Octupolar order in three dimensions
Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, I-20133, Milano, Italy
2 Dipartimento di Matematica, Università di Pavia, via Ferrata 5, I-27100, Pavia, Italy
Accepted: 2 November 2016
Published online: 28 November 2016
Octupolar order in three space dimensions is described by a real-valued, fully symmetric and traceless, third-rank tensor A. The real generalized eigenvalues of A are also the critical values of a real-valued potential defined on the unit sphere by A. Generalized eigenvalues of A and critical points of are equivalent means to describe octupolar order in a molecular assembly according to Buckingham's formula for the probability density distribution. Intuition suggests that would generically have four maxima, corresponding to the most probable molecular orientations, so that a (possibly distorted) tetrahedron would effectively describe A. This paper shows that another generic octupolar state flanks the expected one, featuring three maxima of . The two generic states are divided by a separatrix manifold, which may physically represent an intra-octupolar transition.
Key words: Soft Matter: Liquid crystals
© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg, 2016