Electrochemistry in Nanometer-Wide Electrochemical Cells

2850 Langmuir 2008, 24, 2850-2855 Electrochemistry in Nanometer-Wide Electrochemical Cells Ryan J. White and Henry S. White* Department of Chemistry, UniVersity of Utah, 315 South 1400 East, Salt Lake City, Utah 84112 ReceiVed October 12, 2007. In Final Form: NoVember 29, 2007 The electrochemical properties of an electrochemical cell defined by two concentric spherical electrodes, separated by a 1 to 20-nm-wide gap, and a freely diffusing electrochemically active molecule (e.g., ferrocene) have been investigated by coupling of Brownian dynamics simulations with long-range electron-transfer probability values. The simulation creates a trajectory of a single molecule and calculates the likelihood that the molecule undergoes a redox reaction during each time interval based on a probability-distance function derived from literature first-order kinetic data for a surface-bound ferrocene. Steady-state voltammograms for the single-molecule concentric spherical electrochemical cell are computed and are used to extract a heterogeneous electron-transfer rate for the freely diffusing molecule redox reaction. The Brownian dynamics simulations also indicate that long-range electron transfer, between the redox molecule and electrode, leads to nonsigmoidal-shaped i-e characteristics when the distance between electrodes approaches the characteristic redox tunneling decay length. The long-range electron transfer generates a tunneling depletion layer that results in a potential-dependent diffusion-limited current. Introduction The application of nanometer-scaled electrodes (radii e 20 nm) in experimental investigations of very fast heterogeneous electron-transfer (ET) kinetics and mass-transfer (MT) has been of fundamental interest in modern electroanalytical chemistry. 1-13 A schematic depicting the sequential nature of an electrode reaction is shown in Figure 1, where mass transfer of the molecule to the electrode surface is followed by electron transfer at the electrode surface. To experimentally evaluate a heterogeneous electron-transfer rate constant (k ), the electron-transfer rate must be slower than the mass transfer rate (D/a, where D is the diffusion coefficient and a is the electrode radius), 14 i.e., the condition D/a g k must be fulfilled. This can often be achieved using nanometer-scale electrodes, as D/a scales inversely with the electrode radius, leading to diffusion rates as large as 100 cm/s for a 1-nm-radius electrode. Thus, it is possible to measure heterogeneous rate constants of comparable magnitude, although the difficulty in characterizing such small electrodes limits measurements to values of 10 cm/s or less. Nanometer-scaled electrodes positioned in close proximity to conductive substrates using a scanning electrochemical microscope (SECM), are also * Corresponding author. E-mail: white@chem.utah.edu. Current address: Department of Chemistry and Biochemistry, University of CaliforniasSanta Barbara, Santa Barbara, CA 93106-9510. (1) Smith, C. P.; White, H. S. Anal. Chem. 1993, 65, 3343-3353. (2) He, R.; Chen, D.; Yang, F.; Wu, B. J. Phys. Chem. B 2006, 110, 3262-3270. (3) Morris, R. B.; Franta, D. J.; White, H. S. J. Phys. Chem. 1987, 91, 3559-3564. (4) Watkins, J. J.; Chen, J.; White, H. S.; Abruña, H. D.; Maisonhaute, E.; Amatore, C. Anal. Chem. 2003, 75, 3962-3971. (5) Watkins, J. J.; White, H. S. Langmuir 2004, 20, 5474-5483. (6) Shao, Y.; Mirkin, M. V.; Fish, G.; Kokotov, S.; Palanker, D.; Lewis, A. Anal. Chem. 1997, 69, 1627-1634. (7) Heller, I.; Kong, J.; Heering, H. A.; Williams, K. A.; Lemay, S. G.; Dekker, C. Nano Lett. 2005, 5, 137-142. (8) Penner, R. Μ.; Ηeben, M.; Longin, T. L.; Lewis, N. S. Science 1990, 250, 1118-1121. (9) Sun, P.; Mirkin, M. V. Anal. Chem. 2006, 78, 6526-6534. (10) Tsirlina, G. A.; Petrii, O. A. Russ. Chem. ReV. 2001, 70, 285-298. (11) Menon, V. P.; Martin, C. R. Anal. Chem. 1995, 1920-1928. (12) Krapf, D.; Wu, M.; Smeets, R. M. M.; Zandbergen, H. W.; Dekker, C.; Lemay, S. G. Nano Lett. 2006, 6, 105-109. (13) Bond, A. M.; Henderson, T. L. E.; Mann, D. R.; Mann, T. F.; Thormann, W.; Zoski C. Anal. Chem. 1988, 60, 1878-1882. (14) Gardiner, W. C. Rates and Mechanisms of Chemical Reactions; Benjamin/ Cummings Publishing: Menlo Park, CA, 1969. Figure 1. A schematic representation of the sequential process of diffusion and electron transfer at an electrode surface. The curved line illustrates the dependence of electron-transfer probability, P ET, on the distance from the electrode surface. D is the diffusion coefficient of the molecule (cm 2 /s), a is the electrode radius (cm), and k is the electron-transfer rate constant (cm/s). D/a describes the diffusion rate. advantageous in kinetic studies of fast ET reactions, due to the inverse relationship between mass-transfer rate and separation distance. 15,16 A survey of the current literature suggests that the largest reliable experimental heterogeneous ET rate constants are on the order of 10 cm/s. 3,4,6-8,13 This value corresponds to experimental data for a range of compounds including aromatic organics, 17,18 ferrocene, 13 ferrocenylmethyltrimethylammonium (FcTMA +/0 ), 4 Ru(NH 3 ) 6 3+/2+, 19 and IrCl 6 3-/2-. 5 This apparent experimental (15) Mirkin, M. V.; Richards, T. C.; Bard, A. J. Anal. Chem. 1992, 64, 2293-2302. (16) Mirkin, M. V.; Richards, T. C.; Bard, A. J. J. Phys. Chem. 1993, 97, 7672-7677. (17) Kojima, H.; Bard, A. J. J. Am. Chem. Soc. 1975, 97, 6317-6324. (18) Russel, A.; Repka, K.; Dibble, T.; Ghoroghchian, J.; Smith, J. J.; Fleischmann, M.; Pitt, C. H.; Pons, S. Anal. Chem. 1986, 58, 2961-2964. 10.1021/la7031779 CCC: $40.75 2008 American Chemical Society Published on Web 02/02/2008

Nanometer-Wide Electrochemical Cells Langmuir, Vol. 24, No. 6, 2008 2851 upper limit is 10 3 times smaller than that predicted by Marcus 20 from heterogeneous collision theory and assuming thermal molecule velocities. 21-23 Recently, we reported the use of Brownian dynamics simulations to examine the collision frequency between a redox molecule and a nanometer-sized electrode and described the role that collision frequency plays in determining when an electrochemical reaction becomes kinetically limited. 24 As the electrode radius is reduced, the average number of collisions between a redox molecule and the electrode also decreases, thereby reducing the probability that a successful ET event occurs when the molecule diffuses in vicinity of the electrode. For instance, the average number of collisions that a small molecule, e.g., ferrocene, undergoes each time it visits the electrode surface decreases from 20 000 at a 100-nm-radius electrode to 30 at a 4-nmradius electrode. This result, coupled with a finite probability of ET during each collision, leads to the ET rate limitation at the small electrodes. For example, a rate constant of 5 cm/s requires, on average, 2000 collisions at room temperature for the electron to transfer. Thus, the ET should appear reversible for ferrocene oxidation at a 100-nm radius, where there are many more collisions than required (20 000. 2000), and quasireversible at a 4-nmradius electrode, where there are fewer than required (30 < 2000). These predictions are in good agreement with recent experimental findings. 5 Herein, we report Brownian dynamics simulations of an electrochemical cell in which diffusion of a single molecule between two closely spaced electrodes (<20 nm) is coupled to long-range ET between the redox molecule and one electrode surface. This simulation closely mimics the single-molecule SECM experiment reported by Fan et al. 25 Since ET occurs over distances up to few nanometers, it is anticipated that the comparable dimensions of mass transport and ET may influence the overall behavior of such a small electrochemical cell. To include the influence of a finite ET rate, we employ experimental values of first-order electron-transfer rates for surface-bound redox molecules. The measurement of first-order ET rates associated with redox molecules at a well-defined distance from the electrode surface was pioneered by Chidsey 26 in the early 1990s. The Chidsey experiment and related experiments 27-29 typically employ the self-assembly of redox molecules (e.g., ferrocene-terminated alkylthiols) to produce a well-ordered monolayer with all redox sites located at a specified distance from the electrode. This experiment was further developed by measuring electron-transfer rates as a function of the distance between the electrode and molecule, as reported by Smalley et al. 30 Electron-transfer rates decrease roughly exponentially with increasing distance between the molecule and the electrode, a distance dependence characterized by the decay coefficient for electronic tunneling, β. Typically, β values from the above experiments are 1Å -1. 28 The distance dependence of ET rates (19) Wipf, D. O.; Kristensen, E. W.; Deakin, M. R.; Wightman, R. M. Anal. Chem. 1988, 60, 306-310. (20) Marcus, R. J. Chem. Phys. 1965, 43, 679-701 (21) Einstein, A. Ann. Phys. 1905, 17, 549-560. (22) Einstein, A. Z. Elektrochem. 1907, 13, 41-42 (23) English translations of refs 18 and 19: Einstein, A. InVestigations on the Theory of the Brownian MoVement; Dover Publications: Mineola, NY, 1956. (24) White, R. J.; White, H. S. Anal. Chem. 2005, 77, 214A-220A. (25) (a) Fan, F. F.; Bard, A. J. Science 1995, 267, 871-874. (b) Fan, F. F.; Kwak, J.; Bard, A. J. J. Am. Chem. Soc. 1996, 118, 9669-9675. (26) Chidsey, C. E. D. Science 1991, 251, 919-922. (27) Summer, J. J.; Creager, S. E. J. Phys. Chem. B 2001, 105, 8739-8745. (28) Smalley, J. F.; Finklea, H. O.; Chidsey, C. E. D.; Linford, M. R.; Creager, S. E.; Ferraris, J. P.; Chalfant, K.; Zawodzinsk, T.; Feldberg, S. W.; Newton, M. D. J. Am. Chem. Soc. 2003, 125, 2004-2013. (29) Creager, S. E.; Wooster, T. T. Anal. Chem. 1998, 70, 4257-4263. (30) Smalley, J. F.; Feldberg, S. W.; Chidsey, C. E. D.; Linford, M. R.; Newton, M. D.; Liu, Y. J. Phys. Chem. 1995, 99, 13141-13149. Figure 2. A schematic representation of the electrochemical cell geometry used in the simulations. R 1 and R 2 indicate the radii of the inner and outer electrodes, respectively. The simulations assume that the probability of electron transfer at the inner electrode is a function of distance and potential. The outer electrode is held at a constant potential sufficiently positive to reduce the molecule with unit probability upon collision. reported by Smalley et al. is used in the simulations reported below to compute the ET probability of a molecule as it moves along its random trajectory. Feldberg presented a conceptually similar finite-difference simulation of coupled diffusion and long-range ET for macroscopic electrochemical systems defined by a semi-infinite volume condition (i.e., the counter electrode is placed at an infinite distance). 31 As shown below, the influence of ET on the redox reaction is greatly enhanced when the diffusion length scale approaches ET distances, as occurs at nanometer-scale electrodes. We show, for instance, that the qualitative shape of the steadystate i-e curve of an electrochemical cell is strongly dependent on the ET rate and cell thickness. Methods Simulations. The simulation geometry is two concentric spherical electrodes, as schematically depicted in Figure 2. The solution volume between the two concentric spheres defines the space in which the molecule is able to diffuse. Simulations track the trajectory of the random motion of a single molecule over the course of several microseconds. The simulations were performed using an in-house code written for MatLab (version 6, The MathWorks), which is a text-based mathematical programming language. The diffusion coefficient of the molecule is defined by the Einstein relationship, 21 D ) δ 2 /2τ, where δ and τ are the step distance and step time, respectively. This definition of D is consistent with a three-dimensional random walk, where the net displacement, r ) (δ x2 + δ y2 + δ z2 ) 1/2, for uncorrelated motion in the x, y, and z directions is given by r 2 ) 6Dτ. The values of step time (τ ) 0.2 ps) and length (δ ) 0.02 nm) used in the simulation uniquely satisfy the realistic values of diffusivity, D ) 10-5 cm 2 s -1, and thermal velocity, V 1/2 ) (d/t) 1/2 ) (kt/m) 1/2 ) 10 4 cm/s (k is the Boltzmann constant, T is temperature, and m is the mass ) 3.1 10-22 g), corresponding approximately to measurements using a molecule of mass equal to ferrocene at room temperature. 32,33 (31) Feldberg, S. W., J. Electroanal. Chem. 1986, 198, 1-18. (32) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, 2nd ed.; Wiley & Sons: New York, 2001.

2852 Langmuir, Vol. 24, No. 6, 2008 White and White The probability of ET, either reduction or oxidation, between the redox molecule and the inner electrode is finite and computed during each time step based upon the literature ET rate data. These dependencies are outlined below. Potential- and Distance-Dependent Electron-Transfer Kinetics. The ET system is modeled as a chemically reversible, oneelectron reaction, eq 1 Ox + e - T kox k red Red (1) where k ox and k red are the potential-dependent oxidation and reduction rate constants. These rates are based on the standard ET rate constant (k r,s -1 ) and are described by the expressions and In eqs 2 and 3, F is Faraday s constant, n is the number of electrons transferred, R is the transfer coefficient, and E and E are, respectively, the potential applied at the electrode and the standard reduction potential of the electroactive molecule. Values of n ) 1, R) 0.5, and E ) 0 are employed in the simulations. 32 The above equations are consistent with anodic currents having positive value. Equations 2 and 3 are presented in this section in context of first-order redox kinetics of a molecule located at a specified position within the cell at each time step of the simulation. The value of k r in eqs 2 and 3 is described by eq 4 34 where β and r are the exponential tunneling decay coefficient (cm -1 ) and the distance (cm) from the electrode surface, respectively, and k 0 is the standard ET rate constant (s -1 )atr ) 0. Equation 4 describes the distance-dependent ET rate. Substituting eq 4 into eqs 2 and 3 gives k ox and k red as a function of distance (r) and potential (E). The experimental values of k 0 and β reported by Smalley et al. 30 for surface-bound ferrocene molecules were used to compute ET rates in the simulations. Values of 1 10-8 cm -1 (1 Å -1 ) and 6 10 8 s -1 for β and k 0, respectively, were reported by Smalley et al. by fitting eq 4 to a plot of ln(k r )vsr. Smalley et al. also estimated the heterogeneous electron-transfer rate constant (k ) between a freely diffusing ferrocene molecule and an electrode by integration of eq 4 from the plane of closest approach to an infinite distance from the electrode, yielding a value of 6 cm/s. This value is in good agreement with experimental values for soluble ferrocene 14 and is consistent with the value obtained from our simulations. Probability of Electron Transfer. At each time step, τ, throughout the simulation, a statistical evaluation of the probability of the occurrence of an ET event was performed. The rate of ET was calculated using the oxidation and reduction rate constants as defined by eqs 2-4 at various potentials and distances from the electrode surface. The dimensionless probability of electron transfer (P ET )is derived using first-order rate kinetics. 35 The probability of a molecule undergoing reduction or oxidation during a simulation time step, τ, is given by eqs 5 and 6: and (1 -R)nF k ox ) k r exp[ (E - RT E )] (2) k red ) k r exp [ -RnF RT (E - E ) ] (3) k r ) k 0 exp(-βr) (4) P red ET ) 1 - exp{-k red τ} (5) P ox ET ) 1 - exp{-k ox τ} (6) (33) Rabinowitch, E.; Wood, W. C. Trans. Faraday Soc. 1936, 32, 1381-1387. (34) Li, T. T.; Weaver, M. J. J. Am. Chem. Soc. 1984, 106, 6107-6108. (35) Petrucci, R. H.; Harwood, W. S. General Chemistry, 7th ed.; Prentice Hall: New York, 1997. Figure 3. Plot of P ox. ET (solid lines) as a function of distance from the electrode surface as calculated by eq 6 at various potentials. A τ value of 0.2 ps was used for the calculations. Figure 4. Example Brownian trajectory of a single molecule diffusing between concentric spherical electrodes of 20- and 100- nm radii. Figure 3 shows plots of P ox ET for τ ) 0.2 ps as a function of distance from the electrode surface for different electrode potentials. The plots indicate that, at potentials significantly more positive than E, the probability of electron transfer is essentially unity at distances close to the electrode surface. If the molecule is in the reduced form, there is a finite probability of being oxidized and vice versa. During each given simulation time increment, the probability of either oxidation or reduction was calculated using a subset of eqs 2-6 contingent on the redox state of the molecule. Current Resulting from ET and Diffusion of a Single Molecule. The current through the electrochemical system was computed by counting the number of ET events per unit time. The outer-electrode rate constants were set to k red ) and k ox ) 0 to ensure that the oxidized molecule (e.g., ferrocenium) is reduced immediately when it reaches the outer electrode (thus, conversely, no reaction occurs when the reduced molecule collides with the outer electrode). Longrange electron transfer is ignored at the outer electrode for simplicity. An ET event is defined as an oxidation at the inner electrode that is followed by diffusion of the molecule across the cell and then by reduction at the outer electrode. This definition is consistent with the fact that not every oxidation reaction at the inner electrode will immediately result in a reduction reaction at the outer electrode, for the oxidized molecule can instead be rereduced at the inner electrode before diffusing to the outer electrode. The steady-state electrochemical cell current at a potential E is defined as the product of the number of successful ET events and the elementary charge of an electron (1.6 10-19 C), divided by the total simulation time [) the number of steps step time (τ)]. The current was computed as a function of applied potential to produce current-voltage (i-e) plots. Figure 4 shows an example Brownian trajectory of a single molecule over a period of 1 ns. In this particular example, the molecule is randomly moving within the volume between 20- and 100-nm-radius spherical surfaces. The molecule s trajectory is plotted

Nanometer-Wide Electrochemical Cells Langmuir, Vol. 24, No. 6, 2008 2853 as the displacement (or position) of the molecule from the center of the inner electrode. As is clearly evident from the plot, in this particular example, the molecule collides many times with the outer electrode over the initial 700 ps before eventually diffusing in the vicinity of the inner electrode. Of course, each trip between the inner and outer electrodes, and vice versa, occurs by a unique and random trajectory. An analytical solution, eq 7, for the voltammetric response for the concentric spherical electrode system is readily derived from Fick s laws and the Butler-Volmer equation, for comparison to the simulated values. i ) D k red R 1( R 2 Here, R 1 and R 2 are the inner and outer sphere electrode radii, respectively. In eq 7, k ox and k red represent potential-dependent heterogeneous rate constants described by expressions analogous to eqs 2 and 3, where the standard rate constant, k r, in these expressions is replaced by the standard heterogeneous rate constant, k. The diffusion-limited current (i lim ) is given by eq 8. Results and Discussion Heterogeneous ET Kinetic Rate Constants from Simulated Voltammetric Waves. When using sufficiently small electrodes to experimentally evaluated ET rates, a shift of the half-wave potential, E 1/2, of the voltammetric steady-state response is observed at the onset of kinetic control. 4 The E 1/2 potential is defined as the potential where the voltammetric current is 50% of the diffusion-limited current, and for a chemically and thermodynamically reversible reaction, i.e., large k, E 1/2 is approximately equal to E. The shift in E 1/2 away from E, as the electrode size is reduced, results from the increased energy necessary to drive the reaction at the electrode surface as the number of collisions between the redox molecule and electrode decrease. 32 In general, by fitting kinetic model equations (e.g., the Butler-Volmer equation) to the voltammogram or to the dependence of E 1/2 on radius, a heterogeneous ET rate constant, k, can be extracted from the data. By analogy to real experimental analysis, the voltammetric response of the nanometer-wide electrochemical cell was simulated using experimental first-order rate constants (as described above). The Butler-Volmer based equation describing the overall i-e wave (eq 7) was then fit to the resulting simulated data to obtain the corresponding heterogeneous ET rate constant, k. The simulated steady-state voltammetric response of an electrochemical cell with a 20-nm-radius inner electrode and 30-nm-radius outer electrode is shown in Figure 5. Values for current are calculated from the simulated three-dimensional random motion of a single molecule between the two concentric spheres, taking into account the probability of electron transfer at distances away from the electrode surface. As described above, literature values for the kinetic ET rates for surface-bound ferrocene moieties were employed to calculate P red ox ET and P ET during each step of the simulation. The inner electrode potential, E - E, was varied from -0.25 to 0.5, and k ox and k red were computed using eqs 2 and 3. In Figure 5, each data point represents 10 individual simulations comprising 10 7 steps, and the error bars represent an uncertainty of 1σ. This i-e plot exhibits steadystate sigmoidal behavior expected for the concentric sphere geometry in which the electrodes are separated by nanometer i lim R 2 - R 1) + k ox /k red + 1 i lim ) ( 4πnFCD R 2 - R 1 R 2 R 1 ) (7) (8) Figure 5. A steady-state voltammetric current-voltage (i-e) response at a 20-nm-radius inner electrode. The data points represent simulated current values using a thermal molecular velocity of 10 000 cm/s and D ) 1 10-5 cm 2 s -1. The solid line represents the best-fit voltammetric curve for k ) 7 cm/s determined by a χ 2 test. The dashed line represents the voltammetric wave for a reversible reaction with k ) 10 000 cm/s. Table 1. Simulated Limiting Current (i lim) at an Overpotential of 0.5 V as a Function of Gap Distance Compared to the Analytical Solution Described by Eq 8 i lim (A) gap distance (nm) simulated calculated 20.0 3.6 (( 1.2) 10-13 3.4 10-13 15.0 6.7 (( 1.2) 10-13 6.4 10-13 10.0 1.8 (( 0.2) 10-12 1.5 10-12 9.0 1.7 (( 0.3) 10-12 1.9 10-12 7.0 3.7 (( 0.2) 10-12 3.2 10-12 5.0 7.4 (( 0.4) 10-12 6.3 10-12 3.0 2.1 (( 0.06) 10-11 1.8 10-11 1.0 3.3 (( 0.04) 10-10 1.6 10-10 0.5 3.0 (( 0.02) 10-9 6.4 10-10 dimensions. The limiting current arises from the diffusion-limited motion of the molecule between the inner and outer electrodes. For comparison to the simulated i-e curve, eq 7 is plotted in Figure 5 using k ) 10 4 cm/s (dashed line), which is sufficiently large that it corresponds to the reversible or Nernstian i-e response. 15,16 As discussed above, a shift in the E 1/2 value from E is observed when the reaction becomes partially controlled by the ET kinetics. A 50 mv shift in E 1/2 is observed in the simulated data for this particular geometry (R 1 ) 20 and R 2 ) 30 nm), indicating an ET kinetic limitation, D/a > k. 6,32,36 At low overpotentials, the molecule does not undergo electron transfer every time it visits the electrode surface, resulting in the kinetic overpotential. The solid line in Figure 4 was obtained using eq 7 by varying k until the optimal fit was found, as determined by a weighted χ 2 analysis. The best fit is found using k ) 7.0 cm/s, in good agreement with the value of 6 cm/s from the work by Smalley et al. 30 (the latter was computed from the analytical integration of eq 4 from the plane of closest approach to an infinite distance from the electrode surface). The small discrepancy between the two values may be a consequence of the fact that the simulation treats Ox and Red as point molecules. Also, a value of 10-5 cm 2 /s was used for the diffusion coefficient in the simulations, when the actual value is 2.4 10-5 cm 2 /s in acetonitrile. 32 Voltammetric Response at Small Gap Distances. The cell geometry used to acquire simulation data for Figure 5 comprises two concentric spherical electrodes separated by a 10-nm gap. In this case, the 10-nm separation distance is sufficiently large that any effect of long-range ET on the voltammetric limiting current is negligible. Table 1 shows tabulated results of i lim obtained from the analytical solution (eq 8) and from the simulations for E - E ) 0.5 V at varying gap distances. The (36) Zoski, C. G. Steady-State Voltammetry at Microelectrodes. In Modern Techniques in Electroanalysis; Vanysek, P., Ed.; John Wiley & Sons, Inc.: New York, 1996, pp 241-312.

2854 Langmuir, Vol. 24, No. 6, 2008 White and White simulation is in good agreement with the predicted values for gap distances ranging from 20 to 7 nm. However, when the gap distance is 7 nm or smaller, the current from the simulation is significantly larger than the value computed by eq 8. Figure 6 shows four simulated voltammograms for electrochemical cells consisting of a 20-nm-radius inner electrode separated by gap distances of 10, 3, 1, and 0.5 nm. Each data point of the voltammogram is calculated as an average of 10 10 7 -step simulations, using eqs 2 and 3 to incorporate the potentialand spatial-dependent rate constants. The solid lines represent the analytical expression (eq 7) using k ) 7 cm/s. A significant increase in current relative to the diffusion-limited plateau current is apparent at potentials (E - E ) greater than 0.2 V, for separation distances less than 5 nm. Figure 7B presents histograms showing the spatial dependency of the state of oxidation of the molecule (Ox or Red) at steady state. The histograms were constructed using the data from a single 10 7 -step simulation of a 20-nm-radius inner electrode with a gap distance of 1 nm at E - E ) 0.5 V. In essence, the histogram represents the time-averaged concentration profiles of the molecule in the oxidized and reduced states. Figure 7a is the corresponding voltammogram for this particular geometry. From Figure 7B, it is apparent that the probability of the molecule being in the reduced state (i.e., ferrocene) at distances less than 0.3 nm from the inner electrode is negligibly small. This indicates that long-range electron-transfer efficiently depletes Red at distances up to 0.3 nm from the electrode surface. At this distance, P ox ET ) 0.095 and P red ET ) 10-9 as calculated from eqs 6 and 5. Thus, if the molecule is in the Red state, it has a 10% chance of undergoing electron transfer (being reduced) during a given time step. Due to the nature of random walks, a 10% ET probability during one time step results in a very high probability that ET will occur within a few time steps, since the redox molecule has a tendency to explore the small volume of space centered at about 0.3 nm before wandering away. 37 After being oxidized, the probability of the molecule being rereduced at this distance and overpotential is negligibly small. The nearly exponential rise of i at overpotentials (E - E ) greater than 0.2 V is a consequence of the exponential dependence of k ox on E - E.AsE - E increases, P ox ET increases (see Figure 3), resulting in Red being depleted at distances further away from the inner electrode surface. The consequence of this tunneling depletion layer is that the distance that the molecule is required to diffuse, in order to carry charge back and forth between the inner and outer electrode, is reduced. This shortening of the transport distance is the origin of the potential dependence of the diffusion-limited current. Conclusions Using experimentally measured values of first-order ET rate constants for surface-bound molecules, we have simulated the voltammetric response of a single molecule electrochemical cell in which the thickness approaches the characteristic length of electron tunneling. By coupling the random motion of a molecule with long-range ET probabilities, the simulated voltammetric response displays a shift in E 1/2 that is characteristic of a kinetic limitation. We did not consider the influences of the electric field between the electrode surfaces, 1 near-surface solvent ordering, or the finite size of the molecules, all of which may exert a significant influence on the i-e behavior. 3 Even in the absence of these factors, our results indicate that nonsigmoidal (37). Berg, H. C. Random Walks in Biology; Princeton University Press: Princeton, NJ, 1983 Figure 6. Steady-state voltammetric response using gap distances (A) 11 nm, (B) 3 nm, (C) 1 nm, and (D) 0.5 nm. In each case, the inner electrode radius is 20 nm. The plotted lines represents the best-fit voltammetric response with k ) 7 cm/s for each gap distance. Each current data point is from 10 simulations at 10 7 steps per simulation. Figure 7. Simulations at an electrochemical cell with a 1-nm separation distance. (A) A plot of the steady-state voltammetric response for a 20-nm-radius electrode encapsulated by an outer electrode with a gap distance of 1 nm. The solid line represents the best-fit voltammetric response at the electrode with a k ) 7 cm/s. Each current point is calculated from 10 iterations of a 10 7 -step simulation. (B) Histograms of counts of the spatial position of Ox and Red, i.e., concentration profiles, as a function of distance at E - E of 0.5 V. The data comes from one simulation of 10 7 steps at a 20-nm-radius inner-electrode separated by a gap distance of 1 nm from the outer electrode. steady-state i-e characteristics result from electron transfer over distances that are comparable to the cell thickness. As noted in the Introduction, our simulations mimic the singlemolecule SECM experiment reported by Fan et al., 25 although the simulation is an obvious oversimplification of the experiment. As noted by a reviewer of this paper, as a sharp SECM tip is brought within a few nanometers of a conductive electrode surface, the tip current increases above the diffusion-limited current that is expected from the normal oxidation/reduction cycling of the molecule between the tip and electrode. The current that is in excess of the diffusion-limited value has been interpreted as corresponding to direct electron tunneling between the tip and

Nanometer-Wide Electrochemical Cells Langmuir, Vol. 24, No. 6, 2008 2855 conductive surface. 38 Our simulations suggest that tunneling from the conductive surface to the redox molecule occurs in parallel with direct tunneling between tip and conductive substrate. At intermediate distances, where direct tip-to-conductive-substrate tunneling is small, long-distant tunneling to the redox molecule remains operative and may represent the more significant (38) Fan, F.-R. F.; Mirkin, M. V.; Bard, A. J. J. Phys. Chem. 1994, 98, 1475-1481. tunneling pathway that leads to observations of tip currents in excess of the diffusion limit. Acknowledgment. This work was supported by the National Sciences Foundation (CHE-0616505) and the DoD Multidisciplinary University Research Initiative (MURI) program administered by the Office of Naval Research under Grant N00014-01-1-0757 and the Office of Naval Research. LA7031779