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Nematic Liquid Crystals with Tensorial Order Parameters

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

Abstract

This chapter goes beyond the ELP theory of LCs by modeling the microstructure of the liquid by a number of rank-i tensors \((i=1, \ldots ,n)\) (generally just one) with vanishing trace. These tensors are called alignment tensors or order parameters. When formed as exterior products of the director vector and weighted with a scalar and restricted to just one rank-2 tensor, the resulting mathematical model describes uniaxial LCs. The simplest extensions of the ELP model are theories, for which the number of independent constitutive variables are complemented by a constant or variable order parameter S and its gradient \(\mathrm {grad}\,S\) paired with an evolution equation for it. We provide a review of the recent literature. Two different approaches to deduce LC models exist; they may be coined the balance equations models, outlined already in Chap.  25 for the ELP model, and the variational Lagrange potential models, which, following an idea by Lord Rayleigh (Strutt, Proc Lond Math Soc 4:357–368, 1873, [50]), are extended by a dissipation potential. The two different approaches may lead to distinct anisotropic fluid descriptions. Moreover, it is not automatically guaranteed in either description that the balance law of angular momentum is identically satisfied. The answers to these questions cover an important part of the mathematical efforts in both model classes. Significant conceptual difficulties in the two distinct theoretical concepts are the postulations of explicit forms of the elastic energy W and dissipation function R. Depending upon, how W and R are parametrized, different particular models emerge. Conditions are formulated especially for uniaxial models, which guarantee that the two model classes reduce to exactly corresponding mathematical models.

Keywords

Liquid crystals of tensorial microstructure Balance law approach Variational formulation Alignment tensor model Uniaxial LC theory 

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Copyright information

© 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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