Mechanism of Electrostatic Gating at Conical Glass Nanopore

Sep 24, 2008 - Chemistry Department, UniVersity of Utah, Salt Lake City, Utah 84112, and ... fluxes through the nanometer-scale orifices of conical gl...
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Langmuir 2008, 24, 12062-12067

Mechanism of Electrostatic Gating at Conical Glass Nanopore Electrodes Henry S. White*,† and Andreas Bund*,‡ Chemistry Department, UniVersity of Utah, Salt Lake City, Utah 84112, and Lehrstuhl fuer Physikalische Chemie and Elektrochemie, Technische UniVersitaet Dresden, D-01062 Dresden, Germany ReceiVed June 7, 2008. ReVised Manuscript ReceiVed July 28, 2008 The mechanism of molecule-based electrostatic gating of redox fluxes at conical glass nanopore (GNP) electrodes has been investigated using finite-element simulations. The results demonstrate that the fluxes of cationic redox molecules through the nanopore orifice can be reduced to negligibly small values when the surface charge of the nanopore is switched from a negative to a positive value. Electrostatic charge reversal can be affected by ionization of surface-bound moieties in response to environmental stimuli (e.g., photoionization or acid protonation), but only if the negative charge of the glass is included in the analysis. Numerical simulations of the responses of GNP electrodes are based on a simultaneous solution of the Poisson and Nernst-Planck equations and are in excellent agreement with our previously reported experimental results for electrostatic gating of the fluxes of Ru(NH3)63+ and Fe(bpy)32+ at GNP electrodes with orifice radii between 15 and 100 nm. The gating mechanism is discussed in terms of three components: (1) migration of ionic redox species in the depletion layer adjacent to the electrode surface; (2) migrational transport along the charged pore walls; (3) electrostatic rejection of charged molecules at the pore orifice. The numerical results indicate that all three components are operative, but that ion migration along the pore walls is dominant.

1. Introduction We describe finite-element simulations that provide a quantitative mechanism underlying the electrostatic gating of redox fluxes through the nanometer-scale orifices of conical glass nanopore (GNP) electrodes.1 A GNP is a metal microdisk electrode embedded at the bottom of a truncated conical pore that is synthesized in glass. GNP electrodes are fabricated with pore orifice radii as small as a few nanometers.2 Nanopore sensors, including GNP electrodes and membranes, have been proposed for the detection and analysis of single molecules and particles.3-6 Electrostatic gating refers to the ability to control the flux of molecules from the bulk solution to the electrode surface, through the orifice (Figure 1), by either chemical (e.g., pH) or external (e.g., photons) stimuli that alter the sign and density of electrical charge on the interior and exterior walls of the nanopore. The stimulus causes a shift in the equilibrium position of chemical reactions associated with molecules bound to the surfaces, creating or removing electrical charge. For instance, attachment of an amine (e.g., (3-aminopropyldimethylethoxysilane) to the nanopore surface provides a means to create (or remove) a net positive surface charge by lowering (or raising) the pH of the contacting solution. The electrical charge created by surface reactions has been shown to effectively block the entry (or exit) of charged redox molecules due to strong electrostatic interactions. For instance, * To whom correspondence should be addressed. E-mail: white@ chem.utah.edu (H.S.W.); [email protected] (A.B.). † University of Utah. ‡ Technische Universitaet Dresden. (1) Wang, G.; Zhang, B.; Wayment, J. R.; Harris, J. M.; White, H. S. J. Am. Chem. Soc. 2006, 128, 7679–7686. (2) Zhang, B.; Galusha, J.; Shiozawa, P. G.; Wang, G.; Bergren, A. J.; Jones, R. M.; White, R. J.; Ervin, E. N.; Cauley, C. C.; White, H. S. Anal. Chem. 2007, 79, 4778–4787. (3) Sexton, L. T.; Horne, L. P.; Sherrill, S. A.; Bishop, G. W.; Baker, L. A.; Martin, C. R. J. Am. Chem. Soc. 2007, 129, 13144–13152. (4) Ervin, E. N.; Kawano, R.; White, R. J.; White, H. S. Anal. Chem. 2008, 80, 2069–2076. (5) Kovarik, M. L.; Jacobson, S. C. Anal. Chem. 2008, 80, 657–664. (6) Wong, C. T. A.; Muthukumar, M. J. Chem. Phys. 2007, 126, 164903–6.

Figure 1. Schematic sketch of the glass nanopore. As an example the reduction of Ru(NH3)63+ is shown. The interior surface is electrically neutral or positively charged due to functionalization with an aminosilane that may or may not be protonated depending on the solution pH. The origin of the coordinate system (r, z) used in the finite-element modeling is at the center of the pore orifice.

in a pH neutral solution containing 5 mM Ru(NH3)6Cl3 (without supporting electrolyte), a GNP electrode (30 nm radius orifice) modified with (3-aminopropyldimethylethoxysilane displays a diffusion-limited faradic current of -27 pA for the 1e- reduction of Ru(NH3)63+. When the bulk pH is decreased to 3.0, the current decreases to ca. -2 pA (see Figure 3 in ref 1). This behavior was qualitatively explained by protonation of the amine groups inside the pore, electrostatically repelling Ru(NH3)63+ and preventing its entrance into the conical nanopore. In a previous paper, the photon-gated transport of Fe(bpy)32+ at conical GNPs with orifice radii between 15 and 90 nm was described.7 In this experiment, the interior surface of a GNP was functionalized with a spiropyran (SP) moiety. Upon exposure to UV light, SP is converted in the presence of a weak acid to the protonated merocyanine MEH+. MEH+ is converted back to SP by shining visible light on the GNP. The effect of the photongenerated charges on the limiting oxidation current, ilim, of Fe(bpy)32+ is significant. In the dark (i.e., with the surface-attached (7) Wang, G.; Bohaty, A. K.; Zharov, I.; White, H. S. J. Am. Chem. Soc. 2006, 128, 13553–13558.

10.1021/la801776w CCC: $40.75  2008 American Chemical Society Published on Web 09/24/2008

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molecule in the electrically neutral form, SP), ilim was ca. 8 pA for a 15 nm radius orifice GNP and a concentration of 5 mM Fe(bpy)32+ (no supporting electrolyte). Upon exposure to UV the current dropped below 1 pA (see Figure 4 in ref 7). Exposing the “blocked” electrode to visible light recovered the diffusionlimited voltammetric response. In the presence of excess supporting electrolyte (0.1 M tetrabutylammonium perchlorate, corresponding to a very short Debye length, ∼0.3 nm), exposure to UV light did not trigger the photon gating. GNP electrodes with large pores (2 µm orifice radius) also showed no photon gating. We qualitatively explained these findings as resulting from electrostatic repulsion of Fe(bpy)32+ by the surface-bound photogenerated MEH+.7 In addition to gating the flux of Fe(bpy)32+ from the bulk solution to the electrode, it is also possible to electrostatically “trap” Fe(bpy)32+ inside the nanopore and subsequently release it using the same photochemical reactions. In the current computational study, we have investigated the coupling of the electrostatic potential distribution inside the nanopore with the distributions and fluxes of redox and electrolyte ions. There are three possible mechanisms responsible for the observed gating of redox fluxes: (1) classical migration of redox species in the depletion layer of the electrode (at the bottom of the nanopore); (2) surface conduction along the nanopore glass walls, i.e., migration of redox species within the electrical double layer of the wall; (3) electrostatic rejection of charged molecules at the pore orifice. Our previous interpretation of the gating phenomenon, based only on experimental observations, was that electrostatic rejection of redox molecules by charge near the orifice (component 3) was responsible for the observed effects. While all three components are indeed simultaneously operative, the numerical results reported herein strongly suggest that electrical migration of redox molecules along the insulating glass walls of the nanopore is the dominant mechanism.

2. Numerical Procedure We solve the Poisson and Nernst-Planck equations, eqs 1 and 2, using a commercial software package (COMSOL Multiphysics 3.3a) which is based on the finite-element method. In a previous paper we described the use of COMSOL Multiphysics in investigations of electrical double layer phenomena, including ion rectification in nanopores.8 In eqs 1 and 2, Ji is the flux, Di is the diffusion coefficient, ci is the concentration, and zi is the charge of species i. Φ is the electric potential, ε ) εrε0 is the permittivity of the medium (ε0 is the permittivity of space), and F, R, and T are the Faraday constant, the gas constant, and the absolute temperature, respectively.

∇2Φ ) -

∑ zici

(1)

ziF Dc ∇Φ RT i i

(2)

F ε

Ji ) -Di ∇ ci -

i

There is one flux equation for each species i, and the whole system of equations is coupled via the dependent variables ci(r, z) and Φ (r, z), which vary throughout the computation domain (r and z are the spatial variables). An axisymmetric computation domain was chosen (Figure 1). At the wall carrying the surface charge the maximum element size was set to 0.3 nm (or smaller). This element size proved to be sufficient to resolve the features of the electric double layer. Quadratic Lagrange polynomials were chosen for the shape functions. The temperature T ) 298 K. Relative permittivities of 78 and 36, respectively, were used for aqueous and acetonitrile (CH3CN) solutions. The following description refers to the transport-limited reduction and oxidation of the electroactive species Ru(NH3)63+ and Fe(bpy)32+, (8) White, H. S.; Bund, A. Langmuir 2008, 24, 2212–2218.

respectively, as these reactions were employed in previous electrostatic gating experiments using GNP electrodes. We present a numerical simulation using the finite-element method to rationalize the values of the transport-limited currents for charged species at open and blocked nanopores. Our conjecture is that, in the blocked state, the glass surface is charged positively whereas in the open state it is charged negatively. As will be discussed below the double layer associated with the negative surface charge results in transport of positively charged species along the insulating walls of the nanopore. The diffusion coefficients in aqueous media were assumed as D(Ru(NH3)63+) ) D(Ru(NH3)62+) ) 0.67 × 10-9 m2 s-1, D(K+) ) 1.957 × 10-9 m2 s-1, D(Cl-) ) 2.032 × 10-9 m2 s-1 and those in CH3CN as D(Fe(bpy)33+) ) D(Fe(bpy)32+) ) 0.21 × 10-9 m2 s-1.9 The concentrations of the electroactive species at the electrode surface, cRed,s and cOx,s, were fixed by the following expressions: cRed,s ) (cOx* + ξcRed*)/(µ + ξ) and cOx,s ) cOx* - ξ(cRed,s - cRed*), where ci* is the bulk concentration of species i, ξ ) DRed/DOx is the ratio of the diffusion coefficients, and µ ) exp[nF(E - E°)/(RT)] describes the potential dependence of the surface concentrations (Nernst behavior, where E is the applied potential, E° the standard potential, and n the number of electrons transferred). In the simulations, we set the reaction driving force, E - E°, equal to -1.0 V for the reduction of Ru(NH3)63+ and to +1.0 V for oxidation of Fe(bpy)32+. These potential values ensure that the redox molecules are reduced or oxidized at transport-limited rates. The mean field approach used in this study is based on the Poisson-Nernst-Planck (PNP) equations. When nanometer-scale structures are described, the question arises of whether such continuum equations can still be applied. Comparing experimental and numerical results, Krapf et al.10 concluded that mean field treatments will give incorrect results for electrode radii smaller than 10 nm. They explained that the PNP equations cannot accurately capture the overall molecular dynamics under conditions of high ionic fluxes in small structures. For their experimental conditions [(ferrocenylmethyl)trimethylammonium in 0.5 M aqueous NH4NO3] they found that the experimental values of the current agreed with PNP calculations for current densities less than 0.15 A cm-2. In our case the maximum current density at the pore base is ca. 0.05 A cm-2. Corry et al. demonstrated that the PNP approach breaks down in narrow ion channels that have radii smaller than the Debye length.11 Molecular dynamics simulations of Ho et al.12 indicate that the mobilities of K+ and Cl- in negatively charged silicon nitride pores (radius 0.5-1.6 nm) are reduced by almost an order of magnitude. As the pores discussed in the present paper have radii larger than 15 nm and the Debye length is typically a few nanometers, the use of continuum equations seems to be justified. Another issue of concern in context of our simulations is the effect of the “condensation” of highly charged counterions (|z| g 3) at charged surfaces. There is theoretical and experimental evidence that mean field approaches (such as the Poisson-Boltzmann equation13) are not rigorously applicable under these conditions.14 Refined integral function based models have been proposed which seem to describe the double layer structure more adequately under these conditions.15,16 The attracting forces of a charged surface toward the counterions in the solution can be so strong that seen from a large distance the surface charge appears to have changed sign (charge (9) Morita, M.; Tanaka, Y.; Tanaka, K.; Matsuda, Y.; Matsumura-Inoue, T. Bull. Chem. Soc. Jpn. 1988, 61, 2711–14. (10) Krapf, D.; Quinn, B. M.; Wu, M. Y.; Zandbergen, H. W.; Dekker, C.; Lemay, S. G. Nano Lett. 2006, 6, 2531–2535. (11) Corry, B.; Kuyucak, S.; Chung, S.-H. Biophys. J. 2000, 78, 2364–2381. (12) 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. (13) ThePoisson-BoltzmannequationemergesfromthePoisson-Nernst-Planck equation for equilibrium conditions, i.e., Ji ) 0. (14) Quesada-Pe´rez, M.; Gonza´lez-Tovar, E.; Martı´n-Molina, A.; Lozada´ lvarez, R. ChemPhysChem 2003, 4, 234–248. Cassou, M.; Hidalgo-A (15) Marcelo, L.-C.; Rafael, S.-B.; Douglas, H. J. Chem. Phys. 1982, 77, 5150–5156. (16) Steven, L. C.; Derek, Y. C. C.; Mitchell, D. J.; Barry, W. N. J. Chem. Phys. 1981, 74, 1472–1478.

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inversion or overcharging).14,17 Quesada-Perez et al.14 compare values of the diffuse double layer potential calculated from the PBE with results from a refined integral equation (hypernetted chain/ mean spherical approximation, HNC/MSA). For a 20 mM solution of a 3:1 electrolyte the PBE values start to deviate positively from the HNC/MSA values beyond -20 mC m-2 (planar surface). Charge inversion occurs at ca. -100 mC m-2. As the following discussion involves concentrations and charge densities below these values, we conclude that we operate in a region where the classic mean field approaches, i.e., the Poisson-Boltzmann equation (PBE) and PNP equation, are applicable.

3. Results and Discussion 3.1. Validation of the Numerical Model. We verified that our numerical model (governing equations, boundary conditions, etc.) yields valid numerical results by computing several quantities for which known analytical solutions exist. For equilibrium conditions (no current flow, Ji ) 0) the electric potential Φ within a nanopore defined by surface charge σ is described by the PBE. For a one-dimensional problem (Φ ) Φ(x), planar surface) the solution of the PBE is given by eq 3.18 In eq 3, c0,i is the bulk concentration of species i, Φd is the potential drop across the diffuse double layer (in the bulk Φ ) 0 V), and the function sign(u) returns the sign (-1, 0, or 1) of the argument u.



σ ) sign(Φd)

2RTε

( (

∑ c0,i exp i

) )

ziFΦd -1 RT

(3)

For asymmetric electrolytes, eq 3 cannot be easily inverted to give an analytical expression for Φd(σ). However, it is straightforward to find the root Φd for a given value of σ using standard numerical methods. For this puropose, we used the FindRoot function of the commercial mathematics software package Mathematica 6.0. We performed a simulation for a 15 nm radius pore (depth 300 nm, θ ) 10°) filled with an aqueous solution of 5 mM Ru(NH3)6Cl3 (no supporting electrolyte) in the absence of a redox reaction occurring at the Pt disk electrode (i.e., equilibrium). The potential profile was determined from the simulations along a line segment normal to the pore wall (0, -200 f 48.75, -191.40 nm) (see Figure 1 for a description of the coordinate system). The double layer potential Φd was determined as the difference of the electric potential between the end points of this line. For surface charges of 10-3, 10-2, and 10-1 C m-2 our simulations yielded values for Φd of 2.60, 29.5, and 133 mV, respectively. The values predicted by eq 3 are 2.62, 30.4, and 136 mV, respectively. Thus, the maximum error is less than 3%, indicating that our numerical model captures the equilibrium properties of electric double layers correctly (in addition, the curvature of the nanopore wall is sufficiently small to allow comparison of the numerical results with the expression for the electrical double layer at a flat surface, eq 3). The mass transfer resistance RMT of a conical pore is given by eq 4,19,20 where a is the orifice radius of the pore, L its length, θ its half-cone angle, and D the diffusion coefficient of the electroactive species. The values of θ for glass nanopores are typically between 8° and 12°. For L tan θ . a, RMT is weakly dependent on the pore depth. As the depths of the pores in the experiments were not always known exactly but were sufficiently (17) Grosberg, A. Y.; Nguyen, T. T.; Shklovskii, B. I. ReV. Mod. Phys. 2002, 74, 329. (18) Newman, J. S.; Thomas-Alyea, K. E. Electrochemical Systems, 3rd ed.; John Wiley & Sons: Hoboken, NJ, 2004. (19) Zhang, Y.; Zhang, B.; White, H. S. J. Phys. Chem. B 2006, 110, 1768– 1774. (20) Zhang, B.; Zhang, Y.; White, H. S. Anal. Chem. 2006, 78, 477–483.

Figure 2. Dependence of ilim for the reduction of Ru(NH3)63+ on the nanopore surface charge (σ) for a 15 nm radius nanopore (depth 300 nm, half-cone angle 10°). The bulk concentration of Ru(NH3)6Cl3 is 5 mM. The open symbols are for a solution containing 1 M KCl supporting electrolyte.

deep to satisfy this condition, we used L ) 20a in all simulations, which satisfies the above criterion.21 (The use of this approximation has no qualitative influence on the numerical results.) The diffusion-limited current, idiff, of an electroactive species through a nanopore can be calculated according to idiff ) nFc/RMT.

RMT )

1 L + 4Da πDa(a + L tan θ)

(4)

Figure 2 shows the dependence of the simulated limiting current, ilim, on the charge density, σ, on the pore wall for the reduction of 5 mM Ru(NH3)63+ in aqueous solution at a GNP electrode with a 15 nm radius (length 300 nm, half-cone angle 10°). In the presence of an excess of supporting electrolyte (1 M KCl), the limiting current is -2.88 pA and varies weakly with σ. The excess of a supporting electrolyte assures that the transport of the electroactive species occurs primarily by diffusion (no migration), and that eq 4 can be applied to calculate the limiting current. Furthermore, the supporting electrolyte screens the surface charge (the Debye length in 1 M KCl is 0.31 nm) and thus inhibits electrostatic interactions. The simulated value is in excellent agreement with the calculated value idiff ) -2.92 pA (eq 4) assuming the geometric parameters given in the caption of Figure 2, n ) 1, and c ) 5 mM. We take this agreement between the simulation and the calculated value as a further confirmation that our numerical model is set up correctly and that it accurately describes the flux of ionic species. 3.2. Electrostatic Gating of Redox Molecules at a GNP Electrode. In the absence of supporting electrolyte the simulated value for ilim for Ru(NH3)63+ reduction decreases with increasing positive surface charge (Figure 2). Qualitatively, this is understood as due to the electrical charge on the pore surface electrostatically interacting with positively charged ions, resulting in a decreased flux. The simulated radial concentration distributions of Ru(NH3)63+ at the pore orifice (r ) 15 nm) show that the cations are repelled from the positively charged wall (σ ) 100 mC m-2, Figure 3A). The repelling force has its origin in the positive electric potential associated with the surface charge (Figure 3B). For a neutral surface (σ ) 0) the concentration of the electroactive species at the orifice increases slightly near the wall, a consequence of the hemispherical diffusion profile at the nanopore that manifests itself in an increased flux at the perimeter. For the same reason the concentration of the reduction product, Ru(NH3)62+, which is transported out of the pore (Figure 1) is (21) Zhang, B.; Zhang, Y.; White, H. S. Anal. Chem. 2004, 76, 6229–6238.

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Figure 4. Effect of the surface charge on the axial dependence of the ruthenium species concentration along the centerline (r ) 0) during the transport-limited reduction of Ru(NH3)6Cl3 at a 15 nm radius nanopore (depth 300 nm, half-cone angle 10°). The bulk concentration of Ru(NH3)6Cl3 was 5 mM. The solid line is for Ru(NH3)63+ and the dashed line for Ru(NH3)62+. Black curves are for σ ) 0, and red curves are for σ ) 100 mC m2. The inset shows the line segment (-300, 0 f 300, 0 nm) that corresponds to the x axis of the plot.

Figure 3. Effect of the surface charge on (top) the radial dependence of the ruthenium species concentration and (bottom) the electric potential at the orifice (z ) 0) during the transport-limited reduction of Ru(NH3)63+ at a 15 nm radius nanopore (depth 300 nm, half-cone angle 10°). The bulk concentration of Ru(NH3)6Cl3 was 5 mM. The solid lines are for Ru(NH3)63+ and the dashed lines for Ru(NH3)62+. Black curves are for σ ) 0, and red curves are for σ ) 100 mC m-2. The inset shows the line segment (0, 0 f 0, 15 nm) that corresponds to the x axis of the plot.

slightly decreased at the perimeter, even for σ ) 0. The simulated limiting currents for the neutral pore (σ ) 0) are -4.66 pA (no supporting electrolyte) and -2.89 pA (1 M KCl). The concentration profiles for σ > 0 of Figure 3A can be interpreted as a reduction in the effective cross section for the transport of the electroactive species, by a distance corresponding to the thickness of the electrical double layer. For the uncharged pore the effective orifice radius equals the geometric radius, 15 nm, while for the pore carrying a charge density of σ ) 100 mC m-2, it is ca. 10 nm. Note that the Debye length in 5 mM Ru(NH3)6Cl3 is 1.75 nm, which agrees well with the shape of the concentration profile shown in Figure 3A. The decrease of the effective orifice radius corresponds to an increase of the mass transfer resistance, eq 4, by ca. 60%. Comparison with Figure 2 shows that the decrease of the limiting current is of the same order of magnitude, i.e., ilim ≈ -4.7 nA for 0 e σ < 0.1 mC m-2 and -2.4 nA for σ ) 100 mC m-2 (an approximately 50% decrease in absolute value). The electrostatic repulsion between the positive surface charge at the pore orifice and the positively charged redox molecule results in a factor of ∼2 decrease in the limiting current relative to the diffusion-controlled value. This computed reduction in current is, however, significantly smaller than the 10-fold reduction observed in the experiments. Thus, electrostatic repulsion at the orifice is only partially responsible for the observed gating effect. The simulated concentration profiles along the centerline of the pore (r ) 0) show that, for a charged nanopore, a slight accumulation of the Ru(NH3)62+ species occurs near the electrode surface (Figure 4). As we assumed equality of the diffusion

Figure 5. Effect of the surface charge and supporting electrolyte concentration on the potential profiles along the centerline (r ) 0) during the transport-limited reduction of Ru(NH3)6Cl3 at a 15 nm radius GNP electrode (depth 300 nm, half-cone angle 10°). The bulk concentration of Ru(NH3)6Cl3 was 5 mM. Key: black curve, no surface charge, no supporting electrolyte; red curve, σ ) 100 mC m-2, no supporting electrolyte; green curve, σ ) 100 mC m-2, 1 M KCl. The inset shows the line segment (-300, 0 f 300, 0 nm) that corresponds to the x axis of the plot.

coefficients of Ru(NH3)62+ and Ru(NH3)63+, this accumulation must be related to the reduced effective radius of the positively charged pore and the electrostatic trapping of the positively charged reduction product. Thus, the conically arranged positive surface charge hinders Ru(NH3)63+ from entering the pore, and at the same time it hinders Ru(NH3)62+ from exiting the pore. Figure 2 also shows that, without supporting electrolyte, the absolute values of ilim for Ru(NH3)63+ reduction are considerably higher in comparison to values in the presence of supporting electrolyte as long as the surface charge is below 10 mC m-2. Again, a qualitative explanation can be put forward; the reduction of Ru(NH3)63+ decreases the positive space charge near the electrode and thus generates a negative electric field in the depletion layer (Figure 5), which accelerates the transport of the Ru(NH3)63+ species toward the electrode by migration. In the presence of excess supporting electrolyte, the buildup of electric fields is effectively suppressed (Figure 5). This classical effect at ultramicroelectrodes is well-known and has been discussed extensively in the literature.22-25 Amatore et al. treated the problem quantitatively for a spherical electrode23 and proposed equations to calculate the current enhancement ilim/idiff, where idiff is the current due to diffusion alone, i.e., in the presence of

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excess supporting electrolyte (no migration). Using their approach, we calculate a theoretical enhancement factor of 1.17, significantly smaller than the enhancement computed in our simulations: 4.66/ 2.89 ) 1.61 (Figure 2; values are for σ ) 0 C m-2). The difference is believed to reflect the restricted transport in the complex geometry and surface migration (vide infra), which are very different from bulk solution diffusion and migration assumed in the theory of Amatore et al. With increasing surface charge the ratio of the limiting currents without and with supporting electrolyte becomes smaller (Figure 2). For σ > 20 mC m-2 the retarding effect by electrostatic repulsion at the pore orifice becomes stronger than the enhancement of the migrational transport in the depletion layer, and the absolute values of ilim without supporting electrolyte are smaller than those in the presence of supporting electrolyte. For a positively charged pore (σ ) 100 mC m-2) and no supporting electrolyte, Ru(NH3)62+ accumulates near the electrode surface (see the discussion above and Figure 4). This is why the z component of the electric field, -dΦ/dz, at the electrode surface (see Figure 5 in the region -300 nm < z < -250 nm) is larger than the case for a neutral pore. The preceding results and discussion quantitatively predict the small currents that are observed in the experiments when the pore wall is charged positively. For the reduction of Ru(NH3)63+ at pH 3 (i.e., when the glass surface is positively charged due to protonation of the surface-bound amines), the measured values of ca. -2 pA compare reasonably well with our simulated values for a surface charge of σ ) 100 mC cm-2 (Figure 2). We chose this value of σ as an upper limit for our simulations for the reasons outlined in section 2 (charge inversion, etc.). Furthermore, it is in accordance with estimations of the maximum surface charge associated with surface-bound amines that is physically reasonable for our experimental conditions. However, the measured limiting current in a pH neutral solution is -27 pA, which is approximately 5 times larger than the simulated value (-4.7 pA) for the uncharged pore without supporting electrolyte (Figure 2). Figure 6A shows the radial dependence of the fluxes of Ru(NH3)63+ at a 30 nm radius pore orifice for a wide range of surface charges, including negative ones. A negative surface charge affects the fluxes associated with a positively charged species much stronger than a positive surface charge of the same magnitude (Figure 6A). At σ ) -10 mC m-2, Ru(NH3)63+ moves preferably along the wall. This is the consequence of the formation of a space charge layer at the glass surface that enables diffusional transport of cations. This space charge transport appears to be responsible for anomalously large fluxes when the surface is negatively charged, as discussed below in more detail. In passing, we note that, in solidstate electrochemistry, the conductivity of heterogeneously doped ionic conductors is based on similar surface transport.26 The question arises of whether the presence of negative surface charge at the aminosilane-modified surfaces is a realistic scenario. Our conjecture is that the aminosilane does not bind to all silanol groups on the silica surface. Thus, at pH > 6, the amino groups will be uncharged but there will be a certain fraction of silanol groups which are deprotonated, thus creating a negative surface charge at the glass surface. The experimental data (-27 pA in a 30 nm radius pore, 5 mM Ru(NH3)63+, pH 6.8, no supporting electrolyte; see (22) Smith, C. P.; White, H. S. Anal. Chem. 1993, 65, 3343–3353. (23) Amatore, C.; Fosset, B.; Bartelt, J.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1988, 256, 255–268. (24) Norton, J. D.; White, H. S. J. Electroanal. Chem. 1992, 325, 341–350. (25) Norton, J. D.; White, H. S.; Feldberg, S. W. J. Phys. Chem. 1990, 94, 6772–6780. (26) Bhattacharyya, A. J.; Dolle, M.; Maier, J. Electrochem. Solid-State Lett. 2004, 7, A432-A434.

White and Bund

Figure 6. Effect of the sign of the surface charge on (top) the flux of Ru(NH3)63+ at the orifice and (bottom) the flux at the base of a 30 nm radius nanopore (600 nm depth, half-cone angle 10°). The flux is driven by the transport-limited reduction of RuIII species at the bottom of the pore. The curves show the radial dependence of the flux, where r ) 0 corresponds to the centerline of the pore. Key: black curve, σ ) -10 mC m-2; green curve, no surface charge; red curve, σ ) +100 mC m-2. No supporting electrolyte was present, and the concentration of (NH3)63+ was 5 mM. The insets show the line segments (0, 0 f 0, 30 nm for (top) and -600, 0 f -600, 135.8 nm for (bottom)) corresponding to the x axes of the plots.

Figure 7. Effect of the surface charge on the simulated ilim for Ru(NH3)63+ reduction (5 mM) at the bottom of a 30 nm radius nanopore (depth 600 nm, half-cone angle 10°). The open symbols are for a solution without supporting electrolyte, and the red solid symbols are for 1 M supporting electrolyte. The stars mark the experimental values from ref 1.

Figure 3A in ref 1) would be consistent with a surface charge of ca. -16 mC m-2 (Figure 7). Behrens and Grier27 presented a method for calculating the surface charge of silica. Their theory links the chemical nature of the silica surface (mainly the mass action law for the silanol dissociation) with the Grahame equation28 for the charge density. The adjustable parameters in their theory are the pKA for the dissociation of the silanol groups, the surface density of chargeable sites (Γ), and the Stern layer’s (27) Behrens, S. H.; Grier, D. G. J. Chem. Phys. 2001, 115, 6716–6721. (28) Grahame, D. C. Chem. ReV. 1947, 41, 441–501.

Electrostatic Gating at Conical Glass Electrodes

phenomenological capacity (CSL). Calculated values of the surface charge for a given pH value are mostly sensitive to pKA and Γ. If we use the values given by Behrens and Grier of pKA ) 7.5 and CSL ) 2.9 F m-2, we obtain Γ ≈ 2 × 1018 m-2 for our experimental conditions, viz., pH 6.8 and σ ) -16 mC m-2. Behrens and Grier report Γ ≈ 8 × 1018 m-2 as a generally accepted value for native silica surfaces. That would mean that the total silanol site density is reduced by a factor of 4 due to the silanization procedure. If we assume pKA ) 6.8, we obtain Γ ≈ 5 × 1017 m-2. From this discussion it becomes clear that we can only estimate the order of magnitude of the silanol site density or the grafting density of the aminosilane. Nevertheless, the values of the grafting density are of the correct order of magnitude.29,30 In Figure 7 our experimental values from ref 1 (stars) have been included together with the simulated values to show the good agreement with the simulation. The deviation between the simulated and the experimental currents for the positively charged (i.e., blocked) pore is probably due to experimental uncertainties (pore mouth radius, half-cone angle, concentration, etc.). If the electrolyte contains an excess of supporting electrolyte, the limiting current shows no dependence on the surface charge density (Figure 7), which is in perfect agreement with the experiments (see Figure 4A in ref 1). We will now discuss the experimental results of the photongated transport of Fe(bpy)32+ in CH3CN solutions7 within the framework of our numerical simulations. Using the mass transport resistance RMT ) 5.26 × 1017 s m-3 for a 15 nm radius pore (length 300 nm) and c ) 5 mM yields a theoretical value of ilim ) 0.92 pA. A numerical simulation of an uncharged pore (σ ) 0) with 1 M supporting electrolyte yielded ilim ) 0.90 pA. Without supporting electrolyte, the simulation gives ilim ) 0.56 pA. In contrast to the reduction of Ru(NH3)63+, the limiting current for Fe(bpy)32+ oxidation is lower in the absence of supporting electrolyte because the oxidation of the Fe(bpy)32+ species leads to a buildup of positive charge (Fe(bpy)33+) in the depletion layer at the electrode. The resulting electric field decreases the transport of the positively charged reactant species, Fe(bpy)32+. The above-discussed theory of Amatore predicts a decrease by a factor of 0.88. Again, there is a discrepancy with the simulated results, the latter predicting a decrease of 0.56/0.90 ) 0.62. The simulated current for σ ) 0 is almost 1 order of magnitude smaller than the experimentally observed value of 8 pA at an SP-modified, 15 nm radius nanopore in the dark.7 To explain this large current, we have to assume that the surface carrying the neutral SP moieties is charged negatively. The argument is the same as in the case of the aminosilane-functionalized surface: not every silanol group carries an SP functionality. The simulated data presented in Figure 8 show how ilim decreases as the surface charge varies from negative to positive values. From this plot, we estimate a surface charge density of -16 mC m-2 for the GNP surface charge density in CH3CN in the dark. This is essentially the same value that we found in the case of the aqueous Ru(NH3)63+ solutions, and we have to rationalize the fact that a silica surface in CH3CN solution containing 20 µM trifluoroacetic acid is charged negatively. We note that in the experiments7 we did not take special precautions to exclude all traces of water. Thus, the CH3CN solution contains a small amount of water. To pursue a detailed discussion and estimation of the surface charge according to the theory of Behrens and Grier,27 one would need the following data: the pKA of silica in CH3CN, the pH of the solution, the surface density of ionizable (29) Duchet, J.; Gerard, J.-F.; Chapel, J.-P.; Chabert, B. Compos. Interfaces 2001, 8, 177–187. (30) Genzer, J.; Efimenko, K.; Fischer, D. A. Langmuir 2002, 18, 9307–9311.

Langmuir, Vol. 24, No. 20, 2008 12067

Figure 8. Effect of the surface charge on the simulated ilim for Fe(bpy)32+ oxidation (5 mM) in CH3CN at the bottom of a 15 nm radius nanopore (depth 300 nm, half-cone angle 10°). The open symbols are for a solution without supporting electrolyte, and the red solid symbols are for 1 M supporting electrolyte. The stars mark the experimental values from ref 7.

groups (Γ), and the Stern layer capacity (CSL) of silica in CH3CN. As these data are not known with the needed accuracy, we refrain from an estimation of the silanol site density. Figure 8 also shows that in the presence of excess supporting electrolyte the effect of surface charge on the limiting current is negligible, in accordance with the experiment (see Figure 4 in ref 7). Exposure to UV light triggers the conversion of the neutral SP molecule to the cation, MEH+, and the net charge of the surface becomes positive. In the experiment the current drops below ∼1 pA, which is in excellent agreement with the simulation. The stars in Figure 8 show the experimental data from ref 7, which are in excellent agreement with the simulation. One can see that the shining of UV light triggers a charge inversion from a positive value to -16 mC cm-2.

4. Summary and Conclusion Using a finite-element model to solve the Poisson-NernstPlanck equations, we were able to describe the electrostatic gating mechanism at conical glass nanopores. The gating mechanism results from three components: (1) migration of ionic redox species in the depletion layer adjacent to the electrode surface; (2) migrational transport along the charged pore walls; (3) electrostatic rejection of charged molecules at the pore orifice. The numerical results indicate that all three components are operative, but that ion migration along the pore walls is dominant. For the reduction and oxidation of the multiply charged redox species, Ru(NH3)63+ and Fe(bpy)32+, respectively, the dependence of the limiting current on the surface charge is strongly nonlinear (Figures 7 and 8), which makes the gating mechanism very effective. For aqueous media, the value of the surface charge can be computed using a model connecting the surface chemistry of silica with the Grahame equation. The analysis of the concentration fields and the ionic fluxes in negatively charged pores shows that the positively charged redox species accumulate within the diffuse double layer region at the negatively charged pore wall. Thus, in a small pore (i.e., pore radius on the order of the Debye length) a considerable amount of current flows along the pore surface. Acknowledgment. A.B. gratefully acknowledges financial support from the Deutsche Forschungsgemeinschaft (Heisenberg fellowship). H.S.W. acknowledges financial support by the National Science Foundation (Grant CHE-0616505) and the National Institutes of Health. LA801776W