Sawtooth profile in smectic A liquid crystals

TL;DR

This paper analyzes the formation of chevron patterns in smectic A liquid crystals under magnetic fields using energy minimization and $ ext{Gamma}$-convergence, supported by numerical simulations.

Contribution

It provides a rigorous mathematical analysis of chevron pattern formation in smectic A liquid crystals, including both de Gennes and Chen-Lubensky free energies, with numerical illustrations.

Findings

Chevron structures connect two sphere minima in energy minimizers.

Both free energy models predict similar chevron behavior.

Numerical simulations confirm theoretical predictions.

Abstract

We study the de Gennes free energy for smectic A liquid crystals over $\mathbb{S}^2$-valued vector fields to understand the chevron (zigzag) pattern formed in the presence of an applied magnetic field. We identify a small dimensionless parameter $\epsilon$, and investigate the behaviors of the minimizers when the field strength is of order $\mathcal{O}(\epsilon^{-1})$. In this regime, we show via $\Gamma$-convergence that a chevron structure where the director connects two minimum states of the sphere is favored. We also analyze the Chen-Lubensky free energy, which includes the second order gradient of the smectic order parameter, and obtain the same general behavior as for the de Gennes case. Numerical simulations illustrating the chevron structures for both energies are included.