Disclinations in Limiting Landau–de Gennes Theory

Abstract

In this article we study the low-temperature limit of a Landau-de Gennes theory. Within all S 2 -valued R-axially symmetric maps (see Definition 1.1), the limiting energy functional has at least two distinct energy minimizers. One minimizer has biaxial torus structure, while another minimizer has split-core segment structure on the z-axis.(1). u is isotropic at x if x is a point singularity of u (see Definition 1.2);(2). u is uniaxial at x if two of the three eigenvalues determined by u are identical and different from zero at x;(3). u is biaxial at x if all the three eigenvalues determined by u are different at x.Moreover a vector field is called director field determined by u if its values are normalized eigenvectors corresponding to the largest eigenvalue in λ 1 , λ 2 , λ 3 .With the above definitions, we can define the so-called biaxial torus and split-core line segment.Definition 1.4. Suppose that T is an open torus in B 1 and u is a given 3-vector field on T . T is called biaxial torus with uniaxial core determined by u if u is uniaxial on a closed simple loop in T . Meanwhile u is biaxial in T except the points on the closed simple loop. In this case the torus T is also called a biaxial torus of the augmented map L rus. Given a segment, we have a straight line, denoted by l, passing across the segment. The segment is called split-core segment associated with a 3-vector field u if there is a neighborhood, denoted by O, containing the segment so that u is uniaxial on O X l except the two end-points of the segment. Meanwhile u is isotropic at the two end-points of the segment and biaxial on O " l. Here and in what follows, we use " to denote the set minus. In this case L rus has split-core segment structure on l.

Torus-like Solutions for the Landau-de Gennes Model. Part I: The Lyuksyutov Regime

“…The coexistence property stated in previous result is similar and somewhat related to the one established in the recent interesting paper [35], although the methods employed are completely different. As already commented in more details in [12, Section 7], the analysis in [35] is restricted to the case when the domain is the unit ball Ω = B 1 , the boundary condition is the constant norm hedgehog H given by (1.10) and minimization is restricted to the strictly smaller class of O(2) × Z 2 -equivariant configurations (the extra Z 2 -symmetry corresponding to the reflection along the horizontal plane).…”

“…The coexistence property stated in previous result is similar and somewhat related to the one established in the recent interesting paper [35], although the methods employed are completely different. As already commented in more details in [12, Section 7], the analysis in [35] is restricted to the case when the domain is the unit ball Ω = B 1 , the boundary condition is the constant norm hedgehog H given by (1.10) and minimization is restricted to the strictly smaller class of O(2) × Z 2 -equivariant configurations (the extra Z 2 -symmetry corresponding to the reflection along the horizontal plane). In this restricted class the author performs a clever parametrized constrained minimization which yields in the limit coexistence of "torus" and "split" minimizers, although in a sense weaker than those in [12], of the unconstrained minimization problem in the O(2) × Z 2equivariant class having the same energy.…”

“…Symmetry ansätze have been considered in several other recent papers, e.g. [19,20,1,18,35,2,33,4], in two and three dimensional Landau-de Gennes models. As reviewed in Section 2 , we identify the group S 1 with the subgroup of SO(3) of rotations around the vertical axis of R 3 and we consider the induced action on S 0 given by S 0 ∋ A → RAR t ∈ S 0 .…”

Modelling and computation of liquid crystals

“…The disclination ring solution is a symmetry-breaking configuration. The split core solution contains a +1 disclination line in the centre with two isotropic points at both ends, the existence of which is proved in Yu (2020) under the rotational symmetric assumption. The authors also consider the relaxed radial anchoring condition, which allows s = s(x) to be a free scalar function on ∂Ω.…”

Pattern Formation in Landau-de Gennes Theory

“…To our surprise, there are few theoretical studies on the core structures of the biaxial-ring disclination and the split-coresegment disclination. A first attempt was made in [33]. It shows that there are two families of solutions to (1.2)-(1.3) which can be suitably rescaled so that in the low-temperature limit (a ∞), one family of the rescaled solutions converges to a limiting state with biaxial-ring disclination, while another family of the rescaled solutions converges to a limiting state with split-core-segment disclination.…”