Quantification of Steady-State Ion Transport through Single Conical

Jun 25, 2013 - Juan Liu, Dengchao Wang, Maksim Kvetny, Warren Brown, Yan Li, and Gangli Wang*. Department of Chemistry, Georgia State University, ...
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Quantification of Steady-State Ion Transport through Single Conical Nanopores and a Nonuniform Distribution of Surface Charges Juan Liu, Dengchao Wang, Maksim Kvetny, Warren Brown, Yan Li, and Gangli Wang* Department of Chemistry, Georgia State University, Atlanta, Georgia 30302, United States S Supporting Information *

ABSTRACT: Electrostatic interactions of mobile charges in solution with the fixed surface charges are known to strongly affect stochastic sensing and electrochemical energy conversion processes at nanodevices or devices with nanostructured interfaces. The key parameter to describe this interaction, surface charge density (SCD), is not directly accessible at nanometer scale and often extrapolated from ensemble values. In this report, the steady-state current−voltage (i−V) curves measured using single conical glass nanopores in different electrolyte solutions are fitted by solving Poisson and Nernst−Planck equations through finite element approach. Both high and low conductivity state currents of the rectified i−V curve are quantitatively fitted in simulation at less than 5% error. The overestimation of low conductivity state current using existing models is overcome by the introduction of an exponential SCD distribution inside the conical nanopore. A maximum SCD value at the pore orifice is determined from the fitting of the high conductivity state current, while the distribution length of the exponential SCD gradient is determined by fitting the low conductivity state current. Quantitative fitting of the rectified i−V responses and the efficacy of the proposed model are further validated by the comparison of electrolytes with different types of cations (K+ and Li+). The gradient distribution of surface charges is proposed to be dependent on the local electric field distribution inside the conical nanopore.

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INTRODUCTION Rectified ionic conductance, memresistance and memcapacitance (meaning their magnitudes depend on previous states instead of being constant), and other interesting fundamental properties have been observed in the ion transport (IT) through individual synthetic nanopores and nanochannels connecting two solution reservoirs.1,2 The overall steady-state (SS) or non-SS ionic current signals are limited by the most constrained, and thereby most resistive, nanopore region. This feature makes these nanodevices promising platforms as stochastic analytical sensors and other functional devices.3−13 At the signal limiting nanopore region, both nanoscale geometry (radius, length, etc.) and interface (i.e., Coulombic interactions between the fixed charges on substrate surface and the mobile ions in solution) affect the charge transport process. The resulting signal, often in the format of overall current or conductance measured experimentally, reveals the combined impacts from both nanoscale geometry and interface. While significant advances have been achieved in the fabrication of nanodevices and the characterization of their geometry, it remains a significant challenge to characterize the interfacial features of individual nanodevices such as the density and distribution of the fixed surface charges, let alone their impacts on the transport behaviors. It is even more complicated if the interface is confined inside a nanochannel that limits the direct © 2013 American Chemical Society

accessibility for analysis. Consequently, the impacts of surface effects on the transport signals remain empirical and qualitative, which hinders broad applications. Effective theoretical models that could quantify those experimental observations have yet to be established. A high surface-to-volume ratio in nanodevices generally renders larger effective interfaces and thus enhances the desired properties and functions. Under such a scenario, the surface effects greatly impact the ion transport processes and may even overshadow the geometric conductance. Consequently, the characterized nanogeometry is often found inadequate to quantitatively describe the observed transport behaviors of a certain nanodevice in different solution conditions (i.e., electrolyte concentrations, type of electrolytes, solution pH, etc.). Furthermore, heterogeneous responses from different nanodevices with comparable characterized nanogeometry are frequently encountered. Those types of heterogeneity in nanodevice responses suggest missing parameters in the existing theoretical models. To quantitatively describe or predict the signals in sensing and other analytical applications that frequently employ various solution conditions, an effective Received: March 11, 2013 Revised: June 5, 2013 Published: June 25, 2013 8743

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physical origin of the proposed SCD variation has been lacking. The rationale of why the SCD should be adjusted for (1) different nanodevices with the same type of substrate materials under the same measurement conditions and (2) the same nanodevice in different measurement conditions is basically unaddressed. In this report, the experimental measurements of ion transport through a conical nanopore are correlated with simulation studies. Quantitative fitting of experimental currents at both high and low conductivity states is achieved by the introduction of a gradient SCD distribution on the nanopore interior surface in simulation. A maximum SCD at pore orifice can be determined from fitting the high conductivity state current, while the low conductivity state responses reveal the distribution of the SCD gradient inside the pore. The model efficacy is further demonstrated by predicting the ion transport behavior of the same nanopore device in different types of electrolytes (KCl and LiCl). The origin of the proposed SCD gradient is attributed to the applied electric field localized near the conical nanopore orifice.20 Illustrated in Scheme 1, the SCD

description of surface features, specifically the density and distribution of surface charges on the effective area of individual nanodevices, is required. The physical origin of the fixed charges on SiO2 substrates, as in the case of glass and quartz nanopores and nanochannels, is a consequence of the deprotonation of surface silanol groups. Since the deprotonation reaction involves the separation of charges (H+ and SiO−), the equilibrium is obviously a function of solution ionic strength.14 The degree of deprotonation is determined by the solution pH with respect to the pKa of the surface silanol groups. A certain percentage of those surface silanol groups would be deprotonated at a certain protonation− deprotonation equilibrium. The location of those deprotonated sites is generally considered to be randomly distributed on flat surfaces statistically. If an external electric field is applied normal to the surface, as in the case of chemically modified electrodes, the protonation−deprotonation equilibrium and thus the surface pKa of the functional groups are known to shift accordingly.15−19 If the externally applied electric field is in parallel with the surface, it might not change the overall surface pKa but could alter the distribution of those deprotonation sites. This effect might be negligible at the macroscopic ensemble system but requires scrutiny at nanoscale. In the measurements using nanodevices with asymmetric nanogeometries, the applied electric field can be very high, at the MV/m range within the signal-limiting region. The high applied electric field is believed to affect the distribution of those deprotonation sites and thus lead to nonuniform surface charge distribution.20 This is analogous to the electron transfer reactions driven by the applied electric field in solution at a bipolar electrode, which consists of an isolated electrode on an insulating substrate (with no direct contact with external circuit). Therefore, a uniform SCD value at the substrate− solution interface defined in existing theoretical models might be inadequate to describe the experimental system. Other parameters might be required, which is studied in this report. Because of the surface charges on the nanopore substrate, the SS conductivity of asymmetric nanopore devices deviates from the linear Ohm’s behavior, known as ionic current rectification (ICR).21−23 In multifrequency domain, with a small perturbation by applying a sine potential waveform over a constant dc bias, impedance analysis reveals complex multi-time-constant transport processes.24 The facilitation and inhibition to the overall ion transport by the fixed charges at substrate−solution interfaces induce intriguing memory effects in potential scanning measurements.2,25−27 A common theoretical approach in simulation literature is to construct the nanodevice geometry based on the experimental characterization. The initial value of surface charge density (SCD) at the substrate−solution interface is often defined based on ensemble values. By the variation of the SCD definition, the general trend of ICR has been successfully demonstrated in theoretical analysis for various nanogeometries.28−33 However, the current at the low conductivity side of the rectified i−V curve is overestimated in the simulation using those models in the literature.34 In other words, the experimental current responses at both high and low conductivity states could not be fitted simultaneously for a given nanogeometry using the existing models with constant SCD, especially at low conductivity state. Note the ratio of the simulated current at high and low conductivity states, generally referred as rectification factor, could match the experimental trends well, albeit both high and low conductivity state current values are deviated from experimental ones. Furthermore, the

Scheme 1. A Gradient SCD Distribution inside a Conical Nanoporea

a

A half cross section of a nanopore is drawn with the left boundary representing the nanopore center axis. The external electric field is applied on the top and bottom boundaries in simulation. The applied electric field intensity decreases sharply at the first 1−2 μm inside the nanopore, indicated by the color transition from red to blue.

will decrease from a maximum value at the pore orifice to the bulk value of ca. −1 mC m−2 at a certain depth, where the electric field becomes negligible. To the best of our knowledge, the physical origin is offered for the first time to address why the density and/or the distribution of surface charges inside a nanodevice should be defined and varied corresponding to experimental conditions. Both exponential and linear distribution models are tested for the comparison and validation with literature.35

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EXPERIMENTAL SECTION

Materials. Water (∼18.2 MΩ·cm) is purified by a Barnstead Epure water purification system. Silver wire (99.9985%, diameter 0.5 mm), silver conductive paste, platinum wire (99.95%, diameter 25 μm), and ferrocene were used as received from Alfa Aesar. Tungsten rods (A-M System, Inc.), Corning 8161 glass capillaries (o.d. 1.50 mm, i.d. 1.10 mm, Warner Instruments) were used as received. All other chemicals and materials were from Sigma-Aldrich, with 3-aminopropyldimethylethoxysilane from Gelest Inc. 8744

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Figure 1. (A) Conductivity of a 46 nm radius nanopore in KCl (blue) and LiCl (green) solutions at different concentrations: 10 mM (solid line) and 1 mM (dashed line). Scan rate was at 20 mV/s. (B) The ratio of i+/i− (RF) versus absolute potential amplitude. Conical Nanopore Fabrication and Surface Modification. The fabrication procedure of the glass nanopores followed previous reports.36,37 Briefly, a Pt nanotip is created by electrochemically etching, followed by sealing the Pt nanotip inside a glass capillary. Manual polishing of excess glass from the end leads to the exposure of the Pt nanotip or nanodisk. A through conical glass nanopore is obtained by full removal of the Pt wire via electrochemical etching and mechanical pulling. The nanopore shape replicates that of sharpened Pt nanotip. The interior surface of the nanopores is modified with 3aminopropyldimethylethoxysilane to improve the stability of the current responses. Note the surface coverage is ca. 20%; thus, the surface remains to be negatively charged at ambient pH. The fabrication, characterization, and surface modification of the nanopores have been well documented in previous reports and thus not elaborated.24,36−38 Electrochemical Measurements. The current−potential (i−V) responses (cyclic voltammograms) are measured using a Gamry Reference 600 potentiostat. The potential scan rate is 20 mV/s. Two Ag/AgCl wires are used to control the applied bias. The bias is defined by working versus reference, with the working electrode in the outside solution while reference inside the nanopore. The solution is under ambient pH (ca. pH 6) in all measurements. The radius of the nanopore is determined based on the conductivity at +0.050 to −0.050 V in 1 M KCl solution, at which condition the surface effect is effectively screened by the high concentration of electrolytes and small potential amplitudes (Figure SI-1). Simulation. The simulation was performed by solving Poisson and Nernst−Planck equations using COMSOL Multiphysics 4.0a software. Adaptive triangle mesh was used. The geometry and SCD distribution are represented in Scheme 1. In the simulation, the half-cone angle θ and pore depth are 11° and 10 μm, respectively. Further details are consistent with previous report and provided in the Supporting Information. One key notion is the introduction of a gradient SCD distribution instead of a uniform (constant) SCD on interior surface. The following parameters at room temperature were used (T = 298.13 K): density ρ = 1000 kg/m3, relative permittivity ε = 80, Faraday constant F = 96 485 C/mol. Instead of using the values at infinite dilution, an effective diffusion coefficient Di is used to compute the volume conductivity (reflecting geometry effects) in simulation as previously reported.34 The effective diffusion coefficients of K+ and Cl− in KCl solutions and Li+ and Cl− in LiCl solutions at different concentrations are listed in Tables SI-1 and SI-2 respectively. The correction is based on eqs 1 and 2 using the values obtained from ensemble bulk measurements. κi is the conductivity of species (ion) i which equals to its transference number (ti) multiplies solution conductivity (κ). R, F, and T are the gas constant, Faraday constant, and the absolute temperature, respectively. Zi and Ci represent charge and concentration of species i.

κi = κti

Di =

κiRT |Zi|F 2ci

(2)

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RESULTS Experiments: i−V Responses of Conical Nanopores in Different Electrolytes. Representative experimental i−V curves in Figure 1 are from a 46 nm radius nanopore in different electrolyte solutions. A linear i−V curve in 1 M KCl, shown in Figure SI-1, reflects the volume conductance defined by the nanopore geometry. At lower electrolyte concentrations, electrostatic interactions of the mobile ions with fixed surface charges will intensity. Correspondingly, the i−V curves in Figure 1 deviate from the linear Ohm’s behavior. The nonlinear correlation of the measured ion current with different potential amplitudes and polarities is in agreement with literature reports. The key point is that by comparing the i−V responses from the same nanopore in different electrolyte concentrations and different type of cations, in which the geometry or volume remains constant, quantitative fittings in simulation will allow the elucidation of surface conductance and therefore surface parameters. The high conductivity states appear at positive applied potential because the bias is applied on the outside electrode with the inside electrode as reference. At the same concentration, the current from KCl solution is always larger than that of LiCl due to the larger ion mobility of K+ over Li+. The nonlinear correlation with ionic concentration of either RF (i+/i−) or absolute current amplitude makes quantitative comparison of the measurements under different conditions challenging, which is the prerequisite for the detection of unknown analytes in sensing applications. From Figure 1B, it is obvious that the RF values differ at different potential amplitudes as well as from different ions or ionic strength. For example, i+0.4 V increases by a factor of 3.6 (4.0 nA/1.1 nA) from 1 to 10 mM KCl solution, while i−0.4 V increases by a factor of 3.2 (0.92 nA/0.29 nA). A similar trend can be seen from the LiCl responses. Note that the RF (i+0.4 V/i−0.4 V) of 3.8 (1.1 nA/0.29 nA) in 1 mM KCl is lower than the RF of 4.3 (4.0 nA/0.92 nA) in 10 mM KCl. The RF values are 4.3 and 5.5 in 1 and 10 mM LiCl, respectively. Combined with the linear i−V response, or RF = 1 at 1 M concentration, these results offer direct experimental supports for recent theoretical studies in which the RF values are predicted to decrease at both extremely high and low electrolyte concentrations.25,39

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Simulation of the Measured Current at High and Low Conductivity States. The impacts of the nanopore SCD on the simulated ion transport current are demonstrated and compared with the experimental results in Figure 2. Panel A

In an earlier report, through the variation of a uniform SCD maginitude at the interface, the simulated current at high and low conductivity states displays different responses upon electrolyte concentration variation and could not be fitted simultaneously.34 Those observations suggest that the common practice of adjusting the SCD magnitude in simulation could have oversimplified the physical processes involved. The high and low conductivity state responses might require additional parameters besides a uniform SCD to describe the surface charges in the modeling. This is confirmed in the following discussion, in which the experimental results will be subject to theoretical simulation for quantitative fitting. A predictive model is developed for the quantitative comparison of the i−V responses collected in either different electrolytes or different concentrations from a certain nanodevice and to determine the surface parameters of individual nanodevices. The physical origin of the proposed surface charge distribution is discussed at the end. Simulation Based on Continuum Theory. The measured current signal originates from the combined volume and surface conductivity. For the same nanopore in different electrolyte solutions, the volume conductance is solely determined by the solution conductivity since the geometry remains constant. Therefore, the deconvolution of the volume contribution from the overall current will reveal the surface impacts and vice versa. In theoretical simulation, the respective surface contribution is quantified directly through the introduction of SCD parameter. Aiming at quantitative fitting of the characteristic experimental data shown in Figure 1 and not just to mimic the trend of current rectification qualitatively, three types of SCD definition are employed. In the first type SCD definition, a constant SCD is applied to the nanopore surfaces that affect the ion transport processes. This enables the comparison with most simulation reports, in which the magnitude of SCD is systematically varied to demonstrate ICR effects. A linear SCD gradient is also proposed in the literature to better mimic ICR responses.33 To evaluate the impacts of SCD definition on the fitting and the physical origin of the proposed SCD definition, in the next two scenario, the SCD is defined to have two types of gradient distribution inside the nanopore, as defined by the following equations: exponential SCD: σ(z) = (σ0 − σb)(e−z / τ ) + σb

(τ < 3 μm)

(3)

(4)

Figure 2. Simulation of the experimental current from a 46 nm nanopore in 10 mM KCl with different SCD definitions on the interior surface: (A) a constant SCD uniformly distributed; (B) an exponentially decreased SCD, with σ0 = −100 mC m−2 and the distribution length varied as indicated in the plot; (C) a linearly decreased SCD, with σ0 = −100 mC m−2 and the distribution length varied as indicated in the plot.

σ(z) is the SCD at depth z inside the nanopore. At the pore orifice (z = 0), σ reaches a maximum value σ0. At the pore base (z = 10 μm), σ = σb = −1 mC m−2, a commonly used value from a planar silica surface. τ represents a characteristic distribution length established by the experimental conditions. For τ > 3 μm, in the exponential SCD definition, to be able to define the SCD at the pore base at −1 mC m−2, a practical expression is provided in Equation SI-1. The SCD distribution length τ and the maximum SCD σ0 are the varables adjusted during the fitting process.

shows the results by the variation of a constant SCD value that is defined uniformly throughout the nanopore interior surface. The nanopore geometry used in the simulation is validated by the excellent fitting of 1 M KCl and LiCl experiments shown in Figure SI-1. As the SCD magnitude increases, the simulated current increases drastically at high conductivity states as previously reported. Unlike the simulation literature that mostly focus on the RF (the ratio of current at opposite potentials), the current at low conductivity states is found to remain basically unchanged if the SCD is below ca. −50 mC m−2 and

linear SCD: σ0 − σb ⎧ z (z ≤ τ ) ⎪ σ0 − τ σ (z ) = ⎨ ⎪ −2 (z > τ ) ⎩ σb = −1 mC m

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Figure 3. Optimized simulation of the measured i−V curves with a 46 nm nanopore based on an exponential SCD distribution in (A) KCl and (B) LiCl solutions. The solid curves were measured experimentally, and the symbols were from the simulation. Fitting parameters: σ0 = −100 mC m−2 and τ = 0.4 μm for 10 mM KCl; σ0 = −70 mC m−2 and τ = 1.46 μm for 1 mM KCl; σ0 = −120 mC m−2 and τ = 0.4 μm for 10 mM LiCl; σ0 = −70 mC m−2 and τ = 1.46 μm for 1 mM LiCl.

interactions, leading to an overestimation of low conductivity state current in the simulation. The minor contribution from anion is discussed in the transference number analysis at the end. Quantitative Fitting and Prediction of the Experimental i−V Results. Next we demonstrate that the proposed model can be employed to quantitatively describe and even predict the experimental i−V responses in a range of concentrations and predict the responses of electrolytes with different monovalent cations (main charge carriers) at comparable concentration ranges. Using the σ0 and τ determined at ±0.4 V in each concentration as demonstrated in Figure 2, the simulated i−V results in both 1 and 10 mM KCl solutions are presented in Figure 3A. The simulated current (symbols) based on an exponential SCD distribution fits the experimental data (solid curves) almost perfectly at all tested potentials. The fact that a common SCD distribution could fit the current responses in a range of potentials at high accuracy (less than 5% error) is further addressed in the Discussion section. The fitting with a linear SCD distribution is shown in Figure SI-3. The results from other sized nanopores (26 nm and 110 nm radius) are included in Figure SI-4, which displays similar fitting quality. To further demonstrate the effectiveness of this approach, the experiments using the same nanopore but in LiCl solutions and the corresponding simulations are shown in Figure 3B. By replacing the K+ with Li+ in the simulation (corresponding diffusion coefficient), the simulated current matches experiments well. The fitting parameters σ0 and τ of the optimized simulation are slightly differently upon fitting optimization. Considering the variation of SCD at ±10 mC m−2 corresponds to 0.6 e/10 nm2, which corresponds to an uncertainty of ±1 deprotonated site per 17 nm2, the fitting parameters are consistent under experimental conditions. The differences in σ0 and τ between KCl and LiCl could also indicate the difference in ion size and/or surface binding/adsorption of cations with Si−O− groups. The explanation is supported by the general description that a measured surface pKa is a function of the activity and association coefficient of the counterions, in addition to the intrinsic pKa of the corresponding functional group.15 The fitting results suggest that as electrolyte concentration decreases, σ0 decreases while τ increases. This can be qualitatively explained by the deprotonation of surface silanol groups that involves the separation of charges (H+ and Si−O−).

then slightly increases at higher SCD magnitude as indicated by arrows. It is obvious that in the fitting of the experimental current a discrepancy arises from the overestimation of the low conductivity current. Even though the simulated RF (i+/i−) could match that of the experiments if the SCD magnitude is further increased, both high and low conductivity state currents would be larger than real experimental data. The results suggest that a simple constant SCD could not quantitatively describe the measured ionic current at all bias simultaneously. The simulations with an exponential and a linear SCD gradient are presented in panels B and C, respectively. The σ0 (−100 mC m−2) is determined from the best fitting for high conductivity current in panel A while the distribution length is systematically varied. The change in the simulated high conductivity current appears to be negligible upon the variation of the SCD distribution length (panels B and C). Excitingly, the simulated current at low conductivity states decreases as distribution length is lowered for both exponential and linear SCD gradients. Ultimately, the experimental value is approached at a certain distribution length. This trend is also confirmed using other nanopores, with one example shown in Figure SI-2. Qualitatively, the observation can be explained with the two factors σ0 (SCD at pore orifice) and τ (distribution length) separately. Since the cation is known to be the main charge carriers, the following discussion will mainly focus on cation transport. At high conductivity states (positive bias, outside vs inside), the cations will migrate from outside solution into the pore and then move from the tip to base. This process is limited by the most resistive region along the transport trajectory, which is at the tip orifice. Since the nanopore orifice remain constant during the experiments, the rectified high conductivity current is therefore mainly correlated with the σ0 at the pore orifice. This explains why the simulated high conductivity current remains unaffected with a fixed σ0 upon the variation of the distribution length inside the nanopore. At low conductivity states, the cations will migrate from the base toward the tip. Therefore, the fixed charges along the nanopore interior surface, described by both τ and σ0, will affect the cation flux and thus the detected current. An overestimation of SCD, by defining either a uniform distribution of σ0 or a gradient with longer τ, will introduce more negative charges on the nanopore interior surface in the simulation. Consequently, the cation concentration and cation flux will be arbitrarily increased inside the nanopore due to the electrostatic 8747

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Figure 4. Error analysis of the maximum SCD σ0 at the nanopore orifice and the distribution length τ in the simulation of i−V measurements of a 46 nm GNP in 10 mM KCl: (A) high conductivity state at +0.4 V; (B) low conductivity state at −0.4 V. Different colors represent different σ0: −90 mC m−2 (blue); −100 mC m−2 (red); −110 mC m−2 (green). Solid symbols represent the simulation based on an exponential SCD distribution, and open ones represent a linear SCD distribution.

shown in panel B, a ±10 mC m−2 SCD (blue and green symbols) results in negligible difference of simulated current at each distribution length (i.e., less than 5%). However, the distribution length τ affects the simulated current significantly. At the same σ0 ± dσ (the three types of symbols), shorter τ defines less surface negative charges, which will decrease the local cation concentration and thus its conductivity. Accordingly, the simulated current decreases and matches the measured current at a certain τ value. The analysis of the results in 1 mM KCl follows a similar trend as presented in Figure SI-5. A unique combination of σ0 and τ can therefore be determined for a certain nanopore under the measurement condition following the above approach. Transference Numbers of Cations and Anions at High and Low Conductivity States. The respective contribution of cations and anions to the overall measured current is quantified in Table 1. The cation transference number is

Lower ionic strength will inhibit charge separation, and thus less surface charge is expected in this case.40 The argument is also supported by an earlier report using fluorescence dyes confined inside nanodevices. The fluorescence intensity was found to vary when the SCD was changed from 100 to 2 mC m−2 (Figure 3d of ref 40). The SCD variation is well correlated with the electrolyte concentration dilution from 1 M to 1 mM in this study. With less charge carriers available at lower concentration inside the nanopore, the applied electric field will drop along a longer distribution length and therefore longer τ. The comparison of the simulated and experimental currents at relative low concentration reveals surface contribution to the overall measurements. At high electrolyte concentration, the surface effects are effectively screened by high ionic strength. The simulated current should no longer be affected by σ0 and τ. This is confirmed by the comparison of the simulation with different SCD definitions in 1 M KCl and 1 M LiCl solutions, which are shown in Figure SI-1. Tolerance of the Effective SCD Profile Determined from i−V Measurements. Retrospectively, the quantitative fitting of the experimental current by simulation reveals surface charge parameters at nanoscale interfaces that are very challenging to determine. The σ0 and τ in the proposed model could be noninvasively determined by fitting the high and low conductivity current, respectively. From Figure 2, the simulated high conductivity current appears to be sensitive to σ0 and less dependent on τ while the low conductivity current reflects distribution length τ at a certain σ0 value. The dependence of the simulated current on σ0 and τ is further illustrated in Figure 4. The experimental current under the measurement condition is independent of the simulation parameters and thus plotted as a solid line. To demonstrate the resolution of the simulation (tolerance of the determined σ0 and τ), a ±5% variation from experimental current is added to represent the possible measurement uncertainty (dashed lines). The solid and open symbols represent the simulated current computed with exponential and linear SCD distribution defined on the nanopore interior surface. At +0.4 V bias shown in panel A, a ±10 mC m−2 SCD (0. 6e per 10 nm2) from the optimized SCD of 100 mC m−2 gives ca. 5% variation of the simulated current, regardless of the variation of τ. This suggests that σ0 can be directly determined by the fitting of high conductivity state current with this constant SCD value. At −0.4 V bias

Table 1. Cation Transference Number Based on an Exponential SCD Distributiona KCl

LiCl

conc (M)

H (+0.4 V)

L (−0.4 V)

H (+0.4 V)

L (−0.4 V)

0.001 0.01 1

0.72 0.60 0.49

0.95 0.76 0.49

0.57 0.43 0.30

0.92 0.64 0.30

a

The transference number listed in this table is calculated from the simulations that best fit the experiments in each concentration. H and L represent high and low conductivity states, respectively. tCl− = 1 − tcation.

determined from the simulation with an exponential SCD gradient that best fits the corresponding experimental current in each KCl or LiCl solution. Anion transference number can be obtained by tCl− = 1 − tcation and thus is not listed. As expected, at high electrolyte concentration (1 M), the surface effect is negligible. The K+ transference number (0.49) approximately equals that of Cl− (0.51) due to the similar ion mobility. Because the mobility of Li+ is ca. half of that from Cl−, the Li+ transference number is lower accordingly. As bulk electrolyte concentration decreases, the surface effects become more prominent. Being the counterions compensating the negative surface charges, the cations always have larger 8748

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Figure 5. Electric field intensity distributions (solid lines) for a 46 nm conical nanopore with neutral surface in 10 mM KCl: (A) centerline and (B) a parallel cutline 3 nm away from the interior surface at −0.4 V. The dashed lines are exponential fittings of the electric field intensity profiles. Electric field intensity distributions through the whole pore are shown in inserted panels. Pore depth at 0 and 10 μm correspond to the location of orifice and base, respectively.

transference number than the anions Cl− at both high and low conductivity states. This validates the earlier argument that K+ is the major current carrier at each condition. Therefore, as ion concentration decreases, the surface conductivity contribution increase and correspondingly the cations transference number increases. Interestingly, at low conductivity states (−0.4 V), the K+ transference number increases from 0.49 in 1 M KCl (and 0.30 for Li+ in 1 M LiCl) to almost unity in 1 mM. The observation supports previous theoretical prediction.41 Furthermore, at lower electrolyte concentrations, the transference number of K+ is higher than that at high conductivity states though the current amplitude is lower. Those behaviors can be explained by the electrostatic interactions between mobile solution ions with the fixed negative surface charges. Because of the surface effects, cations will concentrate near the negatively charged surface, while anions will be repelled. The high conductivity states are established by the enrichment of both cations and anions inside the nanopore as extensively reported in the literature.28 Cations are obviously more abundant, giving larger transference number. However, at low conductivity states, the concentrations of both ions are lower than bulk at the nanopore region. The depletion effect is more significant for anions, leading to a much lower current primarily carried by the accessible cations, or approaching unity for cation transference number. The concentration profiles of K+ and Cl− are provided in Figures SI-6 and SI-7.

The distribution of an applied electrical field inside a nanopore with neutral surface (no surface charge) is first presented in Figure 5. This corresponds to a simplified scenario that the surface electric field established by fixed surface charges is negligible. Figure 5 shows the overall computed electric field intensity profiles at low conductivity states (−0.4 V). The electric field intensity profiles at −0.1 V gives a distribution with the same gradient as shown in Figure SI-8. Meanwhile, electric field intensity profiles at high conductivity states (+0.4 and +0.1 V) are included in Figure SI-9. Both along the nanopore centerline in panel A and near the interior surface in panel B, a high electric field (1−2 MV/m) at the orifice is observed due to the taper geometry. Because no surface charge density is defined, there is no surface electric field. The computed field intensity corresponds solely to the applied electric field, which drops primarily within the first 1−2 μm inside the nanopore. The electric field intensity curves display an exponential gradient in approximation by fitting (dashed lines). Note this applied electric field exists regardless of the surface charge definition. The overall electric field, however, is an overlapping effect of the applied electric field plus the surface electric field that is surface charge dependent. The rationale that an exponential SCD gradient inside the nanopore is established in accordance with the applied electric field is illustrated in Scheme 2. Within ca. 1 μm inside the nanopore orifice, the higher applied electric field will enhance the probability of the proton dissociation compared to the lower field region toward the base. If the electric field becomes negligible, the surface reaction is less affected. The SCD should therefore maintain the magnitude comparable to undisturbed surface (i.e., flat surface or similar, reported in the literature). Consequently, the SCD at the nanopore tip should be higher than the SCD at the base which adopts the value at ca. −1 to −10 mC m−2 (−1 mC m−2 used in the simulation). It is worth pointing out that a slight redistribution of the surface deprotonation sites will be sufficient to induce such SCD gradient. As 1 e = −1.6 × 10−19 C, a SCD of −100 mC m−2 corresponds to ca. 6 deprotonated silanol groups per 10 nm2. Taking the total site density of surface silanol groups at 4.9 per nm2, or 49 per 10 nm2,42−44 only ca. 10% variation is expected. This variation corresponds to ca. 0.1 shift in surface pKa at submicrometer scale, making it very challenging to characterize at such spatial resolution. Note the surface pKa and

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DISCUSSION The physical origin of the gradient SCD distribution and the rationale of its variation are proposed in the following discussion. The SCD of silica surface, even planar at macroscopic scale, could vary in a wide range depending on measurement conditions as reported in the literature. Unfortunately, direct characterization of the SCD and SCD distribution (τ and σ0) inside a nanodevice at submicrometer resolution is unavailable to the best of our knowledge. Since the surface charges are from the deprotonation of surface functional groups and this surface reaction process is a function of local electrical field, the gradient SCD distribution inside a nanopore is attributed to the applied electric field and the available charge carriers (electrolytes in solution) inside a conical nanopore. 8749

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Scheme 2. Surface Deprotonation Affected by the Applied Electric Field and the Formation of an Exponential Gradient Surface Charge Distribution.

Figure 6. Electric field intensity along the cutlines parallel to nanopore interior surface and the comparison with the SCD profile (dashed line). The results are from a 46 nm nanopore in 10 mM KCl. The overall electric field Etot is divided by the listed factor for direct comparison of the gradient distribution.

the site density at 4.9 per nm2 are based on statistics. For the dynamic surface chemical process, the probability of the deprotonation at different locations at nanoscale is not accurately reflected in those ensemble descriptions. Furthermore, individual nanopore devices could have different site density of different types of silanol groups (isolated and neighboring ones linked by hydrogen bonding) and unknown distributions of each type at nanoscale.45,46 Therefore, the SCD profile of individual nanodevices could be different fundamentally, which accounts for the heterogeneity in the frequently encountered measurement responses and device functionality. The differences in σ0 and τ of the nanopores analyzed in this paper therefore reflect the heterogeneous distribution of surface functional groups. It is therefore significant to be able to noninvasively address the surface parameters of individual nanodevices for related applications. The electric field near a charged surface is known to be strong at MV/m or higher in most cases. Next, we discuss whether the applied electric field is sufficient to alter the surface charge distribution that establishes the surface electric field. The overall electric field intensity Etot is the sum of the applied and surface electric fields. The surface electric field inside the nanopore, with the vector normal to the interior surface, can be divided into two components normal and parallel to the applied electric field direction separated by the half-cone angle θ of nanopore. Therefore, the applied electric field should compensate or compare with the parallel component that corresponds to ca. 10−20% of the surface electric field intensity (by sin θ). Note the applied electric field only needs to redistribute ca. 10% of the deprotonation sites, or change the probability by 10% inside the nanopore, to establish the proposed SCD gradient. Quantitative comparison is shown in Figure 6. The Eapp is computed with a neutral surface, corresponding to the pure externally applied field. Because the SCD needs to be predefined during the simulation, an exponential SCD profile replicating the distribution of the applied electric field is first introduced. The τ and σ0 are then systematically varied to fit the experimental results as shown in Figure 2. The SCD profile shown is computed using the τ and σ0 that best fit the experimental current responses. To the best of our knowledge, a quantitative correlation between the electric field strength and the shift in the deprotonation probability is not available at

submicrometer scale in literatures. Two Etot curves near the surface (along cutline parallel from the charged surface) are included for comparison, each divided by the listed factor to better illustrate the curve gradient. Noticing the difference in the curvatures of SCD and total electric field is ca. 100s nm, we postulate that optical imaging approaching even surpassing diffraction limit could provide direct evidence of the proposed correlation. The factors affirm that the applied electric field is comparable with the parallel component of the surface electric field. The results within 1 nm from the surface are not discussed due to the unknown interior surface roughness from experiments and the continuum theory being used in the simulation. Similar comparison with electric field along the centerline is provided in Figure SI-10. Shown in Figure SI-11, a linear SCD gradient generates discontinuous sharp transitions at the defined transition point. Since the dimension discussed here is still in the continuum regime (10s−100s nm), and on the basis of the physical picture discussed, we believe the exponential gradient is a better description of the real experiments. Impacts of Nanogeometry and Charge Distribution on External Surface. It is worth evaluating the impacts of finite geometric variation near the nanopore orifice on the aforementioned analysis. In a computational study, the nanopore geometry has been systematically varied by a mathematical description. Higher rectification could be achieved following the ICR trend of RFbullet > RFconical > RFtrumpet.47 The geometry near the nanopore orifice used in this study is arbitrarily adjusted to test whether the systematical larger low conductivity state current is due to the finite uncertainty of nanopore geometry limited by the characterization resolution. Because the overall nanopore geometry has been well characterized in previous reports and validated by the volume conductivity measurements in 1 M electrolyte solutions, the half cone angle was fixed at 11° to maintain the volume conductivity. Note the systematic geometric variation in ref 47 leads to the change of volume conductivity (much smaller overall half-cone angle toward the pore base) and thus is not applicable to this study. With a small variation of the orifice geometry to create bullet-like geometry shown in Figure SI-12, the impacts on the simulation results appear to be negligible. The chipped geometry only gives a 2% current change at +0.4 8750

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work was supported as part of the Fluid Interface Reactions, Structures and Transport (FIRST) Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Basic Energy Sciences.

V (3.77 nA for chipped geometry and 3.85 nA for normal conical geometry). Meanwhile, the effect of SCD distribution of the external surface is also studied. The high electric field at the nanopore orifice will have similar effects following the same rationale discussed earlier. The SCD distribution is determined following the same approach as that inside the nanopore. Accordingly, an exponentially decreased SCD on the exterior surface near the orifice is found to cause ca. 10% variation in the simulated current compared to that from a constant (uniform) SCD distribution (σ0). The difference is comparable with the 5% tolerance and thus not further discussed. The absolute values of the fitting parameters could change slightly, but the trend of the analysis will not be affected.

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CONCLUSION Steady-state ion transport current through a single conical nanopore displays well-known rectification effect. The experimental results are then subject to theoretical simulations by solving Poisson and Nernst−Planck equations. By introducing an exponentially decayed distribution of surface charge density inside a conical nanopore in the finite element simulation, quantitative fittings of the experimental measurements of steady-state ion transport through the nanopore are achieved. The fitting error is within 5% in both high and low conductivity range of the rectified i−V curves, which could not be quantitatively fitted using previous models, especially at the low conductivity states. The two parameters that define the SCD profile are noninvasively determined independently: a SCD maximum at the pore orifice is determined by optimized fitting of the experimental current at high conductivity state; the distribution length of SCD is determined by fitting the low conductivity current. Those parameters obtained from KCl electrolytes are then used to simulate and to predict the transport behaviors of LiCl electrolytes at different concentrations. The simulated LiCl i−V curves find great match with the experimental measurements at ca. 10% accuracy. The mechanism of the SCD gradient formation is proposed based on the physical origin of the surface charges: the deprotonation of surface silanol groups. A correlation is found between the surface charge distribution with the spatial distribution of the applied electric field inside the conical nanopore.

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ASSOCIATED CONTENT

S Supporting Information *

Details of simulation; the effective diffusion coefficient of ions in KCl and LiCl solutions; the i−V measurements and the corresponding simulation under different conditions; simulated electric field and ion distribution; the analysis of the results from a 26 nm nanopore; variation of nanopore geometry. This material is available free of charge via the Internet at http:// pubs.acs.org.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (G.W.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The startup fund provided by Georgia State University is acknowledged. J.L. acknowledges GSU MBD fellowship. This 8751

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