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arXiv · 1701.06761

Octupolar Tensors for Liquid Crystals

Yannan Chen·Liqun Qi·Epifanio G. Virga
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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.

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Yannan Chen, Liqun Qi, Epifanio G. Virga. 2017-04-02. Octupolar Tensors for Liquid Crystals. https://doi.org/10.1088/1751-8121%2Faa98a8

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