Bifurcation Analysis in a Frustrated Nematic Cell

Abstract

Using Landau-de Gennes theory to describe nematic order, we study a frustrated cell consisting of nematic liquid crystal confined between two parallel plates. We prove the uniqueness of equilibrium states for a small cell width. Letting the cell width grow, we study the behaviour of this unique solution. Restricting ourselves to a certain interval of temperature, we prove that this solution becomes unstable at a critical value of the cell width. Moreover, we show that this loss of stability comes with the appearance of two new solutions: there is a symmetric pitchfork bifurcation. This picture agrees with numerical simulations performed by P. Palffy-Muhorray, E.C. Gartland and J.R. Kelly. Some of the methods that we use in the present paper apply to other situations, and we present the proofs in a general setting. More precisely, the paper contains the proof of a general uniqueness result for a class of perturbed quasilinear elliptic systems, and general considerations about symmetric solutions and their stability, in the spirit of Palais' Principle of Symmetric Criticality.

Minimizers of the Landau–de Gennes Energy Around a Spherical Colloid Particle

“…The proof is similar to [22,Lemma 8.3], with slight modifications to deal with the unbounded domain.…”

Liquid Crystals and Their Defects

“…The director n is subjected to antagonistic boundary conditions n(x 1 , x 2 , 0) = ±e 1 , n(x 1 , x 2 , δ) = ±e 3 on the plates, and periodic boundary conditions n(0, x 2 , x 3 ) = n(l 1 , x 2 , x 3 ), n(x 1 , 0, x 3 ) = n(x 1 , l 2 , x 3 ) on the other faces. Similar problems have been considered by many authors using a variety of models (see, for example, [2,13,14,23,30,61,76]). In [6] it is explained how using a Landau -de Gennes model, or molecular dynamics simulations [77], leads for sufficiently small plate separation δ to a jump in the director (defined as in Section 3.2 as the eigenvector of Q corresponding to it largest eigenvalue).…”

“…One indication as to why the one-constant approximation is easier than the general case is that in general the Lagrange multiplier λ(x) corresponding to the pointwise constraint |n(x)| = 1 in the Euler-Lagrange equation (61) for I OF in general depends on second derivatives of n, as can be seen by taking the inner product of (61) with n. However the identity ∆n · n = −|∇n| 2 for |n| = 1 shows that in the one-constant case λ = K|∇n| 2 is an explicit function of ∇n.…”

The Well Order Reconstruction Solution for three-dimensional wells, in the Landau–de Gennes theory

“…We first recall a result from [3] that ensures that the WORS is globally stable with natural boundary conditions on Γ, for arbitrary well heights or all values of . This result follows from a general uniqueness criterion for critical points of functionals of the form (2.5); see, e.g., [14,Lemma 8.2], [3, Lemma 3.2]. The WORS exists for all λ and A < 0 and an immediate consequence is that the WORS is the unique LdG energy minimizer for sufficiently small λ.…”