Theory of the splay nematic phase

Recent experiments have reported a novel splay nematic phase, which has alternating domains of positive and negative splay [1]. To model this phase, researchers have considered a 1D splay modulation of the director field, accompanied by a 1D modulation of polar order. When the flexoelectric coupling between splay and polar order becomes sufficiently strong, the uniform nematic state becomes unstable to the formation of a modulated phase [1,2]. Here, we re-examine this theory in terms of a new approach to liquid crystal elasticity [3], which shows that pure splay deformation is double-splay rather than planar single-splay. Following that reasoning, we propose a structure with a 2D splay modulation of the director field, accompanied by a 2D modulation of polar order, and show that the 2D structure generally has a lower free energy than the 1D structure. We also investigate a generalized flexoelectric coupling between director deformations and tetrahedral order, but still find a lower free energy for the 2D modulation.
[1] A. Mertelj et al., Phys. Rev. X 8, 041025 (2018).
[2] N. Chaturvedi and R. D. Kamien, Phys. Rev. E 100, 022704 (2019).
[3] J. V. Selinger, Liq. Cryst. Rev. 6, 129 (2018).