Electric-field variations within a nematic-liquid-crystal layer

A thin layer of nematic liquid crystal (NLC) across which an electric field is applied is a setup of great industrial importance in liquid crystal display devices. There is thus a large literature modeling this situation and related scenarios. A commonly used assumption is that an electric field generated by electrodes at the two bounding surfaces of the layer will produce a field that is uniform: that is, the presence of NLC does not affect the electric field. In this paper, we use calculus of variations to derive the equations coupling the electric potential to the orientation of the NLC's director field, and use a simple one-dimensional model to investigate the limitations of the uniform field assumption in the case of a steady applied field. The extension of the model to the unsteady case is also briefly discussed.

Figure 1

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

Figure 2

Leading order solutions for small field limit $\mathcal{F}\ll 1$, discussed in Sec. 3b [Eq. (24), with $\widetilde{{\rm Y}}=1$; and Eq. (28)]; and for $O$$\left(1\right)$ field case with ${\rm Y}\gg 1$ (see Sec. 3c) given by Eqs. (29) and (30), for $\mathcal{F}=1$. All solutions use $\varpi =0.25$, ${\alpha }_0=\pi /8$, and ${\alpha }_1=3\pi /8$; anchoring is strong in all cases.

Figure 3

Comparison of the director field and electric potential with uniform and nonuniform field models at large field with weak anchoring: $\mathcal{F}=10$, ${\rm Y}=2.5$, ${\mathcal{A}}_0=5.0$, ${\mathcal{A}}_1=2.4$, ${\alpha }_0=0$, ${\alpha }_1=\pi /2$.

Figure 4

Comparison of the director field and electric potential with uniform and nonuniform field models at small field with (a) ${\rm Y}\sim \mathcal{F}\ll 1$ and weak anchoring: $\mathcal{F}=-0.1$, ${\rm Y}=0.1$, ${\mathcal{A}}_0=5.0$, and ${\mathcal{A}}_1=2.4$; and (b) ${\rm Y}\sim {\mathcal{F}}^{-1}\gg 1$ and strong anchoring: $\mathcal{F}=0.1$, ${\rm Y}=10$, ${\mathcal{A}}_0={\mathcal{A}}_1=1000$. In both cases, ${\alpha }_0=0,{\alpha }_1=\pi /2$.

Figure 5

Comparison of the director field and electric potential with uniform and nonuniform field models at $O\left(1\right)$ field with strong anchoring: $\mathcal{F}=1.0$, ${\rm Y}=10.0$, ${\mathcal{A}}_0={\mathcal{A}}_1=1000.0$, ${\alpha }_0=\pi /8$, ${\alpha }_1=3\pi /8$.

Figure 6

Comparison of the director field and electric potential with uniform and nonuniform field models at $O\left(1\right)$ field with weak anchoring: $\mathcal{F}=1.0$, ${\rm Y}=10.0$, ${\mathcal{A}}_0=5.0$, ${\mathcal{A}}_1=2.4$, ${\alpha }_0=0$, ${\alpha }_1=3\pi /8$.

Figure 7

(a) Deviation of the dimensionless electric potential $\varphi$ from the uniform field case as $\mathcal{F}$ varies: ${\rm Y}=10.0$, ${\mathcal{A}}_0={\mathcal{A}}_1=1000.0$ (strong anchoring). (b) Deviation of the dimensional electric potential ${\varphi }^{*}$ from uniform as the applied voltage $P^{*}$ varies. The ranges of the two plots coincide. Further details are given in the text.

Figure 8

Deviation of the director angle $\theta$ (solid line) and total free energy $J$ (dashed line) as $\mathcal{F}$ varies: ${\rm Y}=10.0$, ${\mathcal{A}}_0={\mathcal{A}}_1=1000.0$ (strong anchoring).

Figure 9

(a) Deviation of the electric potential $\varphi$ from the uniform field case as $\mathcal{F}$ varies: ${\rm Y}=10.0$, ${\mathcal{A}}_0=5.0$, and ${\mathcal{A}}_1=2.4$ (weak anchoring). (b) Deviation of the dimensional electric potential ${\varphi }^{*}$ from uniform as the applied voltage $P^{*}$ varies. The ranges of the two plots coincide. Further details are given in the text.

Figure 10

Deviation of the director $\theta$ (solid line) and total free energy $J$ (dashed line) as $\mathcal{F}$ varies: ${\rm Y}=10.0$, ${\mathcal{A}}_0=5.0$, and ${\mathcal{A}}_1=2.4$ (weak anchoring).