Influence of flexoelectric effect on the Fréedericksz transition in chiral nematic liquid crystals

Electric field driven Frèedericksz transition in chiral nematic (cholesteric) planar liquid crystal cell is studied in the presence of flexoelectric effect. An inhomogeneity of electric field and finiteness of anchoring energy are considered. The extra contribution to the electric field induced by flexoelectricity is taken into account as well as the common contribution arising from the applied voltage $U$. Equilibrium director distribution is obtained in dependence on voltage and flexoelectric coefficients by numerical minimization of the free energy. The threshold voltage was found to decrease with the increase of the flexoelectric coefficients within the range of high flexoelectric coefficients. Thus flexoelectricity promotes the transition. The orientational structure becomes asymmetric about the center of the cell due to flexoelectricity, even in the case of symmetric boundary conditions. The equilibrium structure was also shown to be different for $U>0$ and $U<0$. It was found, that for sufficiently high flexoelectric coefficients the director distribution can be described by a simple function. In this case the planar helicoidal structure transfers into a hybrid-aligned one as the voltage increases. This transformation may form the basis for new flexoelectricity-based switching devices. The nonmonotonic behavior of threshold voltage can be observed in the case of small flexoelectric coefficients. An analytical form of the stability conditions was obtained for the planar helicoidal configuration. We have shown that the Frèedericksz transition can be either continuous or discontinuous depending on the material constants and thickness of the liquid crystal cell.

Figure 1

A sketch illustrating discontinuous (I) and continuous (II) Fréedericksz transitions for $U>0$. (I) The stability areas for the distorted (D) “phase” $U>U^{**}$, and for the undistorted planar (P) “phase” $U<U^{*}$. In the interval PD, $U^{**}<U<U_c$, the D phase is metastable, and in the interval DP, $U_c<U<U^{*}$, the P phase is metastable. (II) The stability intervals for the distorted “phase” $U>U_c$, and for the undistorted planar “phase” $U<U_c$.

Figure 2

Phase diagram. The voltages $U^{*}$, $U_c$, and $U^{**}$ as functions of the mean flexoelectric coefficient $\overline{e}$: dashed line, $U^{**}$; solid line, $U_c$; dashed-dotted line, $U^{*}$. The stability and metastability areas are schematically shown in the insets. The tricritical points are denoted by TP, other notations are the same as in Fig. 1. The lines $U^{*}$, $U_c$, and $U^{**}$ coincide to the right of TPs.

Figure 3

Equilibrium profiles $\theta \left(z\right)$ and $\phi \left(z\right)$ calculated via Eq. (3.17) for $U=1.2$ V. Lines 1 correspond to the LC without flexoelectricity, $\overline{e}=0$; lines 2 are for $\overline{e}={10}^{-4}\mathrm{statC}/\mathrm{cm}$, lines 3 are for $\overline{e}={10}^{-3}\mathrm{statC}/\mathrm{cm}$, lines 4 are for $\overline{e}=3\times {10}^{-3}\mathrm{statC}/\mathrm{cm}$, and lines 5 are for $\overline{e}={10}^{-2}\mathrm{statC}/\mathrm{cm}$ (we use the statcoulomb unit, statC, for electric charge in cgs system). Dashed lines (red online) are calculated via formulas (3.18) and (3.19).

Figure 4

Equilibrium profiles $\theta \left(z\right)$ for $U=\pm 1.2$ V in the case of symmetric anchoring: $W_{\theta }^{\left(1\right)}=W_{\theta }^{\left(2\right)}=1.5\times {10}^{-3}\mathrm{erg}/{\mathrm{cm}}^2$ and $W_{\phi }^{\left(1\right)}=W_{\phi }^{\left(2\right)}=1.5\times {10}^{-4}\mathrm{erg}/{\mathrm{cm}}^2$. The lines' numbers, $\overline{e}$ values, and other material parameters are the same as in Fig. 3, except line 2, which corresponds to $\overline{e}=5\times {10}^{-4}\mathrm{statC}/\mathrm{cm}$. Dashed lines (red online) are calculated via formula (3.18).

Figure 5

Profiles $\theta \left(z\right)$ for continuous (a) and discontinuous (b) transitions. Coinciding lines 1a and 1b refer to the planar helicoidal structure ($U=0.999U_c$) and lines with the number 2 correspond to the distorted structures ($U=1.005U_c$). For “a” profiles $\overline{e}=8\times {10}^{-4}\mathrm{statC}/\mathrm{cm}>{\overline{e}}^{\text{TP}}$ and for “b” profiles $\overline{e}=2\times {10}^{-4}\mathrm{statC}/\mathrm{cm}<{\overline{e}}^{\text{TP}}$, where ${\overline{e}}^{\mathrm{TP}}=5.132\times {10}^{-4}\mathrm{statC}/\mathrm{cm}$. The threshold voltages are (a) $U_c=0.6926\text{V}$, (b) $U_c=0.9396\text{V}$. Other material parameters are the same as in Fig. 3.

Figure 6

The curves $ABCD$ and $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ plotted using formulas (4.21) present the boundary of the stability zone of the planar helicoidal configuration.

Figure 7

The effective coefficient $\tilde{B}$ of the Landau model vs the flexoelectric coefficient $\overline{e}$. The material constants are the same as in Figs. 2 and 6. The circles correspond to the tricritical points.