Abstract

Perm-selective nanochannel could initiate concentration polarization near the nanochannel, significantly decreasing (increasing) the ion concentration in the anodic (cathodic) end of the nanochannel. Such strong concentration polarization can be induced even at moderate buffer concentrations, because of local ion depletion (therefore thicker local Debye layer) near the nanochannel. In addition, fast fluid vortices were generated at the anodic side of the nanochannel due to the non-equilibrium electroosmotic flow (EOF), which was at least ~10X faster than that can be predicted from any equilibrium EOF. This result corroborates the relation among induced electroosmotic flow, concentration polarization, and limiting current behavior.

1. Introduction

Nanofluidic channels thinner than ~50 nm demonstrate unique ion-permselectivity at moderate buffer ionic strengths, which can be utilized for various applications. Wang et. al. have demonstrated nanofluidic protein preconcentrator which can efficiently concentrate proteins and peptides up to 106 fold [1]. Ions near nanochannel were depleted on the anodic side of nanochannels (concentration polarization) and both positively and negatively charged molecules were accumulated at the depletion boundary when coupled with tangential electric field. While the fast accumulation speed of the device was attributed to the induced electroosmotic flow [2] that could be generated by permselective ion current through the nanochannel, a detailed study of such a nonlinear electrokinetic flow is critically needed, for further development of nonlinear electrokinetics theory as well as for the optimization of nanofluidic preconcentrator, which directly utilizes concentration polarization phenomena. To explore this phenomenon with more details, we visualized the both concentration and electrokinetic flow pattern inside and outside the ion depletion region by tracking the fluorescent nanoparticles in-situ. Meanwhile, current measurements were also performed to examine the relationship between over-limiting current and the nonlinear electrokinetic mixing.

2. Theory and Experiment

Classical theory of concentration polarization, which is normally used to explain concentration polarization near electrodes in electrochemistry, can also be useful to understand ion perm-selectivity of nanofluidic channels. As shown in Fig. 1A, the nanochannel array can be considered as a cation-selective membrane. On the anodic side, positive ions can enter the perm-selective nanochannel under a DC bias, while negative ions will be driven away from the nanochannel by the same DC bias. Conditions for overall electroneutrality require the concentrations of both cations and anions to decrease in the anodic side of the channel, creating a concentration gradient. Due to this concentration gradient, preferential cation transport through nanochannel is satisfied across the entire system, while maintaining net zero anion flux. The ion transport and concentration is limited in the boundary region (diffusion layer) near the membrane surface, and the distance is assumed to be about 10~100 μm sitting between the bulk and the membrane. With a fixed diffusion length and bulk concentration, when we increase DC bias, the ion current will increase, therefore, the system will respond by decreasing the local ion concentration on the anodic side of the membrane (a.k.a ion depletion phenomenon). When the concentration approaches zero on the anodic side of the membrane, the system reaches a limiting current, above which no further increase in ion current is possible even with higher voltage applied to the system. However, in reality, significant over-limiting current can be observed in most perm-selective membranes, which is often associated with water dissociation at the vicinity of the membrane. Rubinstein and coworkers, however, argued that water dissociation alone cannot explain the over-limiting current in its entirety, and suggested (theoretically) that there exists a strong convective mixing (electroosmotic flow of the second kind, which destroys the concentration polarization), created by the ‘amplified’ electrokinetic response of fluid layer right next to the membrane. In this case, the electrokinetic response would be amplified because of significantly lowered ions concentration near the nanochannel (membrane), therefore higher ‘local’ zeta potential.

Figure 1 Schematic configuration of single gate ion-preconcentration device and schematic diagram of ion concentration distribution front and back of perm selective material which only let cations pass through. Ions in anodic side were depleted while they were enriched in cathodic side. N+ and N- are the flux of positive and negative ion, respectively, and subscript diff and drift represent the driving forces, diffusion and electrical field, respectively.

3. Results

By imaging microbeads, charged and uncharged dye molecules within the nanochannel/microchannel junction, along with current measurements, basic ion depletion and ion enrichment behavior were studied in SG device as shown in Fig. 2A. Figure 2B plots the time required for the initial depletion boundary to reach the opposite wall of the main microchannel, against buffer concentration and applied voltages. To significantly overlap the electrical double layer inside the nanochannel (40 nm deep), buffer concentration as low as 1 mM would be required. However, the graph clearly showed that the ion depletion was obtained at even higher than 10 mM buffer concentration, which corresponds to only 3 nm double layer thickness. At higher buffer ionic strength, however, it takes longer to reach concentration polarization. This clearly demonstrates that the concentration polarization (ion depletion) is a dynamic process, and the equilibrium Debye length alone cannot adequately describe the phenomena properly. Even when the Debye layer is not thick enough to cover the entire nanochannel depth, ion current through such channel could induce a small amount of additional counterions transported. Then, a weak concentration polarization can be induced, which will decrease ion concentration on the anodic side of the nanochannel. This will in turn lead to increased perm selectivity of nanochannel (via larger local Debye length), therefore eventually to even faster concentration polarization. Therefore, the concentration polarization process in nanochannel is indeed a positive feedback process, with ever decreasing local ion concentration and increasing perm selectivity (thicker local Debye layer thickness). It is also noteworthy that the electric field distribution will also be dependent on the concentration gradient in the system.

Figure 2 (a) The basic ion-enrichment and ion-depletion behavior in SG device. Depletion voltage condition (same voltage at anodic side and ground at cathodic side) was applied. Ion was enriched at cathodic side and depleted at anodic side. Both depletion and enrichment regions were rapidly expanded. (b) The time required for the depletion boundary to reach opposite microchannel wall as functions of applied voltage at anodic side (depletion voltage condition) and buffer concentrations. The values in the box were the equilibrium electrical double layer thickness.

To study the correlation between the concentration polarization and nonlinear current behavior, negatively-charged GFP molecules were injected into microchannels as fluorescent tracers. As shown in Fig. 2C, the current through nanochannels and the depletion length were measured simultaneously. The distribution of GFP molecules and the approximate width of the depletion region (l) were monitored continuously in the microchannel. As shown in Fig. 2C, the current profile can be divided into three regimes similar to ones reported in permselective membrane studies. Following the ohmic regime, the onset of concentration polarization phenomenon concurs when limiting currents were measured. Moreover, if one further increases the applied potential, the depletion region can be further expended and the current can be extended beyond the limiting condition.

Due to the non-equilibrium concentration distribution inside/outside the ion depletion zone, the Smoluchowski slip velocity (~εξE/η) cannot be applied to our system. The electroosmotic velocity at a permselective solid/liquid interface is proportional to the square of applied voltage (|E|^2) and the concentration gradient terms [3]. The non-equilibrium electroosmotic slip velocity induced strong vortex at the permselective interface and we experimentally showed the vortex in SG and DG device as shown in Fig. 3 .  The vortex flow speed was estimated to be usually over 1000 mm/sec, which is at least ~10X higher than that of primary EOF under the same electrical potential as shown in Fig. 3C.Corresponding Peclet number, which is the ratio of convection and diffusion, is over 100 and the experimental condition that present here lies in a high Peclet number region. At the steady state, we can clearly observe the two counter-rotating vortex beside of the nanochannel as shown in Fig. 3A . In DG device, since the ions were depleted through both walls, the four independent vortices were formed in the four divided regions as shown in Fig. 3B .  Such strong vortices would induce fast mixing and destroy any concentration gradient inside the ion depletion region. Since the thickness of the diffusion layer scales as d ~ (xD/v)^0.5 (assuming high Peclet number), with the convective flow velocity v ~ E^2 (x: characteristic length of the system), one can conclude d ~ 1/E and the limiting current i’ = 2FDC/d ~ E. The result complies with the experimental observation we have in Fig. 2C. This observation is compatible with the idea that such a strong convection (in front of permselective membrane) is one of the mechanisms to explain overlimiting currents in the perm-selective membrane.

Figure 3 (a) Fast vortex at steady state in SG device. (b) Four independent strong vortices in DG device. (c) The linear velocity and angular velocity of the vortex as a function of applied voltage (depletion voltage condition). The speed was 10-times or even higher than the equilibrium EOFs. The data was fitted by both the 2nd and 3rd polynomial to reveal the proportionality to E.

This study has several important implications in understanding nanochannel/membrane ion transport. In studying nanochannel ion/molecular transport, concentration changes in micro/nanochannel interface would have significant impact both in field and concentration distribution, and cannot be simply ignored as an ‘edge effect’. Even at moderate buffer ionic strength, strong concentration polarization could be triggered, especially under the DC field. Once generated, concentration polarization near nanochannel (membrane) would lead to strong, nonlinear electrokinetic flow, which in turn would affect ion/molecular transport through the nanochannel. While theoretically challenging, such coupled nature of this problem should be considered or checked in future studies involving permselective nanochannels or nanomembranes.

References

1. Ying-Chih Wang et. al., Anal. Chem. 77, 4293-4299 (2005).
2. S. S. Dukhin, Adv. Colloid Interface Sci. 35, 173-196 (1991).
3. I. Rubinstein and B. Zaltzman, Phys. Rev. E 62, 2238-2251 (2000).