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Correlation of Ion Transport Hysteresis with the Nanogeometry and Surface Factors in Single Conical Nanopores Dengchao Wang, Warren Brown, Yan Li, Maksim M. Kvetny, Juan Liu, and Gangli Wang Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.7b03477 • Publication Date (Web): 04 Oct 2017 Downloaded from http://pubs.acs.org on October 10, 2017

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Analytical Chemistry

Correlation of Ion Transport Hysteresis with the Nanogeometry and Surface Factors in Single Conical Conical Nanopores Dengchao Wang, Warren Brown, Yan Li, Maksim Kvetny, Juan Liu, and Gangli Wang* Department of Chemistry, Georgia State University, Atlanta, GA, 30302 *Email: [email protected]; Tel: 404-413-5507

ABSTRACT: Better understanding in the dynamics of ion transport through nanopores or nanochannels is important for sensing, nucleic acid sequencing and energy technology. In this paper, the intriguing non-zero cross point, resolved from the pinched hysteresis current-potential (i-V) curves in conical nanopore electrokinetic measurements, is quantitatively correlated to the surface and geometric properties by simulation studies. The analytical descriptions of the conductance and potential at the cross point are developed: the cross-point conductance includes both the surface and volumetric conductance; the cross-point potential represent the overall/averaged surface potential difference across the nanopore. The impacts by individual parameter such as pore radius, half cone angle, and surface charges are systematically studied in the simulation that would be convoluted and challenging in experiments. The elucidated correlation is supported by and offer predictive guidance for experimental studies. The results also offer more quantitative and systematic insights in the physical origins of the concentration polarization dynamics in addition to ionic current rectification inside conical nanopores and other asymmetric nanostructures. Overall, the cross point serves as a simple yet informative analytical parameter to analyze the electrokinetic transport through broadly defined nanopore-type devices.

Ion transport (IT) through nanoscale interfaces has been a subject of extensive research due to its fundamental roles in various applications such as the sequencing of nucleic acids1,2, molecular sensing3,4, nanofluidics5, membrane transport6,7, high-efficiency energy harvesting and storage8,9, and fluidic circuits10,11. At nanoscale, strong electrostatic interactions between the charged surface and solution ions lead to many novel IT features which differ sharply from the counterparts at larger dimensions. Nonlinear and dynamic mass transport features have been observed from nanopores, at both transient and steady states.12-19 However, the dynamics of IT at nanoscale is far less explored in comparison to the steadystate IT. Better understanding of the nanoscale IT dynamics is of fundamental interest and essential to advance aforementioned applications mostly exploiting the nanoscale IT under non-steady-state conditions. A representative steady-state IT uniquely observed at nanoscale pores or channels with broken symmetry is ionic current rectification (ICR) 12,20, in which the ionic current under a certain potential is much higher than the one under the same potential amplitude but opposite polarity. The broken symmetry generally involves an asymmetrical geometry and/or inhomogeneous surface charge distribution in the nanopores. The ICR of the nanopores have been extensively investigated experimentally and theoretically.20-23 Many factors have been found to affect the ICR behaviors, such as surface charge density,

electrolyte type and concentration, geometry, potential, pressure, and concentration gradient.24-28 The distribution of charge carriers in solution, i.e. ionic concentration profile near the transport-limiting nanopore orifice region, can be more depleted or enriched by an applied potential, pressure and other factors.29-33 The charge-selective IT redistributes the charge carriers or polarizes the electrical double layer structure in nanopores. This dynamic process, from state one to state two (either state can be further enriched/depleted), depends on the rate of stimulus. Greater insights can be gained by studying the transition from one state to another rather than the approaches mostly focused on steady-state phenomena that provide limited dynamic information. For example, the rectification factor, the ratio of current/conductance at an arbitrary bias magnitude with opposite polarity, decreases at higher scan rates of the sweeping potential.13,34,35 By applying a small AC perturbation with different frequencies, inductive (or negative capacitive phase shift) reveals multi-timeconstant transport processes in impedance analysis by our group.36 Recently, we reported the pinched hysteresis current loops in the conical nanopores when applying a cyclic sweeping potential waveform in experiments and in simulation.37 The rectified current shows strong hysteresis or memory effect in IT that depends on the previous conductance states. Importantly, the pinched hysteresis current loops cross at a non-zero point, which is independ-

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ent of the external potential waveforms, the amplitude and the sweeping rates, when the charging/discharging of the exterior surfaces (glass substrate) is insignificant.37

tion ionic strength will inevitably affect the surface charge profiles. Consequently, impacts by an individual parameter cannot be readily determined.

The applied potential at the cross point measures an overall surface effect across a nanopore. As shown in Scheme 1, in single conical nanopores, the surface electric field has a component (Vs/dl) in the mass transport direction, i.e. along the centerline z direction across the nanopore, in which dl can be considered a function of effective nanopore depth limiting the transport. The surface potential (Vs) superimposes with the external bias (Vapp) and polarizes ion concentration profile. In the specific case/time when the Vapp balance out Vs (Vs + Vapp = 0), a unique steady-state independent of external stimulus (i.e. applied potential) is established. This would correspond to the cross point observed in the current-potential curves where dynamic enrichment or depletion of ions is zero (dCp /dt= 0).

In this paper, the cross-point potential and conductance in the electrokinetic transport through single conical nanopores are quantified, and correlated to surface and geometric factors through numerical simulation by solving Poisson and Nernst-Planck equations. The approach is founded on our earlier reports that the pinched hysteresis current-potential experimental responses can be well reproduced in the simulation.37 By fixing the remaining parameters constant, the key surface and geometry factors are systematically and individually adjusted in the simulation. Analytical expressions of the surface potential and conductance are developed based on the classical double layer theory. These results offer new physical insights and suggest viable routes to better control the transport processes through nanoscale interfaces.

EXPERIMENTAL SECTION Averaged surface potential. In Gouy-Chapman electrical double layer model, the potential distribution normal to an electrode or charged surface in an electrolyte solution can be simplified as:

φ x = φ0 e −κ x (1) Scheme 1. (A) Diagram of a single conical nanopore. The component of surface electric field in the mass transport direction Vs/dl is illustrated with a K+ as mobile charge carrier and negative surface charges on glass substrate. (B) The cross point in the pinched hysteresis current-potential curve (Cp = 0) separates the high and low conductivity states, or the dynamic concentration enrichment (Cp > 0) and depletion (Cp < 0) driven by the potential Vs + Vapp. In other words, the cross point is a very important parameter in several aspects: 1. it is the boundary potential for the external electrical field to induce either dynamic concentration enrichment or depletion inside nanopores; 2. Accordingly, the cross point, instead of the axis origin (0 V, 0 A), should be the separation of the high and low conductivity states of the nanopores in the cases where stimulated transport of matter occurs. It can therefore be thought of as a reference point for calibration/comparison. In addition, the cross-point potential and conductance could serve as descriptive parameters to evaluate the surface effect. This is fundamentally advantageous over the widely-used rectification factor (RF) that lacks physical meaning because it is calculated at arbitrarily selected potential amplitude. Moreover, the RF is not suitable to describe dynamic IT features. Quantitative correlation of the cross point (Vcp, Icp) to the surface and nanogeometry factors is important from fundamental perspective and those aforementioned applications. However, there are significant challenges in experiments because many of those nanopore parameters (i.e. surface and geometry) can be difficult to adjust systematically and are inter-related under a given measurement condition: variation of nanopore geometry or solu-

where ϕ0 is the surface potential. The decay constant κ (m-1) is a characteristic parameter to indicate the thickness of the double layer, 1/κ well known as the Debye length. The Debye length is strongly dependent on the electrolyte concentration c (mol/L). At room temperature and in dilute aqueous solutions with 1:1 electrolyte, it follows this widely used expression:

κ (nm−1 ) = c 0.3 (2) Inside a conical nanopore, surface charges exist on the functional groups (i.e. deprotonation of silanol groups on SiO2 substrate or carboxyl on polymer substrate). The density and distribution of surface charges depends on the surface reaction equilibrium under the given experimental conditions. Accordingly, the surface charge density (SCD) inside a nanopore, correspondingly the surface potential, can be heterogeneous and vary at different location, applied potential, and in different pH or electrolyte solutions.38 The potential profile , inside nanopore, where r is the radial direction and z the axial direction, will depend on the surface potential at the z position . At 293 K in 1:1 electrolyte:

φ x ( r , z ) = φ0 ( z ) e − κ r = φ0 ( z ) e − r

c /0.3

(3)

The expression considers small half cone angle to be consistent with our experimental results (ca. 11 degree) and makes approximation to that of flat surface or in cylinder nanopore geometry. The overall surface effects are indicated by the measured IT across the nanopore. Because the ion flux is known to distribute inhomogeneously away from a charged surface, i.e. higher for counter ions and lower for co-ions near nanopore surface com-


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pared to bulk, an averaged surface potential (ϕavg) along radial direction is necessary to characterize the overall surface effects, especially for the comparison of different sizes (volume) nanopores. The effective surface potential across the nanopore is measured as Vcp: φavg =

∫

r(z)

0

φ0 ( z )e −κ r dx

∫

r(z)

0

=

dx

φ0 ( z ) kr ( z )

(4)

(1 − e− kr ( z ) ) = Vcp

The Eq. 4 simplifies into Eq. 3 and ultimately into a linear relationship between the Vcp and a variable (such as r or √ ) when κr is larger than 2.3 by allowing a ca. 10% error/approximation. In the continuum regime, i.e., nanopore sizes much larger than ion sizes, the Eq. 3 and 4, although derived from 2D planar surface, could still be applied to the 3D conical nanopores as the potential profile is obtained by the (continuous) integration based on the axial symmetry. Surface curvature and vicinity effects in 3D nanopores are reflected by the newly fitted parameter 1/κ as counterpart of the ‘Debye length’ on 2D electrode surface. Cross point conductance. The overall conductance at the cross point (Gcp) is constant, not affected by concentration polarization dynamics. It comprises volumetric conductance Gvol and surface conductance Gs:

Gcp = Icp / Vcp = Gs +Gvol

(5)

For a conical nanopore with a radius r, half cone angle θ, and bulk concentration C0 (molar conductivity is Λm), the volumetric conductance can be estimated from bulk resistance Rbulk based on the ohm’s law: 1 1 1 R vol = ( + ) C 0 Λ m r π tan θ 4 (6) with a small half cone angle θ, 11 degree for example, the cone resistance term, 1/ (π tan θ), is much larger than the access resistance term (1/4). Ignoring the access resistance leaves the volumetric conductance of the nanopore in approximation as:

∇2 (ε0εrφ) = −F∑zici

(10)

i

In Eq. 9 and 10, Di, ci, and zi is the diffusion coefficient, electrolyte concentration, and valence of the ionic species i respectively. F is the Faraday constant, R is the gas constant, T is the temperature, ϕ is the potential, ε0 and εr are the relative permittivity of the vacuum and medium respectively. The electroosmotic flow is ignored here also in most nanopore transport studies because its contribution is found negligible compared to the diffusion and migration components with small SCD and low bias potentials.39 The glass membrane substrate generates a charging current experimentally that can be deconvoluted as reported previously.37 Because we are focusing on the dynamic mass transport processes signaled by the nonzero cross point, the large glass membrane is not included to reduce the demands on the simulation. A timedependent triangular potential waveform with adjustable scan rate v is applied. Supporting information provides details (Table S1). Gradient SCD. A gradient distribution of surface charge density (SCD) along the nanopore is found necessary to match the experimental i–V curves.38 The physical meaning has been proposed in terms of local electric field induced shift in deprotonation along the nanopore. The SCD profile in the nanopores is defined as follows:

σ ( z) = σ 0 e− z/L + σ b

(11) at the nanopore base (large opening), the SCD takes the bulk value σb (−0.001 C/m2). At the pore orifice (z = 0), the applied potential establishes a high electric field that will alter the surface charge distribution. A maximum SCD is defined as σ0 + σb. The SCD then decays exponentially along the z direction toward pore base. The exponential decay gradient SCD is shown in supporting information Scheme S1. The decay constant L is normally within hundreds of nanometers or few microns from the nanopore orifice indicating the distribution length of the high SCD.

Gvol =1 Rvol = C0Λmrπ tanθ (7)

RESULTS AND DISSCUSSION

The surface enhanced conductance (Gs) would be determined by the surface charges normally characterized by the parameter such as density (SCD, σ). Therefore, (8)

G cp = G s + G vol = G s (σ ) + C 0 Λ m r π tan θ

Note at the cross point, there is no dynamic concentration enrichment and depletion. In other words, Gcp is independent of potential scan rate or scan direction. Simulation details The simulation models and boundary conditions established in our previous reports are employed that have successfully reproduced key experimental features.37 The Poisson and Nernst-Planck (PNP) equations are solved with finite element analysis using the commercial software COMSOL Multiphysics 4.3. Ji = −Di∇ci −

zi F Di∇ci∇φ RT

(9)

Pinched i-V hysteresis loops with non-zero cross point. The simulated i–V responses from a 46 nm-radius nanopore in 1 mM KCl at different scan rates are shown in Figure 1. The maximum SCD is − 0.07 C/m2 with a distribution length of 1.5 µm. These parameters are selected that best fit the experimental results from a conical nanopore used in previous report.38 Consistent to the experimental results, pinched hysteresis current loops with a non-zero constant cross point are observed at different scan rates. The much higher scan rates needed in simulation to reproduce experimental features indicate there are missing factors such as molecular level ion-ion, ionsolvent interactions that require further study.37 The directions of the two hysteresis loops can be seen from the enlarge view around the cross point, in which the red arrows indicate the potential sweeping directions. The cross-point potential Vcp (86 mV) represents the averaged surface potential across the nanopore, and the conduct-



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ance Gcp (0.11 nA/ 86 mV) would consist both the surface and volumetric contributions, Gs and Gvol.

Figure 1. The simulated i–V responses from a 46 nmradius nanopore at different scan rates in 1 mM KCl (experimental data in Ref. 38). The inset is an enlarged view near the cross point with the red arrows indicating the potential sweeping direction. The pinched hysteresis loops cross at a non-zero cross point, which is independent of the scan rates. To correlate the cross point with individual nanopore characteristics, in the following sections unless defined otherwise, one of these parameters is varied systematically while all others are kept unchanged: maximum SCD of the nanopore (−0.07 C/m2), ionic strength (1 mM), and nanopore geometry (radius 46 nm and half-cone angle 11º). Since the cross point is independent of the scan rates and potential amplitudes (Figure S1) by excluding the exterior surfaces of the glass membrane substrate, a fixed scan rate is used.37 It is worth mentioning that it is very difficult to correlate the cross points to just one specific nanopore characteristic experimentally, due to the convolution of surface and geometry factors. The argument is based on our earlier attempts to fit experimental results by the systematic variation of simulation parameters.38,40 Another example is included in Figure S2 showing the dependence of surface parameters on ionic strength that remains to be verified by independent experimental methods at the relevant time/space scales. Surface charges effect. Shown in Figure 2, both the potential Vcp and the conductance Gcp at the cross-point increase as the maximum SCD (σ0 + σb) increases. Such dependence is expected since higher SCD results in higher surface potential, and thus the averaged surface potential. Furthermore, a well-defined linear relationship is observed between the Vcp and the calculated surface potential (ϕ0) at various maximum SCD which is in accordance with the relationship defined in Eq 4. The equivalent r0 is about 17.6 nm determined from the slope of 0.47 which depends on the nanogeometry and charge definitions. The surface potential ϕ0 under different SCD values is calculated based on the Grahame equation detailed in supporting information Fig. S3. The physical origin of the total ionic conductance at the cross point is revealed by the respective K+ and Cl‒ contributions shown in the panel B. Clearly, the conductance from K+ flux increases as the negative SCD values increases which constitutes the majority of the total cross

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point conductance. With comparable ion mobility, The K+ and Cl‒ flux is comparable at zero and very low SCDs, which corresponds to the volumetric conductance. Upon increase in SCD, the Cl‒ conductance decreases slightly initially and remains largely consistent. These results, as described by Eq 8, highlight two aspects: 1. significant surface effects can be observed from nanopore dimension much larger than Debye length, and 2. the conductance at the cross point include both the volumetric and surface charge enhanced contributions. Although the total conductance is the same for the forward and backward current branches at the cross point, there are detectable differences for both ions (data not shown). For simplicity that does not affect the concepts discussed in this report, the K+ and Cl‒ conductance values simulated under constant potential at the cross point are used to explain the dependence on surface charges. Interpretation of the dynamics in the respectively K+ and Cl‒ conductance, i.e. from data simulated under sweeping potentials, requires more comprehensive charge analysis that will be separately reported.

Figure 2. Dependence on SCD of (A) cross point potentials Vcp and (B) the total and respective ion conductance at the cross point. The K+ and Cl‒ conductance values were simulated under constant potential at the cross point. The maximum SCD values are varied in 1 mM KCl solution. Ionic strength effect. Due to the electrostatic screening effect, the cross-point potential is expected to decrease under increasing ionic strength. The Vcp and Gcp in different KCl concentrations are shown in Figure 3. An excellent exponential relationship (inset in panel A ) between the Vcp and C1/2 is observed which is in agreement with previous experimental studies.13 This is basically because the SCD remains unchanged and thus the surface potential ϕ0 remains constant in those simulations. According-


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ly, the Vcp should display an exponential relationship with the square root concentration as defined in Eqs. 3&4, similar to the Debye length description of potential profiles in solution near the solid-electrolyte interface. The decrease of Vcp can be explained by the screening of surface potential as the concentration increases. The contributions of cation and anion to the ionic conductance at the cross point vary in different electrolyte concentrations as shown in panel B. At high ionic strength, each conductance is close to its bulk value as illustrated by the dashed lines. This is consistent with the Vcp being close to zero at high ionic strength in panel A. Similar contributions from cations and anions originate from their similar diffusion coefficients. At low concentration regime, the enriched K+ ions that balance the negative surface charges dominate the cross-point conductance. This observation is also in good agreement with the threshold ionic conductance observed from cylinder nanochannels with low ionic strength.26 The Cl‒ conductance at the cross point, on the other hand, displays a linear relationship with respect to the concentration. This bulk behavior confirms the Cl‒ conductance at the cross point is largely volumetric. Starting from this zero-polarization cross point, there are two aspects worthy emphasizing. The first is, while electroneutrality could enable significant enrichment of both cations and anions, the extent of depletion will be limited by the available mobile charge carriers, i.e. anion conductance does not deviate from volumetric much. Secondly, while the conductance at this cross point is the same for the forward and backward scans, the ion transport hysteresis causes the conductivity to be different: the available ions to be re-distributed or transported will depend on the previous states under the varying stimulating potential. The Gcp is therefore an effective parameter to evaluate the respective contributions by surface and volume, just like the commonly used concentration-conductance plots for nanochannels with cylinder geometry: Gcp at low electrolyte concentration directly reveal the surface charges, since Gvol is insignificant and negligible compared to Gs; while Gcp at high ionic strength would be mainly Gvol. Noting that, except for this specific cross point conductance, the conductance values under any other potential values will include the potential-induced ion polarization, which is both dynamic and nonlinear.

Figure 3. (A) The cross-point potential Vcp and (B) the cross-point conductance Gcp as a function of the ionic strength. The conductance data were from a single point simulation at the Vcp. The red curves in panel A are the fittings with Eq 4, and inset in A shows the natural logarithm plot (Eq 3). The dash lines in panel B indicate the volumetric conductance Gvol calculated from Eq 7. Nanopore radius effect. The radius effect on the cross point is included in Figure 4. The cross-point potential decreases as expected upon the increasing pore radius under the same interface properties (SCD and ionic strength). The solid line shows the excellent fitting using Eq 4, with fitting parameters ϕ0 = 140 mV and 1/κ = 29 nm. In 10 mM KCl, Debye length 1/κ is about 3 nm on planar surface. At a SCD of −0.07 C/m2, the calculated ϕ0 (Figure S2) is 127 mV, which is reasonably close to the fitted ϕ0 albeit the 1/κ is significantly larger presumably due to the convergence of the interface inside conical nanopores. Therefore, nanopores with sizes up to tens or even hundreds nanometer, or much larger than the corresponding ‘Debye length’ estimated by the bulk ionic strength, still display significant or detectable surface effects.



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SCD gradient introduced in the simulation, the radius effect on the cross-point potential under constant SCD is shown in the panel C. The total surface charges in the two data sets are maintained to be the same. The same trend can be observed by fitting with the decay constant 1/κ at 8.5 nm using Eq 4. The cross-point potential decreases quickly with respect to the increase of the radius. The difference is due to more evenly distributed charges inside the nanopore toward the base which would lead to less asymmetric ion transport processes, and thus smaller Vcp. This is the same reason that cylindrical channels, though with charged surfaces, do not have rectification in the i-V measurements. Related, under a constant SCD without a gradient, the simulated cross point potential against different SCD values, different ion concentrations along with the i-V curves are included in Fig. S4for reference. Similar i-V features and dependence were obtained. Half cone angle effect. Within a small variation that is within the range of available experimental results, i.e. the fabrication of conical nanopores, the half cone angle (around 10 degree) and the nanopore depth (membrane thickness at microns) should have less impact on the cross-point potential at a given radius. Shown in Figure 5, one can clearly see that the cross-point potential is a weak function of the sharp half cone angles that would be difficult to resolve experimentally. However, the cross-point conductance displays a linear relationship with sin θ. It is easily understandable from volumetric perspective (Eq. 8). Larger nanopore volume when θ increases corresponds to more ions inside the nanopore and thus larger ionic conductance. Meanwhile, the component of the surface electric field in the ion transport direction also increases as θ increases, correspondingly higher polarized concentration and thus larger surface conductance Gs. Figure 4 (A) The cross-point potentials Vcp as a function of the radius at a fixed half cone angle of 11 deg. The red curve is the fitting with Eq 4. The gradient SCD: maximum SCD is −0.07 C/m2 with a distribution length of 1.5 µm. (B) The GM as a function of the radius. The red line is the fitting with Eq 8 and the volumetric conductance calculated from Eq 7 is also offered for comparison. (C) The cross-point potentials as a function of radius at a constant SCD definition. Constant SCD is −0.05 C/m2. The red curve is the fitting with Eq 4.

It is important to point out that in experimental studies, the surface charge parameters (SCD & gradient) depend on ionic strength, nanopore radius, half cone angle, with the nanogeometry characterization subject to heterogeneity and the resolution limits). For example, different solution ionic strength and applied field could change the deprotonation process of surface functional groups, thus result in different SCD distribution on the nanopore substrate. Accordingly, there might be deviations to fit the experimental results quantitatively with the proposed equations.

The linear relationship between the cross-point conductance and nanopore radius shown in panel B can be understood by Eqs. 7&8. The relatively constant difference between the Gcp and Gvol would be the surface contributed conductance Gs. The surface conductance is not affected by the nanopore radius due to the unchanged surface charge definition in the simulation: the induced changes in concentration of the counter-/co- ions remain unchanged, regardless of the pore size. The gradient of surface charges will depend on the electrical field strength at the conical nanopore orifice region experimentally.38 For reference to the simulation literature in which a constant SCD is widely adopted (without a quantitative fitting to experimental results), and in comparison to the results in panel A with an exponential


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Analytical Chemistry izes the dynamic aspects of the mass transport processes through nanopores.

ASSOCIATED CONTENT Supporting Information The details of the simulation, one experimental example, the cross-point potential analysis for a uniformly SCD definition. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author *Email: [email protected]; Tel: 404-413-5507

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENT The support by NSF grant CHE-1610616 is acknowledged.

REFERENCES

Figure 5 The half cone angle effect on the (A) cross-point potential and (B) conductance at small half cone angle values. The dashed lines are added to illustrate the trend, not by mathematic fitting. Geometrically, the Gvol should depend on tan θ, while the Gcp on sin θ. Mathematically, at small θ values, θ=tan θ=sin θ in approximation.

CONCLUSIONS The unique cross points from the pinched hysteresis current loops in ion transport through single conical nanopores are investigated and analyzed. By keeping other parameters constant while varying a single factor systematically, definitive and quantitative correlations are established regarding the dynamic ion transport characteristics inaccessible for experimental studies. We have further demonstrated that the cross-point potential represents the averaged surface potential as proposed in earlier report. Further, the cross-point conductance is shown to include both volumetric and, in many cases, primarily the diffuse layer conductance without any polarization. The dependence of the cross-point potential and conductance on nanopore size, cone angle, solution ionic strength and surface charges can be quantitatively described by the models developed from classical double layer theory. The surface curvature effects as well as the vicinity/overlapping of the EDL in a 3D nanopore are reflected by the fitted ‘Debye length’ parameter that is much larger than that from a flat 2D surface under comparable conditions. Overall, the potential and conductance at the cross point, which eliminate the dynamic polarization effect, will allow a more meaningful quantification of the surface effects in the nanopore-type devices that better character-

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