On the Dynamics of the Weak Fréedericksz Transition for Nematic Liquid Crystals

Abstract

We propose an implicit finite-difference method to study the time evolution of the director field of a nematic liquid crystal under the influence of an electric field with weak anchoring at the boundary. The scheme allows us to study the dynamics of transitions between different director equilibrium states under varying electric field and anchoring strength. In particular, we are able to simulate the transition to excited states of odd parity, which have previously been observed in experiments, but so far only analyzed in the static case.

Weak anchoring effects in smectic-A Fréedericksz transitions

“…Thus, the instability with the lowest critical field is the one corresponding to the even symmetry of the layers (28). By analogy with nematics, other symmetries may become locally stable for larger applied field [1,3].…”

A convergent finite difference scheme for the variational heat equation

“…If however one of the grid points coincides with a zero of c(u 0 (x)), we get another solution which corresponds to a classical solution of (7), "glued together" at the zeros of c(u 0 (x)) with Dirichlet boundary conditions. Interpreted as solutions of (1), this type of solutions shows clearly that the gradient is unbounded.…”

“…In this paper we investigate the Cauchy problem (1) u t = c(u)(c(u)u x ) x , x ∈ Ω, t > 0 u(x, 0) = u 0 (x),…”

“…The director field is a unit vector field that gives the average direction of the molecules at each point. To arrive at equation (1), we assume the simplified setting of a uniaxial nematic with no flow and a director field n that lies in the x − y plane and varies only in x-direction. Then the director can be described by an angle u as n = (cos(u), sin(u), 0).…”

A convergent numerical scheme for a model of liquid crystal dynamics subjected to an electric field

“…where ǫ 2 := ǫ a ǫ −1 ⊥ and f = −ρ free /(ǫ 0 ǫ ⊥ ), that can be augmented with Dirichlet boundary conditions (1.7) ϕ(t, x) = g(t, x), x ∈ ∂Ω, t ≥ 0, for the potential. To account for non-equilibrium situations such as the dynamic transition between two equilibrium states, e.g., the Freedericksz transition [19,20,37,2], one adds inertial and damping terms to the equations for the director field (which can be obtained as the Euler-Lagrange equations from a suitable action functional) [21,33], since the liquid crystal molecules move on a slower time scale compared to the electric field:…”