Quasi-mesoscopic model for ferroelectric switching in the chevron geometry

Abstract

We present a theory of ferroelectric liquid crystal switching which combines elements of standard macroscopic continuum theories with mesoscopic Landau–de Gennes chevron theories. The macroscopic elements of the theory apply in the chevron arms, and are subject to a boundary condition at the chevron interface. This boundary condition can be derived from an anchoring energy associated with the director discontinuity at the chevron tip. The anchoring energy, which corresponds to the degree to which the cone mismatch condition is not satisfied, is calculated using the mesoscopic Landau–de Gennes theory. In the combined theory the frequently used cone-matching condition emerges as a thick cell limit. We are able to calculate a free energy associated with the imposition of a field on particular configurations. There follows a switching phase diagram determining the conditions for thresholdless and bistable switching. We further show that the time dependence of the switching process is determined by the slower bulk relaxation dynamics rather than by the fast chevron surface dynamics.

Notes

^†This term is different from the original Vaupotič et al. [14] (D|(n×∇)² Φ|²) formulation and has been proposed by Chen and Lubensky, where. In the level of approximation to be used here it would have made little difference to the results using either formulation, and would have only served to complicate the calculations.

^†Typical ferroelectric liquid crystal parameters are P=300×10⁶ C m⁻², B=10⁶ J m⁻³, K=10⁻¹¹ J m⁻¹, δ=0.2 rad, θ=0.25 rad, B_i =10⁻¹² N, L=3×10⁻⁶ m, E=10⁷ V m⁻¹, η _B=0.01 kg m⁻¹ s⁻¹.

^‡Using a trial solution for ϕ(ρ) it is possible to obtain the free energy in terms of the bulk and surface contributions, i.e. F[ϕ _s]=F _B[ϕ _s]+F _s[ϕ _s]. By relating this free energy through a dissipation function we may write. The limiting time scale for the relaxation is proportional to η _B, as η _B≫η _s.