# Thin film liquid crystals with oblique anchoring and boojums

### Stan Alama

Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada### Lia Bronsard

Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada### Dmitry Golovaty

Department of Mathematics, University of Akron, Akron, OH 44325, United States of America

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## 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 angle $\alpha \in (0,\frac{\pi }{2})$, they serve to reduce the winding of the phase by steps of 2*α* and $(2\pi −2\alpha )$ 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.

## Cite this article

Stan Alama, Lia Bronsard, Dmitry Golovaty, Thin film liquid crystals with oblique anchoring and boojums. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 4, pp. 817–853

DOI 10.1016/J.ANIHPC.2020.02.002