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Transport-induced inversion of screening ionic charges in nanochannels

This work reveals a counter-intuitive but basic process of ionic screening in nano-fluidic channels. Steady-state numerical simulations and mathematical analysis show that, under significant longitudinal ionic transport, the screening ionic charges can be locally inverted in the channels: their charge sign becomes the same as that of the channel surface charges. The process is identified to originate from the coupling of ionic electro-diffusion transport and junction 2-D electrostatics. This finding may expand our understanding on ionic screening and electrical double layers in nano-channels. Furthermore, the charge inversion process results in a body-force torque on channel fluids, which is a possible mechanism for vortex generation in the channels and their nonlinear current-voltage characteristics.

Nevertheless, more detailed account of the electrostatics, particularly at the junctions of nano-channels and micro-reservoirs, have been found important when electrokinetic transport is significant 10,14,15,21 . The existence of the junction space charge regions and their extending into the reservoirs were revealed and proposed to explain the over-limiting ionic currents 10,11,14 . At such junctions, counter-ions diffuse into the reservoirs and leave behind underscreened channels where the surface charges are not fully screened. They are analogous to semiconductor junctions, and nano-fluidic junction devices have been proposed accordingly 21,22 . The impact of transport on nano-channel's ion selectivity has been revealed 3 . Recently, a semi-analytical capillary pore model based on local quasi-equilibrium has been developed for electrolyte transport through nanopores 27 . For side-gated nanopores, the transport-induced suppression of ionic screening, i.e., the descreening effect, has also been numerically investigated 28,29 . Thus far, a commonly-accepted basic concept on ionic screening is that the screening ionic charges should have an opposite sign to that of the surface charges, since the former are induced to counteract the effect of the latter. This paper reports that, under significant ionic transport, the sign of the screening ionic charges can be locally inverted in the nano-channels and become the same as that of the surface charges. The study is based on numerical simulations and mathematical analysis of the continuum transport model (Poisson-Nernst-Planck and Stokes). Although counter-intuitive, the charge inversion process is found not a peculiarity, but a rather general effect resulting from the coupling of the ionic electro-diffusion transport and junction electrostatics. The finding may expand our understanding on ionic screening in nano-channels as well as imply device applications. Particularly, it indicates that the ion selectivity of nano-channels is no longer solely defined by the channel surface charges and can even be locally inverted by longitudinal ionic transport.
Furthermore, the impact of the charge inversion process on the fluid transport, particularly its relation to channel vortex generation and nonlinear conductance, is investigated.
We firstly model a typical nano-channel as schematically shown in Fig. 1(a). The channel is a nano-slit geometry with a negative surface charge density . The channel connects two micro-reservoirs filled with a KCl solution of bulk concentration . An electrical bias, , is applied between the two reservoirs and generates an ion current, . Self-consistent numerical simulations are conducted based on the steady-state Poisson-Nernst-Planck-Stokes (PNP-S) equations 28,29 (see Supporting Information). The fluid transport is modeled as a low-Reynoldsnumber flow with hydrophilic, no-slip boundary condition 30  Information. The charge inversion phenomenon is evident in the cylindrical channels as well, but its scaling with channel dimension is quantitatively different from that of the nano-slits due to their different geometries.
The ionic charge inversion can directly impact the transport properties of nanochannels. In particular, it produces a possible mechanism for vortex generation inside the nano-channels. It is known that the product of the net ion charges and electric field determines the electrical body force on fluids. When driven by significant biases, the longitudinal electric field points to the same direction throughout the channel. Consequently, the charge inversion region exerts a body force opposite to that at the surface, thus applying a torque to the channel fluid. In fact, electro-osmosis has been shown to alter the ion distributions in nanochannels with weakly coupled EDLs 31 ; its effect on ionic charge distribution is expected. Nonetheless, the following analysis intends to identify the root cause of charge inversion and therefore will be based on the PNP model.
To reveal the root cause, a simplified structure (Fig. 1b)  According to Gauss's law, this corresponds to net space charges extending to both the channel and the reservoirs.
To further analyze the extending effect, we can average the PNP equations along the transversal direction and obtain the following 1-D homogenized equations: where is the solution permittivity, mobility, diffusivity, and cation and anion flux densities, respectively. The overbar denotes that the corresponding quantity is averaged over the transversal direction. The ion fluxes, , are constant due to continuity. In deriving these 1-D equations, the assumption, , has been made, which is valid for weak EDL overlap (i.e. ) 13 or under large biases. The average net charge density includes contributions from both ion and surface charges. It is inside the channel ( ) and in the reservoirs. Accordingly, the entire structure consists of three segments, intrinsic reservoir/positive channel/intrinsic reservoir, as illustrated in Fig. 1c. Within each segment, the following equation for can be readily derived: , where is the net current density. Here, and . As increases, the analytical expression starts to deviate from simulation results due to higher-order effects.
In particular, is no longer constant in the channel. But it still remains negative in an extended portion of the channel.
In the next, we examine the essential role of the second key factor, i.e. the We then compare Eq. 4 with the average 1-D Poisson's equation (Eq. 2) and can readily find that, because their left-hand-side terms approach each other, should approach as well. Since is already shown above to be negative due to the extended under-screening, it therefore follows that also tends to be negative in the channel, i.e. the ionic charge inversion occurs locally along the centerline.
It is worth emphasizing that, in the above analysis, what approaches is , not . In the process of under-screening, is positive but smaller than , and is negative. To verify the analysis, the local and average net charges, and , are compared in Fig. 5. As increases, it is evident that the former approaches the latter more closely.  34,35 . That analysis relied on the condition that the micro-reservoir walls also have significant surface charges 34 . In our previous 31 and present studies, we have respectively modeled the cases with and without micro-reservoir surface charges. In both cases, the vortex is generated inside the channel, indicating that it is not essentially caused by EOF bending at the corners.
An experimental observation of intra-channel vortices was previously reported by visualizing the nano-colloid transport within nano-channels 19 . In that work, streamlines in opposite directions at one end of the channel can be observed, indicating the formation of a vortex pair. It was also observed that the nanocolloids pack and unpack in the channel periodically. Whether the charge inversion process is responsible for the observed intra-channel vortices in that work remains to be further examined. Since the packed nano-colloid structure may alter the channel electric field 19 , the model would need to self-consistently account for the effects of the packed nano-colloids, and it would be interesting to explore how the nano-colloids can modulate the charge inversion process.
In a previous work on concentration polarization, the positive feedback between depleted ion concentration and increased local Debye length has been studied 6 .
However, we note that the increased Debye length at the ion depletion side does not fully account for the extended under-screening, as evident from the perturbation analysis in Methods. In this case, the ion concentration is only slightly perturbed from (1mM), so the local Debye length is still very close to its bulk value (10nm). But our results clearly show that net space charges can extend to the entire channel ( long), far beyond the local Debye length. This is consistent with the previously studied descreening effect, in which the field effect is shown to extend beyond the limit of Debye lengths 29,36 .
The under-screening process is closely related to the extending of space charges into the reservoirs, which was previously proposed to explain the overlimiting currents and the instable generation of vortices 10,11,14 . However, the space charges studied in those works were located in the reservoirs and positive for cation-selective channels. Therefore, they were still based on the common ionic screening picture. In another closely related study, the nano-channel's ion selectivity was found to change with ionic transport, but the applied bias was not high enough to reach charge inversion due to strong EDL overlap of the channels 3 . Overall, the charge inversion effect in this paper has not been revealed in those related works.
More insight into the cause of the charge inversion process may be obtained by making a comparison with semiconductor junctions as schematically shown in The charge inversion process has also been shown to apply a body-force torque on the fluid, thus producing a mechanism for vortex generation inside the channels.
Computational Methods:       Fig. 1b. The PNP model is used. Two quantities are plotted, the average net charges (including both ion and surface charges), , and the local ion charges at the centerline, . Both quantities are normalized to . Results are shown for two biases, 5V and 12V.