Abstract.
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 \(\Phi\) defined on the unit sphere \(\mathbb{S}^{2}\) by A. Generalized eigenvalues of A and critical points of \(\Phi\) are equivalent means to describe octupolar order in a molecular assembly according to Buckingham's formula for the probability density distribution. Intuition suggests that \(\Phi\) 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 \(\Phi\). The two generic states are divided by a separatrix manifold, which may physically represent an intra-octupolar transition.
Graphical abstract
Similar content being viewed by others
References
L.G. Fel, Phys. Rev. E 52, 702 (1995)
L.G. Fel, Phys. Rev. E 52, 2692 (1995)
T.C. Lubensky, L. Radzihovsky, Phys. Rev. E 66, 031704 (2002)
H.R. Brand, H. Pleiner, P. Cladis, Eur. Phys. J. E 7, 163 (2002)
T. Niori, T. Sekine, J. Watanabe, T. Furukawa, H. Takezoe, J. Mater. Chem. 6, 1231 (1996)
D.R. Link, G. Natale, R. Shao, J.E. Maclennan, N.A. Clark, E. Körblova, D.M. Walba, Science 278, 1924 (1997)
H. Pleiner, H.R. Brand, Eur. Phys. J. E 37, 11 (2014)
E.G. Virga, Eur. Phys. J. E 38, 1 (2015)
A.D. Buckingham, Discuss. Faraday Soc. 43, 205 (1967)
C. Zannoni, in The Molecular Physics of Liquid Crystals, edited by G. Luckhurst, G. Gray (Academic Press, 1979) pp. 51--83
L. Qi, J. Symbolic Comput. 40, 1302 (2005)
L. Qi, J. Math. Anal. Appl. 325, 1363 (2007)
G. Ni, L. Qi, F. Wang, Y. Wang, J. Math. Anal. Appl. 329, 1218 (2007)
D. Cartwright, B. Sturmfels, Linear Algebra Appl. 438, 942 (2013)
X. Zheng, P. Palffy-Muhoray, electronic-Liquid Crystal Communications (2007), http://www.e-lc.org/docs/2007_02_03_02_33_15
S. Romano, Phys. Rev. E 74, 011704 (2006)
S. Romano, Phys. Rev. E 77, 021704 (2008)
J.J. Stoker, Differential Geometry (Wiley-Interscience, New York, 1969)
J.C. Maxwell, Philos. Mag. 2, 233 (1870)
A. Cayley, Philos. Mag. 18, 264 (1859)
D. Hilbert, S. Cohn-Vossen, Geometry and the Imagination, 2nd edn. (AMS Chelsea Publishing, Providence, 1999)
S.S. Turzi, J. Math. Phys. 52, 053517 (2011)
L. Qi, J. Math. Anal. Appl. 325, 1363 (2007)
F.W. Byron, jr., R.W. Fuller, Mathematics of Classical and Quantum Physics (Dover, Mineola, N.Y., 1992), unabridged, corrected republication of the work first published in two volumes by Addison-Wesley, 1969 (Vol. 1) and 1970 (Vol. 2)
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Rights and permissions
About this article
Cite this article
Gaeta, G., Virga, E.G. Octupolar order in three dimensions. Eur. Phys. J. E 39, 113 (2016). https://doi.org/10.1140/epje/i2016-16113-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epje/i2016-16113-7