Effect of geometry on concentration polarization in realistic heterogeneous permselective systems

This study extends previous analytical solutions of concentration polarization occurring solely in the depleted region, to the more realistic geometry consisting of a three-dimensional (3D) heterogeneous ion-permselective medium connecting two opposite microchambers (i.e., a three-layer system). Under the local electroneutrality approximation, the separation of variable methods is used to derive an analytical solution of the electrodiffusive problem for the two opposing asymmetric microchambers. The assumption of an ideal permselective medium allows for the analytic calculation of the 3D concentration and electric potential distributions as well as a current-voltage relation. It is shown that any asymmetry in the microchamber geometries will result in current rectification. Moreover, it is demonstrated that for non-negligible microchamber resistances, the conductance does not exhibit the expected saturation at low concentrations but instead shows a continuous decrease. The results are intended to facilitate a more direct comparison between theory and experiments, as now the voltage drop is across a realistic 3D and three-layer system.

Figure 1

Schematics describing the 3D geometry of the three-layer system consisting of a straight permselective medium connecting two opposite asymmetric microchambers.

Figure 2

Schematics describing the 2D geometry (i.e., $w=W_1=W_3$) of the three-layer system consisting of a straight permselective medium connecting two opposite asymmetric microchambers. The electrodiffusive boundary conditions have also been added for clarity. In most of the subsequent analysis, the 2D geometry shall be used for demonstration purposes.

Figure 3

The 2D (i.e., $w=W_1=W_3$) concentration distribution plot obtained from numerical simulation for the following geometry: $L_1=L_3=1,d=0.5,H_1=0.4,H_3=0.2,h=0.01,N=25,\delta ={10}^{-3}$ at $V=2$ exhibiting concentration polarization, i.e., depletion and enrichment within regions 1 and 3, respectively.

Figure 4

(a) The concentration and (b) electric potential profiles along the bottom surface $y=0$ are compared between theory and numerical simulations $\left(\delta ={10}^{-3}\right)$ for the asymmetric microchamber geometry depicted in Fig. 3. The inset of part (a) shows the concentration of the permselective region, which is substantially higher than the reservoirs with a fixed counterion concentration for the ideal case (theory) and a slight concentration gradient for the numerical simulation.

Figure 5

$I$-$V$ curves for a 2D device (i.e., $w=W_1=W_3=1$) with symmetric microchambers, $L_1=L_3=1$, $H_1=H_2=H$. The length of the nanoslot, $d=0.5,$ and the average volumetric concentration, $N=25,$ are kept constant. (a) $H$ is varied while the permselective region height is kept constant, $h=0.01$. The inset shows that the average current density $\overline{i}=I/HW$. (b) The microchamber height is kept constant at $H=0.4$ and $h$ is varied. The inset shows the current density $i=I/hw$.

Figure 6

Comparison of various conductance (per unit width) asymptotic models vs the fully coupled numerical simulation for a symmetric microchamber geometry of a 2D device (i.e., $\tilde{w}={\tilde{W}}_1={\tilde{W}}_3=\tilde{W}$), ${\tilde{L}}_1={\tilde{L}}_3=\tilde{d}/3=100\mu \mathrm{m},{\tilde{H}}_1={\tilde{H}}_3=10\mu \mathrm{m},\tilde{h}=190\mathrm{nm},\tilde{N}=0.76{\mathrm{mol}/\mathrm{m}}^3$. A clear divergence from saturation of the conductance at low concentrations is obtained when the microchamber resistance is not negligible.

Figure 7

(a) $I$-$V$ and (b) rectification curves for a 2D device (i.e., $w=W_1=W_3=1$) consisting of microchambers with $L_1=L_3=1,h=0.01,d=0.5,H_1=0.4,N=25,$ and varying $H_3$.