Liquid crystal defects in the Landau–de Gennes theory in two dimensions — Beyond the one-constant approximation

Abstract

We consider the two-dimensional Landau-de Gennes energy with several elastic constants, subject to general k-radially symmetric boundary conditions. We show that for generic elastic constants the critical points consistent with the symmetry of the boundary conditions exist only in the case k = 2. In this case we identify three types of radial profiles: with two, three of full five components and numerically investigate their minimality and stability depending on suitable parameters.We also numerically study the stability properties of the critical points of the Landau-de Gennes energy and capture the intricate dependence of various qualitative features of these solutions on the elastic constants and the physical regimes of the liquid crystal system.

Revisiting the Two-Dimensional Defect-Free Azimuthal Nematic Equilibrium on an Annulus

“…In particular, the stability/instability of these solutions as a function of their winding number or degree, has been analyzed carefully in [20,21]. Our results are not subsumed by the results in [16,[20][21][22], particularly since these papers focus on a disk and we have a geometrical parameter, ρ, which in turn implies that we lose the crucial monotonicity properties exploited in these previous works. Further, our asymptotic estimates remain of independent interest.…”

“…Of course, the relevance of our simplified 2D approach in a 3D framework is not immediately clear, although equivalent 2D LdG models have been used with some success in a batch of experimental and theoretical papers [14,24,28] to model severely confined approximately two-dimensional systems. In [16,22], the authors refer to the B = 0 case of the LdG energy in (1) as the very low-temperature limit, and the defect-free state defined in Section 3.1 is an exact solution of the LdG Euler-Lagrange equations with B = 0. We make the connection between the idealized 2D LdG approach and the full 3D LdG setting more precise in Section 4 by relating the defect-free state in (11)- (14) to a radially symmetric (u, v)-type solution introduced in [16] at a fixed temperature A = −B 2 /(3C).…”

“…In [20], the authors prove that such solutions are unstable on R 2 and in [22], the authors study the stability of these solutions in terms of an anisotropy parameter M , the disc radius R and specific values of β in (68). We do not make specific comparisons to the results in [20,22] because M = 0 in our case and our domain is a re-scaled annulus (not a disc) with a dimensionless geometrical parameter ρ.…”

“…In [20], the authors prove that such solutions are unstable on R 2 and in [22], the authors study the stability of these solutions in terms of an anisotropy parameter M , the disc radius R and specific values of β in (68). We do not make specific comparisons to the results in [20,22] because M = 0 in our case and our domain is a re-scaled annulus (not a disc) with a dimensionless geometrical parameter ρ. More generally, we note that solutions of the form Q = u(x, y)E 1 + w(x, y)E 2 + v(x, y)E 5 have been studied in [6] as minimizers of the Landau-de Gennes energy on 2D bounded simply-connected domains, Ω, for topologically non-trivial Dirichlet conditions in the limit of vanishing elastic constant.…”

“…Finally, we relate our results in the restricted 2D LdG framework to exact equilibria in the fully three-dimensional (3D) LdG framework. We do this in terms of the radially symmetric LdG solutions, that are exact solutions of the Euler-Lagrange equations associated with a one-constant version of the 3D LdG energy on coaxial concentric cylinders, introduced in [16] and analyzed in detail in [22]. In particular, the stability/instability of these solutions as a function of their winding number or degree, has been analyzed carefully in [20,21].…”