Suppression of nano-channel ion conductance by electro-osmotic flow in nano-channels with weakly overlapping electrical double layers

This theoretical study investigates the nonlinear ionic current-voltage characteristics of nano-channels that have weakly overlapping electrical double layers. Numerical simulations as well as a 1-D mathematical model are developed to reveal that the electro-osmotic flow (EOF) interplays with the concentration-polarization process and depletes the ion concentration inside the channels, thus significantly suppressing the channel conductance. The conductance may be restored at high electrical biases in the presence of recirculating vortices within the channels. As a result of the EOF-driven ion depletion, a limiting-conductance behavior is identified, which is intrinsically different from the classical limiting-current behavior.


Introduction
Nano-fluidic channels have important applications in membrane technologies 1-3 , analytical sample preparation [4][5][6][7][8] , current rectification [9][10][11] , and field-effect gating [12][13][14][15] . Understanding the ionic current-voltage characteristics in such devices has been a focus of extensive research efforts. The majority of the works pertain to channels that have strong overlap between the electrical double layers (EDLs), i.e. the channel height is comparable to or even smaller than the Debye screening length, . In such a classical regime, a remarkable electrical characteristics is the limiting current behavior, where the ionic current approaches a limiting value at elevated voltage biases; its cause is well-understood as from the concentration polarization (CP) process 2,4,[16][17][18][19] . There are other effects that are still under active study, such as the extended space charge layers, the vortex formation and their relation to overlimiting currents 5,[20][21][22][23][24][25] . In general, the electro-osmotic flow (EOF) inside the nano-channels does not play a significant role in the classical regime 26 .
More recently, nano-channels with weak EDL overlap have attracted great research interest for their unique device characteristics and relaxed constraints on fabrication 10,13,[27][28][29][30][31] . In this new regime, both unipolar and ambipolar ion transport processes coexist; the EOF inside the nano-channels can be significant and has been experimentally used for DNA translocation modulation 13,32 , current rectification 31 , and enhanced molecular binding 33 . Its impact on the nonlinearity of ionic currents has also been briefly discussed in a numerical study for nanopores that are voltage-gated by embedded side electrodes 27,28 . In a previous work by Mani et al. 6 , the interplay of the EOF and CP processes at the interface of micro-channel and nano-channel has been studied comprehensively; it has been particularly shown that, under the non-propagation CP condition, the ion concentration inside the nano-channel is suppressed by the localized CP effect. In the present work, we aim to numerically examine the nonlinear current-voltage characteristics in nano-channels with weak EDL overlap and reveal a limiting-conductance behavior that is intrinsically different from the limiting-current behavior classical regime. It will be shown that this unique characteristics can be explained by the general theory of Mani et al. 6 Furthermore, a refined 1-D mathematical model will be developed to give a simple yet accurate account of the limiting conductance.

Results and Discussions
A basic nano-channel structure as schematically shown in Fig. 1(a) is studied without the loss of generality. The cation-selective nano-channel has a fixed surface charge density, , and connects two micro-reservoirs filled with KCl solution of a bulk ion concentration, . Only the top-half of the channel is modeled for symmetry consideration, and the y-dimension is assumed infinite.
An electrical bias, , is applied between the two reservoirs and generates an ion current, . Here, the EOF flows from left to right, thus defining the channel entrance and exit, respectively. Our study pertains to the regime of weak EDL overlap, i.e. , where is the channel half-height.
It is noted that there are two common types of nano-fluidic channels 12 : 3D nanopores/nano-tubes and 2D nano-slits. This study models the nano-slit structures by assuming that the channel width is much greater than the channel height. Nonetheless, the nonlinear conductance effect revealed in this paper is rather general and also occurs in cylindrical nano-pore structures. Additional The simulated current-voltage curves are shown in Fig. 1(b) for , , , . The bulk ion concentration is 1mM, corresponding to a of 10nm. The surface charge density, , is set to a typical value of . For the PNP model, the differential conductance slightly decreases with increasing . This is expected from the classical limiting-current theory: the moderate ion selectivity from the weak EDL overlap results in a weak limiting-current behavior 16 . In contrast, the PNP-S model produces a strong nonlinear I-V curve: a severe decrease in the differential conductance is observed at moderate biases; as increases further ( in this particular device), the differential conductance rapidly restores to a high value that is even greater than that of PNP. To further illustrate this effect, PNP-S simulations are carried out by artificially increasing the fluid viscosity from its nominal value to and ( Fig. 1(b)). As the fluid flow is suppressed by the viscosity increase, the I-V curves become less nonlinear and approach to that of PNP.
The simulated fluid flow patterns in the channel region are plotted in Fig. 2 for two specific values representing the conductance suppression (5V) and restoration (12V) stages, respectively. Inside the entrance reservoir, a vortex can be seen for both biases, and the recirculating magnitude increases from 5V to 12V. This type of vortices is known for sharp EOF transition from the microreservoir to the nano-channel 6,26 . The most remarkable difference between the two biases is the flow pattern change inside the channel. At 5V, the flow lines exhibit a typical EOF velocity profile that is parallel to the channel surface. In contrast, a recirculating vortex is generated inside the channel at 12V.
The impact of the flow pattern on ion concentration is also examined in Fig. 2.
Here, we define the mean ion concentration , where and are the cation and anion concentrations, respectively. The plotted quantity is obtained by subtracting the distribution of PNP from its PNP-S counterpart.
At 5V, the ion concentration is significantly depleted inside the channel. In contrast, the change is small at 12V. These observations confirm that the conductance suppression and restoration are intrinsically related to the fluid patterns.
The effect of ion depletion inside the nano-channel has been accounted for in the general theory on CP propagation by Mani et al. 6 Qualitatively, it can be understood by considering the interplay of the CP and EOF processes. The classical theory of CP does not account for EOF and states that the ion depletion occurs inside the entrance reservoir 19 . For channels with weak EDL overlap studied in this work, however, there exists an ambipolar portion (where by definition) of the channel that connects the reservoirs, as schematically shown in the inset of Fig. 3(a). The ambipolar ion transport is of the convectiondiffusion type: the EOF can drive the ion depletion zone from the reservoir into the channel and thus dramatically suppresses its conductance. This depletion process is alleviated when the vortex occurs to mix the ion solution inside the channel.
To further quantitatively analyze the EOF-induced ion depletion process, a 1-D model is developed for long channels following the approach by Dydek et al. 21  The 1D model assumes sufficiently strong EOF so that the limiting condition is reached. This would require infinitely high as approaches zero.
In the limiting condition, the overall conductance is expressed as , Eq. (8) where the channel conductance is , and the reservoir conductance is . In Fig. 5(a), we re-plot the simulated I-V curve of the PNP-S model from Fig. 3(a). The asymptote is plotted using the limiting conductance calculated from Eq. (8). As a reference, another asymptote is plotted using the classical expression, , for the channel conductance at low biases 26 . Clearly, the ion conductance approaches the predicted limiting value at high biases. More data of the limiting conductance are plotted in Fig. 5(b) for varying surface charge density. Quantitative agreement between the calculations and simulations is observed, except for the disparity at the zero bias as explained above.
We emphasize that the quantity approaching a limiting value here is the conductance , as clearly shown in Fig. 5(a), rather than the current as commonly studied 26 . The limiting-current and limiting-conductance processes are both related to the concentration polarization, but they are intrinsically different. In the former, the ion depletion occurs in the entrance reservoir, and the diffusion therein limits the overall conductance. In the latter, the ion depletion is driven into the channel, which becomes the conductance bottleneck.
Our analysis is focused on the ion depletion process, since it is the cause of the nonlinearity in the first place. A quantitative model on the vortex generation and conductance restoration is beyond the scope of this study. We instead note that the over-limiting current and associated vortices (commonly referred to those generated in the reservoirs) are still actively studied from aspects such as hydrodynamic instability 23,24 and EOF back-flow 21 . It is also known that the variation of the channel height can lead to vortex generation both inside the channel and at the openings 8 . In the present study, there exists a variation in the screening length, particularly near the channel openings. Whether the channel vortex observed in this study shares one of those generation mechanisms or has a different origin remains to be further investigated.

Conclusions
In summary, the nonlinear current-voltage characteristics of nano-channels with weak EDL overlap are numerically studied. It is shown that the EOF drives the ion depletion zone into the channels and suppresses the ion conductance. The conductance may be restored at high electrical biases due to the occurrence of