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A Continuum Approach to Liquid Crystals—The Ericksen–Leslie–Parody Formulation

  • Kolumban HutterEmail author
  • Yongqi Wang
Chapter
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

Abstract

Liquid crystals (LCs) are likely the most typical example of a polar medium of classical physics, in which the balance of angular momentum is a generic property, not simply expressed as a symmetry requirement of the Cauchy stress tensor. They were discovered in the second half of the nineteenth century. Liquid crystals are materials, which exhibit fluid properties, i.e., they possess high fluidity, but simultaneously exhibit crystalline anisotropy in various structural forms. We present an early phenomenological view of the behavior of these materials, which conquered a tremendous industrial significance in the second half of the twentieth century as liquid crystal devices (LCD) (Sect. 25.1). The theoretical foundation as a continuum of polar structure was laid in the late 1950s to 1990s by Ericksen, Leslie, Frank , and Parodi , primarily for nematic LCs by postulating their general physical conservation laws, hydrostatics, and hydrodynamics, thus, illustrating their connection with nontrivial balance laws of angular momentum (Sect. 25.2). This is all done by treating nematics as material continua equipped with continuous directors (long molecules), which by their orientation induce a natural anisotropy. The thermodynamic embedding (Sect. 25.3) is performed by employing an entropy balance law with nonclassical entropy flux and the requirement of Euclidian invariance of the constitutive quantities, which are assumed to be objective functions of the density, director, its gradient, and velocity, as well as stretching, vorticity, temperature, and temperature gradient. This is specialized for an incompressible LC with directors of constant length (Sect. 25.4). Constitutive parameterizations with an explicit proposal of the free energy as a quadratic polynomial of the director and its gradient (according to Frank ) are reduced to obey objectivity. Based on this, the objective form of the free energy is derived (Appendix 25.A), as are the linear dissipative Cauchy stress, director stress, and heat flux vector for the cases that the Onsager relations are fulfilled. The chapter ends with the presentation of shear flow solutions in a two-dimensional half-space and in a two-dimensional channel.

Keywords

Phenomenology Balance laws Hydrodynamics Thermodynamics of LCs Directors Nematics Free energy Parodi relation Onsagerism 

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.c/o Laboratory of Hydraulics, Hydrology, GlaciologyETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringDarmstadt University of TechnologyDarmstadtGermany

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