Communications in Mathematical Sciences

Volume 12 (2014)

Number 6

Analysis and simulations of the Chen-Lubensky energy for smectic liquid crystals: onset of undulations

Pages: 1155 – 1183

DOI: https://dx.doi.org/10.4310/CMS.2014.v12.n6.a7

Authors

Carlos J. García-Cervera (Department of Mathematics, University of California at Santa Barbara)

Sookyung Joo (Department of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia, U.S.A.)

Abstract

We study the Chen-Lubensky energy to investigate layer undulations in smectic liquid crystals in response to an applied magnetic field. In earlier work [C.J. García-Cervera and S. Joo, Arch. Rat. Mech. Anal., 203(1), 1–43, 2012], the authors obtained an asymptotic expression of the unstable modes and a sharp estimate of the critical field using the Landau-de Gennes model for smectic A liquid crystals. In this paper, we extend our theory to the Chen-Lubensky energy, which includes a second order smectic order parameter gradient. Analysis based on $\Gamma$-convergence theory and bifurcation theory provide the estimate of the critical field and frequency of the undulations. Furthermore, we present a new numerical formulation of fourth order partial differential equations. With this formulation, the fourth order system reduces to a second order equation with a constraint, which resembles the incompressible Navier-Stokes equations from fluid dynamics. We use this method to illustrate the presence of layer undulations near the critical field and to confirm that the results from our analysis agree with these numerical simulations. We also use asymptotic analysis to determine the structure of the domain wall at high fields under the assumption that the layer density is constant.

Keywords

Chen-Lubensky, smectic liquid crystals, undulations

2010 Mathematics Subject Classification

34E10, 35B30, 35B35, 65M06

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Published 20 March 2014