Bifurcation properties of nematic liquid crystals exposed to an electric field: Switchability, bistability, and multistability

Bistable liquid crystal displays (LCDs) offer the potential for considerable power savings compared with conventional (monostable) LCDs. The existence of two (or more) stable field-free states that are optically distinct means that contrast can be maintained in a display without an externally applied electric field. An applied field is required only to switch the device from one state to the other, as needed. In this paper we examine the basic physical principles involved in generating multiple stable states and the switching between these states. We consider a two-dimensional geometry in which variable surface anchoring conditions are used to control the steady-state solutions and explore how different anchoring conditions can influence the number and type of solutions and whether or not switching is possible between the states. We find a wide range of possible behaviors, including bistability, tristability, and tetrastability, and investigate how the solution landscape changes as the boundary conditions are tuned.

Figure 1

Sketch showing the setup and summarizing the key parameters in the dimensionless coordinates.

Figure 2

The 3D region of $({\mathcal{A}}_0,{\mathcal{A}}_1,{\alpha }_0^{\left(0\right)})$ parameter space within which two-way switching is achieved for the 1D model, showing the particular points $P_1$, $P_2$, and $P_3$ used in our investigations of Sec. 3. All points lying within the solid shaded region give two-way switching under the given switching protocol. The grayscale (color online) corresponds to the value of the anchoring angle at $z=0$.

Figure 3

Explanation of symbols used in the switching results that follow below. The notation within curly brackets denotes which steady states exist at a given point in parameter space. The statement $n_i\to n_j$ denotes that switching occurs from state ${\mathit{n}}_i$ to ${\mathit{n}}_j$ and $n_i\leftrightarrow n_j$ denotes that two-way switching occurs between states ${\mathit{n}}_i$ and ${\mathit{n}}_j$.

Figure 4

The three steady states (a) ${\mathit{n}}_1$, (b) ${\mathit{n}}_2$, and (c) ${\mathit{n}}_3$, corresponding to the point $(L,\delta )=(4,0.7)$ in Fig. 5.

Figure 5

Switching results when perturbing the 1D case represented by point $P_1$, $({\mathcal{A}}_0,{\mathcal{A}}_1,{\alpha }_0^{\left(0\right)})=(5.41,2.45,1.40)$, according to Eq. (16). Symbols are defined in the global legend of Fig. 3.

Figure 6

Bifurcation diagrams representing stable steady states obtained when perturbing the 1D case represented by point $P_1$, $({\mathcal{A}}_0,{\mathcal{A}}_1,{\alpha }_0^{\left(0\right)})=(5.41,2.45,1.40)$, for the cases (a) $L=0.5$ and (b) $L=4$.

Figure 7

Switching results when perturbing the 1D case represented by point $P_2$, $({\mathcal{A}}_0,{\mathcal{A}}_1,{\alpha }_0^{\left(0\right)})=(5.50,2.30,1.46)$, according to Eq. (16). Symbols are defined in the global legend of Fig. 3.

Figure 8

Bifurcation diagrams representing stable steady states obtained when perturbing the 1D case represented by point $P_2$, $({\mathcal{A}}_0,{\mathcal{A}}_1,{\alpha }_0^{\left(0\right)})=(5.50,2.30,1.46)$, for the cases (a) $L=0.5$ and (b) $L=4$.

Figure 9

The four steady states (a) ${\mathit{n}}_1$, (b) ${\mathit{n}}_2$, (c) ${\mathit{n}}_3$, and (d) ${\mathit{n}}_4$, corresponding to the point $(L,\delta )=(6,0.6)$ in Fig. 7.

Figure 10

Switching results when perturbing the 1D case represented by point $P_3$, $({\mathcal{A}}_0,{\mathcal{A}}_1,{\alpha }_0^{\left(0\right)})=(4.85,2.10,1.46)$, according to Eq. (16). Symbols are defined in the global legend of Fig. 3.

Figure 11

Bifurcation diagrams representing stable steady states obtained when perturbing the 1D case represented by point $P_3$, $({\mathcal{A}}_0,{\mathcal{A}}_1,{\alpha }_0^{\left(0\right)})=(4.85,2.10,1.46)$, for the cases (a) $L=0.5$ and (b) $L=4$.

Figure 12

Switching results when perturbing the 1D case represented by point $P_1$, $({\mathcal{A}}_0,{\mathcal{A}}_1,{\alpha }_0^{\left(0\right)})=(5.41,2.45,1.40)$, according to Eq. (17). The domain length is fixed at $L=0.5$. Symbols are defined in the global legend of Fig. 3.

Figure 13

Switching results when perturbing the 1D case represented by point $P_2$, $({\mathcal{A}}_0,{\mathcal{A}}_1,{\alpha }_0^{\left(0\right)})=(5.50,2.30,1.46)$, according to Eq. (17). The domain length is fixed at $L=0.5$. Symbols are defined in the global legend of Fig. 3.

Figure 14

Switching results when perturbing the 1D case represented by point $P_3$, $({\mathcal{A}}_0,{\mathcal{A}}_1,{\alpha }_0^{\left(0\right)})=(4.85,2.10,1.46)$, according to Eq. (17). The domain length is fixed at $L=1$. Symbols are defined in the global legend of Fig. 3.

Figure 15

Switching results when perturbing the 1D case represented by point $P_1$, $({\mathcal{A}}_0,{\mathcal{A}}_1,{\alpha }_0^{\left(0\right)})=(5.41,2.45,1.40)$, according to Eq. (18). The lower boundary only is perturbed and $\delta$ and $L$ vary. Symbols are defined in the global legend of Fig. 3.

Figure 16

Switching results when perturbing the 1D case represented by point $P_2$, $({\mathcal{A}}_0,{\mathcal{A}}_1,{\alpha }_0^{\left(0\right)})=(5.50,2.30,1.46)$, according to Eq. (18). The lower boundary only is perturbed and $\delta$ and $L$ vary. Symbols are defined in the global legend of Fig. 3.

Figure 17

Switching results when perturbing the 1D case represented by point $P_3$, $({\mathcal{A}}_0,{\mathcal{A}}_1,{\alpha }_0^{\left(0\right)})=(4.85,2.10,1.46)$, according to Eq. (18). The lower boundary only is perturbed and $\delta$ and $L$ vary. Symbols are defined in the global legend of Fig. 3.