Reorientation of smectic a liquid crystals by magnetic fields

Abstract

We consider the de Gennes' smectic A free energy with a complex order parameter in order to study the influence of magnetic fields on the smectic layers in the strong field limit as well as near the critical field. In previous work by the authors [6], the critical field and a description of the layer undulations at the instability were obtained using $\Gamma$-convergence and bifurcation theory. It was proved that the critical field is lowered by a factor of $\sqrt{\pi}$ compared to the classical Helfrich Hurault theory by using natural boundary conditions for the complex order parameter, but still with strong anchoring condition for the director. In this paper, we present numerical simulations for undulations at the critical field as well as the layer and director configurations well above the critical field. We show that the estimate of the critical field and layer configuration at the critical field agree with the analysis in [6]. Furthermore, the changes in smectic order density as well as layer and director will be illustrated numerically as the field increases well above the critical field. This provides the smectic layers' melting along the bounding plates where the layers are fixed. In the natural case, at a high field, we prove that the directors align with the applied field and the layers are homeotropically aligned in the domain, keeping the smectic order density at a constant in $L^2$.