Electro-osmosis in nematic liquid crystals

We derive a mathematical model of a nematic electrolyte based on a variational formulation of nematodynamics. We verify the model by comparing its predictions to the results of the experiments on the substrate-controlled liquid-crystal-enabled electrokinetics. In the experiments, a nematic liquid crystal confined to a thin planar cell with surface-patterned anchoring conditions exhibits electro-osmotic flows along the “guiding rails” imposed by the spatially varying director. Extending our previous work, we consider a general setup which incorporates dielectric anisotropy of the liquid-crystalline matrix and the full set of nematic viscosities.

Figure 1

Sketch of charge separation: (a),(d) in the cell $A$; (b),(e) in the cell $B$; (c),(f) in the cell $C$. White arrows show the direction of the triggered electro-osmotic flow of the liquid crystal. The key parameters ${\lambda }_{\sigma }={\sigma }_{\parallel }/{\sigma }_{\perp }$ and ${\lambda }_{\varepsilon }={\varepsilon }_{\parallel }/{\varepsilon }_{\perp }$ are defined as the ratios of the nematic conductivity and permittivity along and perpendicular to the director, respectively. (a)–(c): ${\lambda }_{\varepsilon }-{\lambda }_{\sigma }<0$; (d)–(f): ${\lambda }_{\varepsilon }-{\lambda }_{\sigma }>0$.

Figure 2

Nondimensional charge concentration: (a),(d) in the cell $A$; (b),(e) in the cell $B$; (c),(f) in the cell $C$. (a)–(c): ${\lambda }_{\sigma }=1.4, {\lambda }_{\varepsilon }=1$ (dashed line) and ${\lambda }_{\sigma }=3, {\lambda }_{\varepsilon }=1$ (solid line); (d)–(f): ${\lambda }_{\sigma }=1.4, {\lambda }_{\varepsilon }=4$ (dashed line) and ${\lambda }_{\sigma }=3, {\lambda }_{\varepsilon }=4$ (solid line).

Figure 3

Nondimensional flow velocity: (a),(d) in the cell $A$; (b),(e) in the cell $B$; (c),(f) in the cell $C$. Plots (a)–(c) correspond to the fixed anisotropy ${\lambda }_{\varepsilon }=1, {\lambda }_{\sigma }=1.4$ and varying viscosity: ${\tilde{\eta }}_1=0.6, {\tilde{\eta }}_2=2.5$, and ${\tilde{\eta }}_{12}=0.08$ (solid line); ${\tilde{\eta }}_1=1, {\tilde{\eta }}_2=1$, and ${\tilde{\eta }}_{12}=0$ (dashed line); ${\tilde{\eta }}_1=5.1, {\tilde{\eta }}_2=1.3$, and ${\tilde{\eta }}_{12}=0.7$ (dotted line). Plots (d)–(f) correspond to the fixed viscosity ${\tilde{\eta }}_1=0.6, {\tilde{\eta }}_2=2.5$, and ${\tilde{\eta }}_{12}=0.08$ and varying anisotropy: ${\lambda }_{\varepsilon }={\left(1.4\right)}^{-1}, {\lambda }_{\sigma }=1$ (solid line); ${\lambda }_{\varepsilon }=1, {\lambda }_{\sigma }=1.4$ (dashed line).

Figure 4

Nondimensional concentration of the positive ions in the cell $A$ found from (49) and (50) numerically (points) and analytically (solid lines). (a): ${\lambda }_{\varepsilon }=1, {\lambda }_{\sigma }=1.5$. (b): ${\lambda }_{\varepsilon }=1.5, {\lambda }_{\sigma }=1$. $F=47$ and $G=63$ correspond to the experiment [11], where $E=24\mathrm{mV}/\mu \mathrm{m}, \overline{c}={10}^{19}{\mathrm{m}}^{-3}, L=50\mu \mathrm{m}$, and ${\varepsilon }_{\perp }=6$.

Figure 5

Comparison between the numerical (dashed line) and asymptotic (solid line) solutions in the cell $C$. (a) Nondimensional charge concentration $Q^C=c^{+}-c^{-}$ and (b) nondimensional flow velocity. Both plots were obtained under the assumption that $\eta =1, {\lambda }_{\varepsilon }=1, {\lambda }_{\sigma }=1.4, F=47$, and $G=63$.

Figure 6

Behavior of the nondimensional concentration $c^{+}$ found numerically (solid line) and analytically (dashed line) in the cell $A$ when the nematic is extremely anisotropic. (a): ${\lambda }_{\varepsilon }=1, {\lambda }_{\sigma }=400$; (b): ${\lambda }_{\varepsilon }=50, {\lambda }_{\sigma }=1$. Both plots were obtained under the assumption that $F=47$ and $G=63$.