Mechanisms to splay-bend nematic phases

While twist-bend nematic phases have been extensively studied, the experimental observation of two dimensional, oscillating splay-bend phases is recent. We consider two theoretical models that have been used to explain the formation of twist-bend phases—flexoelectricity and bond orientational order—as mechanisms to induce splay-bend phases. Flexoelectricity is a viable mechanism, and splay and bend flexoelectric couplings can lead to splay-bend phases with different modulations. We show that while bond orientational order circumvents the need for higher order terms in the free energy, the important role of nematic symmetry and phase chirality rules it out as a basic mechanism.

Figure 1

The figure shows (a) twist-bend and (b) splay-bend structures. The twist-bend structure has molecules rotating about the $z$ direction while maintaining a constant angle with it. In the splay-bend texture, molecules oscillate along the $z$ direction in two dimensions.

Figure 2

Schematics of the splay-bend phase in the ansatz in (a) Eq. (4) and (b) Eq. (7) showing the maximum angle $\theta$. The modulations are parallel to the average nematic director direction in the left schematic, and perpendicular to it in the right.

Figure 3

Plots of ${\overline{f}}_{\mathrm{splay}}$ and ${\overline{f}}_{\mathrm{bend}}$ as a function of $\theta$ at the free energy minimizing values of $q$ and $p$. The plots clearly show that both free energies have a minimum at a nontrivial value of $\theta$, implying that the splay-bend phase is a possibility with both couplings. The parameter values are $K_1=1.5$, $K_3=2$, $\gamma =40$, $b=2$, and $t=10$.