Abstract
By analyzing elastic theory for nematic liquid crystals, we distinguish three regimes of elastic constants. In one regime, the Ericksen inequalities are satisfied, and the ground state of the director field is uniform. In a second regime, certain necessary inequalities are violated, and the free energy is thermodynamically unstable. Between those possibilities, there is an intermediate regime, where the Ericksen inequalities are violated but the system is still stable. Remarkably, lyotropic chromonic liquid crystals are in the intermediate regime. We investigate the nonuniform structure of the director field in that regime, show that it depends sensitively on system geometry, and discuss the implications for lyotropic chromonic liquid crystals.
Similar content being viewed by others
References
Oseen, C.W.: The theory of liquid crystals. Trans. Faraday Soc. 29, 883 (1933). https://doi.org/10.1039/tf9332900883
Frank, F.C.: I. Liquid crystals. On the theory of liquid crystals. Discuss. Faraday Soc. 25, 19 (1958). https://doi.org/10.1039/df9582500019
Nehring, J., Saupe, A.: On the elastic theory of uniaxial liquid crystals. J. Chem. Phys. 54, 337 (1971). https://doi.org/10.1063/1.1674612
Nehring, J., Saupe, A.: Calculation of the elastic constants of nematic liquid crystals. J. Chem. Phys. 56, 5527 (1972). https://doi.org/10.1063/1.1677071
Ericksen, J.L.: Inequalities in liquid crystal theory. Phys. Fluids 9, 1205 (1966). https://doi.org/10.1063/1.1761821
Selinger, J.V.: Interpretation of saddle-splay and the Oseen-Frank free energy in liquid crystals. Liq. Cryst. Rev. 6, 129–142 (2018). https://doi.org/10.1080/21680396.2019.1581103
Machon, T., Alexander, G.P.: Umbilic lines in orientational order. Phys. Rev. X 6, 011033 (2016). https://doi.org/10.1103/PhysRevX.6.011033
Virga, E.G.: Uniform distortions and generalized elasticity of liquid crystals. Phys. Rev. E 100, 052701 (2019). https://doi.org/10.1103/PhysRevE.100.052701
Sadoc, J.F., Mosseri, R., Selinger, J.V.: Liquid crystal director fields in three-dimensional non-Euclidean geometries. New J. Phys. 22, 093036 (2020). https://doi.org/10.1088/1367-2630/abaf6c
Pollard, J., Alexander, G.P.: Intrinsic geometry and director reconstruction for three-dimensional liquid crystals. New J. Phys. 23, 063006 (2021). https://doi.org/10.1088/1367-2630/abfdf4
da Silva, L.C.B., Efrati, E.: Moving frames and compatibility conditions for three-dimensional director fields. New J. Phys. 23, 063016 (2021). https://doi.org/10.1088/1367-2630/abfdf6
Nayani, K., Chang, R., Fu, J., Ellis, P.W., Fernandez-Nieves, A., Park, J.O., Srinivasarao, M.: Spontaneous emergence of chirality in achiral lyotropic chromonic liquid crystals confined to cylinders. Nat. Commun. 6, 8067 (2015). https://doi.org/10.1038/ncomms9067
Davidson, Z.S., Kang, L., Jeong, J., Still, T., Collings, P.J., Lubensky, T.C., Yodh, A.G.: Chiral structures and defects of lyotropic chromonic liquid crystals induced by saddle-splay elasticity. Phys. Rev. E 91, 050501 (2015). https://doi.org/10.1103/PhysRevE.91.050501
Fu, J., Nayani, K., Park, J.O., Srinivasarao, M.: Spontaneous emergence of twist and the formation of a monodomain in lyotropic chromonic liquid crystals confined to capillaries. NPG Asia Mater. 9, 393 (2017). https://doi.org/10.1038/am.2017.84
Javadi, A., Eun, J., Jeong, J.: Cylindrical nematic liquid crystal shell: effect of saddle-splay elasticity. Soft Matter 14, 9005 (2018). https://doi.org/10.1039/C8SM01829D
Paparini, S.: Mathematical Models for Chromonic Liquid Crystals. PhD thesis, Università degli Studi di Pavia (2021) http://hdl.handle.net/10281/362343
Paparini, S., Virga, E.G.: Stability Against the Odds: The Case of Chromonic Liquid Crystals (2022). https://doi.org/10.48550/arxiv.2203.12576
Burylov, S.V.: Equilibrium configuration of a nematic liquid crystal confined to a cylindrical cavity. J. Exp. Theor. Phys. 85, 873–886 (1997). https://doi.org/10.1134/1.558425
Dozov, I.: On the spontaneous symmetry breaking in the mesophases of achiral banana-shaped molecules. Europhys. Lett. 56, 247–253 (2001). https://doi.org/10.1209/epl/i2001-00513-x
Shamid, S.M., Dhakal, S., Selinger, J.V.: Statistical mechanics of bend flexoelectricity and the twist-bend phase in bent-core liquid crystals. Phys. Rev. E 87, 052503 (2013). https://doi.org/10.1103/PhysRevE.87.052503
Rosseto, M.P., Selinger, J.V.: Theory of the splay nematic phase: single vs. double splay. Phys. Rev. E 101, 052707 (2020). https://doi.org/10.1103/PhysRevE.101.052707
Selinger, J.V.: Director deformations, geometric frustration, and modulated phases in liquid crystals. Annu. Rev. Condens. Matter Phys. 13, 49–71 (2022). https://doi.org/10.1146/annurev-conmatphys-031620-105712
Rosseto, M.P., Selinger, J.V.: Modulated phases of nematic liquid crystals induced by tetrahedral order (2021). 2112.15218
Meiri, S., Efrati, E.: Cumulative geometric frustration in physical assemblies. Phys. Rev. E 104, 054601 (2021). https://doi.org/10.1103/PhysRevE.104.054601
Nych, A., Fukuda, J-i., Ognysta, U., Žumer, S., Muševič, I.: Spontaneous formation and dynamics of half-skyrmions in a chiral liquid-crystal film. Nat. Phys. 13, 1215 (2017). https://doi.org/10.1038/nphys4245
Duzgun, A., Selinger, J.V., Saxena, A.: Comparing skyrmions and merons in chiral liquid crystals and magnets. Phys. Rev. E 97, 062706 (2018). https://doi.org/10.1103/PhysRevE.97.062706
Ettinger, S., Dietrich, C.F., Mishra, C.K., Miksch, C., Beller, D.A., Collings, P.J., Yodh, A.G.: Rods in a lyotropic chromonic liquid crystal: emergence of chirality, symmetry-breaking alignment, and caged angular diffusion. Soft Matter, 20–23 (2022). https://doi.org/10.1039/D1SM01209F
Acknowledgements
This work was supported by National Science Foundation Grant DMR-1409658.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Long, C., Selinger, J.V. Violation of Ericksen Inequalities in Lyotropic Chromonic Liquid Crystals. J Elast 153, 599–612 (2023). https://doi.org/10.1007/s10659-022-09899-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-022-09899-z