Rectification of Concentration Polarization in Mesopores Leads To

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Rectification of Concentration Polarization in Mesopores Leads to High Conductance Ionic Diodes and High Performance Osmotic Power Chih-Yuan Lin, Cody Combs, Yen-Shao Su, Li-Hsien Yeh, and Zuzanna S. Siwy J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.8b13497 • Publication Date (Web): 03 Feb 2019 Downloaded from http://pubs.acs.org on February 3, 2019

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Journal of the American Chemical Society

Rectification of Concentration Polarization in Mesopores Leads to High Conductance Ionic Diodes and High Performance Osmotic Power

Chih-Yuan Lin,1,2,3 Cody Combs,1 Yen-Shao Su,2 Li-Hsien Yeh,2,* Zuzanna S. Siwy1,4,5* 1Department

of Physics and Astronomy, University of California, Irvine, California 92697, United States

2Department

of Chemical Engineering, National Taiwan University of Science and Technology, Taipei 10607, Taiwan

3Department

of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan

4Department

of Biomedical Engineering, University of California, Irvine, California 92697, United States

5Department

of Chemistry, University of California, Irvine, California 92697, United States

Abstract Nanopores exhibit a set of interesting transport properties that stem from interactions of the passing ions and molecules with the pore walls. Nanopores are used e.g. as ionic diodes and transistors, biosensors, and osmotic power generators. Using nanopores is however disadvantaged by their high resistance, small switching currents in nA range, low power generated, and signals that can be difficult to distinguish from the background. Here, we present a mesopore with ionic conductance reaching S that rectifies ion current in salt concentrations as high as 1 M. The mesopore is conically shaped and its region close to the narrow opening is filled with high molecular weight poly-L-lysine. In order to elucidate the underlying mechanism of ion current rectification (ICR), a continuum model based on a set

*

Corresponding Authors: [email protected], Tel. 949-824-8290; [email protected], Tel. +886-2-27376942

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of Poisson-Nernst-Planck and Stokes-Brinkman equations was adopted. The results revealed that embedding the polyelectrolyte in a conical pore leads to rectification of the effect of concentration polarization (CP) that is induced by the polyelectrolyte, and observed as voltage polarity dependent modulations of ionic concentrations in the pore, and consequently ICR. Our work reveals the link between ICR and CP, significantly extending the knowledge how charged polyelectrolytes modulate ion transport on nano and meso-scales. The osmotic power application is also demonstrated with the developed polyelectrolyte-filled mesopores, which enable a power of up to ~120 pW from one pore, which is much higher than the reported values using single nanoscale pores.

Keywords:

Ion

Current

Rectification;

Concentration

High-Conductance Diodes, Osmotic Power Generation.


Polarization;

Poly-L-lysine;

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1. Introduction Nanopores attracted a great deal of interest due to their ability to control ionic and molecular transport of charged and neutral analytes. As an example, a nanopore with negatively charged walls in external electric field transports mostly cations if the pore radius is comparable to the Debye screening length.1 Ionic selectivity at high external fields leads to another nanoscale effect called concentration polarization (CP), in which a depletion zone is created at one pore entrance, and a region with enhanced ionic concentrations is formed at the other entrance.2-4 The local accumulation of ions in some conditions produces self-sustained ion current oscillations with frequencies reaching even 200 Hz.5 Tuning nanopore charge or/and shape further changes transport properties of these systems, and is the basis for ion current rectification (ICR) and transistor-like means of controlling ionic and molecular transport.6-11 Even uncharged nanopores can control transport on the nanoscale; e.g. nanopores with hydrophobic walls are completely closed for ionic and molecular transport at low pressure and electric potential differences, but open up when a threshold stimulus is applied.12-15 In addition, nanopores can be decorated with chemical recognition agents, such as antibodies/antigens and DNA among others, creating the basis for sensors and new routes to separation processes.16-23 All these effects rely on (i) the interactions of the passing analyte with the pore mouth and the pore walls as well as (ii) minimal transport through the unselective middle of the pore. Nanoscale pores are, however, characterized by a high ionic/molecular resistance.24 Consequently, ion current through the majority of nanopore systems have conductances in nS range. Low magnitudes of currents are advantageous since they do not lead to Joules heating, but are problematic when small changes of currents are to be monitored especially in low salt concentrations.



Recently, a series of publications appeared, which showed that some of the transport effects previously thought to be restricted to nanopores could also be induced in mesoscale and even microscale structures. Ionic selectivity and CP were for example observed in a 100 m in height channel partly filled with a cation selective hydrogel.4 The effect of CP was further used to create ionic diodes and transistors based on microfluidic channels integrated with ion selective membranes.25 ICR previously reported for nanopores was also observed in conically shaped mesopores26-28 and micropores29 as well as macroscopic asymmetric shapes of graphene oxide.30 In this manuscript, we show how voltage-dependent modulation of ICR with hundreds of nA switching currents can be achieved via placement of an anion selective poly-L-lysine in a conically shaped mesopore. The polyelectrolyte is expected to create a porous ‘plug’ that partly fills the pore volume, and modulates local electrochemical potential of the system. For one voltage-polarity, the recorded currents feature a plateau followed by a nonlinear current increase; ion currents for the opposite voltage polarity are nearly voltage-independent. A similar plateau and steep current increase with voltage was reported before for ion exchange membranes and nanofluidic channels,31-32 but the transition between the two regions in current-voltage curves occurred for significantly higher voltages than in the system reported here. Current-voltage curves of the poly-L-lysine filled pores described in this manuscript are explained by an interplay between the polyelectrolyte induced CP, and the asymmetric geometry of a conically shaped pore. The poly-L-lysine modified pores create an unprecedentedly robust ionic diodes, which rectify the current even in 1 M KCl. Our work presents a strategy to prepare ionic rectifiers with hundreds of nA switching currents, and to overcome the limiting current typical for ion selective systems. The pore system is also expected to be directly applicable in emerging applications such as osmotic power generation and chemical sensors, among others.19-20,

23, 33-35

To demonstrate this, we provide the first


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experimental results showing that an extremely high osmotic power of up to ~120 pW (corresponding to a density of ~945 W/m2) can be achieved by a single mesopore. Thus, these experiments significantly extend the knowledge of osmotic power from nanoscale to mesoscale pores. The experiments were performed with single mesopores prepared in the polyethylene terephthalate (PET) films by the track-etching technique.36-37 The pores are characterized by high density of surface charge due to the presence of carboxyl groups,38 thus we expected high molecular weight poly-L-lysine would be readily adsorbed to the surfaces and fill the pore volume.39-40

2. Experimental Section 2.1. Modification of Conical Pores. Single conical pores were fabricated in 12 μm thick ion-tracked PET membranes (GSI, Darmstadt, Germany) by the asymmetric wet chemical etching process, as described previously.27, 37 The film thickness decreased to ~11 μm during the etching.37 The pore geometry was characterized by scanning electron microscopy (SEM, JEOL JSM-7900F) at an acceleration voltage of 15 kV. A 10μL of 0.01 % poly-L-lysine solution (Mw=150,000 - 300,000, Sigma-Aldrich) was then dropped onto the side of membrane that contained the large opening of the pore, and air dried overnight. The membranes were rinsed in de-ionized water before measurements. The same etching and modification procedure was adopted for making a multi-pores membrane having the density of 106 pores/cm2. 2.2. Current Measurement. Single pore PET membranes were mounted between two chambers of a home-made conductivity cell; each chamber had a volume of ~ 1 mL. Current-voltage characteristics were recorded with a Keithley 6487 picoammeter/voltage source (Keithley Instruments, Cleveland, OH), and a pair of Ag/AgCl electrodes (A-M Systems, Sequim, WA). The working electrode was on the base side of the conical pore while



the tip side was grounded. The voltage was changed between −2 V and +2 V with 0.1 V steps. 2.3. Numerical Modeling. The ion transport through a pore that was partly filled with poly-L-lysine was modeled by the coupled Poisson-Nernst-Planck and Stokes-Brinkman equations.41-43 The charges carried by the poly-L-lysine were space distributed and included in the Poisson equation. Solving the Stokes and Brinkman equations allowed us to include the effect of the electroosmotic (EOF) flow as well as the effect of additional friction that stems from polymer chains present in the pore.43-44 These highly coupled equations were numerically solved by commercial finite element package, COMSOL Multiphysics, on a high-performance computer. A more detailed description for the governing equations and the associated boundary conditions are provided in Supporting Information. Aforementioned model has been previously verified for describing the ion transport in a nanopore functionalized with polyelectrolytes.44-46 Since the computational effort is very expensive when considering the polyelectrolyte, the modeling was performed for a conically shaped pore having tip opening diameter of 400 nm and length of 2 m (versus 11 m in experiments). The influences of the length of the zone filled with poly-L-lysine (Figure S7a) and the pore length (Figure S7b) on the current-voltage curves are shown as well. These figures clearly reveal that the rectification behavior remains qualitatively the same if the pore length and poly-L-lysine region are sufficiently large, implying that the 2 m long nanopore used in the modeling is a good model system for our experiments.

3. Results and Discussion To fabricate mesopores filled with poly-L-lysine, a droplet of the polyelectrolyte solution was placed on the side of the membrane with the large opening of the conical pores (Figure 1a). Scanning electron microscopy (SEM) characterization of a multipore membrane showed 6 ACS Paragon Plus Environment

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that the diameters of the tip and base opening of the pores we fabricated were ~400 nm and ~2 m , respectively, and the poly-L-lysine plug was indeed located at the region close to the narrow opening (Figure 1b). Note that it is difficult to observe the complete and clear image of poly-L-lysine near the surface of the tip opening, due to the damage imparted by electron beam particularly when we focus and magnify the image. We expected that filling a mesopore with poly-L-lysine would have two effects .47 (i) The steric effect would lead to partial blocking of the current compared to an empty pore. (ii) The charge effect that stems from the positive charge of poly-L-lysine will cause the polyelectrolyte-modified region of the pore to contain mostly anions; the charge effect could be described as a finite volume charge.

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Figure 1. Anomalous ion transport in a mesopore partly filled with poly-L-lysine. (a) Schematic of the system under consideration. (b) SEM images of the tip and base sides of a pore in a multipore membrane containing 106 pores/cm2 before (top) and after (bottom) poly-L-lysine modification. (c-f) Experimental I  V

curves of a single PET conically

shaped pore with a tip opening diameter of 400 nm in various KCl solutions before (black circles) and after (red squares) poly-L-lysine modification. (g) Rectification degree at 2V for various KCl concentrations. (h) Dependence of the measured current on KCl bulk concentration at pH 6 and +2 V for the same PET pore.

Figures 1c-f show current-voltage curves through a single conically shaped pore with an opening diameter of 400 nm before and after exposing it to a solution of high molecular weight poly-L-lysine from the large opening of the pore. The pore before and after modification was examined in KCl solutions at concentrations between 1 M and 1 mM, and pH values between pH 3 and pH 6. As prepared pore exhibited linear I-V curves, in accordance with the pore radius being more than a hundred times larger than the Debye screening length. Introducing poly-L-lysine into the pore changed the pore transport characteristics in all conditions examined including 1 M KCl (Figures 1c-f, S1). Moreover, in all salt concentrations and pH values at which the polyelectrolyte was positively charged, the pore rectified such that positive currents were higher than negative currents. The ratio of the currents, called rectification degree, reached ~25 in 100 mM KCl, and ~8 in 1 M (Figures 1g, S2). In general, rectification degrees at pH 3 (red columns) were found higher than those at pH 6 (blue columns), which we believe stems from a higher charge density of poly-L-lysine in the more acidic condition. Measurements in 1 M also revealed that the polyelectrolyte indeed had a steric effect on ion transport, because in these conditions, poly-L-lysine modified pore exhibited lower currents than as prepared pore (Figure 1h and Figure S1e). The


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hindrance of ion current in 1 M KCl could result from a possible shrinkage of the polyelectrolyte expected at high ionic strengths.48-49

We also realized that the I-V curves of poly-L-lysine modified pores could not be simply described by the rectification degree calculated at the highest voltage. It is because at intermediate voltages, between 0 V and +1 V, the curves contained a plateau that was especially pronounced at low salt concentrations and pH 3 (Figures 1c, e, f); in this voltage range, rectification degree reached an anomalous minimum (Figure S2). With further increase of positive voltages above +1V, the current increased non-linearly, enhancing the rectification. A similar non-linear current-voltage characteristic with a plateau followed by a steep current increase was reported before for ion selective membranes as well as individual nanochannels subjected to high electric fields.2, 32 The plateau is well-known to be caused by CP, but the origin of the nonlinear current increase has been a topic of a debate.32 In the system of single nanochannels, the current increase for high voltages was explained via non-linear electrokinetic flow.2,

50-51

Water splitting and changes in ionic selectivity have

been proposed as other possible mechanisms for the current increase.52-56 We believe that due to much lower voltages used in this study, a yet different physical mechanism is responsible for the effect, as detailed below.

Our experimental findings of ICR and current plateau were reproduced with few other independently prepared mesopores. Figure S3 shows recordings for another 440 nm in diameter pore.

In order to provide further evidence that ICR and the presence of plateau are linked to high charge density of the polyelectrolyte, experiments were also performed at pH 12, thus at



conditions at which poly-L-lysine is expected to be negatively charged.40 In agreement with expectations, at pH 12 the poly-L-lysine modified pore rectified ion current in the opposite direction indicating that at these conditions cations were the majority carriers (Figure S4). Note that no plateau was observed in this case due to lower charge density of poly-L-lysine at pH 12 compared to pH 3. Considering possible applications of the rectifying meso-pore system, we first demonstrated that a multipore membrane containing 106 pores per cm2 subjected to the same modification with poly-L-lysine also rectified the current. Figure S5 shows that such membrane rectifies even in 1 M KCl with rectification degree of ~5. As the next step, we significantly increased the amount of poly-L-lysine solution placed on the membrane surface and found that when the poly-L-lysine completely covered the pores on the base side, and did not successfully penetrate into the pore interior (Figure S6a), no rectification was observed (Figure S6b). These experiments provide further evidence supporting our claim that the rectification shown in Figure 1 is due to poly-L-lysine filling the pores and creating a plug at the tip side.

To further understand the mechanisms underlying the shape of the current-voltage curves recorded, we modeled the ion transport in a conical pore with tip diameter of 400 nm by considering the presence of the poly-L-lysine as a region with set volume charge (see details in the Supporting Information). We assumed the poly-L-lysine zone started at the pore tip and had length between 100 nm and 500 nm. All results shown in the main manuscript were obtained for 200 nm thick polyelectrolyte zone. Ion current, electric potential, and local ion concentrations were found by solving the coupled Poisson-Nernst-Planck and Stokes-Brinkman equations.41-43 In order to make the system numerically tractable, the pore modeled had length of 2 m, and opening angle of 4 degrees.37 As shown in Figure 2, the


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predictions from our model agree well with the experimental observations: the simulated current-voltage curves exhibit a significant rectification as well as the ion current plateaus. 60

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Figure 2. (a,b) Simulated I  V curves of a single conically shaped pore that was partially filled with poly-L-lysine. Inset in (a) denotes the schematic view of simulated system. The polyelectrolyte zone was 200 nm long and placed at the narrow opening of the pore. Two bulk concentrations, 10 mM and 100 mM in (a) and (b), respectively as well as two volume charge densities,  PE , of poly-L-lysine were considered.57 The tip diameter of the pore was 400 nm. (c, d) Axial variations of total ionic concentration for positive and negative voltages at Cb  10 mM and  PE  6.4 106 C/m3 . The region highlighted in blue corresponds to the zone that contains poly-L-lysine.

We also plotted axial variations of total ionic concentration for various voltages. Figures 11 ACS Paragon Plus Environment


2c and 2d reveal that the cumulative concentration within the pore ( z  0 nm) at positive voltages is significantly larger than that at negative voltages, providing explanation for the ICR observed in our experiments. For positive voltages above 0.4 V, ionic concentrations in the polyelectrolyte region increase with the increase of voltage; for negative voltages, ionic concentrations are nearly voltage-independent. The voltage dependent number of ions in the pore provides explanation for the voltage-dependent rectification behavior shown in Figure S2. We also would like to point to the gradient of the total ionic concentration in the pore that is created at positive voltages: ionic concentration at the base side of the poly-L-lysine region (z > 200 nm) is significantly larger than that at the tip opening ( z  0 nm). Origin of the gradient as well as its consequences for ionic transport are discussed below. The simulated current-voltage curves predicted both ICR and the presence of the current plateau at lower voltages, as observed in the experiments. Similar to experimental findings (Figure 1), Figures 2a and 2b reveal that the presence of the plateau is more pronounced for higher charge density in the polyelectrolyte region, which would correspond to acidic conditions of pH 3. Figure S7 shows that the findings are not sensitive to the thickness of the polyelectrolyte layer and occur for all thicknesses considered between 100 nm and 600 nm. In order to provide evidence that the pores we prepared indeed contained a region filled with poly-L-lysine, we considered few other pore systems with different arrangements of the polyelectrolyte inside the pore. First, we modeled a conical pore whose walls were homogeneously covered with a layer of poly-L-lysine (Figures S8).29 In this case, the simulated current-voltage curves were always linear even if the poly-L-lysine layer was 150 nm thick and carried high charge density of  PE  1.6  107 C/m3 (Figure S8b). Secondly, we modeled two pores shapes, conical (Figure S9) and cylindrical (Figure S10), whose middle regions were occupied with a poly-L-lysine ‘plug’. For both pores, the simulated current-voltage curves were of S-type with well-defined plateaus (Figures S9a,b, S10a,b) that 12 ACS Paragon Plus Environment

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extended until the maximum voltage examined of 2V.

In the conical pore, only weak

rectification properties were predicted in accordance with nearly voltage-independent ionic concentrations (Figures S9c,d, S10c,d); I-V curves for the cylindrical pore were symmetric. The above numerical study provided evidence that the pore systems considered in this manuscript indeed contained a zone, which was filled with the polyelectrolyte and was located close to the narrow opening of the cone. As the next step, we analyzed in detail local concentrations of potassium and chloride ions in the polyelectrolyte region as well as at the two adjacent zones, at the tip and base side, respectively (Figure 3). The modeling confirmed the expected anion selectivity of the region filled poly-L-lysine. For positive voltages, the majority carriers - chloride ions - are moving from the tip towards the base of the pore; ionic concentrations of both chloride and potassium ions at the region closer to the tip (z = 0 nm) are lower than these on the opposite side of the polyelectrolyte plug (z > 200 nm). For negative voltages, the region with depleted concentrations of both ions occurs in the pore at the polyelectrolyte side that faces the base, suggesting the local ionic concentrations are modulated by CP. Figure 3b shows the electric potential difference across the poly-L-lysine region,  , as a function of external voltage. Note that  is the resultant potential difference that stems from the externally applied field as well as CP modulated local conductivity of the solution adjacent to the polyelectrolyte region.58 Since the CP-induced electric field is opposite in its direction to the external field, the effective electric field acting on ions is reduced when CP effect becomes significant, resulting in reduced  . Figure 3b shows that

 increases linearly with increasing applied voltage at 100 mM (red circles), thus at conditions at which the current plateau is weak or absent.  , however, deviates appreciably from the linear relationship at 10 mM (black squares), especially for voltages above 0.4 V, consistent with our experimental findings. These results support our predictions that the CP 13 ACS Paragon Plus Environment


effect is more significant in lower salt concentrations. The proposed mechanism of the competition between the applied electric field and the CP-induced electric field is illustrated in Figure 3c.

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Figure 3. Numerical modeling of ionic concentrations in a mesopore shown in Figure 2. (a) Distributions of concentrations of cations and anions along the pore axis for various voltages at Cb  10 mM and  PE  6.4 106 C/m3 . Dashed and dotted lines denote the results for K+ and Cl ions, respectively. The region highlighted in blue corresponds to the part of the pore that was filled with poly-L-lysine. (b) Difference in the electric potential difference,  , across the poly-L-lysine region as a function of positive voltages for the case shown in Figure 2a (  PE  6.4 106 C/m3 ). (c) Schematic illustration of the competition between the applied 14 ACS Paragon Plus Environment

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(external) electric field, Eapp , and the CP-induced (internal) electric field, Ecp . J an and

J ca denote the fluxes of anions and cations, respectively.

Finally, we looked in detail at the origin of the nonlinear increase of ion current and rectification for voltages above ~0.5 V. Note that the current increase was reproduced in our modeled system, which did not include redox reactions or nonlinear fluid flows; 2, 32 thus we concluded the ICR effect stemmed from an interplay between CP and pore geometry. Indeed, at positive voltages, CP enhances ionic concentrations in the pore (z > 200 nm) in a strongly voltage-dependent fashion; note, the depletion seen at the pore entrance (z = 0 nm) is much weaker than the concentration enhancement (Fig. 2c). For the opposite voltage polarity, the depletion in the pore is largely voltage-independent, and similar in magnitude to the ion concentration enhancement on the other side of the polyelectrolyte region - at the pore entrance (Fig. 2d). The current rectification is therefore a consequence of rectification of the CP effect by the conical shape of the pore. As demonstrated above, the poly-L-lysine filled mesopores exhibit ionic selectivity and ion current rectification, thus we probed application of the system in osmotic power conversion to convert Gibbs free energy in the form of salinity gradients into electricity. Concentration of salt on the tip side of the mesopore was kept constant at 500 mM, analogous to the seawater; the other side of the pore was in contact with 1 mM, 10 mM or 100 mM KCl (Figure 4a). The subsequent osmotic power analysis was based on subtraction of the contribution from the redox potential, Ered , on the electrodes59 (Table S1). In general, the redox potentials obtained from the single un-modified mesopore are close to those obtained from the Nernst equation. In a 500 mM/1 mM KCl gradient, for example, the diffusion potential, Ediff , and the osmotic current, I osm , can be extracted from the intercepts on the voltage and current axes of the pure osmotic power contribution curve, respectively (Figure 15 ACS Paragon Plus Environment


4b). Figure 4c shows that the diffusion potential (osmotic current density) increased from 31 mV to 84 mV (40 kA/m2 to 45 kA/m2) as the salinity gradient increased from 5 to 500-fold. This result is significant and surprising, because it was previously predicted that mesoscale pores were unable to be applied in harvesting salinity gradient power due to negligible ionic selectivity.60

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Figure 4. Osmotic power conversion demonstrated in a mesopore filled with poly-L-lysine. (a) Schematic of the system under consideration; 500 mM KCl was placed on the side of the membrane with the narrow opening of ~400 nm, the other side of the membrane was in contact with 1 mM, 10 mM, or 100 mM KCl. (b) Current-voltage characteristics of a single PET conically shaped mesopore filled with poly-L-lysine recorded in a 500 mM/1 mM KCl gradient at pH 6. The pure osmotic power contribution (blue line with symbols) can be evaluated from the original total current (black line with symbols) by subtracting the 16 ACS Paragon Plus Environment

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contribution of the redox potential ( Ered ) on the electrodes, which can be obtained from the single un-modified mesopore. Ediff and I osm are the diffusion potential and osmotic current, respectively. (c) Generated osmotic current density and diffusion potential, and (d) osmotic power density as a function of KCl concentration gradient.

We then estimated the osmotic power from the developed polyelectrolyte-filled mesopore under a series of salinity gradients, using the following procedure. Figure 4d and Table S1 2 reveal that the osmotic power densities, calculated by ( I osm  Ediff ) / 4 A , A   Rtip , and

Rtip  200 nm , were 305, 750, and 945 W/m2 for the 5-fold, 50-fold, 500-fold concentration

ratios, respectively. All of the above power densities are several hundreds times higher than previously reported values (generally in the range of 1~5 W/m2) found for nanoporous membranes.35,

61-62

In order to facilitate comparison with results for single pores, the

maximum value of 945 W/m2 from the developed mesopore system corresponds to a power of ~120 pW, which is also significantly higher than reported for other single nanoscale pores (Table S2).59, 63-65

4. Conclusions In summary, our experimental and modeling results merge effects of asymmetric geometry of a mesopore and CP to achieve a meso-scale ionic rectifier. The system we built is characterized with high ionic conductance reaching S and rectification in high ionic strength of 1 M. Our findings were explained by examining voltage-dependence of local ionic concentrations and potential drops within the poly-L-lysine region. The high conductance ionic rectifiers could be applied in systems where high switching currents are required. The system of a conical mesopore with an ionic selective zone also offers means to overcome



CP-induced current limitation. A possible application of the poly-L-lysine filled mesopores in salinity energy conversion was demonstrated as well. A power as high as ~120 pW (corresponding to a density of 945 W/m2) was achieved from a single pore placed in contact with 500 mM and 1 mM KCl solutions, offering a new avenue for harvesting salinity gradients.

Supporting Information The Supporting Information is available free of charge on the ACS Publications website. Additional experimental and modeling results as well as details of numerical modeling.

Acknowledgements This research was supported in part by the National Science Foundation (CBET-1803262) and the Ministry of Science and Technology, Taiwan (MOST 105-2221-E-011-170-MY3, 106-2918-I-224-003 and 107-2622-E-011-023-CC2) for L.H.Yeh. We acknowledge GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany for providing tracked membranes.

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Rectification of Concentration Polarization in Mesopores Leads To

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