Dimension reduction for the Landau-de Gennes model in planar nematic thin films

TL;DR

This paper employs $3$-convergence to analyze how the Landau-de Gennes model simplifies for thin nematic liquid crystal films, emphasizing the role of surface energy and boundary conditions in the limit of vanishing thickness.

Contribution

It introduces a rigorous dimension reduction framework for the Landau-de Gennes model in thin films, accounting for complex boundary conditions and surface effects.

Findings

Surface energy significantly influences minimizer structure in thin films.

Convergence results depend on specific parameter regimes.

Surface anchoring conditions enforce eigenvector orientations of the nematic tensor.

Abstract

We use the method of $\Gamma$-convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film in the limit of vanishing thickness. In this asymptotic regime, surface energy plays a greater role and we take particular care in understanding its influence on the structure of the minimizers of the derived two-dimensional energy. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and the strong Dirichlet boundary conditions on the lateral boundary of the film. The constants in the weak anchoring conditions are chosen so as to enforce that a surface-energy-minimizing nematic $Q$-tensor has the normal to the film as one of its eigenvectors. We establish a general convergence result and then discuss the limiting problem in several parameter regimes.