Ion Current Rectification at Nanopores in Glass Membranes

Jan 29, 2008 - Poisson, and Navier-Stokes equations for the two-ion (K+ and Cl-) system. Fixed surface charge on both the internal and external glass ...
0 downloads 0 Views 199KB Size
Download PDF
2212

Langmuir 2008, 24, 2212-2218

Ion Current Rectification at Nanopores in Glass Membranes Henry S. White† and Andreas Bund*,‡ UniVersity of Utah, Chemistry Department, Salt Lake City, Utah 84112, and Dresden UniVersity of Technology, Department of Physical Chemistry and Electrochemistry, D-01062 Dresden, Germany ReceiVed September 24, 2007. In Final Form: NoVember 28, 2007 The origin of ion current rectification observed at conical-shaped nanopores in glass membranes immersed in KCl solutions has been investigated using finite-element simulations. The ion concentrations and fluxes (due to diffusion, migration, and electroosmotic convection) were determined by the simultaneous solution of the Nernst-Planck, Poisson, and Navier-Stokes equations for the two-ion (K+ and Cl-) system. Fixed surface charge on both the internal and external glass surfaces that define the pore structure was included to account for electric fields and nonuniform ion conductivity within the nanopores and electric fields in the external solution near the pore mouth. We demonstrate that previous observations of ion current rectification in conical-shaped glass nanopores are a consequence of the voltage-dependent solution conductivity in the vicinity of the pore mouth, both inside and outside of the pore. The simulations also demonstrate that current rectification is maximized at intermediate bulk ion concentrations, a combination of (i) the electrical screening of surface charge at high concentrations and (ii) a fixed number of charge-carrying ions in the pore at lower concentration, which are physical conditions where the voltage dependence of the conductivity disappears. In addition, we have quantitatively shown that electroosmotic flow gives rise to a significant but small contribution to current rectification.

1. Introduction Analogous to biological channels and pores, synthetic nanopores often carry surface charges giving rise to interesting electrical properties.1-4 It has been shown that asymmetric nanopores can transport potassium ions against a concentration gradient if stimulated by external field fluctuations.5 Furthermore, they can be used to detect and count particles6 and molecules such as porphyrins7 and DNA.8-10 Charge is transported through the pore by the flux of ions, and these moving charges interact with the immobile surface charges.11 An interesting feature associated with small charged conical pores is ion current rectification (ICR), or the departure of the experimentally measured current-voltage curve from ohmic behavior. In other words, the magnitude of current through the nanopore at negative potentials is greater or less than the current at positive potentials. It is generally accepted that ICR occurs in asymmetric nanopores when the characteristic length scale (e.g., the pore diameter) is of the order of the Debye screening length or smaller. Of course, geometric asymmetry is an intrinsic feature of conically shaped * Corresponding author. E-mail: [email protected]. † University of Utah. ‡ Dresden University of Technology. (1) Siwy, Z. S.; Apel, P.; Baur, D.; Dobrev, D. D.; Korchev, Y. E.; Neumann, R.; Spohr, R.; Trautmann, C.; Voss, K.-O. Surf. Sci. 2003, 532-535, 1061-1066. (2) Siwy, Z. S.; Heins, E.; Harrell, C. C.; Kohli, P.; Martin, C. R. J. Am. Chem. Soc. 2004, 126, 10850-10851. (3) Ramı´rez, P.; Aguilella-Arzo, M.; Alcaraz, A.; Cervera, J.; Aguilella, V. M. Cell Biochem. Biophys. 2006, 44, 287-312. (4) Ho, C.; Qiao, R.; Heng, J. B.; Chatterjee, A.; Timp, R. J.; Aluru, N. R.; Timp, G. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 10445-10450. (5) Siwy, Z. S.; Fulinski, A. Am. J. Phys. 2004, 72, 567-574. (6) Lee, S.; Zhang, Y.; White, H. S.; Harrell, C. C.; Martin, C. R. Anal. Chem. 2004, 76, 6108-6115. (7) Heins, E. A.; Siwy, Z. S.; Baker, L. A.; Martin, C. R. Nano Lett. 2005, 5, 1824-1829. (8) Schiedt, B.; Healy, K.; Morrison, A. P.; Neumann, R.; Siwy, Z. S. Nucl. Instrum. Methods B 2005, 236, 109-116. (9) Harrell, C. C.; Choi, Y.; Horne, L. P.; Baker, L. A.; Siwy, Z. S.; Martin, C. R. Langmuir 2006, 22, 10837-10843. (10) Smeets, R. M. M.; Keyser, U. F.; Krapf, D.; Wu, M. Y.; Dekker, N. H.; Dekker, C. Nano Lett. 2006, 6, 89-95. (11) Lavalle´e, M.; Szabo, G. In Glass Microelectrodes; Lavalle´e, M., Schanne, O. F., He´bert, N. C., Eds.; Wiley: New York, 1969.

systems. It has been shown that ICR can be induced in geometrically symmetric capillaries containing an asymmetric charge distribution.12-14 ICR is interesting from a fundamental point as well as for microfluidic applications. There is still some controversy about the origins of the effect. Siwy and colleagues15,16 proposed that ICR results from a ratchet mechanism that is based on the asymmetry of the electric potential barrier inside the pore. Woermann17-19 proposed a model that is based on an inhomogeneous conductivity near the pore orifice (i.e., transition region, a more detailed discussion is given below). The inhomogeneity is a consequence of the asymmetric fluxes of the cations and anions. The phenomenological model of Wei et al.20 assumes that the pore mouth may be envisioned as comprising a thin membrane whose ionic transference numbers are different from those of the bulk solution. The different solid angles on the outside and inside of the pore mouth result in asymmetric current flows. This asymmetry is described with a heuristic factor that relates the fluxes of the anion and the cation and does not take into consideration the electrical properties of the system (e.g., surface charge density, screening length, etc.). Another open question is whether electroosmotic flow (EOF) in charged conical nanopores contributes to ICR. EOF arises from the passage of current near charged surfaces, and can enhance or depress the flux of an ion species. From the concentration dependence of the ICR, Vlassiouk et al. concluded that EOF plays a minor role in the mechanism of ICR at a nanofluidic diode with a charge pattern.21 However, no detailed quantitative discussion was given. (12) Kosinska, I. D. J. Chem. Phys. 2006, 124, 244707-7. (13) Karnik, R.; Duan, C.; Castelino, K.; Daiguji, H.; Majumdar, A. Nano Lett. 2007, 7, 547-551. (14) Fulinski, A.; Kosinska, I. D.; Siwy, Z. S. Europhys. Lett. 2004, 67, 683689. (15) Siwy, Z. S.; Gu, Y.; Spohr, H. A.; Baur, D.; Wolf-Reber, A.; Spohr, R.; Apel, P.; Korchev, Y. E. Europhys. Lett. 2002, 60, 349-355. (16) Siwy, Z. S. AdV. Funct. Mater. 2006, 16, 735-746. (17) Woermann, D. Nucl. Instrum. Methods B 2002, 194, 458-462. (18) Woermann, D. Phys. Chem. Chem. Phys. 2003, 5, 1853-1858. (19) Woermann, D. Phys. Chem. Chem. Phys. 2004, 6, 3130-3132. (20) Wei, C.; Bard, A. J.; Feldberg, S. W. Anal. Chem. 1997, 69, 4627-4633. (21) Vlassiouk, I.; Siwy, Z. S. Nano Lett. 2007, 7, 552-556.

10.1021/la702955k CCC: $40.75 © 2008 American Chemical Society Published on Web 01/29/2008

Ion Current Rectification at Nanopores

Langmuir, Vol. 24, No. 5, 2008 2213

Daiguji et al.22 estimated the contribution of EOF to the ionic transport in 30 nm nanofluidic channels to be less than 10% but they also pointed out that EOF increases considerably with increasing surface charge. In the majority of the models describing ICR, the possibility of EOF has been neglected. A physical description of the transport properties of a charged nanopore starts from the flux equations (Nernst-Planck equation, eq 1) for each ionic species.

Ji ) -Di∇ci -

ziF D c ∇Φ + ciu RT i i

(1)

Here, Ji, Di, ci, and zi are, respectively, the flux, diffusion constant, concentration, and charge of species i. Φ and u are the local electric potential and fluid velocity, and F, R, and T are the Faraday constant, the gas constant, and the absolute temperature, respectively. Ji, ci, Φ, and u are position-dependent quantities. The relationship between the electric potential and ion concentrations is described by the Poisson equation, eq 2,

∇2Φ ) -

F 

∑i zici

(2)

where  is the dielectric constant of the medium. The flow distribution is described by the Navier-Stokes equation, eq 3, describing pressure and electrical force driven flow.

u∇u )

1 F

(-∇p + η∇2u - F

( )

∑i zici ∇Φ)

(3)

In eq 3, F and η are the density and viscosity of the fluid, and p is the pressure. The system of coupled eqs 1 to 3 must be solved for the given geometry using appropriate boundary conditions. The solution yields the concentration fields ci(r) for all species, the potential Φ(r), velocity u(r), and pressure p(r) distributions in the fluid. Given the complexity of asymmetric nanopore systems, an analytical solution of eqs 1-3 will not be possible. Instead, one has to resort to numerical solutions, e.g., using finite-difference or finite-element methods. The system is also generally simplified by assuming steady-sate conditions (∇‚Ji ) 0) and incompressible fluid flow (∇‚u ) 0). We recently reported the numerical solution of eq 1 using the finite element method to model the voltammetric behavior of a truncated cone-shaped nanopore electrode.23 Numerical simulations of the electric transport properties of conical nanopores have been performed by Cervera et al.24,25 They solved eqs 1 and 2 neglecting hydrodynamic effects (u ) 0). Using the surface charge on the pore wall (σ) as the only adjustable parameter, they were able to model the current-voltage behavior for conical pores in PET observed earlier by Siwy and co-workers. Their simulations yielded a value of σ ∼ -1 e/nm2, in agreement with the surface charge of PET. Cervera et al. also could confirm the shape of the electric potential barrier suggested by Siwy. Daiguji and co-workers13,22,26 numerically modeled the ionic transport in rectangular nanofluidic channels based on eqs 1 to 3. Their results impressively show how ionic currents can be controlled by local modifications of the surface charge, i.e., the concept of an asymmetric charge distribution. (22) Daiguji, H.; Yang, P.; Majumdar, A. Nano Lett. 2004, 4, 137-142. (23) Zhang, B.; Zhang, Y.; White, H. S. Anal. Chem. 2004, 76, 6229-6238. (24) Cervera, J.; Schiedt, B.; Ramı´rez, P. Europhys. Lett. 2005, 71, 35-41. (25) Cervera, J.; Schiedt, B.; Neumann, R.; Mafe, S.; Ramirez, P. J. Chem. Phys. 2006, 124, 104706. (26) Daiguji, H.; Oka, Y.; Shirono, K. Nano Lett. 2005, 5, 2274-2280.

Glasses,20,23 silicon nitride4,10 and polymers such PET15 or poly(imide)27 have been used as the matrix materials for synthetic pores. The charge of glass and silica surfaces in highly diluted electrolytes (10-6 M, pH 6) is in the order of -0.5 mC/m2 corresponding to -0.003 e/nm2 28 (e is the electric charge of a proton). The magnitude of σ increases with increasing ionic strength and increasing pH. Furthermore, σ depends on the number of active surface sites (Si-OH in the case of glass) and the pKa of the corresponding dissociation reaction.28 For silicon nitride in KCl solutions, the experimental results of Ho et al.4 are consistent with a surface charge between -0.01 and -0.1 C/m2 (-0.06 to -0.6 e/nm2). The surface charge inside track-etched pores in polymer membranes (PET) has been reported to be in the order of -0.16 C/m2 (-1 e/nm2).14,15,24 In the present paper, we report numerical solutions of the eqs 1 to 3. While most previous simulations have been performed using a finite-difference algorithm, we decided to use the finiteelement method as it is better suited for systems with irregular geometry.29 The accuracy of the numerical methods and software was examined by simulation of the ion distributions and fluxes for several simpler problems involving electrical double layers where known analytical solutions exist. Our analysis of the potential, concentration, and flux distribution provides insight into the origin and mechanism of ICR. Additionally, inclusion of eq 3 in our analysis allows a quantitative estimate of the contribution of the EOF to ICR.

2. Numerical Procedure The set of coupled equations was solved using the commercial software package COMSOL 3.3 Multiphysics. The multiphysics mode proved to be extremely useful for our purpose because it readily allows solutions of the coupled governing equations for arbitrary geometry. The current was calculated using weak constraints. The weak constraints feature in COMSOL Multiphysics implements constraints by using finite elements on the constraint domain for the Lagrange multipliers, and by solving for the Lagrange multipliers along with the original problem. This approach allows the calculation of very accurate fluxes.30 Some more details on the numerical procedure are given in the Supporting Information. A sketch of the computation domain for the conically shaped nanopore is shown in Figure 1. The boundary conditions for eqs 1 to 3 are summarized in Table 1, using the numbers 1-7 to designate the surfaces defining the model system. The conical pore is symmetric along the axis of the pore (surface 1), with a small orifice that opens to a reservoir of solution. In addition to considering charge on the wall of the nanopore (surface 3), the influence of charge on the surface exterior to the pore orifice (surface 4) was also included. It is important to include the effect of charge on this external surface in modeling transport through nanopores in glass membranes31 or nanopipettes,20 as this surface has the same chemistry as the interior wall of the nanopore. We note that the magnitude of the entrance/exit mass transfer resistances at the small orifice may be comparable to interior resistance of a conical nanopore, and can thus significantly influence the overall transport of ions. The length of surface 4 (27) Siwy, Z. S.; Dobrev, D.; Neumann, R.; Trautmann, C.; Voss, K. Appl. Phys. A 2003, 76, 781-785. (28) Behrens, S. H.; Grier, D. G. J. Chem. Phys. 2001, 115, 6716-6721. (29) Chapra, S. C.; Canale, R. P. Numerical Methods for Engineers, 2nd ed.; McGraw-Hill: New York, 1988. (30) COMSOL Multiphysics User’s Guide; Comsol AB: Stockholm, 2006; p 572. (31) White, R. J.; Zhang, B.; Daniel, S.; Tang, J. M.; Ervin, E. N.; Cremer, P. S.; White, H. S. Langmuir 2006, 22, 10777-10783.

2214 Langmuir, Vol. 24, No. 5, 2008

White and Bund

Figure 1. Sketch of the computation domain for the conically shaped nanopores. The numbers designate the boundary conditions that are summarized in Table 1. Only the right half of this full 2π cylindrical representation is used for the numerical simulation. Line 1 is the axis of symmetry. Table 1. Boundary Conditions for the Numerical Solution of Equations 1-3 in the Computation Domain Sketched in Figure 1 surface 1 2 3, 4 5, 6 7

a

Nernst-Planck eq (eq 1) axial symmetry constant concentration ci ) c0 insulation n‚Ji ) 0a insulation n‚Ji ) 0a constant concentration ci ) c0

Poisson eq (eq 2)

Navier-Stokes eq (eq 3)

axial symmetry constant potential Φ ) Φinside surface charge n‚∇Φ ) -σ/ea zero charge n‚∇Φ ) 0 constant potential Φ)0

axial symmetry constant pressure p)0 no slip u)0 no slip u)0 constant pressure p)0

n means the surface normal vector.

was equal to twenty times the pore mouth radius. Surfaces 5 and 6 that partially define the reservoir are assumed to be insulating surfaces. Two surfaces (2 and 7) correspond to the electrodes used to apply a bias voltage across the nanopore. These surfaces are held at constant potential, with the surface 7 in the reservoir designated as ground. The concentration of electrolyte at these surfaces is held constant at the bulk value. The electrode (surface 7) on the small orifice side of the nanopore is placed in the reservoir, away from the orifice, in order to accurately simulate entrance/exit effects at this small opening (vide infra). The electrode (surface 2) on the side of the nanopore with the large opening is placed directly at the position of this opening (i.e., the base of the nanopore), as the entrance/exit mass transport resistance at the large opening is negligibly small. An adaptive mesh refinement was used to optimize the mesh size geometry. The adjustment algorithm refined the mesh at the charged wall down to a value of 1 nm, sufficient for accurately

Figure 2. (a) Comparison of the simulated spatial variation of the electric potential (open symbols), Φ(x), in front of a charged wall (σ ) -1 mC/m2) with the analytical solution (solid lines) (eq 4). The value of Φ has been normalized with respect to its value at x ) 0. Three different concentrations of KCl (i.e., three different Debye lengths) are compared: (O) 1, (0) 10, (4) 100 mM KCl. The inset shows a sketch of the computation domain. (b) Comparison of the simulated spatial variation of the concentrations of K+ (O) and Cl(0), ci(x), adjacent to a charged wall (σ ) -1 mC/m2) with the analytical solution (solid lines) (eq 5). The bulk concentration is 1 mM KCl.

resolving the features of the electric double layer. Quadratic Lagrange polynomials were chosen as the element functions. The computation domain assumed a room-temperature aqueous KCl solution, for which the following parameters were used: T ) 298 K, D(K+) ) 1.957×10-9 m2/s, D(Cl-) ) 2.032×10-9 m2/s, relative dielectric constant  ) 80, η ) 10-3 Pa s, and F ) 103 kg/m3.

3. Results and Discussion 3.1. Validation of the Numerical Model. In this section, we briefly present numerical results for two problems involving electrical double layer structures for which analytical solutions exist. By this approach we verify the accuracy of the numerical model used to evaluate ICR in conical nanopores. (i) Diffuse Double Layer Adjacent to a Flat Surface. The concentration field and the electric field were simulated in a rectangular (500 × 200 nm2) computation domain with a surface charge density σ )-1 mC/m2 (-6.24×10-3 e/nm2) at the left wall (see inset of Figure 2a). At equilibrium (i.e., no current flow, Ji ) 0, u ) 0) the potential distribution normal to a charged wall can be compared with the analytical expression developed

Ion Current Rectification at Nanopores

Langmuir, Vol. 24, No. 5, 2008 2215

in the framework of the Gouy-Chapman theory.32,33 For a symmetric electrolyte of valence z the potential is given by eq 4,33

Φ(x) )

2RT 1 - K exp(-x/λ) ln F 1 + K exp(-x/λ)

(4)

where x is the distance from the wall, λ ) [RT/(2z2F2c)]1/2 is the Debye length, K ) Q/[2 + (4 + Q2)], Q ) -zλFσ/(RT), and σ is the charge density on the wall. As shown in Figure 2a, the agreement between simulation and the analytical solution is excellent. The simulated values for Φ(x ) 0) are -13.63, -4.36, and -1.37 mV in 1, 10, and 100 mM KCl, respectively. The corresponding values calculated from eq 4 are -13.72, -4.38, and -1.39 mV. The equilibrium distribution of the ionic species follows a Boltzmann distribution, eq 5. This can be easily shown by setting Ji ) 0 and u ) 0 in eq 1 followed by spatial integration of ci and Φ, to yield

(

ci(x) ) ci,0 exp -

)

ziFΦ(x) RT

(5)

where ci,0 is the bulk concentration of species i. Figure 2b shows that the finite-element model also correctly predicts the doublelayer concentration fields. (ii) EOF in a Straight Nanopore. To verify if the finite-element model correctly describes EOF in small structures, we solved eqs 1 to 3 for a 50 nm radius capillary of 10 µm length, separating two cylindrical reservoirs (radius 2 µm, length 2 µm) at the nanopore entrance and exit. The z-directed velocity profile within the nanopore is given by

Vz(r) ) -

λσEz ηI1(r0/λ)

(I0(r0/λ) - I0(r/λ))

(6)

where Ez the z component of the electric field, r0 the capillary radius, and Ii are modified Bessel functions of the first kind, of order i.33 In our simulations, a charge density of σ ) -1 mC/m2 was assigned to the capillary surface and face of the reservoirs wall facing the nanopore. The electrolyte was 1 mM KCl and a voltage of -0.5 V was applied between the ends of the reservoirs. Figure 3 shows a comparison of the EOF velocity profiles obtained from the simulation and predicted by eq 6.33 For the calculation of Vz(r) according to eq 6 the z component of the electric field, Ez, is needed. We used Ez ) -5 × 104 V/m, which is in agreement with the geometry (applied voltage -0.5 V, capillary length 10 µm) and the results of our numerical simulations. The poorer agreement between analytical and numerical values for the EOF example presented in Figure 3, relative to results for a static electrical double layer (Figure 2), is due to the increased computation load associated with simulating both mass and momentum transport, demanding a coarser mesh. Having verified our model for the above geometries, the next step is to apply the finite-element model to the computation of the concentration, flux and potential distributions within a conical nanopore. To the best of our knowledge no analytical solution exists for eqs 1 to 3 for this geometry. However, in the absence of surface charge or in solutions of high ionic strength (i.e., shielding of the surface charge) conical nanopores display ohmic behavior,16,20 with a resistance is given by eq 7,31,34 (32) Grahame, D. C. Chem. ReV. 1947, 41, 441-501. (33) Newman, J. S.; Thomas-Alyea, K. E. Electrochemical Systems, 3rd ed.; John Wiley & Sons: Hoboken, NJ, 2004. (34) Zhang, B.; Zhang, Y.; White, H. S. Anal. Chem. 2006, 78, 477-483.

Figure 3. Comparison of the simulated value (open symbols) of the electro-osmotic flow in a straight 50 nm radius capillary (10 µm length) with the analytical solution (solid line) (eq 6). The surface charge of the capillary is σ ) -1 mC/m2, and the bulk electrolyte concentration is 1 mM KCl. The center of the capillary is at r ) 0. The simulated data correspond to the midpoint of the capillary (i.e., 5 µm from the reservoirs). The inset shows a sketch of the computation domain.

Rp )

L 1 + πκr1(r1 + L tan Θ) 4κr1

(7)

where L is the length of the pore, r1 is the radius of the small opening (mouth or orifice), κ is the conductivity of the electrolyte, and Θ is the half cone angle. For L tan Θ . r1, eq 7 reduces to Rp ) 1/(κr1)[1/4 + 1/(π tan Θ)], a convenient expression for the determination of the size of the pore orifice provided that rectification is absent. The solid line plotted in Figure 4 shows the simulated i(E) curve in 1 M KCl for a conical pore without surface charge and with r1 ) 5 nm, L ) 400 nm, and Θ ) 12°. The simulated response is ohmic with a slope corresponding to Rp ) 23 MΩ (for comparison, eq 7 yields an analytical value of 22.2 MΩ). Decreasing the KCl concentration results in a proportional increase in the simulated resistance. The difference in Rp obtained by simulation and analytical solution is less than 5% in all cases. 3.2. Numerical Simulation of ICR at Conical Nanopores. We now turn our attention to conical nanopores with surface charge. In the initial results described below, the effect of EOF was not included in the simulation (i.e., u ) 0 in the computation domain). The influence of EOF on the nanopore electrical behavior is addressed in a later section. The dashed lines plotted in Figure 4 shows how the introduction of surface charge on the internal and/or external pore surfaces affects the i(E) curve of a 50 nm mouth radius nanopore in 1 mM KCl. As noted above, in the absence of any surface charge, the i(E) curve is a straight line whose inverse slope is given by eq 7. Applying σ ) -1 mC/m2 to either the interior nanopore surface (surface 3, Figure 1) and/or around the mouth of the pore (surface 4, Figure 1) makes the i(E) curve asymmetric. The sign convention used in the present paper is that the applied bias is equal to the electrostatic potential of the inside electrode versus the outside electrode (i.e., E ) φinside vs φoutside, see Figure 1). In other words, a positive bias means that the electrode defined by surface 2 (Figure 1) is the anode and the electrode defined by surface 7 is the cathode. In an experimental setup with silver/silver chloride electrodes, the anodic and cathodic reactions would be Ag + Clf AgCl + e- and AgCl + e- f Ag + Cl-, respectively. Thus for a positive bias Cl- ions are transferred from the outside

2216 Langmuir, Vol. 24, No. 5, 2008

White and Bund

Figure 4. Effect of surface charge on the i(E) curve of a 50 nm mouth radius nanopore (base radius 2.2 µm, length 10 µm, half cone angle 12°) in 1 mM KCl. (-) no surface charge, (--) -1 mC/m2 on the wall, (‚‚‚) -1 mC/m2 around the mouth (-‚-) -1 mC/m2 on the wall and around the mouth. Electroosmotic flow was ignored in these simulations.

electrode (surface 7) to the inside electrode (surface 2) and at the same time K+ ions move in the other direction. Figure 4 shows that negative charge on the pore surface results in the current at negative biases being larger than the current at positive bias (rectification). This finding is in qualitative agreement with the reports of Wei et al.20 and White et al.31 for ICR at nanopores constructed from glass whose surface is negatively charged. As a quantitative measure of ICR, one can define a rectification ratio, r(E) ) |i(-E)/i(E)|. For the data shown in Figure 4, the r(0.5 V) values are 1.17 (charge on the external surface surrounding the orifice), 1.18 (charge on the interior surface of the nanopore), and 1.38 (charge on both the internal and external surfaces). As expected, the presence of charge on both the wall and around the mouth has the strongest effect. Similarly, positive charge on the pore surface yields i(E) curves with larger currents at positive biases (data not shown). This effect of the sign of the surface charge on ICR is experimentally well documented.2,16,35 The following discussion refers to simulations where charge resides on both the interior pore wall and the surface exterior to the mouth of the pore, if not otherwise noted. It is instructive to examine the potential and concentration profiles near the pore orifice in order to identify possible mechanisms of ICR. Figure 5 shows the electrical potential distribution along the centerline axis of a 50 nm mouth radius pore with a surface charge -1 mC/m2 for applied biases of +0.5, 0, and -0.5 V. First, it is noteworthy that a surface charge of -1 mC/m2, which is typical of glass surfaces, generates a very small electrostatic potential at the nanopore orifice (