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

Carlos J. García-Cervera, Sookyung Joo
2014 Communications in Mathematical Sciences  
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
more » ... smectic order parameter gradient. Analysis based on Γ-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.
doi:10.4310/cms.2014.v12.n6.a7 fatcat:ze7xldlci5cdjpwdb35z7gqowu