Theory of the splay nematic phase: Single versus double splay

Recent experiments have reported a novel splay nematic phase, which has alternating domains of positive and negative splay. To model this phase, previous studies have considered a one-dimensional (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. Here we reexamine this theory in terms of a recent approach to liquid crystal elasticity, which shows that pure splay deformation is double splay rather than planar single splay. Following that reasoning, we propose a structure with a two-dimensional (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.

Figure 1

Schematic representations of proposed structures for the splay nematic $\left(N_S\right)$ phase: (a) single splay, (b) double splay. In both cases, yellow represents regions of positive polar order and positive splay, and blue represents regions of negative polar order and negative splay.

Figure 2

Numerical results for (a) the polar order parameter $P\left(x\right)$ and (b) the director orientation $\theta \left(x\right)$, as functions of the scaled coordinate $x$, for several values of the scaled temperature $\tilde{\mu }$, at fixed parameters $\tilde{\nu }=5$ and $K_{11}/K_{33}=8$.

Figure 3

Scaled wavelength as a function of scaled temperature $\tilde{\mu }$, for fixed parameters $\tilde{\nu }=5$ and $K_{11}/K_{33}=8$. The solid red line denotes the single-splay modulation, and the dotted blue line represents the double-splay modulation.

Figure 4

Average scaled free energy as a function of scaled temperature $\tilde{\mu }$, for fixed parameters $\tilde{\nu }=5$ and $K_{11}/K_{33}=8$. The solid red line represents the single-splay modulation, dotted blue line the double-splay modulation, and dashed black line the polar phase. The insets show the crossover points, with first-order transitions between double splay and single splay.

Figure 5

Numerical results for the polar order parameter $P(x,y)$ for two values of the scaled temperature, (a) $\tilde{\mu }=0.9$ and (b) $\tilde{\mu }=-0.1$. In both cases, $\tilde{\nu }=5$ and $K_{11}/K_{33}=8$.

Figure 6

Phase diagram for our model, as a function of the scaled temperature $\tilde{\mu }$ and the ratio $K_{11}/K_{33}$, for fixed parameter $\tilde{\nu }=5$.