Electromigration Current Rectification in a Cylindrical Nanopore Due

Departments of Mechanical and Aerospace Engineering, Electrical Engineering, Chemical Engineering, and Arizona Institute of Nanoelectronics, Arizona S...
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Anal. Chem. 2009, 81, 3128–3133

Electromigration Current Rectification in a Cylindrical Nanopore Due to Asymmetric Concentration Polarization Jung-Yeul Jung,†,⊥ Punarvasu Joshi,‡,| Leo Petrossian,‡,| Trevor J. Thornton,‡,| and Jonathan D. Posner*,†,§ Departments of Mechanical and Aerospace Engineering, Electrical Engineering, Chemical Engineering, and Arizona Institute of Nanoelectronics, Arizona State University, Tempe, Arizona 85287 We present experimental measurements of electromigration current through a single cylindrical nanopore. A single cylindrical nanopore with 175 nm diameter was fabricated in silicon in series with two micropores with 2 and 100 µm diameters. Thick electrical double layers (EDLs) (Ka ≈ 1) exhibit current rectification due to asymmetric concentration polarization while thinner EDLs show nearly symmetric conductance. After the electric field is turned off, electrical current is measured and observed due to redistribution of ions in the concentration polarization layer. Nanopores, nanopore membranes, and nanochannels have been used for various applications such as ion pumps,1,2 power generation,3,4 single DNA detection, trapping and sequencing,5-11 monitoring of biochemical reactions,12 biomolecule preconcentration,13,14 detection of protein ion channel blocking events,15 and * To whom correspondence should be addressed. E-mail: Jonathan.Posner@ asu.edu. Phone: 480-965-1799. ⊥ Current address: Department of Mechanical Engineering, Kyung Hee University, Gyeonggi-do, South Korea. † Department of Mechanical and Aerospace Engineering. ‡ Department of Electrical Engineering. § Department of Chemical Engineering. | Center for Solid State Electronics Research. (1) Nishizawa, M.; Menon, V. P.; Martin, C. R. Science 1995, 268, 700. (2) Siwy, Z.; Fulinski, A. Phys. Rev. Lett. 2002, 89, 198103. (3) van der Heyden, F. H. J.; Bonthuis, D. J.; Stein, D.; Meyer, C.; Dekker, C. Nano Lett. 2006, 6, 2232. (4) van der Heyden, F. H. J.; Bonthuis, D. J.; Stein, D.; Meyer, C.; Dekker, C. Nano Lett. 2007, 7, 1022. (5) Chang, H.; Kosari, F.; Andreadakis, G.; Alam, M. A.; Vasmatzis, G.; Bashir, R. Nano Lett. 2004, 4, 1551. (6) Kim, Y. R.; Min, J.; Lee, I. H.; Kim, S.; Kim, A. G.; Kim, K.; Namkoong, K.; Ko, C. Biosens. Bioelectron. 2007, 22, 2926. (7) Sigalov, G.; Comer, J.; Timp, G.; Aksmentiev, A. Nano Lett. 2008, 8, 56. (8) Fologea, D.; Brandin, E.; Uplinger, J.; Branton, D.; Li, J. Electrophoresis 2007, 28, 3186. (9) Gershow, M.; Golovchenko, J. A. Nat. Nanotechnol. 2007, 2, 775. (10) Zhao, Q.; Comer, J.; Dimitrov, V.; Yemenicioglu, S.; Aksimentiev, A.; Timp, G. Nucleic Acids Res. 2008, 36, 1532. (11) Zhao, Q.; Sigalov, G.; Dimitrov, V.; Dorvel, B.; Mirsaidov, U.; Sligar, S.; Aksimentiev, A.; Timp, G. Nano Lett. 2007, 7, 1680. (12) Karnik, R.; Castelino, K.; Fan, R.; Yang, P.; Majumdar, A. Nano Lett. 2005, 5, 1638–1642. (13) Zhang, Y.; Timperman, A. Analyst 2003, 128, 537–542. (14) Kim, S. J.; Wang, Y. C.; Lee, J. H.; Jang, H.; Han, J. Phys. Rev. Lett. 2007, 99, 044501. (15) Ervin, E. N.; Kawano, R.; White, R. J.; White, H. S. Anal. Chem. 2008, 80, 2069.

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nanofluidic bipolar transistors.16 These studies often monitor the electromigration current through the nanopore in an aqueous solution. Electromigration current rectification and diodelike behavior have also been observed in nanopores and nanochannels. Ion rectification is possible with imposed electric fields (electromigration)6,17-25 or through chemical gradients similar to biological ion pumps.2,26 Electromigration ion rectification has been demonstrated in systems with overlapping electrical double layers (EDLs) and asymmetric surface charge in both symmetric18,19 and asymmetric20 nanopore geometries. The rectification in these systems is due to ion concentration and depletion at the intersection of the charged surfaces in the nanopore. There are also several reports that cylindrical, chemically modified nanopores show ion current rectification; however, the mechanism was not explained.5,6 The electrokinetic behavior of nanoscale geometries has been recently reviewed by Schoch, Han, and Renaud.27 In this study, we present the rectification of ion electromigration current in a single, high aspect ratio nanopore with uniform surface charge. The nanopore is in series with two micropores with 2 and 100 µm diameters. Pores and channels with dimensions on the order of the electric double layer (EDL) thickness have been shown to be ion-selective.12,27 The ion- (or perm) selectivity is a result of the exclusion of co-ions in the diffuse electric double layer that surrounds a charged surface. Ion selectivity in nanoscale pores is well established with a long history in membrane science.28-30 In this work the nanopore is negatively charged and (16) Kalman, E. B.; Vlassiouk, I.; Siwy, Z. S. Adv. Mater. 2008, 20, 293. (17) Siwy, Z.; Heins, E.; Harrell, C. C.; Kohli, P.; Martin, C. R. J. Am. Chem. Soc. 2004, 126, 10850–10851. (18) Karnik, R.; Duan, C. H.; Castelino, K.; Daiguji, H.; Majumdar, A. Nano Lett. 2007, 7, 547. (19) Daiguji, H.; Oka, Y.; Shirono, K. Nano Lett. 2005, 5, 2274. (20) Vlassiouk, I.; Siwy, Z. S. Nano Lett. 2007, 7, 552. (21) Siwy, Z.; Apel, P.; Baur, D.; Dobrev, D. D.; Korchev, Y. E.; Neumann, R.; Spohr, R.; Trautmann, C.; Voss, K. O. Surf. Sci. 2003, 532, 1061. (22) Wei, C.; Bard, A. J.; Feldberg, S. W. Anal. Chem. 1997, 69, 4627. (23) Apel, P. Y.; Korchev, Y. E.; Siwy, Z.; Spohr, R.; Yoshida, M. Nucl. Instrum. Methods Phys. Res., Sect. B 2001, 184, 337. (24) Cervera, J.; Schiedt, B.; Ramirez, P. Europhys. Lett. 2005, 71, 35. (25) Chen, P.; Mitsui, T.; Farmer, D. B.; Golovchenko, J.; Gordon, R. G.; Branton, D. Nano Lett. 2004, 4, 1333. (26) Siwy, Z.; Kosinska, I. D.; Fulinski, A.; Martin, C. R. Phys. Rev. Lett. 2005, 94, 048102. (27) Schoch, R. B.; Han, J.; Renaud, P. Rev. Mod. Phys. 2008, 80. (28) Raman, L. P.; Cheryan, M.; Rajagopalan, N. Chem. Eng. Prog. 1994, 90, 68. (29) Bowen, W. R.; Mohammad, A. W.; Hilal, N. J. Membr. Sci. 1997, 126, 91. 10.1021/ac900318j CCC: $40.75  2009 American Chemical Society Published on Web 03/19/2009

Figure 1. Micrographs and scheme of nanopore structure: (A) optical micrograph of a micropore, (B) scanning electron micrograph of the cross section of a nanopore, and (C) the cross-sectional scheme of a single pore structure (not to scale). The electrodes are shown in their “normal” configuration.

the pore has excess counterions (H+ and K+) relative to the bulk and excludes co-ions (OH- and Cl-) relative to the bulk. The selectivity of the nanopore increases when the EDLs become more overlapped and described by decreasing the electrokinetic radius κa (where κ is the reciprocal Debye length and a is the pore radius). In this work, the current rectification can be attributed to asymmetric concentration polarization that results from a combination of asymmetric geometry and electric double layer thicknesses that are on the same order as the nanopore radius. The cylindrical nanopores exhibit increasing rectification with decreasing κa. After generation of a concentration polarization by application of an electrical bias, current with exponential decays are measured with no bias applied. These zero bias currents are consistent with diffusive and electromigration fluxes driven by the concentration and potential gradients of the concentration polarization. This rectification is distinct from previous reports of diodelike behavior using asymmetric surface charge. The observations reported here are due to the combined effects of ion-selectivity, concentration polarization, and asymmetric geometry. NANOPORE FABRICATION AND EXPERIMENTS The nanopores are fabricated on silicon-on-insulator (SOI) wafers using two lithography steps and three etch steps.31,32 The first step defines and etches the nanopore on the top side of the device layer of the SOI wafer. This is followed by the alignment and dry etching of a large pore (∼100 µm) from the backside of the handle wafer to reach the buried oxide layer. The buried oxide is removed by wet etching through the nanopore which results in a roughly 2 µm diameter region that connects the front and backside pores. The detailed fabrication process is described in detail elsewhere.31,32 Figure 1 shows schematics and micrographs of a single nanopore structure. Figure 1A shows an optical micrograph of the backside micropore of 100 µm diameter with (30) Hu, K.; Dickson, J. M. J. Membr. Sci. 2006, 279, 529. (31) Petrossian, L.; Wilk, S. J.; Joshi, P.; Hihath, S.; Goodnick, S. M.; Thornton, T. J. J. Microelectromech. Syst. 2007, 16, 1419. (32) Petrossian, L.; Wilk, S. J.; Joshi, P.; Hihath, S.; Posner, J. D.; Goodnick, S. M.; Thornton, T. J. Solid-State Electron. 2007, 51, 1391. (33) Burgreen, D.; Nakache, F. R. J. Phys. Chem. 1964, 68, 1084.

Figure 2. Currentsvoltage (I-V) characteristics of a 175 nm diameter single nanopore filled with 100 µM KCl. The black data points shows the “normal” case which means the positive electrode is located on the nanopore side of the wafer, while the red data points shows the “inverted” case for which the positive electrode is located on the side of the wafer containing the 100 µm diameter pore. The voltage sweeps start at -10 V at 0 s and increases to 10 V over a 19 min period. Each applied voltage is maintained for a time ∆t ) 1 min, and the voltage increment is ∆V ) 1 V.

the ∼2.0 µm diameter interconnect region. Figure 1B shows a scanning electron micrograph (SEM) of the cross section of a typical 40 nm nanopore. The larger region below the nanopore is the ∼2.0 µm interconnect region. Figure 1C shows a crosssectional schematic of the pore structure (not to scale). The pores used in this work are 175 nm in diameter as measured using SEM cross sections of neighboring pores on the same wafer. The pore is mounted in a two piece Teflon chamber and sealed with two Teflon O-rings. We measure an apparatus seal resistance of 775 GΩ using a device with a 400 nm silicon membrane without a pore etched in the top. This seal resistance is several orders of magnitude greater than the resistance of the nanopores measured in this work suggesting that the currents measured are due to electromigration current through the pore and not leakage through the silicon or around the wafer. Electrical measurements were made using reversible Ag/AgCl electrodes. The electrodes were prepared from 1 mm diameter silver wire dipped in 5% sodium hypochlorite (NaOCl) solution for 15 min before every measurement. A source-measure meter (Keithley 236, Cleveland, Ohio) was used to apply the voltage to the electrode and to measure the electromigration current through the nanopore. RESULTS AND DISCUSSION We measured the current responses to applied voltage from -10 to 10 V with 1 V increments and a dwell time of 1 min. Here we examine the directional effects of electromigration current by performing measurements with the positive electrode above the front surface (“normal” case) and with the positive electrode below the back side (“inverted” case) as illustrated in the insets of Figure 2. Four KCl solutions were examined 0.1, 1, 10, and 100 mM which results in values of κa of 3, 9, 29, 91, where κ is the inverse Debye length λd. Figure 2 shows the current-voltage (I-V) characteristics of a single 175 nm diameter nanopore filled with 100 µM KCl solution. Here we plot the measured current as a function of voltage. We start the scan at an applied voltage of V ) -10 V at t ) 0. After 1 min, the voltage is increased by 1 V and this is repeated every minute to a maximum voltage of 10 V. The black data points shows the normal case, and the red data points shows Analytical Chemistry, Vol. 81, No. 8, April 15, 2009

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Figure 3. (A) I-V characteristics (normal electrode configuration) of a 65 nm diameter nanopore filled with KCl electrolyte for various ionic strengths and characteristic electrokinetic radii: (4) 100 mM κa ) 91, (1) 10 mM κa ) 29, (O) 1 mM κa ) 9, (b) 0.1 mM κa ) 3. (B) I-V characteristic of the same nanopore for KCl concentrations of 100 µM (κa ) 3) and 10 mM KCl (κa ) 29). At κa ) 29, the current is nearly symmetric. When the electrokinetic radius is reduced to κa ) 3, the current is asymmetric and exhibits current rectification.

the inverted case. In the normal case with -10 V applied it takes nearly 40 s for the current to reach a steady ion current. In the normal case, the current starts at -12 nA and reaches -5 nA after 40 s (difference of 7 nA) while the inverted case needs less than 20 s to go from an initial current of -14 nA to a final current of -12.5 nA (difference of nearly 1.5 nA). In the normal case with a negative bias applied, the direction of the positive ion flux is from the micropore side to the nanopore side. Figure 3A shows the I-V characteristics of the nanopore at various KCl concentrations with the normal electrode configuration. This figure was compiled from data similar to that shown in Figure 2. Each current value in Figure 3A was calculated by averaging the steady state current (typically the last 40 s) at each applied bias. It appears from Figure 3A that there is nearly symmetric I-V characteristics at all ion concentrations. At moderate ionic strength (C e 10 mM or κa e 29), the current is relatively Ohmic, increasing linearly with applied bias. At the highest ion strengths of 100 mM (κa ) 91), we observe significant nonlinearity, which is typically attributed to Joule heating in microscale systems. Joule heating in nanoscale pores is unlikely due to the high surface to volume ratio. A one-dimensional heat transfer analysis of our nanopore suggests that Joule heating is not significant even in the worst case conditions of 100 mM KCl (κa ) 91 and σ ) 15 mS/cm) and largest applied bias (10 V). Details of the Joule heating analysis is given in the Supporting Information. Figure 3B shows the I-V characteristic of both the 100 µM (κa ) 3) and the 10 mM (κa ) 29) conditions with reduced y-axis limits. This plot shows significant asymmetry across the zerovolt plane for κa ) 3 and thus ion rectification. Figure 3B suggests that the ion rectification depends on the electrokinetic radius κa which describes the size of the diffuse electric double layer thickness relative to the pore radius. We attribute the rectification to asymmetric concentration polarization due to the geometry of the nanopore and micropore in series. As κa decreases, the diffuse electric double layers become more overlapped and the co-ions throughout the pore are depleted and the counterions are enriched above their free solution concentrations. The precise potential and ion distributions within charged pores with low κa values (thick or overlapping EDLs) have been reported by various groups, although there is still 3130

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significant debate on the appropriate boundary (or volume) conditions to be used to predict these distributions.19,33-42 The Debye layer thickness is simply defined as the distance from the charged wall where the electrical potential drops to 1/e of the wall potential (assuming symmetric binary electrolyte and wall potentials less than the thermal voltage kT/e).43 The co-ion depletion and counterion enrichment extend well beyond the EDL thickness, so that we expect that we will see finite EDL effects for moderate κa values. For low values of κa, the nanopore consists of mostly counterions which results in unipolar, ion-selective electromigration current of only these counterions. Ion selectivity in nanoscale pores is well established with a long history in membrane science.28-30 Concentration polarization is a complex of effects related to the formation of concentration gradients of ionic species in the electrolyte solution adjacent to a chargeselective interface upon the passage of electrical current normal to that interface. Recently, the Han research group reported experimental results that the diffusion layer extended significantly (up to several millimeters), depending on the strength of the external electric field and nonlinear electrokinetic mixing at the junction of a 40 nm deep nanochannel and a microchannel.14 They also reported the generation of fast fluid vortices at the anodic side of the nanochannel due to the nonequilibrium electro-osmotic flow (EOF), which was at least 10 times faster than predicted from any equilibrium EOF. Together with Han, we recently showed (34) (35) (36) (37) (38) (39) (40) (41)

(42)

(43) (44)

Morrison, F. A.; Osterle, J. F. J. Chem. Phys. 1965, 43, 2111. Rice, C. L.; Whitehea, R. J. Phys. Chem. 1965, 69, 4017. Hildreth, D. J. Phys. Chem. 1970, 74, 2006. Levine, S.; Marriott, J. R.; Robinson, K. J. Chem. Soc., Faraday Trans. II 1975, 71, 1. Daiguji, H.; Yang, P.; Majumdar, A. Nano Lett. 2004, 4, 137. Hughes, B. T.; Berg, J. M.; James, D. L.; Ibraguimov, A.; Liu, S.; Temkin, H. Microfluid. Nanofluid. 2008, 5, 761–774. Bhattacharyya, S.; Zheng, Z.; Conlisk, A. T. J. Fluid Mech. 2005, 540, 247– 267. Mani, A.; Zangle, T. A.; Santiago, J. G. On the propagation of concentration polarization from microchannel-nanochannel interfaces part I. Analytical model and characteristic analysis. Langmuir 2009, 25, 3898–3908. Zangle, T. A.; Mani, A.; Santiago, J. G. On the propagation of concentration polarization from microchannel-nanochannel interfaces part II. Numerical and experimental study. Langmuir 2009, 25, 3909–3916. Hunter, R. J. Foundations of Colloid Science, 1st ed.; Clarendon Press: Oxford, U.K., 1987. CRC Handbook of Chemistry and Physics, 76th ed.; CRC Press: New York, 1995.

Figure 4. Schematic of the pore with concentration polarization in the “normal” electrode configuration. Positive charges migrate down through the pore and accumulate in the bottom region. Negative ions on the top side are driven up toward the positive electrode resulting in a region depleted of ions. On the bottom side, negative ions are excluded from passing through the ion-selective nanopore and accumulate in the bottom region along with the positive ions that pass from the top side through the nanopore resulting in a concentrated region. Outside the depleted and concentrated regions (denoted by a dashed line), the ions regain the concentrations of the bulk electrolyte. The diffusive flux limits the replenishment ions on the top side so that a decrease in the electrical conductance is observed. The asymmetry in the conductance shown in Figure 3b is due to the confined bottom geometry. When the electrodes are reversed (e.g., “normal” electrode with negative bias or “inverted” electrode with positive bias) the bottom region is depleted and the top side is concentrated. Since the bottom region is confined, the diffusive replenishment of the ions is reduced and thereby the current flux. This results in asymmetric current and rectification. The approximate electric field lines are also shown and indicate a radial pattern on the top side while the serial micropore structures on the bottom side confine the electric field and result in a more linear distribution. This asymmetry in the electric fields may also contribute to the ion rectification.

that electrokinetic instabilities can develop due to the coupling of electric fields with sharp concentration gradients that result from concentration polarization.28 As long as diffusion or advection can supply the pore with adequate counterions, the current remains linear with applied voltage exhibiting Ohmic behavior. If diffusion or advection is inadequate to provide the sufficient counterions, then the ion concentration decreases resulting in less electromigration current or so-called concentration polarization. Our nanopore structure consists of three pores in series with decreasing diameter from 100 µm to 2 µm to 175 nm. At 100 µM ionic strength, the electrokinetic radius of the nanopore is κa ) 3, suggesting significant overlap of the EDLs and counterion ion selectivity. When a positive bias is applied with the normal electrode configuration, positive ions migrate through the nanopore. The counterions are replenished from the bulk electrolyte in the nearly infinite, front side reservoir. This results in Ohmic behavior shown in Figures 2 and 4B. The backside of the pore is connected in series to the bulk solution with 2 and 100 µm diameters pores with approximately 2 aL and 3.5 pL volumes, respectively. When a negative bias is applied (or positive bias with inverse electrodes), the current flux at each applied field is diminished because the finite reservoirs are quickly depleted of positive ions and reach a diffusion limiting regime. Once the counterions are depleted from the reservoir, the electromigration

current is limited by the diffusive flux of ions from the bulk into the 2 and 100 µm diameter reservoirs. This concentration polarization is shown schematically in Figure 4. Note also that the electric field lines and ion electromigration path resembles a radial pattern on the top side of the pore and linear on the bottom side due to the confinement of the microscale pores. This asymmetry in the electric field and migration path may also contribute to the asymmetry in the current flux. This geometric asymmetry coupled with finite EDL thickness results in electromigration current rectification as shown in Figure 3B. We also measure the current with zero applied voltage after the 20 min scans conducted to produce Figure 5. Figure 5A shows the zero-volt current for both the normal and inverted cases. The ion currents are as large as 140 nA and decrease exponentially to zero. The normal case requires nearly 8.5 min to reach zero current and the inverted case takes more than 30 min to reach zero current. After concentration polarization through the nanopore, time is required to re-equilibrate the ion distribution. Figure 5B shows the current as a function of time for zero applied voltage bias immediately after all measurements of Figure 3A (i.e., all cases of the “normal” configuration for various ion strengths). The time to reach the zero current increases with decreasing κa. Assuming a nearly exponential decay in the current magnitude, we can express the time-dependent current as I/I0 ) 1 - exp[-t/τ(κa)]. Here, I0 is the initial current at t ) 0, and τ is the characteristic time to reach steady state. Figure 5C shows the extracted characteristic time constants as a function of the electrolyte concentration, expressed as κa in this plot. The characteristic equilibration time decreases with increasing ion strength and electrokinetic radius. For κa e 29, the characteristic time decreases exponentially with κa as τ ) - 70 log(κa) + 345. The characteristic time to redistribute the ions after concentration polarization also decreases with increasing κa. We expect this for two reasons. First, the concentration polarization is greater for smaller κa since the pore is more ion selective. In addition, larger κa for a fixed pore diameter suggests higher electrolyte concentration which carries more ions and results in a more rapid redistribution of ions. Figure 5B shows that several minutes are required for ions to redistribute across the nanopore and re-equilibrate after significant concentration polarization. We can estimate the total charge due to concentration polarization by integrating the current in Figure 4B as Q)

∫

∞

0

I(t) dt

(1)

A plot of the total charge Q as a function of the electrokinetic radius κa is shown in Figure 6. This plot shows that the depleted charges in micropore side are inversely proportional to logarithmic κa. It is clear that as the electric double layers become more overlapped, the strength of concentration polarization increases and so does the total charge separation across the pore. The time required to redistribute the ions across the nanopore may be due to several physical phenomena. Here we briefly present scaling analysis for several phenomena that may be important in redistribution of the ions, namely, diffusion, diffusioosmosis, and concentration polarization induced electromigration. The RC time constant of the nanopore is estimated as 125 ms and too rapid to be responsible for the current decay. The Analytical Chemistry, Vol. 81, No. 8, April 15, 2009

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Figure 5. (A) Zero-volt current after measurements of Figure 2. The black line is the normal case, and the red line is the inverted one. Enriched cations due to ion polarization flow through the nanopore for re-equilibrating the ion distribution. (B) Zero applied voltage bias current after measurements of Figure 3 for normal electrode configuration at various concentrations and respective electrokinetic radii. As the electrokinetic radius decreases, it takes more time for the ion distribution to equilibrate. (C) Characteristic exponential time constant τ of ion current extracted from part B. The time constant τ decreases with increasing κa. For low to moderate electrokinetic radii (κa ) 3-29), the time constant varies as a function of κa as τ ) - 70 ln(κa) + 345. (fit shown as a dashed line.)

nm. Using this estimate, we calculate a diffusive time scale of roughly τ ∼ 2500 min which is 3 orders of magnitude larger than the measured time scale. The characteristic areas, lengths, and concentrations are chosen to represent the physics of diffusion and result in the smallest possible diffusion time scale (i.e., largest area A, largest concentration difference dc, smallest diffusion distance dx). This simple estimation suggests that the redistribution of the charges is not due to diffusion alone. The flux due to diffusioosmosis can be estimated as j)

Figure 6. Calculated total charge transferred across pore during redistribution. The charge is calculated by integrating the current in Figure 5B.

capacitance of our system was measured using impedance spectroscopy as 1 nF while the resistance varied with the electrolyte concentration but is roughly 125 MΩ. With division of the total charge by Faraday’s constant F and the volume of the backside micropore, V, we can obtain an estimate for the enrichment concentration as Q/FV. For the κa ) 3 case, the enrichment is ∼1.5 mM which suggests a concentration polarization that is roughly 15 times that of the original electrolyte concentration of 100 µM. Assuming dilute, Fickian diffusion across the nanopore with flux j ) D dc/dx, we can estimate the time required to reach equilibrium as τ∝

Q jAF

(2)

where A is the cross-sectional area of the nanopore. We estimate the diffusive flux with the diffusivity of potassium ions DK+ ) 1.5 × 10-5 cm2 s-1.44 A linearized estimate of the concentration gradient is simply the concentration difference dc ∼ 1.5 mM divided by the nominal distance of the diffusion process, namely, the thickness of the nanopore dx ∼ 400 (45) Ma, H. C.; Keh, H. J. J. Colloid Interface Sci. 2007, 313, 686–696.

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2RuT ηκ2

c∇c

(3)

where Ru is the ideal constant, T is the temperature, η is the fluid viscosity, and 3c is the concentration gradient.45 Using the time scale eq 2, we estimate the time scale for diffusioosmosis as 735 min, also much greater than the measured characteristic time. Electromigration mass flux can be estimated by the NernstPlank equation, given as j ) cνzFE

(4)

where ν is the mobility, z is the valence, and E is the local electric field strength. We estimate the mobility of potassium using the Nernst-Einstein equation as 6 × 10-13 mol s kg-1. The concentration polarization results in charge separation across the nanopore. The electric field that develops due to the concentration polarization should be on the order of the applied electric field required to generate the concentration polarization which is 1 V over a distance of 400 nm, the thickness of the nanopore. Using the time scale eq 2, the time required to reach equilibrium is ∼75 min. These simple estimates suggest that the redistribution of ions after concentration polarization is most likely due to electromigration of ions across the nanopore, although the time scale is nearly an order of magnitude larger than observed. Ultimately, the redistribution of ions after concentration polarization may be drive by a combination of diffusion, diffusioosmosis, and electromigration. Figure 7 shows the rectification properties of the pore as a function of the electrokinetic radius for an applied voltage of 3 V.

series. At higher applied fields, the current becomes unsteady (results not reported here) suggesting onset of the overlimiting regime and fluid stirring in extended space charge region.46,47 The stirring mechanisms are attributed to nonequilibrium electroosmotic slip such as electroosmotic flow of the second kind48 and electroconvection.49-52 We also observe that the nanopores may take more than 10 min to redistribute the ions and reach equilibrium after being driven with dc fields. Ion rectification and ion redistribution are important effects for applications in membrane science, nanofluidics devices, as well as nanopore based sequencing and sensors.

Figure 7. Current rectification ratio as a function of the electrokinetic radius κa for a 175 nm diameter single nanopore. The rectification decreases with increasing electrokinetic radius because the pore becomes less ion selective.

We calculate the rectification ratio as |I(V+)|/|I(V-)|, where I is the current at a given applied bias V, and the superscripts denote the positive and the negative voltages. Figure 7 shows that the rectification decreases by a factor of 2 when increasing κa from 3 to 91. This is expected since the nanopore becomes less ion selective at larger κa and the electromigration current is more symmetric. CONCLUSIONS Here we report ion current rectification and exponentially decaying redistribution elecromigration current that are a strong function of the electrokinetic radius κa. We show evidence of asymmetric concentration polarization due to relatively thick EDLs coupled with asymmetric geometries. At moderate ion concentrations (1 < C < 10 mM) and electrokinetic radii (9 < κa < 29) we observe nearly Ohmic behavior. At low concentrations (C ) 0.1 mM, κa ) 3), we observe ion rectification due to asymmetric concentration polarization of the nanopore and micropores in

ACKNOWLEDGMENT J.-Y.J was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (Grant KRF-2007-357-D00027). J.D.P acknowledges the support of an NSF CAREER Award (Grant No. CBET0747917) with William Wendell Schultz as the grant monitor. The authors thank Michael Goryll and Marco Saraniti for helpful discussions. SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

Received for review February 10, 2009. Accepted February 20, 2009. AC900318J (46) Manzanares, J. A.; Kontturi, K.; Mafe, S.; Aguilella, V. M.; Pellicer, J. Acta Chem. Scand. 1991, 45, 115. (47) Rubinstein, I.; Zaltzman, B.; Pretz, J.; Linder, C. Russ. J. Electrochem. 2002, 38, 853. (48) Dukhin, S. S. Adv. Colloid Interface Sci. 1991, 35, 173–196. (49) Rubinstein, I.; Zaltzman, B. Phys. Rev. E 2000, 62, 2238. (50) Rubinstein, I.; Zaltzman, B.; Pundik, T. Phys. Rev. E 2002, 65, 041507. (51) Rubinstein, I.; Zaltzman, B.; Lerman, I. Phys. Rev. E 2005, 72, 011505. (52) Zaltzman, B.; Rubinstein, I. J. Fluid Mech. 2007, 579, 173.

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