Abstract

Historically, many liquid-crystal devices and experiments have involved low-molecular-weight nematic liquid crystals, in supra-micron-size confinements, with coupled electric fields. In such settings, equilibrium orientational properties can be modeled most effectively using the Oseen-Frank elastic theory coupled with the equations of electrostatics. In this (mostly) expository talk, we will discuss some of the issues that arise in the mathematical and numerical treatment of such classical models. These issues include the intrinsic minimax nature of such models, which arises from the negative-definite way in which the electrostatic potential enters the free energy functional and which can also arise when Lagrange multipliers are used to enforce the pointwise unit-vector constraints on the liquid-crystal director field, as well as the complications this indefiniteness adds to the assessment of local stability of equilibria. We will also discuss the anomalous behavior that can be exhibited at the thresholds of certain electric-field-induced instabilities because of the nature of the coupling between the director field and the electric field. In addition, we will contrast the macroscopic Oseen-Frank model with the mesoscopic Landau-de Gennes model in such contexts.