Octupolar Tensors for Liquid Crystals

Abstract

A third-order three-dimensional symmetric traceless tensor, called the \emph{octupolar} tensor, has been introduced to study tetrahedratic nematic phases in liquid crystals. The octupolar \emph{potential}, a scalar-valued function generated on the unit sphere by that tensor, should ideally have four maxima capturing the most probable molecular orientations (on the vertices of a tetrahedron), but it was recently found to possess an equally generic variant with \emph{three} maxima instead of four. It was also shown that the irreducible admissible region for the octupolar tensor in a three-dimensional parameter space is bounded by a dome-shaped surface, beneath which is a \emph{separatrix} surface connecting the two generic octupolar states. The latter surface, which was obtained through numerical continuation, may be physically interpreted as marking a possible \emph{intra-octupolar} transition. In this paper, by using the resultant theory of algebraic geometry and the E-characteristic polynomial of spectral theory of tensors, we give a closed-form, algebraic expression for both the dome-shaped surface and the separatrix surface. This turns the envisaged intra-octupolar transition into a quantitative, possibly observable prediction. Some other properties of octupolar tensors are also studied.

References in corpus (3)

• Generalized liquid crystals: giant fluctuations and the vestigial chiral order of $I$, $O$ and $T$ matter
• Octupolar order in two dimensions
• Classification of point-group-symmetric orientational ordering tensors

Cited by in corpus (7)

• Liquid Crystal Distortions Revealed by an Octupolar Tensor
• Spectral Analysis of Piezoelectric Tensors
• Eigenvectors of tensors - A primer
• Transposes, L-Eigenvalues and Invariants of Third Order Tensors
• Lifting map for ordered surfaces
• Irreducible Function Bases of Isotropic Invariants of A Third Order Three-Dimensional Symmetric and Traceless Tensor
• The symmetries of octupolar tensors