Existence of Surface Smectic States of Liquid Crystals

TL;DR

This paper proves the existence of a surface smectic state in liquid crystals by analyzing the Landau-de Gennes model, showing wave function localization near boundaries and deriving boundary energy asymptotics.

Contribution

It constructs critical points with boundary-localized wave functions and determines boundary energy asymptotics, revealing a surface smectic state in the model.

Findings

Wave functions localize near the boundary in the asymptotic limit.

Boundary energy asymptotics are derived for critical points.

Existence of surface smectic states is established.

Abstract

The Landau-de Gennes model of liquid crystals is a functional acting on wave functions (order parameters) and vector fields (director fields). In a specific asymptotic limit of the physical parameters, we construct critical points such that the wave function (order parameter) is localized near the boundary of the domain, and we determine a sharp localization of the boundary region where the wave function concentrates. Furthermore, we compute the asymptotics of the energy of such critical points along with a boundary energy that may serve in localizing the director field. In physical terms, our results prove the existence of a surface smectic state.