Sawtooth Profile in Smectic A Liquid Crystals

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 $\varepsilon$, and investigate the behaviors of the minimizers when the field strength is of order $\mathcal{O}(\varepsilon^{-1})$. 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.