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Elastic energy of liquid crystals in convex polyhedra

A Majumdar·JM Robbins·M Zyskin
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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.

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A Majumdar, JM Robbins, M Zyskin. 2004-10-01. Elastic energy of liquid crystals in convex polyhedra. https://doi.org/10.1088/0305-4470/38/34/c01

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