Enhanced transport of ions by tuning surface properties of the nanochannel

Motivated by recent observations of anomalously large deviations of the conductivity currents in confined systems from the bulk behavior, we revisit the theory of ion transport in parallel-plate channels and also discuss how the wettability of a solid and the mobility of adsorbed surface charges impact the transport of ions. It is shown that depending on the ratio of the electrostatic disjoining pressure to the excess osmotic pressure at the walls two different regimes occur. In the thick channel regime this ratio is small and the channel effectively behaves as thick, even when the diffuse layers strongly overlap. The latter is possible for highly charged channels only. In the thin channel regime the disjoining pressure is comparable to the excess osmotic pressure at the wall, which implies relatively weakly charged walls. We derive simple expressions for the mean conductivity of the channel in these two regimes, highlighting the role of electrostatic and electrohydrodynamic boundary conditions. Our theory provides a simple explanation of the high conductivity observed experimentally in hydrophilic channels, and allows one to obtain rigorous bounds on its attainable value and scaling with salt concentration. Our results also show that further dramatic amplification of conductivity is possible if hydrophobic slip is involved, but only in the thick channel regime provided the walls are sufficiently highly charged and most of the adsorbed charges are immobile. However, for weakly charged surfaces the massive conductivity amplification due to hydrodynamic slip is impossible in both regimes. Interestingly, in this case the moderate slip-driven contribution to conductivity can monotonously decrease with the fraction of immobile adsorbed charges. These results provide a framework for tuning the conductivity of nanochannels by adjusting their surface properties and bulk electrolyte concentrations.

Figure 1

Schematic representation of the system that is coupled with a bulk electrolyte reservoir characterized by the Debye length ${\lambda }_D$. The planar walls are located at $z=\pm H/2$ and separated by distance $H$. Their surface charge is created by adsorbed ions and is neutralized by the diffuse ions in the gap. Anions and cations are denoted with dark and white circles. The distribution of an electrostatic potential $\varphi \left(z\right)$ (top) is nonuniform and is qualitatively different for thick (left) and thin (right) nanochannels. An applied electric field $E$ induces an electro-osmotic velocity of a solvent, $V\left(z\right)$, and a current of density, $j\left(z\right)$ (bottom), which depend on the hydrodynamic slip length $b$ and on the fraction of immobile adsorbed ions, $\mu$. The averaged conductivity of the nanochannel is related to its averaged current density $J$ via Ohm's law

Figure 2

Electrostatic potential computed for a CP channel of $H=100$ nm and ${\varphi }_s=5$. From top to bottom, $H/{\lambda }_D\simeq 0.1$, 1, and 10.

Figure 3

Surface potential as a function ${\lambda }_D/{\ell }_{GC}$ calculated numerically using $H/{\lambda }_D\simeq 0.1,$ 1, and 10 (from top to bottom). Solid and open circles show calculations from Eqs. (7) and (19).

Figure 4

Current density $J$ vs electric field $E$ computed for channels of $H=100$ nm by imposing different electrostatic and electrohydrodynamic boundary conditions. The solid line shows numerical results for a hydrophobic CP channel obtained using ${\varphi }_s=4, \mu =0.5, b=100$ nm, and $c_{\infty }=1\times {10}^{-3}$ mol/L. Alternate lines are computed for hydrophilic CC channels of ${\ell }_{GC}=1$ nm using $c_{\infty }=1\times {10}^{-3}$ mol/L (dashed) and $c_{\infty }=5\times {10}^{-6}$ mol/L (dash dotted). Solid and open circles are calculations from Eqs. (35) and (38), correspondingly. Squares show predictions of Eq. (46).

Figure 5

$K_0$ as a function of $c_{\infty }$ computed for CC channels of $H=100$ nm (solid curves) using ${\ell }_{GC}=2$ and 300 nm (from top to bottom). The dotted line shows $K^{\infty }$ calculated from Eq. (31). Solid circles show predictions of Eq. (35), and open circles show calculations from Eq. (38). Solid squares are obtained from Eq. (39). The big triangles mark the points of ${\ell }_{Du}/H=1$.

Figure 6

The data sets of Fig. 5 reproduced, and plotted as $K_0/K^{\infty }$ vs $c_{\infty }$.

Figure 7

$K_0$ as a function of $c_{\infty }$ computed for CP channels of $H=100$ nm (solid curves) using ${\varphi }_s=5$ and 2 (from top to bottom). The dotted line shows $K^{\infty }$ calculated from Eq. (31). Solid circles show predictions of Eq. (35); open circles show calculations from Eq. (38). Solid squares are obtained from Eq. (39). The big triangles mark the points of ${\ell }_{Du}/H=1$.

Figure 8

The data sets of Fig. 7 reproduced and plotted as $K_0/K^{\infty }$ vs $c_{\infty }$.

Figure 9

Conductivity amplification due to hydrophobicity, $K/K_0$, calculated for the channel of $H=100$ nm using $b=100$ nm and $\mu =1$, and fixed ${\varphi }_s=4$ (solid curve) or fixed ${\ell }_{GC}=2$ nm (dashed curve). Solid circles are obtained using Eqs. (33) and (35). Open circles are calculations from Eq. (42). The big triangles mark the points of ${\ell }_{Du}/H=1$.

Figure 10

$K/K_0$ computed at fixed $c_{\infty }={10}^{-4}$ mol/L for the same hydrophobic CC (dashed curve) and CP (solid curve) channels as in Fig. 9, but for a case where only a fraction of $\mu$ adsorbed at the walls charges is immobile. The circles are calculations from Eqs. (33) and (35).

Figure 11

The same as in Fig. 10, but computed for a case where ${\ell }_{GC}=5$ nm is larger and $b=10$ (solid curve) and 1 nm (dash-dotted curve) is much smaller so that $K/K_0$ exhibits a minimum shifted towards a larger $\mu$ or has no minimum. The circles are calculated from Eqs. (33) and (35).

Figure 12

Local current density calculated numerically for the same ${\varphi }_s$ and $H/{\lambda }_D$ as in Fig. 2 using $\mu =1$ and $b=0$.