Discontinuous Order Parameters in Liquid Crystal Theories

Abstract

The paper is concerned with various issues surrounding the mathematical description of defects in models of liquid crystals, drawing on experience from solid mechanics. The roles played by a suitable choice of function space and by the growth properties of the free-energy density are highlighted. Models in which the director can jump across surfaces are formulated, and their relevance for nematic elastomers, order reconstruction and smectic A thin films discussed.

Liquid Crystals and Their Defects

“…As described, for example, in [4,6,18], it is not sufficient to specify the energy functional, as part of the model is also the function space in which minimizers (and critical points etc) are to be sought. The larger this function space, the wilder potential singularities of minimizers and critical points may be.…”

“…Then W α (n, ∇n) → W (n, ∇n) as α → 0. As shown in [6], and assuming the Ericksen inequalities, W α satisfies the growth conditions…”

“…A necessary and sufficient condition that f satisfies (70), (71) is that (see [5,6] and for a related result [86]) In the following subsections we consider various situations in which planar discontinuities of n and/or models such as (69) are potentially of interest, referring the reader to [6] for more details.…”

“…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). Also in [6] it is shown that for a special choice of W and f in (69) the minimum of I is attained in SBV (Ω δ , S…”

“…Following [5,6], and motivated by similar models from fracture mechanics (see [43,25]), let us explore whether it might be reasonable to consider…”