Flexoelectricity in chiral nematic liquid crystals as a driving mechanism for the twist-bend and splay-bend modulated phases

We present a continuum theoretical model describing the impact of chirality on nematic systems with large flexoelectricity. As opposed to achiral materials, where only one type of the modulated structure can exist in a given material, the model predicts that chirality can stabilize several modulated phases, which have already been observed experimentally [A. Zep et al., J. Mater. Chem. C 1, 46 (2013)].

Figure 1

The three possible twist structures in orthorhombic nematic liquid crystals. The orientation of the blocks is defined by three perpendicular unit vectors: $\mathbf{n}$ (blue solid line), $\mathbf{l}$ (red dotted line), and $\mathbf{m}=\mathbf{n}\times \mathbf{l}$ (green dashed line).

Figure 2

The splay-bend (top) and twist-bend (bottom) modulation superposed over the chiral helix.

Figure 3

(a) The relative change in the wave-vector magnitude ($\Delta \tilde{q}$) in the twist-bend (dashed blue line) and splay-bend (solid red line) modulated structures as a function of the cell thickness $L$ given in units of the modulation length $2\pi /q_0$ [$x=L{q}_0/\left(2\pi \right)$]. (b) The relative reduction in energy ($\Delta \tilde{F}=\Delta F/F_0$) as a function of the cell thickness for the twist-bend ($\Delta {\tilde{F}}^{\left(\text{tb}\right)}$, blue dashed line) and splay-bend ($\Delta {\tilde{F}}^{\left(\text{sb}\right)}$, red solid line) modulated structures. Parameter values: $k_{31}=-1$, $\Delta \mu K/{\lambda }^2={10}^{-3}$. Inset: the value of $\mu$ at which the modulated structure becomes stable. Blue (dashed) line: ${\mu }_0^{\left(\text{tb}\right)}$. Red solid line: ${\mu }_0^{\left(\text{sb}\right)}$.