Electroosmotic flow reversal and ion selectivity in a soft nanochannel

Abstract

This article deals with the modulation of electroosmotic flow (EOF) and transport of ionic species through the parallel plate soft nanochannel. The charged rigid walls of the channel are covered by diffuse polyelectrolyte layer (PEL) which entraps immobile charges. A diffuse distribution of the polymer segment density and charge density is assumed. A nonlinear model based on the Poisson-Nernst-Planck equations coupled with the Darcy-Brinkman equations is adopted. Going beyond the widely employed Debye-H$\ddot {u}$ckel linearization, we adopt a sophisticated numerical tool to study the effect of pertinent parameters on the modulation of EOF through the soft nanochannel. Several interesting key features including the flow reversal, occurrence of zero flow rate, and perm selectivity are studied by regulating the charges entrapped within the diffuse PEL and the surface charge distributed along the channel wall. The results indicate that the channel can be cation-selective, anion-selective, and non-selective based on the nature of the charges within the PEL and wall charge. We have also identified the parameter range for validity of the linearized model for the case of step-like PEL.

Appendix

We now consider a step-like PEL coating of thickness d₁ in which the monomers are uniformly distributed (α = 0). We assume the charge density of the channel walls as well as the immobile charges entrapped within the PEL are low enough so that we can invoke the Debye-H$\ddot {u}$ckel approximation to linearize the Poisson-Boltzmann equation. Under these assumptions, the governing equations for the electric potential and the velocity field can be written in nondimensional form as

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

(A.1)

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

(A.2)

Here, Q _f = (κ _s h)²/2 is the scaled charge inside the PEL and d₁ = d/h is nominal thickness of the PEL. The above equations are solved by using the following boundary conditions

$$ \left\{\begin{array}{ll} \frac{d \phi}{dy}= 0,~ \frac{d u}{dy}= 0;~~~ \text{at}~ y = 1/2 & \\ \frac{d \phi}{dy}=-\sigma^{*},~u = 0~~~ \text{at}~ y = 0 & \end{array} \right. $$

(A.3)

A symmetry condition is considered along the central line (y = 1/2) of the channel. The continuity conditions along the PEL-electrolyte interface (y = d₁) can be expressed as

$$ \left\{ \begin{array}{ll} \phi|_{y=d_{1}^{+}}=\phi|_{y=d_{1}^{-}} & \\ u|_{y=d_{1}^{+}}=u|_{y=d_{1}^{-}} & \\ \frac{d\phi}{dy}|_{y=d_{1}^{+}}=\frac{d\phi}{dy}|_{y=d_{1}^{-}} & \\ \frac{du}{dy}|_{y=d_{1}^{+}}=\frac{du}{dy}|_{y=d_{1}^{-}} & \end{array} \right. $$

(A.4)

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

$$ \phi(y){}={}\left\{ \begin{array}{ll} \frac{zQ_{f}}{(\kappa h)^{2}} \left[ 1- \frac{\sinh kh(\frac{1}{2}-d_{1})\cosh(\kappa hy)}{\sinh \frac{\kappa h}{2}}\right] + \frac{\sigma^{*}}{\kappa h} \frac{\cosh \kappa h (\frac{1}{2}-y)}{\sinh \frac{\kappa h}{2}}; & 0 \leq y \leq d_1 \\ \frac{1}{\sinh(\frac{\kappa h}{2})}\left[ \frac{z Q_{f}\sinh(\kappa hd_{1})}{(\kappa h)^{2}} {}+{} \frac{\sigma^{*}}{\kappa h} \right]\cosh \kappa h(\frac{1}{2}{}-{}y);{} & d_1 {}\leq{} y {}\leq 1/2 \end{array} \right. $$

(A.5)

$$ u(y)=\left\{ \begin{array}{ll} C_{1} \cosh(\beta y)+C_{2} \sinh(\beta y)+\\ C_{3}+C_{4} \cosh(\kappa hy)+ C_{5}\sinh(\kappa hy); & 0 \leq y \leq d_1 \\ C_{6} \cosh \kappa h(\frac{1}{2}-y)+C_{7}; & d_1 \leq y \leq 1/2 \end{array} \right. $$

(A.6)

The coefficients C _i (i = 1, 2,..7) are given by

$$ \left\{ \begin{array}{ll} C_{1}=-(C_{3}+C_{4}) & \\ C_{2}=\frac{ -\beta C_{1} \sinh(\beta d_{1}) -C_{6}\kappa h \sinh \kappa h (\frac{1}{2}-d_{1})-\kappa h C_{4} \sinh (\kappa hd_{1})-\kappa h C_{5} \cosh(\kappa hd_{1})}{\beta \cosh(\beta d_{1})} & \\ C_{3}=-\frac{z Q_{f}}{\beta^{2}} & \\ C_{4}=\frac{\frac{1}{\tanh(\frac{\kappa h}{2})}\left[ \sigma^{*} \kappa h- z Q_{f}\frac{\sinh \kappa h(\frac{1}{2}-d_{1})}{cosh(\frac{\kappa h}{2})} \right]}{(\kappa h)^{2}-\beta^{2}} & \\ C_{5}=\frac{-\sigma^{*} \kappa h}{(\kappa h)^{2}-\beta^{2}} & \\ C_{6}=\frac{1}{\sinh(\frac{\kappa h}{2})}\left[ \frac{z Q_{f}\sinh(\kappa hd_{1})}{(\kappa h)^{2}}+\frac{\sigma^{*}}{\kappa h} \right] & \\ C_{7}{}=-C_{6} \cosh \kappa h(\frac{1}{2}{}-{}d_{1}){}+{}C_{1}\cosh(\beta d_{1}){}+{}C_{2} \sinh(\beta d_{1})+\\ ~~~~~~~~~~ C_{3}+ C_{4} \cosh(\kappa hd_{1}){}+ C_{5} \sinh(\kappa hd_{1})& \end{array}\right. $$

(A.7)

The expression (A.6) for the axial velocity component involves the scaled parameter β = λ₀h, which provides a measure of the permeability of the PEL medium. The hydrodynamic screening length $\lambda _{0}^{-1}$ is related to the permeability $\kappa _{p}=\lambda _{0}^{-2}$ of the PEL. Duval and his group ([7,8,9,10]) considered $\lambda _{0}^{-1}$ in the range 5 to 100 nm for polymer coated channels. For this range of $\lambda _{0}^{-1}$, the non-dimensional parameter β may vary between 1 and 20 when the channel height h = 100 nm.

The present closed form solutions are different from the expressions obtained by Matin and Ohshima [19, 20]. It may be noted that Matin and Ohshima [19] considered an uncharged wall, whereas in the present study, the channel walls are considered to have a non-zero surface charge density. In addition, Matin and Ohshima [19, 20] obtained the closed form solution by considering a positive sign in the second term of l.h.s of Eq. (A.2). The scaled velocity u _c along the center of the channel (y = 1/2) can be obtained as

$$\begin{array}{@{}rcl@{}} u_{c}&=&C_{3}\left[ 1- \frac{1}{\cosh(\beta d_{1})} \right] \\ &&+C_{4}\left[ \cosh(\kappa h d_{1})-\frac{1}{\cosh(\beta d_{1})} \left( 1+ \frac{\kappa h}{\beta}\sinh(\beta d_{1}) \sinh(\kappa h d_{1}) \right) \right]\\ &&+C_{5}\left[ \sinh(\kappa h d_{1})- \frac{\kappa h}{\beta} \tanh(\beta d_{1})\cosh(\kappa h d_{1}) \right] \\ &&+C_{6}\left[ 1- \cosh \kappa h(\frac{1}{2}-d_{1})-\frac{\kappa h}{\beta} \tanh(\beta d_{1})\sinh \kappa h(\frac{1}{2}-d_{1}) \right] \end{array} $$

(A.8)

where the coefficients C₃, C₄, C₅, are C₆ are given in (A.7), which involves the scaled surface charge density (σ^∗ = σh/ε _e ϕ₀) of the channel walls and the immobile charge density entrapped within the PEL (Q _f = (κ _s h)²/2). A detailed discussion on the derivation of these parameters (i.e., σ^∗, $\kappa _{s}^{-1}$) are already made in “Mathematical model” section. An expression for the axial velocity u _c at the central line of the channel for an uncoated channel can be derived from (A.8) by setting the values of the parameters d₁ and Q _f as zero i.e.,

$$ u_{c}=\frac{\sigma^{*}}{\kappa h} \frac{1-\cosh(\frac{\kappa h}{2})}{\sinh(\frac{\kappa h}{2})}. $$

(A.9)

This expression is the same as derived by several authors for EOF in a uncoated slit microchannel (e.g., Masliyah and Bhattacharjee [18]). In Fig. 8a, we have presented the the central line velocity as obtained by (A.8) as a function of Debye-Huckel parameter κh for different values of the PEL charge when β = 1. We found that an increase in κh decreases the value of u _c and it approaches a constant as the Debye length becomes sufficiently thinner than h, i.e., κh becomes large. In addition, we have also presented the the central line velocity as a function of the softness parameter β for higher value of κh(= 100). Increase in β decreases the flow penetration across the PEL and it leads to decrease in the magnitude of u _c .

Fig. 8

Variation of axial velocity at central line of the channel (u _c ) with a κh for β = 1 and b β for κh = 100. The results are shown for surface charge density σ = − 1 mC/m² of the channel wall at different PEL charge density. The nominal thickness of the step-like PEL (α = 0) is considered to be d₁ = 0.1