Algebraic theory of linear viscoelastic nematodynamics Part 1: Algebra of nematic operators

TL;DR

This paper develops an algebraic framework for nematic operators in linear viscoelastic nematodynamics, revealing a noncommutative group structure and subgroups relevant to thermodynamic stability and soft deformation modes.

Contribution

It introduces a comprehensive algebraic theory of nematic operators, identifying their group structure and subgroups, which advances understanding of their mathematical properties in nematic liquid crystals.

Findings

Existence of a noncommutative group of nematic operators

Identification of a subgroup of thermodynamically stable operators

Algebraic structure of nematic soft deformation modes

Abstract

This first part of the paper develops algebraic theory of linear anisotropic, six-parametric nematic "N-operators" build up on the additive group of traceless second rank 3D tensors. These operators have been implicitly used in continual theories of nematic liquid crystals and weakly elastic nematic elastomers. It is shown that there exists a noncommutative, multiplicative group N6 of N-operators build up on a manifold in 6D space of parameters. Positive N-operators, which in physical applications holds thermodynamic stability constraints, form a subgroup of group N6 on a more complicated manifold in parametric space. A three-parametric, commutative transversal-isotropic subgroup S3 < N6 of positive symmetric nematic operators is also briefly discussed. The special case of singular, non-negative symmetric N-operators reveals the algebraic structure of nematic soft deformation modes.