Bistable director alignments of nematic liquid crystals confined in frustrated substrates

We studied in-plane bistable alignments of nematic liquid crystals confined by two frustrated surfaces by means of Monte Carlo simulations of the Lebwohl-Lasher spin model. The surfaces are prepared with orientational checkerboard patterns, on which the director field is locally anchored to be planar yet orthogonal between the neighboring blocks. We found the director field in the bulk tends to be aligned along the diagonal axes of the checkerboard pattern, as reported experimentally [J.-H. Kim et al., Appl. Phys. Lett. 78, 3055 (2001)]. The energy barrier between the two stable orientations is increased, when the system is brought to the isotropic-nematic transition temperature. Based on an elastic theory, we found that the bistability is attributed to the spatial modulation of the director field near the frustrated surfaces. As the block size is increased and/or the elastic modulus is reduced, the degree of the director inhomogeneity is increased, enlarging the energy barrier. We also found that the switching rate between the stable states is decreased when the block size is comparable to the cell thickness.

Figure 1

(a) Plots of the scalar nematic order parameter in the bulk $S_{\mathrm{b}}$ (black circles) and on the surface $S_{\mathrm{w}}$ (red open squares) with respect to the temperature. (b) Plot of the elastic modulus $K$ (black circles) of the nematic phase with $T$. $S_{\mathrm{w}}^2/K$ is also plotted with red open squares.

Figure 2

Dependencies of the stored energy per unit area with respect to ${\varphi }_{\mathrm{t}}$ at ${\theta }_{\mathrm{t}}=\pi /2$ in (a), and to ${\theta }_{\mathrm{t}}$ at ${\varphi }_{\mathrm{t}}=\pi /4$ in (b). The liquid crystal is confined in a type I cell of $D=8$ and $H=16$. The temperature is changed.

Figure 3

(a) Plot of the energy difference $\mathrm{\Delta }\mathcal{E}$ per unit area against the temperature. The cell thickness is $H=8$ (type I) and the block size $D$ is changed. (b) Plot of the energy difference per unit area against the block size. The temperature is $T/T_{\mathrm{IN}}=0.89$ and the cell thickness is changed.

Figure 4

Snapshots of the $xx$ and $xy$ components of the tensorial order parameter $Q_{\mu \nu }$ in the type I cells of the thickness $H=8$. The anchoring direction on the bottom surface ($z=0$) is patterned like the checkerboard, while that on the top surface ($z=9$) is homogeneously along ${\widehat{\mathit{n}}}_{+}$. The temperature is $T/T_{\mathrm{IN}}=0.89$. The block size is $D=8$ in (a) and $D=64$ in (b). Only the snapshots in a small area (${128}^2$) are shown.

Figure 5

Profiles of the inhomogeneity of the nematic order parameter $I\left(z\right)$ along the cell thickness $z$. (a) The cell thickness is $H=32$ (type I) and the temperature is $T/T_{\mathrm{IN}}=0.89$. The block size is increased from $D=1$ to $D=64$. (b) The cell thickness is $H=32$ and the block size is fixed to $D=16$. The temperature $T$ is changed.

Figure 6

(a) Dependencies of the averaged order parameter on the block size. The cell thickness is $H=16$ (type II). The temperature is changed. (b) Schematic pictures of the director field near the midplane in the type II cell of $D\gg H$.

Figure 7

Time sequences of the averaged nematic order along ${\widehat{\mathit{n}}}_{+}$ in the in-plane switching processes. At $t=0$, the director field is completely aligned along ${\mathit{n}}_{+}$. In time intervals $1\times {10}^4\le t<2\times {10}^4$ and $3\times {10}^4\le t<4\times {10}^4$, the external field is applied along ${\widehat{\mathit{n}}}_{-}$ and ${\widehat{\mathit{n}}}_{+}$, respectively. The strength of the external field $e$ is changed. The temperature is $T/T_{\mathrm{IN}}=0.89$, and the type II cell of $D=16$ and $H=16$ is employed.

Figure 8

(a) Time sequences of the averaged nematic order parameter $\langle Q_{xy}\rangle$ in the switching process. Before $\mathrm{\Delta }t=0$, the director field is aligned along ${\widehat{\mathit{n}}}_{+}$ in average. After $\mathrm{\Delta }t=0$, the in-plane external field is applied along ${\widehat{\mathit{n}}}_{-}$. The temperature is $T/T_{\mathrm{IN}}=0.89$ and the cell thickness is $H=16$ (type II). The strength of the external field $e$ is changed. (b) Plots of the characteristic time $\tau$ of the switching process with respect to the block size $D$. $\tau$ is defined as $\langle Q_{xy}(\mathrm{\Delta }t=\tau )\rangle =0$. The temperature is $T/T_{\mathrm{IN}}=0.89$ and the thickness of type II cell is changed.

Figure 9

Snapshots of the nematic order parameter $Q_{xy}(x,y)$ at $z=8$ in the type II cell of $H=16$. The director field in yellow regions is along ${\widehat{\mathit{n}}}_{+}$ and that in blue regions is along ${\widehat{\mathit{n}}}_{-}$. The temperature is $T/T_{\mathrm{IN}}=0.89$ and the block size $D$ is changed.