Laser Nanofluidics of Liquid Crystals

Abstract

Several scenarios of formation of hydrodynamic flows in nanoscale planar-oriented liquid-crystal (POLC) channels are described by numerical methods within nonlinear generalization of the classical Ericksen–Leslie theory, which allows for consideration of thermomechanical contributions both to the expression for shear stress and the equation of entropy balance. A vortex flow can eventually be formed in a nanoscale POLC channel as a result of the formation of both temperature gradient ∇T (in the initially uniformly heated POLC channel under focused laser irradiation) and director field gradient $\nabla {\mathbf{\hat {n}}}$ (under a static electric field arising in the natural way at the LC phase/solid interface) and due to the interaction between ∇T and $\nabla {\mathbf{\hat {n}}}$.

APPENDIX

The balance equation of torque per unit volume of the LC phase has the form

$${{{\mathbf{T}}}_{{{\text{el}}}}} + {{{\mathbf{T}}}_{{{\text{elast}}}}} + {{{\mathbf{T}}}_{{{\text{vis}}}}} + {{{\mathbf{T}}}_{{{\text{tm}}}}} = 0,$$

where T_el = $- {{\epsilon }_{0}}{{\epsilon }_{a}}{\mathbf{E}}$ × ${\mathbf{\hat {n}}}({\mathbf{E}}$ · ${\mathbf{\hat {n}}})$, T_elast = $\frac{{\delta {{\mathcal{W}}_{{{\text{elast}}}}}}}{{\delta {\mathbf{\hat {n}}}}}$ × ${\mathbf{\hat {n}}}$, T_vis = $\frac{{\delta {{\mathcal{R}}^{{{\text{wis}}}}}}}{{\delta {{{{\mathbf{\hat {n}}}}}_{{,t}}}}}$ × ${\mathbf{\hat {n}}}$, and T_tm = $\frac{{\delta {{\mathcal{R}}^{{{\text{tm}}}}}}}{{\delta {{{{\mathbf{\hat {n}}}}}_{t}}}}$ × ${\mathbf{\hat {n}}}$ are, respectively, the electric, elastic, viscous, and thermomechanical contributions to the torque balance. Here, ${{{\mathbf{\hat {n}}}}_{{,t}}}$ ≡ $\frac{{d{\mathbf{\hat {n}}}}}{{dt}}$ is the material derivative of the vector ${\mathbf{\hat {n}}}$ = ${{n}_{x}}{\mathbf{\hat {i}}}$ + ${{n}_{z}}{\mathbf{\hat {k}}}$.

The balance equation of linear momenta per unit volume of the LC phase can be written as

$$\rho \frac{{d{\mathbf{v}}}}{{dt}} = \nabla \cdot \sigma ,$$

where $\frac{{d{\mathbf{v}}}}{{dt}}$ = $\frac{{\partial {v}}}{{\partial t}}$ + $u{{{v}}_{{,x}}}$ + $w{{{v}}_{{,z}}}$, σ = σ^el + σ^vis + σ^tm – $\mathcal{P}\mathcal{E}$ is the complete expression for the ST consisting of the elastic $\left( {{{\sigma }^{{{\text{e}}{{{\text{l}}}^{{^{{^{{}}}}}}}}}}} \right.$ = $- \frac{{\partial {{\mathcal{W}}_{{{\text{el}}}}}}}{{\partial \nabla {\mathbf{\hat {n}}}}}$ · $\left. {^{{^{{^{{^{{}}}}}}}}{{{(\nabla {\mathbf{\hat {n}}})}}^{{\rm T}}}} \right)$, viscous $\left( {{{\sigma }^{{{\text{vi}}{{{\text{s}}}^{{^{{^{{}}}}}}}}}}} \right.$ = $\left. {\frac{{\delta {{\mathcal{R}}^{{{\text{wis}}}}}}}{{\delta \nabla {\mathbf{v}}}}} \right)$, and thermomechanical $\left( {{{\sigma }^{{{\text{t}}{{{\text{m}}}^{{^{{^{{}}}}}}}}}}} \right.$ = $\left. {\frac{{\delta {{\mathcal{R}}^{{{\text{tm}}}}}}}{{\delta \nabla {\mathbf{v}}}}} \right)$ contributions to the ST. Here, $\mathcal{R}$ = ${{\mathcal{R}}^{{{\text{vis}}}}}$ + ${{\mathcal{R}}^{{{\text{tm}}}}}$ + ${{\mathcal{R}}^{{{\text{th}}}}}$ is the total Rayleigh dissipation function; ${{\mathcal{W}}_{{{\text{el}}}}}$ = $\frac{1}{2}[{{K}_{1}}(\nabla$ · ${\mathbf{\hat {n}}}{{)}^{2}}$ + ${{K}_{3}}({\mathbf{\hat {n}}}$ × ∇ × ${\mathbf{\hat {n}}}{{)}^{2}}]$ is the elastic energy density; K₁ and K₃ are, respectively, the longitudinal and transverse elasticity coefficients; $\mathcal{P}$ is the hydrostatic pressure in the LC system; $\mathcal{E}$ is the unit tensor; and ${{\mathcal{R}}^{{{\text{vis}}}}}$ = ${{\alpha }_{1}}({\mathbf{\hat {n}}}$ · D_s · ${\mathbf{\hat {n}}}{{)}^{2}}$ + ${{\gamma }_{1}}({{{\mathbf{\hat {n}}}}_{t}}$ – D_a · ${\mathbf{\hat {n}}}{{)}^{2}}$ + $2{{\gamma }_{2}}({{{\mathbf{\hat {n}}}}_{t}}$ – D_a · ${\mathbf{\hat {n}}}$)(D_s · ${\mathbf{\hat {n}}}$ – (${\mathbf{\hat {n}}}$ · D_s · ${\mathbf{\hat {n}}}$)${\mathbf{\hat {n}}}$) + α₄D_s: D_s + (α₅ + α₆)(${\mathbf{\hat {n}}}$ · D_s · D_s · ${\mathbf{\hat {n}}}$), $\frac{1}{\xi }{{\mathcal{R}}^{{{\text{tm}}}}}$ = (${\mathbf{\hat {n}}}$ · ∇T)D_s: M + ∇T · D_s · M · ${\mathbf{\hat {n}}}$ + (${\mathbf{\hat {n}}}$ · ∇T)(${{{\mathbf{\hat {n}}}}_{t}}$ – D_a · ${\mathbf{\hat {n}}}$ – 3D_s · ${\mathbf{\hat {n}}}$ + 3(${\mathbf{\hat {n}}}$ · D_s · ${\mathbf{\hat {n}}}$)${\mathbf{\hat {n}}}$) · M · ${\mathbf{\hat {n}}}$ + ${\mathbf{\hat {n}}}$(∇v)^T · M · ∇T + $\frac{1}{2}({\mathbf{\hat {n}}}$ · D_s · ${\mathbf{\hat {n}}}$)∇T · M · ${\mathbf{\hat {n}}}$ + ${{{\mathbf{\hat {n}}}}_{t}}$ · M · ∇T + $\frac{1}{2}{{\mathcal{M}}_{0}}\nabla T$ · ∇v · ${\mathbf{\hat {n}}}$ + (${\mathbf{\hat {n}}}$ · ∇T) ${{\mathcal{M}}_{0}}({\mathbf{\hat {n}}}$ · D_s · ${\mathbf{\hat {n}}}$) + $\frac{1}{2}{{\mathcal{M}}_{0}}{{{\mathbf{\hat {n}}}}_{t}}$ · ∇T, and ${{\mathcal{R}}^{{{\text{th}}}}}$ = $\frac{1}{T}$(${{\lambda }_{{||}}}({\mathbf{\hat {n}}}$ · ∇T))² + ${{\lambda }_{ \bot }}(\nabla T$ – ${\mathbf{\hat {n}}}({\mathbf{\hat {n}}}$ · ∇T)²) are, respectively, the viscous, thermomechanical, and thermal contributions to the complete expression for the Rayleigh function $\mathcal{R}$. Here, α₁–α₆ are the Leslie viscosity coefficients; γ₁(T) and γ₂(T) are the rotational viscosity coefficients of the LC system; ξ is the thermomechanical constant; λ_|| and λ_⊥ are the thermal conductivities of the LC system along and across the direction of the director ${\mathbf{\hat {n}}}$, respectively. Tensors D_s = $\frac{1}{2}[\nabla {\mathbf{v}}$ + ${{(\nabla {\mathbf{v}})}^{{\text{T}}}}]$ and D_a = $\frac{1}{2}[\nabla {\mathbf{v}}$ – ${{(\nabla {\mathbf{v}})}^{{\text{T}}}}]$ are, respectively, the symmetric and asymmetric contributions to the strain rate tensor, M = $\frac{1}{2}[\nabla {\mathbf{\hat {n}}}$ + ${{(\nabla {\mathbf{\hat {n}}})}^{{\text{T}}}}]$, and ${{\mathcal{M}}_{0}}$ = ∇ · ${\mathbf{\hat {n}}}$ is the scalar invariant of tensor M.

The heat-conduction equation describing a change in the temperature field T(x, z, t) under the action of the heat flux q through the upper POLC-channel boundary has the form

$$\rho {{C}_{P}}\frac{{dT}}{{dt}} = - \nabla \cdot {\mathbf{Q}},$$

where Q = $- T\frac{{\delta \mathcal{R}}}{{\delta \nabla T}}$ is the heat flux in the LC system and C_P is the specific heat coefficient.