Biaxial escape in nematics at low temperature

Abstract

In the present work, we study minimizers of the Landau-de Gennes free energy in a bounded domain Ω ⊂ R 3 . We prove that at low temperature minimizers do not vanish, even for topologically non-trivial boundary conditions. This is in contrast with a simplified Ginzburg-Landau model for superconductivity studied by Bethuel, Brezis and Hélein. Merging this with an observation of Canevari we obtain, as a corollary, the occurence of biaxial escape: the tensorial order parameter must become strongly biaxial at some point in Ω. In particular, while it is known that minimizers cannot be purely uniaxial, we prove the much stronger and physically relevant fact that they lie in a different homotopy class.

Uniaxial versus biaxial character of nematic equilibria in three dimensions

“…An immediate consequence is that global energy minimizers cannot be purely uniaxial, as also stated in [10] where the authors prove that global LdG minimizers must have at least a point of maximal biaxiality.…”

“…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].…”

“…The uniform convergence gives a fairly good picture away from the singular set of the limiting harmonic map. The next step is to adapt arguments from [10] to prove that the norm of a global LdG minimizer converges uniformly to a constant value, everywhere on (not merely away from singularities). In Theorem 1 (iii), we provide a rigorous proof of the norm convergence in the t → ∞ limit, yielding a rigorous justification of the common Lyuksyutov constraint in the literature.…”

“…In this spirit, one may refer to our limit as the Lyuksyutov limit. In [10], the authors prove the global LdG minimizers have at least a point of maximal biaxiality, in the t → ∞ limit, but do not comment on the number or location of such points. We appeal to a topological result in [8] to deduce the existence of at least a point of maximal biaxiality and a point of uniaxiality near each singular point of the limiting map, in the t → ∞ limit; maximal biaxiality occurs when Q has a vanishing eigenvalue (see [26,38]).…”