Q-tensor model of twist-bend and splay nematic phases

The twist-bend nematic phase $\left(N_{\mathrm{TB}}\right)$ is characterized by a conically twisting director and by a dramatic softening of the bend elastic constant before the transition to the $N_{\mathrm{TB}}$ phase. In the recently found splay nematic phase ($N_S$) the splay elastic constant tends to zero, resulting in a splay modulation perpendicular to the director. We model both phases with a single Q-tensor free energy including a term that breaks the degeneracy between the splay and bend elastic constant, and a flexoelectric term coupling the divergence of the Q-tensor with polarization. The $N_{\mathrm{TB}}$ or $N_S$ phase is obtained by a change of sign of one elastic parameter. Measured elastic constants show that the $N-N_{\mathrm{TB}}$ transition is mainly driven by the increase of the nematic order, while the $N_S$ transition is due to flexoelectric coupling.

Figure 1

(a) Schemes of the CB9CB molecule and the twist-bend nematic phase. (b) Temperature dependence of the splay (red squares), the twist (green circles), and the bend (blue triangles) elastic constants in the nematic phase above the $N-N_{\mathrm{TB}}$ phase transition for CB9CB. Inset: Temperature dependence of the anisotropy of the index of refraction.

Figure 2

Dependence of the bend (blue triangles), the twist (green circles), and the splay (red squares) elastic constants for CB9CB divided by ${\left(\mathrm{\Delta }n/\mathrm{\Delta }{n}_0\right)}^2$ on $\mathrm{\Delta }n$. $\mathrm{\Delta }{n}_0$ is the value of $\mathrm{\Delta }n$ at $T_{N-N_{\mathrm{TB}}}$.

Figure 3

(a) Schemes of the RM734 molecule and the splay nematic phase. (b) Temperature dependence of the splay (red squares), the twist (green circles), and the bend (blue triangles) elastic constants in the nematic phase above the $N-N_s$ phase transition, data from Ref. [7]. Inset: Temperature dependence of the anisotropy of the index of refraction.