Concentration polarization, surface currents, and bulk advection in a microchannel

We present a comprehensive analysis of salt transport and overlimiting currents in a microchannel during concentration polarization. We have carried out full numerical simulations of the coupled Poisson-Nernst-Planck-Stokes problem governing the transport and rationalized the behavior of the system. A remarkable outcome of the investigations is the discovery of strong couplings between bulk advection and the surface current; without a surface current, bulk advection is strongly suppressed. The numerical simulations are supplemented by analytical models valid in the long channel limit as well as in the limit of negligible surface charge. By including the effects of diffusion and advection in the diffuse part of the electric double layers, we extend a recently published analytical model of overlimiting current due to surface conduction.

Figure 1

A sketch of the axisymmetric 2D system studied in this work. A microchannel of normalized length and radius unity connects a reservoir to the left to a cation-selective membrane to the right. To the right of the membrane is another reservoir, but this part of the system is only modeled through boundary conditions. The diffuse double layer adjoining the wall is shown as a shaded (blue) area and the arrows indicate a velocity field deriving from electro-diffusio-osmosis with back-pressure.

Figure 2

Sketch of the full physical system including both reservoirs of equal salt concentration. An electric potential difference $V_0$ is applied between the reservoirs, and the changes in electrochemical and electrical potential across the membrane and adjoining Donnan layers are indicated.

Figure 3

Sketch indicating the two regions in the boundary layer model. In the bulk region (lightly shaded) the boundary-driven velocity field $u$ (black line), the salt concentration profile $c$ (gray line), and the bulk advection $\langle cu\rangle$ are shown. In the boundary region (shaded and top zoom-in) the excess ion concentrations $c_{\pm }^{}-\overline{c}$, the velocity field $u$, and the surface advection $\langle (c_{\pm }^{}-\overline{c})u\rangle$ are shown.

Figure 4

Normalized charge density ${\rho }_{\mathrm{el}}^{{}_{}}/{\rho }_{\mathrm{el}}^{{}_{}}\left(0\right)$ (full) and potential $\varphi /\varphi \left(0\right)$ (dashed) as a function of distance from a charged wall for ${\overline{\lambda }}_{\mathrm{D}}^{}=0.01$ and ${\rho }_{\mathrm{s}}^{}=1$. The Gouy length ${\overline{\lambda }}_{\mathrm{G}}^{}$ and the Debye length ${\overline{\lambda }}_{\mathrm{D}}^{}$ are indicated.

Figure 5

The electro-osmotic flow $u_{\mathrm{eo}}^{\mathrm{u}}$ and the electro-osmotic flow $u_{\mathrm{eo}}^{\mathrm{up}}$ with backpressure for ${\rho }_{\mathrm{s}}^{}=10$ and ${\overline{\lambda }}_{\mathrm{D}}^{}=0.05$. The effective boundary velocity $u_{\mathrm{eo},\mathrm{bnd}}^{\mathrm{up}}$ is also indicated.

Figure 6

Directions of increase of the various mechanisms of overlimiting current. Bulk advection increases with $\alpha$ and ${\rho }_{\mathrm{s}}^{}$ and decreases with ${\overline{\lambda }}_{\mathrm{D}}^{}$. Surface conduction and surface advection increases with ${\rho }_{\mathrm{s}}^{}$, and bulk conduction through the ESC increases with $\alpha {\overline{\lambda }}_{\mathrm{D}}^{}$.

Figure 7

$I\text{−}V$ characteristics for $\alpha =0.05,{\overline{\lambda }}_{\mathrm{D}}^{}=\{0.0001,0.001,0.01,0.1\}$, and ${\rho }_{\mathrm{s}}^{}=\{0.001,0.01,0.1,1\}$. The full (black) line show the characteristics obtained from the FULL model. The dashed (red) curves are obtained from the BNDF model. The (blue) dash-dot curves are from the ASC model, and the (green) dash-diamond curves are from the ASCA model. The (gray) long-dash-short-dash curves are obtained from the ABLK model. The background patterns indicate the dominant overlimiting conduction mechanism. The (green) cross-hatched pattern indicate that surface advection and surface conduction are the dominant mechanisms. The (blue) vertically hatched pattern indicate that surface conduction without surface advection is the dominant mechanism. The (red) horizontally hatched pattern indicate that bulk advection is the dominant mechanism. The (gray) skew-hatched pattern indicate that bulk conduction through the ESC is the dominant mechanism. Intermediate cases are indicated with mixed background patterns.

Figure 8

Same $I\text{−}V$ characteristics as Fig. 7, except that here $\alpha =0.2$ instead of $0.05$.

Figure 9

For a system with ${\overline{\lambda }}_{\mathrm{D}}^{}=0.0001,{\rho }_{\mathrm{s}}^{}=0.01,\alpha =0.2$, and $V_0=60$ is plotted (a) cation current ${\mathit{J}}_{+}$, (b) velocity $\mathit{u}$, and (c) salt concentration $c$. For a system with ${\overline{\lambda }}_{\mathrm{D}}^{}=0.001,{\rho }_{\mathrm{s}}^{}=0.1,\alpha =0.05$, and $V_0=60$ is plotted the (d) cation current ${\mathit{J}}_{+}$, (e) velocity $\mathit{u}$, and (f) salt concentration $c$. The fields are obtained from the BNDF model, and the colors indicate the relative magnitude (black, low; white, high) of the fields within each panel, while arrows represent vector fields.

Figure 10

(a) $I\text{−}V$ characteristics highlighting the role of the surface current for bulk advection. $\alpha =0.2,{\overline{\lambda }}_{\mathrm{D}}^{}=0.0001$, and ${\rho }_{\mathrm{s}}^{}=0.001$. The dashed (red) curve is obtained from the BNDF model and the dash-diamond (green) curve is from the ASCA model. The dotted (purple) curve is obtained from the BNDS model, in which the surface current has been artificially removed while the electro-diffusio-osmotic slip velocity is kept. (b) Same as in (a) but with ${\rho }_{\mathrm{s}}^{}=0.1$.

Figure 11

(a) The overlimiting current $J_{+}^{\mathrm{overlim}}$ obtained from the BNDF model for ${\overline{\lambda }}_{\mathrm{D}}^{}=0.0001,\alpha =0.05$, and ${\rho }_{\mathrm{s}}^{}=\{0.0001,0.0002,0.0003,0.0004,0.0005\}$ plotted versus $V_0$. (b) Same as in (a) but plotted versus $1-x_0$.