The Role of Inertia and Dissipation in the Dynamics of the Director for a Nematic Liquid Crystal Coupled with an Electric Field

Abstract

We consider the dynamics of the director in a nematic liquid crystal when under the influence of an applied electric field. Using an energy variational approach we derive a dynamic model for the director including both dissipative and inertial forces.A numerical scheme for the model is proposed by extending a scheme for a related variational wave equation. Numerical experiments are performed studying the realignment of the director field when applying a voltage difference over the liquid crystal cell. In particular, we study how the relative strength of dissipative versus inertial forces influence the time scales of the transition between the initial configuration and the electrostatic equilibrium state.

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

“…In the previous experiment, the director field only moved in the x − y-plane, and was so constrained into one direction that is perpendicular to the electric field. However, in this experiment, the director field moves in the whole S 2 and can align perpendicular to the electric field in a 2-dimensional subspace of R 3 which is what appears to happen and also leads to more dynamic behavior before the director field relaxes to an equilibrium state, see Figure 7 for a plot of the evolution of the quantities (5.1) and (5.2) for α = 0.5 and α = 3. Again, the electric field does not appear to be perturbed by the director field very much, which could however also be due to our choice of parameters ǫ 1 and ǫ 2 .…”

“…As we will see, for system (1.9), this is not necessary to achieve stability or convergence of the numerical scheme. Besides that, works in which the unit length constraint on the director field is preserved, concern simpler systems, such as the wave or heat map flow or the Landau-Lifshitz-Gilbert equation [7, 1, 5, 8, 6, 10, 9, 23], or 1D settings [2,3]. The numerical methods of [2,3] have not been shown to be convergent, to the best of our knowledge.…”

“…Besides that, works in which the unit length constraint on the director field is preserved, concern simpler systems, such as the wave or heat map flow or the Landau-Lifshitz-Gilbert equation [7, 1, 5, 8, 6, 10, 9, 23], or 1D settings [2,3]. The numerical methods of [2,3] have not been shown to be convergent, to the best of our knowledge. Therefore, our contribution is the construction of a constraint-preserving and convergent discretization of (1.9).…”

A convergent finite difference scheme for the variational heat equation

“…For the last term in (20), let m, n be such that t + τ ∈ [t n , t n+1 ) and t ∈ [t m , t m+1 ). Using the BV bound on z from Lemma 3.2, we get For z ∆t , we will use a version of Kružkov's interpolation lemma [12, p. 208, Lemma 4.11], which gives continuity in time if for all t 1 , t 2…”

“…We are therefore interested in the degenerate case of (1) where c is allowed to vanish at some points, i.e., if c is given by (2), in the case that k 1 = 0 or k 2 = 0. Solutions of degenerate parabolic equations are not necessarily smooth or unique, therefore new concepts of solutions, e.g., weak solutions, entropy solutions, or viscosity solutions are required.…”