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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:
[email protected]. † 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, 32623270. (3) Morris, R. B.; Franta, D. J.; White, H. S. J. Phys. Chem. 1987, 91, 35593564. (4) Watkins, J. J.; Chen, J.; White, H. S.; Abrun˜a, 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, PET, on the distance from the electrode surface. D is the diffusion coefficient of the molecule (cm2/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(NH3)63+/2+,19 and IrCl63-/2-.5 This apparent experimental (15) Mirkin, M. V.; Richards, T. C.; Bard, A. J. Anal. Chem. 1992, 64, 22932302. (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
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upper limit is ∼103 times smaller than that predicted by Marcus20 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 ( 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 χ2 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 cm2/s was used for the diffusion coefficient in the simulations, when the actual value is 2.4 × 10-5 cm2/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 ilim 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.
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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 10 107-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 107-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, PoxET ) 0.095 and PredET ) ∼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 kox on E - E°. As E - E° increases, PoxET 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 E1/2 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
White and White
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 107 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 107-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 107 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
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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, 14751481.
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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 N0001401-1-0757 and the Office of Naval Research. LA7031779
