Electrically driven nematic flow in microfluidic devices containing a temperature gradient

Fluid pumping principle has been developed utilizing the interaction, on the one hand, between the electric field $\mathbf{\text{E}}$ and the gradient $\mathbf{\nabla }\widehat{\mathbf{\text{n}}}$ of the director's field, and, on the other hand, between the $\mathbf{\nabla }\widehat{\mathbf{\text{n}}}$ and the temperature $\mathbf{\nabla }T$ gradient arising in a homogeneously aligned liquid crystal (HALC) microfluidic channel. Calculations, based upon the nonlinear extension of the classical Ericksen-Leslie theory, with accounting the entropy balance equation, show that due to the coupling among the $\mathbf{\nabla }T$, $\mathbf{\nabla }\widehat{\mathbf{\text{n}},}$ and $\mathbf{\text{E}}$ in the HALC microfluidic channel the horizontal flow $\mathbf{\text{v}}=v_x\widehat{\mathbf{\text{i}}}=u\widehat{\mathbf{\text{i}}}$ may be excited. The direction and magnitude of $\mathbf{\text{v}}$ is influenced both by the heat flux $\mathbf{\text{q}}$ across the microfluidic channel and the strength of the electric field $\mathbf{\text{E}}$. The results of calculations showed that the dependence of the maximum value of the equilibrium velocity distribution $|u_{\max }(E/E_{\mathrm{th}})|$ across the LC channel versus electric field $E/E_{\mathrm{th}}$ is characterized by maximum value at $E/E_{\mathrm{th}}=2.0$. In the case when the electric field $E\gg E_{\mathrm{th}}$, the horizontal flow of the LC material completely stops and a novel mechanism of converting of the electric field in the form of the kinklike wave reorientation of the director field $\widehat{\mathbf{\text{n}}}$ can be excited in the LC channel.

Figure 1

The coordinate system used for theoretical analysis. The $x$ axis is taken as being parallel to the director directions on the lower and upper surfaces, $\theta (z,t)$ is the angle between the director $\widehat{\mathbf{\text{n}}}$ and the unit vector $\widehat{\mathbf{\text{k}}}$, respectively. Both the electric field $\mathbf{\text{E}}$ and the unit vector $\widehat{\mathbf{\text{k}}}$ are directed normal to the horizontal surfaces of the LC channel.

Figure 2

Plot of the evolution of the polar angle $\theta (z,{\tau }_k)$ to its equilibrium distribution across the HALC microfluidic channel, under the effect of the electric field $E/E_{\text{th}}=2.0$, at different times ${\tau }_k=\mathrm{\Delta }\tau \left(k-1\right)(k=1,...,11)$, respectively. The curves shown on the left-hand side (a) correspond to the case when $\mathrm{\Delta }\chi =0.0$, whereas the curves shown on the right-hand side (b) correspond to the case when $\mathrm{\Delta }\chi =0.0162$, respectively.

Figure 3

Plot of the evolution of the polar angle $\theta (z,{\tau }_k)$ to its equilibrium distribution across the HALC microfluidic channel, both under the effect of the electric field $E/E_{\text{th}}=2.0$ and the heat flux $\mathbf{\text{q}}$, at different times ${\tau }_k=\mathrm{\Delta }\tau \left(k-1\right)(k=1,...,11)$, respectively. The curves shown on the left-hand side (a) correspond to case I, when $\mathbf{\text{q}}=q\widehat{\mathbf{\text{k}}}$, whereas the curves shown on the right-hand side (b) correspond to case II, when $\mathbf{\text{q}}=-q\widehat{\mathbf{\text{k}}}$, respectively.

Figure 4

Plot of the evolution of the dimensionless velocity field $u(z,{\tau }_k)$ to its equilibrium distribution across the HALC microfluidic channel, under the effect of the electric field $E/E_{\text{th}}=2.0$, at different times ${\tau }_k=\mathrm{\Delta }\tau \left(k-1\right)(k=1,...,11)$, respectively. The curves shown on the left-hand side (a) correspond to case I, when $\mathbf{\text{q}}=q\widehat{\mathbf{\text{k}}}$, whereas the curves shown on the right-hand side (b) correspond to case II, when $\mathbf{\text{q}}=-q\widehat{\mathbf{\text{k}}}$, respectively.

Figure 5

Same as described in the caption of Fig. 4, but a plot of the evolution of the temperature field $\chi (z,\tau )$ to its equilibrium distribution across the HALC microfluidic channel.

Figure 6

Plot of the evolution of the polar angle $\theta (z,{\tau }_k)$ (a) and the dimensionless velocity field $u(z,{\tau }_k)$ (c) to their equilibrium distributions across the HALC microfluidic channel, both under the effect of the electric field $E/E_{\text{th}}=8.0$ and the heat flux $\mathbf{\text{q}}=q\widehat{\mathbf{\text{k}}}$. (b), (d) Same as in panels (a) and (c), respectively, but there is no temperature gradient $\mathbf{\nabla }\chi =0.0$.

Figure 7

Same as described in the caption of Fig. 6, but the value of the electric field $E/E_{\text{th}}$ is equal to 10.0

Figure 8

(a) Plot of the equilibrium dimensionless velocity $u(z,{\tau }_{11})=u_{\text{eq}}\left(z\right)$ vs. the dimensionless space coordinate $z$, both under the effect of the heat flux $\mathbf{\text{q}}=q\widehat{\mathbf{\text{k}}}$, directed from the lower cooler to the upper hotter restricted surfaces, and the electric field $E/E_{\text{th}}=2.0$ (curve 1), 4.0 (curve 2), 8.0 (curve 3), and 10.0 (curve 4). (b) Dependence of the maximum value of the absolute equilibrium velocity $|u_{\max }(E/E_{\mathrm{th}})|$ on the value of the electric field $E/E_{\mathrm{th}}$, for the number of regimes, cases B (curve 1), BB (curve 2), and BBB (curve 3).