to t 0. As the wave number of this distortion is also q, the net effect is a rotation of the optic axis by a small angle relative to t 0. There is no threshold for this effect when the dielectric anisotropy ϵ a is greater than ϵ a > - 8πe2/K. e is the flexoelectric coefficient and K is an elastic constant. When E || t 0 and ϵ a > 0, it is well-known that this director configuration can be created by boundary conditions. In which case, above a critical field, E c, the cholesteric helix transforms to a uniform director field with n || t 0, without q→0 continuously and without introducing defects. As this is similar to solutions presented here when E ⊥ n but ϵa ≤ 0, the suggestion is that flexoelectricity could mediate a similar commensurate defect free vanishing of the cholesteric helix in this case. When ϵ a ≥ 0, the conclusion is that a defect free transformation of the cholesteric helix to a uniform director field with n || E requires the assistance of induced flows.