Nonlinear analysis of flexodomains in nematic liquid crystals

We investigate flexodomains, which are observed in planar layers of certain nematic liquid crystals, when a dc voltage $U$ above a critical value $U_c$ is applied across the layer. They are characterized by stationary stripelike spatial variations of the director in the layer plane with a wave number $p\left(U\right)$. Our experiments for different nematics demonstrate that $p\left(U\right)$ varies almost linearly with $U$ for $U>U_c$. That is confirmed by a numerical analysis of the full nonlinear equations for the director field and the induced electric potential. Beyond this numerical study, we demonstrate that the linearity of $p\left(U\right)$ follows even analytically, when considering a special parameter set first used by Terent'ev and Pikin [Sov. Phys. JETP 56, 587 (1982)]. Their theoretical paper serves until now as the standard reference on the nonlinear analysis of flexodomains, since it has arrived at a linear variation of $p\left(U\right)$ for large $U\gg U_c$. Unfortunately, the corresponding analysis suffers from mistakes, which in a combination led to that result.

Figure 1

Plot of $p_{\text{min}}-p_c$ given by Eq. (30) as function of $\varepsilon =u/u_c-1$ for different values of $\alpha$.

Figure 2

The downward shift ${\mathrm{\Sigma }}_m$ [Eq. (31)] of $p_{\text{min}}-p_c$ as function of $\varepsilon =u/u_c-1$ for different values of $\alpha$.

Figure 3

Phase diagram of flexodomains: neutral curve ${\varepsilon }_N\left(p\right)$, Eckhaus curves ${\varepsilon }_E\left(p\right),$ and ${\varepsilon }_{\text{min}}\left(p\right)$ curves for $\alpha =0$ and $\alpha =8$.

Figure 4

Shadowgraph images of flexodomains recorded at two different applied voltages for the nematic Phase 4. (a) $U^{\text{exp}}-U_c^{\text{exp}}=7$ V; (b) $U^{\text{exp}}-U_c^{\text{exp}}=27$ V. The length of the scale bar is $50\mu \mathrm{m}$, the double arrow indicates the initial director orientation ${\mathit{n}}_0$. The cell thickness is $d=10.8\mu \mathrm{m}$.

Figure 5

The reduced wave number of flexodomains $p-p_c$ (in units of $\pi /d$) as function of $U-U_c$ (in volts) for various nematics: (a) Phase 4, (b) Phase 5, (c) 1OO8, and (d) $7\mathrm{P}\text{−}{\mathrm{CF}}_2\mathrm{OODBP}$. The open circles correspond to the experimental data, where $U$ corresponds to $U^{\text{exp}}$ and $p$ to $p\left(U^{\text{exp}}\right)$. The straight lines present the corresponding theoretical curves of $p_{\text{min}}\left(U\right)$ for the material parameters from Table 1 and $e_1+e_3=0,\pm 10$. Furthermore, the corresponding TPA curves (dashed-dotted) are plotted as well.

Figure 6

Solutions $f\left(z\right)$ of Eq. (24) for $u=8$ ($\varepsilon =3$), $p=p_{\text{min}}\left(u\right)$, and different values of $\alpha$. Since $f\left(z\right)$ is mirror symmetric about $z=0$, it is not shown in the interval $0\le z\le \pi /2$.