Enhanced electroosmotic flow and ion selectivity in a channel patterned with periodically arranged polyelectrolyte-filled grooves

Abstract

An enhanced electroosmotic flow through a surface-modulated microchannel is considered. The microchannel is modulated by periodically arranged rectangular grooves filled with polyelectrolyte materials. The flat surface of the walls is maintained at a constant charge density. A nonlinear model based on the Poisson–Nernst–Planck equations coupled with the Darcy–Brinkman–Forchheimer equation in the polyelectrolyte region and Navier–Stokes equations in the clear fluid region is adopted. Going beyond the widely employed Debye–Hückel linearization, we adopt numerical methods to elucidate the effect of pertinent parameters on electroosmosis in the patterned channel. The patterned microchannel results in an enhancement in the average EOF by creating an intrinsic velocity slip at the polyelectrolyte–liquid interface. An analytical solution of the EOF for a limiting case in which the groove width is much higher than the channel height is obtained based on the Debye–Hückel approximation. This analytical solution is in good agreement with the present numerical model when a low charge density and a thin Debye layer are considered. We have also established an analogy between the EOF in a polyelectrolyte-filled grooved-channel with the EOF in which the grooves are replaced by the charged slipping planes.

Appendix 1

We now consider a slit channel of scaled height $1+d_1$ with one wall coated with step-like polyelectrolyte of thickness $d_1$. The distribution of monomers in PE is assumed to be uniform. We assume the charge density residing along the channel wall as well the immobile charges entrapped within the PE are low enough so that we can invoke the Debye–Hückel approximation to linearize the Poisson–Boltzmann equation. Under these assumptions, the governing equations for the electric potential as well as velocity field can be written in non-dimensional form as

$$\begin{aligned}&\left\{ \begin{array}{ll} \frac{{\text {d}}^2 \phi }{{\text {d}}y^2}= (\kappa H)^2 \phi - Q_{f}; &{} {-d_1 \le y \le 0} \\ \frac{{\text {d}}^2 \phi }{{\text {d}}y^2}= (\kappa H)^2 \phi ; &{} {0 \le y \le 1} \end{array} \right. \end{aligned}$$

(15)

$$\begin{aligned}&\left\{ \begin{array}{ll} \frac{{\text {d}}^2 u}{{\text {d}}y^2} - (\kappa H)^2 \phi -\beta ^2 u=0; &{} {-d_1 \le y \le 0} \\ \frac{{\text {d}}^2 u}{{\text {d}}y^2} - (\kappa H)^2 \phi =0; &{} {0 \le y \le 1} \end{array} \right. \end{aligned}$$

(16)

Here $Q_{\text {f}}=\rho _{\text {f}}H^2/\varepsilon _{\text {e}}\phi _0$ is the scaled charge density inside the PE and $d_1 = d/H$ is nominal thickness of the PE. The above equations are solved using following boundary conditions

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{{\text {d}} \phi }{{\text {d}}y}=0, \quad u=0; \quad {\text {at }} y=- \ d_1 &{} \\ \frac{{\text {d}} \phi }{{\text {d}}y}=\sigma^* , \quad u=0; \quad {\text {at }} y=1 &{} \end{array} \right. \end{aligned}$$

(17)

The continuity conditions along the PE–electrolyte interface (i.e., $y= 0$) can be expressed as (Matin and Ohshima 2015, 2016)

$$\begin{aligned} \left\{ \begin{array}{ll} \phi |_{y=0^+}=\phi |_{y=0^-} &{} \\ u|_{y=0^+}=u|_{y=0^-} &{} \\ \frac{{\text {d}}\phi }{{\text {d}}y}|_{y=0^+}=\frac{{\text {d}}\phi }{{\text {d}}y}|_{y=0^-} &{} \\ \mu \frac{{\text {d}}u}{{\text {d}}y}|_{y=0^+}={\tilde{\mu }}\frac{{\text {d}}u}{{\text {d}}y}|_{y=0^-} &{} \end{array} \right. . \end{aligned}$$

(18)

Here $\mu$ and ${\tilde{\mu }}$ are the viscosity of the fluid and porous medium, respectively. In this study, we have taken a dilute polyelectrolyte gel so that the viscosity of these two media can be considered to be equal, i.e., $\mu = {\tilde{\mu }}$.

The closed form solution for the electrostatic potential and axial velocity can be obtained as

$$\begin{aligned} \phi (y)=\, & {} \left\{ \begin{array}{ll} \frac{\sigma^* }{\kappa H} \frac{\cosh \kappa H (d_1+y)}{\sinh \kappa H(1+d_1)}+ \frac{Q_{\text {f}}}{(\kappa H)^2} \left[ 1- \frac{\sinh (\kappa H)\cosh \kappa H(d_1+y)}{\sinh \kappa H(1+d_1)}\right] ; &{} {-d_1 \le y \le 0} \\ \frac{\sigma^* }{\kappa H} \frac{\cosh \kappa H (d_1+y)}{\sinh \kappa H(1+d_1)}+ \frac{Q_{\text {f}}}{(\kappa H)^2} \frac{\sinh (\kappa Hd_1)\cosh \kappa H(1-y)}{\sinh \kappa H(1+d_1)}; &{} { 0 \le y \le 1} \end{array} \right. \end{aligned}$$

(19)

$$\begin{aligned} u(y)=\, & {} \left\{ \begin{array}{ll} C_4 \left[ \cosh (\beta y)+\frac{\sinh (\beta y)}{\tanh (\beta d_1)}\right] -C_5\left[ 1+\frac{\sinh (\beta y)}{\sinh (\beta d_1)}\right] \\ -C_6\left[ \frac{\sinh (\beta y)}{\sinh (\beta d_1)} +\cosh \kappa H(d_1+y)\right] \\ +C_7\left[ \frac{\sinh (\beta y)}{\sinh (\beta d_1)} +\cosh \kappa H(d_1+y)\right] ; &{} {-d_1 \le y \le 0 } \\ C_1 \left[ \cosh \kappa H(d_1+y)-\cosh \kappa H(1+d_1)\right] \\ +C_2\left[ \cosh \kappa H(1-y)- 1 \right] -C_3(1-y); &{} {0 \le y \le 1} \end{array} \right. . \end{aligned}$$

(20)

The coefficients $C_i (i=1,2,\ldots ,7)$ are given by

$$\begin{aligned} \left\{ \begin{array}{ll} C_1=\frac{\sigma^* }{\kappa H \sinh \kappa H (1+d_1)} &{} \\ C_2=\frac{Q_{\text {f}}}{(\kappa H)^2} \frac{\sinh (\kappa Hd_1)}{\sinh \kappa H(1+d_1)} &{} \\ C_5=\frac{Q_{\text {f}}}{\beta ^2} &{} \\ C_6=\frac{Q_{\text {f}}}{(\kappa H)^2-\beta ^2} \frac{\sinh (\kappa H)}{\sinh \kappa H(1+d_1)} &{} \\ C_7=\frac{\sigma^* \kappa H}{(\kappa H)^2-\beta ^2} \frac{1}{\sinh \kappa H(1+d_1)} &{} \\ C_3=C_1 [ \cosh (\kappa Hd_1)-\cosh \kappa H(1+d_1) ]+C_2[ \cosh (\kappa H)-1 ]\\ -C_4+C_5+C_6\cosh (\kappa Hd_1)-C_7\cosh (\kappa Hd_1) &{} \\ \tilde{C_4}=C_1 [\cosh (\kappa H d_1)-\cosh \kappa H(1+d_1)+\kappa H \sinh (\kappa Hd_1) ]\\ +C_2\left[ \cosh (\kappa H)-1-\kappa H \sinh (\kappa H)\right] +C_5\left[ 1+\frac{\beta }{\sinh (\beta d_1)}\right] \\ +C_6\left[ \cosh (\kappa H d_1)+\frac{\beta }{\sinh (\beta d_1)}+\kappa H \sinh (\kappa Hd_1)\right] \\ -C_7\left[ \cosh (\kappa H d_1)+\frac{\beta }{\sinh (\beta d_1)}+\kappa H \sinh (\kappa Hd_1)\right] \\ C_4=\frac{\tilde{C_4}}{\left[ 1+\frac{\beta }{\tanh (\beta d_1)}\right] } &{} \end{array} \right. . \end{aligned}$$

(21)

The surface potential ($\phi _{\text {s}}$) and slip velocity ($u_{\text {s}}$) along the PE–electrolyte interface (i.e., $y =0$) can be written as

$$\begin{aligned} \phi _{\text {s}}= \frac{\sigma^* }{\kappa H} \frac{\cosh (\kappa H d_1)}{\sinh \kappa H(1+d_1)}+ \frac{Q_{\text {f}}}{(\kappa H)^2} \frac{\sinh (\kappa Hd_1)\cosh (\kappa H)}{\sinh \kappa H(1+d_1)} \end{aligned}$$

(22)

$$\begin{aligned} u_{\text {s}}={\varLambda }_{\text {s}} \frac{{\text {d}}u_s}{{\text {d}}y}, \end{aligned}$$

(23)

where ${\varLambda }_{\text {s}}=\frac{C_8}{ C_1\left[ \kappa H \sinh (\kappa H d_1)+ \cosh (\kappa H d_1)- \cosh \kappa H (1+d_1) \right] +C_2\left[ \cosh (\kappa H)-\kappa H \sinh (\kappa H)-1 \right] -C_8}$ and

$$\begin{aligned} C_8=C_4-C_5-C_6\cosh (\kappa H d_1)+C_7\cosh (\kappa H d_1). \end{aligned}$$