Elastic energy of liquid crystals in convex polyhedra

Abstract

We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants. For a right rectangular prism and a large class of topologies, we derive upper bounds by introducing test configurations constructed from local conformal solutions of the Euler-Lagrange equation. The ratio of the upper and lower bounds depends only on the aspect ratios of the prism. As the aspect ratio is varied, the minimum-energy conformal state undergoes a sharp transition from being smooth to having singularities on the edges.

Tangent unit-vector fields: Nonabelian homotopy invariants and the dirichlet energy

“…The derivation of the bounds (12) and (13) involve combinatorial-group-theoretic arguments and nontrivial explicit constructions. For conformal and anticonformal topologies, Corollary 1 coincides with results given in [11]. For nonconformal topologies, Corollary 1 is a sharp improvement of estimates obtained in [12], which are equivalent to σ |w σ |π ≤ E(H) ≤ 9 σ |w σ |π.…”

“…Theorems 1 and 2 follow from new methods compared to our previous work in [11,12]. The derivation of the bounds (12) and (13) involve combinatorial-group-theoretic arguments and nontrivial explicit constructions.…”

“…As stated in Section 1, homotopy classes in C T O, S 2 are classified as being either conformal, anticonformal or nonconformal. Conformal and anticonformal homotopy classes are studied in detail in [11,12]. For these homotopy classes, ∆(H) = 0 by definition (see (10)) and consequently, the infimum Dirichlet energy is bounded from below by E(H) ≥ σ |w σ | i.e.…”

Tangent unit-vector fields: Nonabelian homotopy invariants and the Dirichlet energy

“…Using arguments from [11], one can show that differentiable maps are dense in (22)) and d ν (s σ ) = −w σ (cf (23)), (46) follows immediately for H conformal or anticonformal (cf (8)). For H nonconformal, (46) is equivalent to (cf (10))…”

“…As stated in Section 1, homotopy classes in C T O, S 2 are classified as being either conformal, anticonformal or nonconformal. Conformal and anticonformal homotopy classes are studied in detail in [11,12]. For these homotopy classes, ∆(H) = 0 by definition (see (10)) and consequently, the infimum Dirichlet energy is bounded from below by E(H) ≥ σ |w σ | i.e.…”