Forces and variational compatibility for equilibrium liquid crystal director models with coupled electric fields
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O R I G I NA L A RT I C L E
Eugene C. Gartland Jr.
Forces and variational compatibility for equilibrium liquid crystal director models with coupled electric fields
Received: 10 October 2019 / Accepted: 20 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Expressions are obtained for force and couple densities and stress tensors in macroscopic models for nematic liquid crystals in electric fields. The coupling between the liquid crystal orientational properties and the electric field is taken into account via a free energy of Oseen–Frank type expressed as an integral functional of the director field and the electric potential field. The variational model here also allows for a gravitational field and a magnetic field, and the differences among these three common types of force fields (gravitational, magnetic, and electric) are discussed. Also included in the free energy is a surface anchoring potential, and its effect on boundary traction and couple stress is explored. The electric field is assumed to arise from electrodes held at constant potential. Flexoelectric effects are included, and as a consequence, the material is no longer a linear dielectric medium. It is shown that the equilibrium solutions of the Euler–Lagrange equations satisfy appropriate expressions of force balance and torque balance, obtained from a virtual work principle. The development here builds from theories of Ericksen related to the notion of “variational compatibility,” and this connection is made. Comparisons are also made to the extensive literature in physics and continuum mechanics on electromagnetic field/matter interaction. Keywords Liquid crystal · Oseen–Frank model · Surface anchoring energy · Electric field · Flexoelectricity · Stress tensor 1 Introduction Our interest is in macroscopic continuum models for the orientational properties of materials in a liquid crystal phase, a complex partially ordered fluid phase exhibited by certain materials in certain parameter ranges. Such models are used at the scales of typical devices and experiments involving these kinds of materials. Our primary objective is to obtain expressions for the various stress tensors, couple stress tensors, and boundary tractions in situations in which the material is subjected to an electric field. These tensors are required to describe conservation laws in the hydrodynamic theory, and they are sometimes needed to model experiments. Electric fields have been used for decades to control liquid crystals, and treating these fields as analogous to the better-understood magnetic fields often produces adequate approximations. There are, however, aspects of the coupling between electric fields and liquid crystal orientational properties that have lacked complete understanding, especially with respect to forces and stresses, and especially with respect to situations in which flexoelectric polarization needs to be taken into account. The history of the macroscopic continuum theory of liquid crystals has its origins in the works of O
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