Critical points and symmetries of a free energy function for biaxial nematic liquid crystals
Critical points and symmetries of a free energy function for biaxial nematic liquid crystals
We describe a general mean field model for the free energy function for a homogeneous medium of mutually interacting molecules, based on the formalism for a biaxial nematic liquid crystal set out by Katriel et al (1986) in an influential paper in Liquid Crystals 1 and subsequently called the KKLS formalism. The free energy is expressed as the sum of an entropy term and an interaction (Hamiltonian) term. Using the language of group representation theory we identify the order parameters as averaged components of a linear transformation, and characterize the full symmetry group of the entropy term in the liquid crystal context as a wreath product SO(3) wr Z2. The symmetry-breaking role of the Hamiltonian, pointed out by Katriel et al, is here made explicit in terms of centre manifold reduction at bifurcation from isotropy. We use tools and methods of equivariant singularity theory to reduce the bifurcation study to that of a D3-invariant function on R2, ubiquitous in liquid crystal theory, and to describe the 'universal' bifurcation geometry in terms of the superposition of a familiar swallowtail surface controlling uniaxial equilibria and another less familiar surface controlling biaxial equilibria. In principle this provides a template for all nematic liquid crystal phase transitions close to isotropy, although further work is needed to identify the absolute minima that are the critical points representing stable phases.
1483-1537
Chillingworth, David
39d011b7-db33-4d7d-8dc7-c5a4e0a61231
May 2015
Chillingworth, David
39d011b7-db33-4d7d-8dc7-c5a4e0a61231
Chillingworth, David
(2015)
Critical points and symmetries of a free energy function for biaxial nematic liquid crystals.
Nonlinearity, 28 (5), .
(doi:10.1088/0951-7715/28/5/1483).
Abstract
We describe a general mean field model for the free energy function for a homogeneous medium of mutually interacting molecules, based on the formalism for a biaxial nematic liquid crystal set out by Katriel et al (1986) in an influential paper in Liquid Crystals 1 and subsequently called the KKLS formalism. The free energy is expressed as the sum of an entropy term and an interaction (Hamiltonian) term. Using the language of group representation theory we identify the order parameters as averaged components of a linear transformation, and characterize the full symmetry group of the entropy term in the liquid crystal context as a wreath product SO(3) wr Z2. The symmetry-breaking role of the Hamiltonian, pointed out by Katriel et al, is here made explicit in terms of centre manifold reduction at bifurcation from isotropy. We use tools and methods of equivariant singularity theory to reduce the bifurcation study to that of a D3-invariant function on R2, ubiquitous in liquid crystal theory, and to describe the 'universal' bifurcation geometry in terms of the superposition of a familiar swallowtail surface controlling uniaxial equilibria and another less familiar surface controlling biaxial equilibria. In principle this provides a template for all nematic liquid crystal phase transitions close to isotropy, although further work is needed to identify the absolute minima that are the critical points representing stable phases.
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Published date: May 2015
Organisations:
Pure Mathematics
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Local EPrints ID: 361542
URI: http://eprints.soton.ac.uk/id/eprint/361542
ISSN: 0951-7715
PURE UUID: 3941f72e-0c8b-4798-83e8-ad9733f5587c
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Date deposited: 27 Jan 2014 14:26
Last modified: 14 Mar 2024 15:52
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