# Thin Film Liquid Crystals with Oblique Anchoring and Boojums

## TL;DR

This paper investigates the formation and distribution of boundary and interior defects, called boojums, in thin-film nematic liquid crystals with oblique anchoring, using variational analysis, numerical simulations, and energy considerations.

## Contribution

It introduces a variational framework for analyzing oblique anchoring in nematic liquid crystals and characterizes the occurrence and pairing of boojums as boundary defects.

## Key findings

- Strong anchoring leads to interior defects.
- Weak anchoring results in boundary boojums.
- Defect locations are determined by a Renormalized Energy.

## Abstract

We study a two-dimensional variational problem which arises as a thin-film limit of the Landau-de Gennes energy of nematic liquid crystals. We impose an oblique angle condition for the nematic director on the boundary, via boundary penalization (weak anchoring.) We show that for strong anchoring strength (relative to the usual Ginzburg-Landau length scale parameter,) defects will occur in the interior, as in the case of strong (Dirichlet) anchoring, but for weaker anchoring strength all defects will occur on the boundary. These defects will each carry a fractional winding number; such boundary defects are known as "boojums". The boojums will occur in ordered pairs along the boundary; for nematic director with angle oblique to the normal vector, they serve to reduce the winding of the phase in two steps, in order to avoid the formation of interior defects. We determine the number and location of the defects via a Renormalized Energy and numerical simulations.