Uniaxial symmetry in nematic liquid crystals
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 5, pp. 1125-1144.

Within the Landau–de Gennes theory of liquid crystals, we study theoretically the equilibrium configurations with uniaxial symmetry. We show that the uniaxial symmetry constraint is very restrictive and can in general not be satisfied, except in very symmetric situations. For one- and two-dimensional configurations, we characterize completely the uniaxial equilibria: they must have constant director. In the three dimensional case we focus on the model problem of a spherical droplet with radial anchoring, and show that any uniaxial equilibrium must be spherically symmetric. It was known before that uniaxiality can sometimes be broken by energy minimizers. Our results shed a new light on this phenomenon: we prove here that in one or two dimensions uniaxial symmetry is always broken, unless the director is constant. Moreover, our results concern all equilibrium configurations, and not merely energy minimizers.

DOI : 10.1016/j.anihpc.2014.05.006
Mots clés : Liquid crystals, Uniaxial symmetry, Biaxial escape, Radial symmetry
@article{AIHPC_2015__32_5_1125_0,
     author = {Lamy, Xavier},
     title = {Uniaxial symmetry in nematic liquid crystals},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1125--1144},
     publisher = {Elsevier},
     volume = {32},
     number = {5},
     year = {2015},
     doi = {10.1016/j.anihpc.2014.05.006},
     mrnumber = {3400444},
     zbl = {1345.49002},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.05.006/}
}
TY  - JOUR
AU  - Lamy, Xavier
TI  - Uniaxial symmetry in nematic liquid crystals
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
SP  - 1125
EP  - 1144
VL  - 32
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2014.05.006/
DO  - 10.1016/j.anihpc.2014.05.006
LA  - en
ID  - AIHPC_2015__32_5_1125_0
ER  - 
%0 Journal Article
%A Lamy, Xavier
%T Uniaxial symmetry in nematic liquid crystals
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 1125-1144
%V 32
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2014.05.006/
%R 10.1016/j.anihpc.2014.05.006
%G en
%F AIHPC_2015__32_5_1125_0
Lamy, Xavier. Uniaxial symmetry in nematic liquid crystals. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 5, pp. 1125-1144. doi : 10.1016/j.anihpc.2014.05.006. http://www.numdam.org/articles/10.1016/j.anihpc.2014.05.006/

[1] S. Mkaddem, E. Gartland, Fine structure of defects in radial nematic droplets, Phys. Rev. E 62 (2000), 6694 -6705

[2] S. Kralj, E.G. Virga, S. Žumer, Biaxial torus around nematic point defects, Phys. Rev. E 60 (1999), 1858 -1866

[3] E. Penzenstadler, H.-R. Trebin, Fine structure of point defects and soliton decay in nematic liquid crystals, J. Phys. France 50 (1989), 1027 -1040

[4] A. Sonnet, A. Kilian, S. Hess, Alignment tensor versus director: description of defects in nematic liquid crystals, Phys. Rev. E 52 (1995), 718 -722

[5] G. De Luca, A. Rey, Ringlike cores of cylindrically confined nematic point defects, J. Chem. Phys. 126 (2007), 094907

[6] G. De Luca, A. Rey, Point and ring defects in nematics under capillary confinement, J. Chem. Phys. 127 (2007), 104902

[7] P. Palffy-Muhoray, E. Gartland, J. Kelly, A new configurational transition in inhomogeneous nematics, Liq. Cryst. 16 (1994), 713 -718

[8] F. Bisi, E. Gartland, R. Rosso, E. Virga, Order reconstruction in frustrated nematic twist cells, Phys. Rev. E 68 (2003), 021707

[9] M. Ambrožič, F. Bisi, E. Virga, Director reorientation and order reconstruction: competing mechanisms in a nematic cell, Contin. Mech. Thermodyn. 20 (2008), 193 -218 | MR | Zbl

[10] L. Madsen, T. Dingemans, M. Nakata, E. Samulski, Thermotropic biaxial nematic liquid crystals, Phys. Rev. Lett. 92 (2004), 145505

[11] B. Acharya, A. Primak, S. Kumar, Biaxial nematic phase in bent-core thermotropic mesogens, Phys. Rev. Lett. 92 (2004), 145506

[12] H. Trebin, The topology of non-uniform media in condensed matter physics, Adv. Phys. 31 (1982), 195 -254 | MR

[13] R. Palais, The principle of symmetric criticality, Commun. Math. Phys. 69 (1979), 19 -30 | MR | Zbl

[14] N. Schopohl, T. Sluckin, Hedgehog structure in nematic and magnetic systems, J. Phys. France 49 (1988), 1097 -1101

[15] C. Morrey, Multiple Integrals in the Calculus of Variations, Springer (1966) | MR | Zbl

[16] X. Lamy, Bifurcation analysis in a frustrated nematic cell, arXiv:1310.6920 (2013) | MR | Zbl

[17] A. Majumdar, A. Zarnescu, Landau–de Gennes theory of nematic liquid crystals: the Oseen–Frank limit and beyond, Arch. Ration. Mech. Anal. 196 (2010), 227 -280 | MR | Zbl

[18] F. Lin, On nematic liquid crystals with variable degree of orientation, Commun. Pure Appl. Math. 44 (1991), 453 -468 | MR | Zbl

[19] A.-G. Cheong, A. Rey, Texture dependence of capillary instabilities in nematic liquid crystalline fibres, Liq. Cryst. 31 (2004), 1271 -1284

[20] C. Chan, G. Crawford, Y. Gao, R. Hurt, K. Jian, H. Li, B. Sheldon, M. Sousa, N. Yang, Liquid crystal engineering of carbon nanofibers and nanotubes, Carbon 43 (2005), 2431 -2440

[21] K. Jian, R. Hurt, B. Sheldon, G. Crawford, Visualization of liquid crystal director fields within carbon nanotube cavities, Appl. Phys. Lett. 88 (2006), 163110

[22] P. Cladis, M. Kléman, Non-singular disclinations of strength s=+1 in nematics, J. Phys. France 33 (1972), 591 -598

[23] F. Bethuel, H. Brezis, B. Coleman, F. Hélein, Bifurcation analysis of minimizing harmonic maps describing the equilibrium of nematic phases between cylinders, Arch. Ration. Mech. Anal. 118 (1992), 149 -168 | MR | Zbl

[24] T. Lopez-Leon, A. Fernandez-Nieves, Drops and shells of liquid crystal, Colloid Polym. Sci. 289 (2011), 345 -359

[25] S. Kralj, E. Virga, Universal fine structure of nematic hedgehogs, J. Phys. A, Math. Gen. 34 (2001), 829 | MR | Zbl

[26] X. Lamy, Some properties of the nematic radial hedgehog in the Landau–de Gennes theory, J. Math. Anal. Appl. 397 (2013), 586 -594 | MR | Zbl

[27] R. Ignat, L. Nguyen, V. Slastikov, A. Zarnescu, Stability of the vortex defect in the Landau–de Gennes theory for nematic liquid crystals, C. R. Math. 351 (2013), 533 -537 | MR | Zbl

[28] D. Henao, A. Majumdar, Symmetry of uniaxial global Landau–de Gennes minimizers in the theory of nematic liquid crystals, SIAM J. Math. Anal. 44 (2012), 3217 -3241 | MR | Zbl

[29] D. Henao, A. Majumdar, Corrigendum: symmetry of uniaxial global Landau–de Gennes minimizers in the theory of nematic liquid crystals, SIAM J. Math. Anal. 45 (2013), 3872 -3874 | MR | Zbl

Cité par Sources :