Symmetry of Uniaxial Global Landau--de Gennes Minimizers in the Theory of Nematic Liquid Crystals

Abstract

Let B (0, R 0 ) ⊂ R 3 denote a three-dimensional spherical droplet of radius R 0 > 0, centered at the origin. Let S 0 denote the set of symmetric, traceless 3 × 3 matrices i.e.where M 3×3 is the set of 3 × 3 matrices. The corresponding matrix norm is defined to be [12]and we will use the Einstein summation convention throughout the paper.We work with the Landau-de Gennes theory for nematic liquid crystals [7] whereby a liquid crystal configuration is described by a macroscopic order parameter, known as the Q-tensor order parameter. Mathematically, the Landau-de Gennes Q-tensor order parameter is a symmetric, traceless 3 × 3 matrix belonging to the space S 0 in (1). The liquid crystal energy is given by the Landau-de Gennes energy functional and the associated energy density is a nonlinear function of Q and its spatial derivatives [7,17]. We work with the simplest form of the Landau-de Gennes energy functional that allows for a first-order nematic-isotropic phase transition and spatial inhomogeneities as shown below [12] -Here, L > 0 is a small material-dependent elastic constant, |∇Q| 2 = Q ij,k Q ij,k ( note that Q ij,k = ∂Q ij ∂x k ) with i, j, k = 1 . . . 3 is an elastic energy density and f B : S 0 → R is the bulk energy density. For our purposes, we take f B to be a quartic polynomial in the Q-tensor invariants as shown below -

Uniaxial versus biaxial character of nematic equilibria in three dimensions

“…Our second theorem generalizes the results in [15] to arbitrary 3D geometries with topologically non-trivial Dirichlet conditions. There exists a global LdG energy minimizer in the restricted class of uniaxial Q-tensors, for all material constants and temperatures.…”

“…Appealing to topological arguments, we prove that any uniaxial critical point of the LdG energy, subject to the hypotheses of Proposition 2.1, has an isotropic point (with Q = 0) near each singular point of the limiting harmonic map, as t → ∞. We then proceed with a local version of the global analysis in [15], equipped with certain energy quantization results for harmonic maps [6] and blow-up techniques, to deduce the local radial-hedgehog (RH) profile near each isotropic point and the instability of the RH profile for large t suffices to conclude the argument. In [20], the author rules out uniaxial critical points in one and two dimensional domains but the RH solution is an example of a semi-explicit uniaxial critical point (excluding an isolated isotropic point) on a 3D droplet and hence, uniaxial critical points in 3D remain of interest.…”

“…In both cases, we consider classical solutions of (30) that satisfy the energy bound (22) (this follows from the fact that any minimizing limiting harmonic map Q 0 belongs toĀ, so it can be used as a trial function). As done in [10,15] …”

“…In recent years, mathematicians have turned to the analysis of the celebrated Landau-de Gennes (LdG) theory for nematic liquid crystals, particularly in various asymptotic limits: see for example [8,10,15,24] which is not an exhaustive list but are directly relevant to our paper. The LdG theory is a variational theory with an associated energy functional, defined in terms of a macroscopic order parameter, known as the Q-tensor order parameter [11,21,38].…”

“…[5,25,31]), whereas the LdG variable is a five-dimensional map, Q : R 3 → R 5 . A uniaxial Q-tensor has two degenerate non-zero eigenvalues and hence, three degrees of freedom and in this case, there is broader scope for transferable methodologies (see for example, [15]). A biaxial Q-tensor has three distinct eigenvalues with five degrees of freedom and maximal biaxiality corresponds to a vanishing eigenvalue.…”