Simulation of cholesteric blue phases using a Landau–de Gennes theory: Effect of an applied electric field

We investigate numerically static and dynamic properties of cholesteric blue phases. Our study is based on a Landau–de Gennes theory describing the orientational order of a liquid crystal in terms of a second-rank tensor. To find the shape and size of the unit cell conforming to the minimum of the free energy, we let the geometrical parameters characterizing the unit cell relax in the course of the time evolution via a simple relaxational equation. We investigate the effect of an electric field on the structure of cholesteric blue phases. We study how the deformation of the unit cell in response to the electric field $\mathit{E}$ depends on the strength and direction of the electric field and the original structure of cholesteric blue phases. Our results qualitatively agree with the experimental findings. Although in a weak field, the strain tensor is proportional to $E^2$ as previously argued, for a moderate field the distortion is no longer proportional to $E^2$ and can be even nonmonotonic with respect to $E^2$. Furthermore, we investigate the kinetic processes of the deformation, rearrangement, and extinction of disclination lines under a strong electric field. We show that the kinetics of disclination lines is highly complicated and sensitively depends on the initial structure of blue phases, the direction of the electric field, and the sign of dielectric anisotropy ${ϵ}_{\text{a}}$. In most cases, a strong field aligns the liquid crystals in a uniform (positive ${ϵ}_{\text{a}}$) or helical (negative ${ϵ}_{\text{a}}$) manner without disclination lines. However, for negative ${ϵ}_{\text{a}}$ and the direction of the electric field parallel to the body diagonal of the unit cell, disclination lines do not disappear and form a two-dimensional hexagonal lattice.

Figure 1

Illustration of the definitions of $L_{\parallel }$ and $L_{\stackrel{̸}{\parallel }}$ for the cases with $\mathit{E}$ parallel to the [100], [110], and [111] directions.

Figure 2

(Color online) Plots of $\left(L_{\parallel }^2-L_0^2\right)/L_0^2$ and $\left(L_{\stackrel{̸}{\parallel }}^2-L_0^2\right)/L_0^2$ for (a) and (b) $O_8^{-}$ and (c) and (d) $O_2$ structures, when the field is applied along the [100] direction. (b) and (d) are magnified plots of (a) and (c), respectively, for small $\left|\widetilde{ϵ}\right|$. Solid lines with $+$ symbols (red) represent $\left(L_{\parallel }^2-L_0^2\right)/L_0^2$, and dashed lines with $\times$ symbols (green) represent $\left(L_{\stackrel{̸}{\parallel }}^2-L_0^2\right)/L_0^2$.

Figure 3

(Color online) Plots of (a) and (b) $\left(L_{\parallel }^2-L_0^2\right)/{\left(\sqrt{2}{L}_0\right)}^2$ for $\mathit{E}\parallel \left[110\right]$ and (c) and (d) $\left(L_{\parallel }^2-L_0^2\right)/{\left(\sqrt{3}{L}_0\right)}^2$ for $\mathit{E}\parallel \left[111\right]$. (b) and (d) are magnified plots of (a) and (c), respectively, for small $\left|\widetilde{ϵ}\right|$. Solid lines with square (◻) symbols (magenta) and dashed lines with cross $(\times )$ symbols (green) represent the results for $O_8^{-}$ and $O_2$ structures, respectively.

Figure 4

(Color online) Time evolution of the $O_8^{-}$ structure after the application of an electric field parallel to the [100] direction. The parameter $\widetilde{ϵ}$ is (a) 5 and (b) $-5$. The numbers indicate the rescaled time $t/{\tau }_{\chi }$ after the application of the electric field. The direction of the field is along the horizontal direction in the plane of paper as shown in the figure. We note that at $t/{\tau }_{\chi }=0.796$ in (a) and $t/{\tau }_{\chi }=3.860$ in (b), no disclination line is present and the alignment of the director is uniform along the field direction in the former, and helical in the latter.

Figure 5

(Color online) The same as Fig. 4 for the $O_2$ structure. Note that in (b) at $t/{\tau }_{\chi }=0.275$, two disclination lines are not connected at the center of the figure. At $t/{\tau }_{\chi }=0.550$ in (a) and $t/{\tau }_{\chi }=0.822$ in (b), no disclination line is present and the alignment of the director is uniform along the field direction in the former, and helical in the latter.

Figure 6

(Color online) The same as Fig. 4 for the $O_8^{-}$ structure with the electric field along the [110] direction. Note that in (a) at $\tau =0.396$, the viewpoint is changed so that the [111] axis is set perpendicular to the plane of paper to make clear the winding of a disclination line that was along the [111] direction in the absence of the electric field. At $t/{\tau }_{\chi }=0.797$ in (a) and $t/{\tau }_{\chi }=15.597$ in (b), no disclination line is present and the alignment of the director is uniform along the field direction in the former, and helical in the latter.

Figure 7

(Color online) The same as Fig. 4 for the $O_2$ structure with the electric field along the [110] direction. Note that in (a) at $t/{\tau }_{\chi }=0.138$, two disclination lines are not connected at the center of the figure. At $t/{\tau }_{\chi }=0.550$ in (a) and $t/{\tau }_{\chi }=7.037$ in (b), no disclination line is present and the alignment of the director is uniform along the field direction in the former, and helical in the latter.

Figure 8

(Color online) The same as Fig. 4 for the $O_8^{-}$ structure with the electric field along the [111] direction. The [111] axis along the field direction is perpendicular to the plane of paper. At $t/{\tau }_{\chi }=0.798$ in (a), no disclination line is present and the alignment of the director is uniform along the field direction.

Figure 9

(Color online) The same as Fig. 4 for the $O_2$ structure with the electric field along the [111] direction. In (a), the [111] axis along the field direction is perpendicular to the plane of paper. In (b), the direction of the [111] axis is indicated in each figure. Figures at $t/{\tau }_{\chi }=0$, 9.141 and 24.328 are drawn from one fixed viewpoint, and those at $t/{\tau }_{\chi }=0.275$, 1.382 and 5.809 are drawn from another fixed viewpoint. At $t/{\tau }_{\chi }=0.688$ in (a), no disclination line is present and the alignment of the director is uniform along the field direction.

Figure 10

(Color online) Director profile in a plane whose normal is along the [111] direction at $\tau /{\tau }_{\chi }=0.476$ in Fig. 8a. The positions of twist disclination lines are highlighted by crosses $(\times )$. The director configuration around a twist disclination is given schematically in the nail picture.