Microfluidics of liquid crystals induced by laser radiation

Several scenarios for the formation of hydrodynamic flows in microsized hybrid aligned nematic (HAN) channels, based on the appropriate nonlinear extension of the classical Ericksen-Leslie theory, supplemented by thermomechanical correction of the shear stress and Rayleigh dissipation function, as well as taking into account the entropy balance equation, are analyzed. Detailed numerical simulations were performed to elucidate the role of the heat flux $\mathbf{\text{q}}$ caused by laser radiation focused on the lower boundary of the equally warmed up the HAN channel containing a monolayer of azobenzene with the possibility of a trans-cis and cis-trans conformational changes in formation of the vortex flow $\mathbf{\text{v}}$. It is shown that a thermally excited vortex flow is maintained with motion in a positive sense (clockwise) in the vicinity of the orientation defect at the lower boundary of the HAN channel caused by the trans-cis and cis-trans conformational changes. In the case of the same HAN channel, but without the azobenzene monolayer at the lower boundary, the heat flux $\mathbf{\text{q}}$ can also produce the vortical flow in the vicinity of the laser spot at the lower boundary, directed in a negative sense (counterclockwise). At that, the second vortex is characterized by a much slower speed than the vortical flow in the first case.

Figure 1

The coordinate system used for theoretical analysis. The $x$-axis is taken as being parallel to the director directions on the upper surface, and $\theta (x,z,t)$ is the angle between the director $\widehat{\mathbf{\text{n}}}$ and the unit vector $\widehat{\mathbf{\text{k}}}$, respectively. Both the heat flux $\mathbf{\text{q}}=q_z\widehat{\mathbf{\text{k}}}$ and the unit vector $\widehat{\mathbf{\text{k}}}$ are directed normal to the horizontal boundaries of the LC channel. $2L$ is the size of the laser spot.

Figure 2

(a) The initial distribution of the polar angle $\theta (-1.0\le x\le 1.0,z=-1.0,\tau =0)$ in the vicinity of the laser spot ($\mathrm{\Delta }=2l=0.1$). (b) The stationary distribution of the polar angle $\theta (-1.0\le x\le 1.0,z=-0.98,\tau ={\tau }_{\mathrm{in}})$ (in rad) along the width ($-1.0\le x\le 1.0$) of the HAN channel in the vicinity of the lower ($z=-0.98$) boundary.

Figure 3

(a) The stationary distribution of the polar angle $\theta (x=0,-1.0\le z\le 1.0,\tau ={\tau }_{\mathrm{in}})$ (in rad) across the middle part ($x=0$) of the HAN channel. (b) Fragment of the stationary distribution of the director field ${\widehat{\mathbf{\text{n}}}}_{st}(x,z)=\widehat{\mathbf{\text{n}}}(x,z,\tau ={\tau }_{\mathrm{in}})$ in the vicinity of the laser spot. The Gaussian spot size $\mathrm{\Delta }=2l=0.1$ and ${\tau }_{\mathrm{in}}={10}^{-4}$.

Figure 4

(a) The relaxation of the dimensionless temperature $\chi (x,z=-0.98,\tau )$ to its stationary distribution $\chi (x,z=-0.98,\tau ={\tau }_{\mathrm{in}})$ along the fragment of the width ($-1.0\le x\le 1.0,z=-0.98$) of the HAN channel, at different times ${\tau }_1={10}^{-7}(\sim 1\text{ns})$ [curve (1)], ${\tau }_2=1.6\times {10}^{-6}(\sim 2.8\mu \text{s})$ [curve (2)], ${\tau }_3=6.4\times {10}^{-6}(\sim 11.2\mu \text{s})$ [curve (3)], ${\tau }_4=2.56\times {10}^{-5}(\sim 45\mu \text{s})$ [curve (4)], and ${\tau }_5={\tau }_{\mathrm{in}}={10}^{-4}(\sim 0.18\text{ms})$ [curve (5)]. (b) The same as in (a), but the relaxation of $\chi (x=0,z,\tau )$ to its stationary distribution $\chi (x=0.0,z,\tau ={\tau }_{\mathrm{in}})$ across the middle part ($x=0$) of the HAN channel.

Figure 5

(a) The relaxation of the vertical component of the velocity $w(x,z=-0.98,\tau )$ to its stationary distribution $w(x,z=-0.98,\tau ={\tau }_{\mathrm{in}})$ along the fragment of the width ($-0.15\le x\le 0.15,z=-0.98$) of the HAN channel, at different times ${\tau }_1={10}^{-7}(\sim 1\text{ns})$ [curve (1)], ${\tau }_2=1.6\times {10}^{-6}(\sim 2.8\mu \text{s})$ [curve (2)], ${\tau }_3=6.4\times {10}^{-6}(\sim 11.2\mu \text{s})$ [curve (3)], ${\tau }_4=2.56\times {10}^{-5}(\sim 45\mu \text{s})$ [curve (4)], and ${\tau }_5={\tau }_{\mathrm{in}}={10}^{-4}(\sim 0.18\text{ms})$ [curve (5)]. (b) The same as in (a), but for the horizontal component of the velocity $u(x,z=-0.98,\tau )$.

Figure 6

(a) Distribution of the velocity field $\mathbf{\text{v}}$ in the microsized HAN channel with the orientational defect located at $-l\le x\le l,z=-1.0$, when the heating occurs during ${\tau }_{\mathrm{in}}={10}^{-4}(\sim 0.18\text{ms})$, whereas the values of the dimensionless heat flux coefficient ${\mathcal{Q}}_z$ is 0.18 ($\sim 1.5\times {10}^{-3}\text{mW}/\mu {\text{m}}^2$), respectively. Here 1 mm of the arrow length is equal to $1.8\mu \mathrm{m}/\mathrm{s}$. (b) The same as in (a), but without the orientational defect. Here 1 mm of the arrow length is equal to 1.3 nm/s.

Figure 7

(a) The stationary distribution of the polar angle $\theta (-10.0\le x\le 10.0,z=-0.98,\tau ={\tau }_{\mathrm{in}})$ along the width ($-10.0\le x\le 10.0$) of the HAN channel, in the vicinity of the lower ($z=-0.98$) boundary. (b) The stationary distribution of the polar angle $\theta (x=0.0,-1.0\le z\le 1.0,\tau ={\tau }_{\mathrm{in}})$ (in rad) across the middle part ($x=0.0$) of the HAN channel.

Figure 8

(a) The relaxation of the dimensionless temperature $\chi (x,z=-0.98,\tau )$ to its stationary distribution $\chi (x,z=-0.98,\tau ={\tau }_{\mathrm{in}})$ along the fragment of the width ($-1.0\le x\le 1.0,z=-0.98$) of the HAN channel, at different times ${\tau }_1={10}^{-7}(\sim 1\text{ns})$ [curve (1)], ${\tau }_2=1.6\times {10}^{-6}(\sim 2.8\mu \text{s})$ [curve (2)], ${\tau }_3=6.4\times {10}^{-6}(\sim 11.2\mu \text{s})$ [curve (3)], ${\tau }_4=2.56\times {10}^{-5}(\sim 45\mu \text{s})$ [curve (4)], and ${\tau }_5={\tau }_{\mathrm{in}}={10}^{-4}(\sim 0.18\text{ms})$ [curve (5)], respectively. (b) The same as in (a), but the relaxation of $\chi (x=0.0,z,\tau )$ to its stationary distribution $\chi (x=0.0,z,\tau ={\tau }_{\mathrm{in}})$ across the middle part ($x=0.0$) of the HAN channel.

Figure 9

(a) Plot of the relaxation of the vertical component of the velocity $w(x,z=-0.98,\tau )$, in case B, to its stationary distribution $w(x,z=-0.98,\tau ={\tau }_{\mathrm{in}})$ along the fragment of the width ($-1.0\le x\le 1.0,z=-0.98$) of the HAN channel, at different times ${\tau }_1={10}^{-7}(\sim 1\text{ns})$ [curve (1)], ${\tau }_2=1.6\times {10}^{-6}(\sim 2.8\mu \text{s})$ [curve (2)], ${\tau }_3=6.4\times {10}^{-6}(\sim 11.2\mu \text{s})$ [curve (3)], ${\tau }_4=2.56\times {10}^{-5}(\sim 45\mu \text{s})$ [curve (4)], and ${\tau }_5={\tau }_{\mathrm{in}}={10}^{-4}(\sim 0.18\text{ms})$ [curve (5)], respectively. (b) The same as in (a), but for the horizontal component of the velocity $u(x,z=-0.98,\tau )$.