The Dynamical Behavior of Liquid Crystals: A Continuum Description through Generalized Brackets

Abstract

Abstract In this paper, we systematically develop a hierarchy of constitutive equations at various levels of abstraction for describing the dynamics of incompressible liquid-crystalline systems. We also introduce a corresponding hierarchy of generalized bracket formulations. The simplest brackets, in terms of the direction vector, are capable of generating the Leslie-Ericksen (LE) theory in both the standard form of the equations (inertial) as well as the form in which the inertia associated with the rotation of the director is neglected. In parallel, a second, more-complex class of brackets is introduced, in terms of a structural tensor, which generates a theory that is more general than the LE theory, in that it can describe more-diverse phenomena, such as biaxiality and phase transitions. As particular cases, the generalized equations reduce to the LE equations or the more recent theory of Ericksen under the uniaxial approximation. Thus the collection of the governing equations of the generalized theory represents an inherently consistent set, which covers all the range from the director (LE-type) theories and the order-parameter (Doi-type) theories.