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.

Keywords: 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}, publisher = {Elsevier}, volume = {32}, number = {5}, year = {2015}, pages = {1125-1144}, doi = {10.1016/j.anihpc.2014.05.006}, zbl = {1345.49002}, mrnumber = {3400444}, language = {en}, url = {http://www.numdam.org/item/AIHPC_2015__32_5_1125_0} }

Lamy, Xavier. Uniaxial symmetry in nematic liquid crystals. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 5, pp. 1125-1144. doi : 10.1016/j.anihpc.2014.05.006. http://www.numdam.org/item/AIHPC_2015__32_5_1125_0/

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

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

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

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

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

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

, ,[7] A new configurational transition in inhomogeneous nematics, Liq. Cryst. 16 (1994), 713 -718

, , ,[8] Order reconstruction in frustrated nematic twist cells, Phys. Rev. E 68 (2003), 021707

, , , ,[9] Director reorientation and order reconstruction: competing mechanisms in a nematic cell, Contin. Mech. Thermodyn. 20 (2008), 193 -218 | MR 2438262 | Zbl 1170.76304

, , ,[10] Thermotropic biaxial nematic liquid crystals, Phys. Rev. Lett. 92 (2004), 145505

, , , ,[11] Biaxial nematic phase in bent-core thermotropic mesogens, Phys. Rev. Lett. 92 (2004), 145506

, , ,[12] The topology of non-uniform media in condensed matter physics, Adv. Phys. 31 (1982), 195 -254 | MR 686631

,[13] The principle of symmetric criticality, Commun. Math. Phys. 69 (1979), 19 -30 | MR 547524 | Zbl 0417.58007

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

, ,[15] Multiple Integrals in the Calculus of Variations, Springer (1966) | MR 202511 | Zbl 0142.38701

,[16] Bifurcation analysis in a frustrated nematic cell, arXiv:1310.6920 (2013) | MR 3275223 | Zbl 1316.76009

,[17] Landau–de Gennes theory of nematic liquid crystals: the Oseen–Frank limit and beyond, Arch. Ration. Mech. Anal. 196 (2010), 227 -280 | MR 2601074 | Zbl 1304.76007

, ,[18] On nematic liquid crystals with variable degree of orientation, Commun. Pure Appl. Math. 44 (1991), 453 -468 | MR 1100811 | Zbl 0733.49005

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

, ,[20] Liquid crystal engineering of carbon nanofibers and nanotubes, Carbon 43 (2005), 2431 -2440

, , , , , , , , ,[21] Visualization of liquid crystal director fields within carbon nanotube cavities, Appl. Phys. Lett. 88 (2006), 163110

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

, ,[23] Bifurcation analysis of minimizing harmonic maps describing the equilibrium of nematic phases between cylinders, Arch. Ration. Mech. Anal. 118 (1992), 149 -168 | MR 1158933 | Zbl 0825.76062

, , , ,[24] Drops and shells of liquid crystal, Colloid Polym. Sci. 289 (2011), 345 -359

, ,[25] Universal fine structure of nematic hedgehogs, J. Phys. A, Math. Gen. 34 (2001), 829 | MR 1826027 | Zbl 1015.82036

, ,[26] Some properties of the nematic radial hedgehog in the Landau–de Gennes theory, J. Math. Anal. Appl. 397 (2013), 586 -594 | MR 2979597 | Zbl 1310.76011

,[27] Stability of the vortex defect in the Landau–de Gennes theory for nematic liquid crystals, C. R. Math. 351 (2013), 533 -537 | MR 3095101 | Zbl 1276.35030

, , , ,[28] Symmetry of uniaxial global Landau–de Gennes minimizers in the theory of nematic liquid crystals, SIAM J. Math. Anal. 44 (2012), 3217 -3241 | MR 3023409 | Zbl 1270.35030

, ,[29] 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 3144795 | Zbl 1283.35006

, ,