Inequalities for Elastic Constants in Nematic Liquid Crystals

In the early years of liquid crystal physics, Ericksen derived a set of inequalities for the elastic constants of nematic liquid crystals [1], which can be written as K₁₁ – K₂₄ > 0, K₂₂ – K₂₄ > 0, K₃₃ > 0, and K₂₄ > 0.  However, recent experiments have shown that lyotropic chromonic liquid crystals violate the inequality on K₂₂ – K₂₄ [2].  Motivated by this discrepancy, we re-examine the inequalities, using a recent reformulation of elasticity theory [3].  This analysis shows that the Ericksen inequalities are stronger than necessary for thermodynamic stability.  Rather, thermodynamic stability only requires the weaker set of inequalities K₁₁ > 0, K₂₂ > 0, and K₃₃ > 0.  In the regime between the strong and weak inequalities, the liquid crystal’s director field becomes nonuniform, but it is still thermodynamically stable.  We compare these results with experiments on lyotropic chromonic liquid crystals.

 

[1] J. L. Ericksen, Inequalities in Liquid Crystal Theory, Phys. Fluids 9, 1205 (1966).

[2] Z. S. Davidson, L. Kang, J. Jeong, T. Still, P. J. Collings, T. C. Lubensky, and A. G. Yodh, Chiral Structures and Defects of Lyotropic Chromonic Liquid Crystals Induced by Saddle-Splay Elasticity, Phys. Rev. E 91, 050501 (2015).

[3] J. V. Selinger, Interpretation of Saddle-Splay and the Oseen-Frank Free Energy in Liquid Crystals, Liq. Cryst. Rev. 6, 129 (2018).