Topology and bistability in liquid crystal devices

We study nematic liquid crystal configurations in a prototype bistable device—the post aligned bistable nematic (PABN) cell. Working within the Oseen-Frank continuum model, we describe the liquid crystal configuration by a unit-vector field $\mathbf{n}$, in a model version of the PABN cell. First, we identify four distinct topologies in this geometry. We explicitly construct trial configurations with these topologies which are used as initial conditions for a numerical solver, based on the finite-element method. The morphologies and energetics of the corresponding numerical solutions qualitatively agree with experimental observations and suggest a topological mechanism for bistability in the PABN cell geometry.

Figure 1

The PABN device [4].

Figure 2

The post cross-section, $L_p\times L_p$, and the cell cross-section, $L_c\times L_c$, where $L_c=2L_p$.

Figure 3

The model geometry with $L_c=1$, $L_p=0.5$, and $H=3$. When translated to physical units, these correspond to $L_c=1\mu \mathrm{m}$, $L_p=500\mathrm{nm}$, and $H=3\mu \mathrm{m}$, respectively.

Figure 4

(a) The high-tilt state. (b) The low-tilt state.

Figure 5

(a) The most symmetric choice. (b) Admissible choice of lower symmetry. (c) Necessarily contains a planar defect.

Figure 6

(Color online) (a) The profile on the bottom substrate. (b) The profile on the top face of the post.

Figure 7

The four vertical edges.

Figure 8

(Color online) (a) The $T$ profile. (b) The $P_1$ profile. (c) The $P_2$ profile. (d) The $P_3$ profile. The scaled vertical coordinate in each plot is $z∕h$, where $h$ is the post height and $0⩽z∕h⩽1$.

Figure 9

(Color online) The $T$ profile on the faces, up to the height of the post $0⩽z⩽h$, where $z$ is in $\mu \mathrm{m}$. The vertical coordinate is the dimensionless quantity—$z∕h$ where $0⩽z∕h⩽1$.

Figure 10

(Color online) The $P_1$ profile on the post faces.

Figure 11

(Color online) The $P_2$ profile on the post faces.

Figure 12

(Color online) The $P_3$ profile on the post faces.

Figure 13

(Color online) A periodically extended domain containing two posts. This is the bottom cross section, with the filled regions corresponding to the post bases. The bold line indicates the cross section $y=L_c∕4$, which is plotted in Figs. 14, 15, 16, 17.

Figure 14

(Color online) This plot is the projection of the $T$ profile onto the plane $y=L_c∕4$, i.e., only the $x$ and $z$ components are shown here. The dots, in particular, correspond to points where the solution is oriented in the $y$ direction. The horizontal coordinate is the dimensionless quantity $x∕L_c$ and the vertical coordinate is $z∕h$.

Figure 15

(Color online) The projection of the $P_1$ profile onto the plane $y=L_c∕4$.

Figure 16

(Color online) The projection of the $P_2$ profile onto $y=L_c∕4$.

Figure 17

(Color online) The projection of the $P_3$ profile onto $y=L_c∕4$. Note that we only plot up to height $z=3h∕2$ because the profiles have similar structures in the remainder of the cell.

Figure 18

(Color online) The solution energies $E$ in the Oseen-Frank case with $K_1=4$, $K_2=2$, and $K_3=6$ versus the scaled post height $h∕L_c$.

Figure 19

(Color online) The solution energies $E$ in the one-constant approximation with $K_1=K_2=K_3=2$ versus the scaled post height $h∕L_c$.

Figure 20

(Color online) Computing ${ϵ}_1$ and ${ϵ}_3$ for the unequal constant case in Fig. 18 as a function of the scaled post height $h∕L_c$.