Effect of advection on transient ion concentration-polarization phenomenon

Here, we studied the effect of advection on the transient ion concentration-polarization phenomenon in microchannel-membrane systems. Specifically, the temporal evolution of the depletion layer in a system that supports net flow rates with varying Péclet values was examined. Experiments complemented with simplified analytical one-dimensional semi-infinite modeling and numerical simulations demonstrated either suppression or enhancement of the depletion layer propagation against or with the direction of the net flow, respectively. Of particular interest was the third-species fluorescent dye ion concentration-polarization dynamics which was further explained using two-dimensional numerical simulations that accounted for the device complex geometry.

Figure 1

(a) Top-view schematics of the microchannel-Nafion membrane device consisting of a main microchannel, that supports net flow, with a membrane embedded at its bottom wall and connected to two side microchannels. (b) Side-view scheme (section A-A) of the main microchannel. (c) Magnified microscopic image (top view) of the Nafion membrane embedded at the bottom of the microchannel.

Figure 2

Microscopic images of the temporal evolution of the ICP propagation in response to a constant applied current (see movie in the Supplemental Material [30]) for (a) $\mathrm{Pe}=+24$ and (c) $\mathrm{Pe}=-22$. The Nafion membrane is delineated by red dashed lines and $x=0$ signifies the anodic membrane-microchannel interface. In the experiments, the measured limiting current density without flow $(\mathrm{Pe}=0)$ was applied. The temporal evolution of the area-averaged 1D concentration profiles for (b) $\mathrm{Pe}=+24$ and (d) $\mathrm{Pe}=-22$. Both 1D numerical and analytical models assumed an ideal permselective membrane. In the analytical and numerical 1D models for $\mathrm{Pe}=+24$, an overlimiting current density equal to the theoretical 1D, steady-state, no-flow limiting current density $j_{\lim }\left(\mathrm{Pe}=0\right)=2FD{c}_0/\phantom{j_{\lim }\left(\mathrm{Pe}=0\right)=2FD{c}_{0}L}L$ was dictated. In the analytical and numerical 1D models for $\mathrm{Pe}=-22$, an overlimiting current density of factor 1.4 larger than the limiting current density, i.e., $j_{\lim }{\left(\mathrm{Pe}=-22\right)}_{\mathrm{theor}}\approx 22(2FD{c}_0/L)$, was dictated. Experimental fluorescent dye (third species) normalized intensity, $c_{3\mathrm{rd}}/c_{3\mathrm{rd}\_0}$: solid line; analytical 1D, salt $\left(c_1+c_2\right)/\phantom{\left(c_1+c_2\right)2c_0}2c_0$: dashed line; numerical 1D, third species $c_{3\mathrm{rd}}/c_{3\mathrm{rd}\_0}$ $[{\mathrm{D}}_{3\mathrm{rd}}=0.38\times {10}^{-9}({\mathrm{m}}^2/\mathrm{s})]$: squares; numerical 1D, salt $\left(c_1+c_2\right)/\phantom{\left(c_1+c_2\right)2c_0}2c_0$: dotted line.

Figure 3

Numerical 2D one- and two-sided model [(d,c), respectively] results of the cross-section-averaged normalized salt $\left[\left(c_1+c_2\right)/2c_0\right]$ and third species [$c_{3\mathrm{rd}}/c_{3\mathrm{rd}\_0}$; ${\mathrm{D}}_{3\mathrm{rd}}=0.38\times {10}^{-9}({\mathrm{m}}^2/\mathrm{s})$] concentration profiles at the anodic side of the microchannel at a normalized time of $t=6.25\times {10}^{-3}$. The applied current density was higher than the theoretical 2D limiting current density without flow ($5\left[j_{\lim }{\left(\mathrm{Pe}=0\right)}_{2\mathrm{D}/\mathrm{theor}}\right]$), with dominant pressure-driven flow, (a) $\mathrm{Pe}=+50$ and (b) $\mathrm{Pe}=-41$. The inset in (b) shows the analytical 1D one-sided model salt enrichment when applying the aforementioned current density with $\mathrm{Pe}=-41$.

Figure 4

Numerical 2D two-sided model (side view of our experimental setup) results of the normalized salt $[\left(c_1+c_2\right)/2c_0]$ (a–c) and third species ($c_{3\mathrm{rd}}/c_{3\mathrm{rd}\_0}$; $D_{3\mathrm{rd}}=0.38\times {10}^{-9}[{\mathrm{m}}^2/\mathrm{s}]$); (d–f) concentration distributions for various Pe values at an applied current density of $10\left[j_{\lim }{\left(\mathrm{Pe}=0\right)}_{2\mathrm{D}/\mathrm{theor}}\right]$ (where the value of the limiting current in the 2D case was obtained numerically to correspond to a complete salt depletion at the membrane interface) and at $t_{\mathrm{norm}}=1.5625\times {10}^{-2}$.

Figure 5

Analytical 1D model scheme.

Figure 6

Numerical 2D model geometry ($L$ is the channel length, $h$ is the channel height, and $L_m$ is the membrane width).