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Langmuir 1999, 15, 5158-5163
Effect of Electron-Transfer Rate and Reorganization Energy on the Cyclic Voltammetric Response of Redox Adsorbates Michael J. Honeychurch* Inorganic Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QR, United Kingdom Received February 17, 1999. In Final Form: May 5, 1999 The theory of cyclic voltammetry of redox adsorbates based on the nonadiabatic kinetic models of Levich and Dogonadze (GMLD kinetics) is re-examined. It is shown that if generalized predictions of cyclic voltammetric behavior are sought, the model is applicable only over a small range of reorganization energies. It is well-known that at high reorganization energies Tafel plots simulated using both Butler-Volmer and GMLD kinetics give similar results; however, cyclic voltammograms simulated from GMLD kinetics at high reorganization energies do not reduce to the generalized Butler-Volmer model for cyclic voltammogram of redox adsorbates derived by Laviron (J. Electroanal. Chem. 1979, 101, 19). Two methods of solution to the cyclic voltammogram equation are considered: a finite difference numerical solution and an analytical approximation. The analytical approximation enables a more rapid simulation of cyclic voltammograms. Methods for applying the model to real systems are suggested which include an equation to estimate the reorganization energy based on the cyclic voltammogram peak potentials and the standard rate constant for the redox system.
Introduction The traditional approach to interpretation of electrode kinetics has been to invoke the Butler-Volmer equation.1 In recent years other approaches have been adopted which have been based on the quantum mechanical approaches to electrode kinetics primarily derived by Levich2,3 and Dogonadze,4 which extended the earlier semiclassical work of Marcus.5-7 These approaches to electron transfer are often referred to as being “modern”, but a brief review of the history of the development of these models dates their origin with Gurney8 at around the same time as the development of the Butler-Volmer equation.9 Using the Gurney-Marcus-Levich-Dogonadze (GMLD) equation for an electron transfer at an electrode, the equation for the current includes the possibility of electron transfers from all the energy levels in the electrode8
ired,ox ) eA
∫∫ c(l) W(l,) F() f(() d dl
(1)
where A is the electrode area, e is the electron charge, c(l) * E-mail:
[email protected]. Fax: +44 (0) 1865 272690. (1) Greef, R.; Peat, R.; Peter, L. M.; Pletcher, D.; Robinson, J. Instrumental Methods in Electrochemistry; Ellis Horwood Ltd.: Chichester, U.K., 1985. (2) Levich, V. G. Present State of the Theory of Oxidation-Reduction in Solution (Bulk and Electrode Reactions). In Advances in Electrochemistry and Electrochemical Engineering; Delahay, P., Ed.; John Wiley & Sons: New York, 1966; Vol. 4, pp 249-372. (3) Levich, V. G. Kinetics of Reactions with Charge Transport. In Physical Chemistry; an Advanced Treatise; Eyring, H., Ed.; Academic Press: New York, 1970; Vol. 9B, pp 985-1074. (4) Dogonadze, R. R. Theory of Molecular Electrode Kinetics. In Reactions of Molecules at Electrodes; Hush, N. S., Ed.; Wiley: New York, 1971; pp 135-227. (5) Marcus, R. A. J. Chem. Phys. 1956, 24, 966-978. (6) Marcus, R. A. J. Chem. Phys. 1965, 43, 679-701. (7) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265322. (8) Gurney, R. W. Proc. R. Soc. 1931, A134, 137-154. (9) Bockris, J. O. M.; Khan, S. U. M. Surface Electrochemistry. A Molecular Level Approach; Plenum Press: New York, 1993.
is the concentration at a distance x from the electrode, W(l,) is the transition probability of an electron going from the metal to an acceptor or from the donor to the metal, f() is the Fermi energy, and F() is the density of electronic states on the electrode surface. The equation is integrated over all distances l and all energies to arrive at the current (rate). For a molecule attached to the surface at a fixed distance, the rate of electron transfer under nonadiabatic conditions, that is, weak electronic coupling, is
(
kf,b ) kmax
RT 4πNAλ
)
1/2
1 × ∫-∞∞1 + exp(x)
{[
exp -
]
}
2 NAλ ( Fη RT -x dx (2) RT 4NAλ
where η is the overpotential (driving force), NA is Avogadro’s number, λ is the reorganization energy in electronvolts, x is a dimensionless integration variable, and kmax is the maximum value of the rate constant given by
kmax )
4π2FHAB2 h
(3)
In eq 3 F is the density of states in the metal, which is assumed to be independent of the electrode potential, h is Planck’s constant, and HAB is the electronic coupling matrix element which describes the electronic coupling of the reactant’s electronic state with the products. For this discussion it is assumed that for a nonadiabatic electron transfer the electronic coupling is RT eq 2 can be simplified to17
anodic sweep:
ψc )
1.50 4.09 × 10-3
[
NAλ ( Fη 2
x
]
1 RTNAλ
(10)
A comparison between the kf,b values calculated by eqs 2 and 10 for λ ) 1.0 eV is shown in Figure 1. The agreement between the two values becomes greater as Fη f NAλ, as expected. Equation 8 can now be solved by using this approximation for kf,b. The solution is
RT k (t) × ψc,a = xOi,Ri nFν f,b NAλRT dkf,b(t) (λ ( η) exp kf,b(t) (11) 2 dη ν νF
[
(
]
)
where xOi and xRi are the initial mole fractions of oxidized and reduced (xRi ) 1 - xOi) adsorbate at the beginning of the cathodic (xOi) and anodic (xRi) sweeps respectively, and dkf,b(t)/dη is
[
]
(NAλ ( Fη)2 1 exp πNAλRT 4NAλRT
x
Fkmax dkf,b(t) )dη 2
(12)
While the use of eq 10 enables an analytical solution for the dimensionless current to be derived, insertion of eq 10 into eq 9 unfortunately does not yield an analytical solution for the peak potential in terms of λ, kmax, and ν. An alternative method to the analytical approximation is to perform a finite difference simulation of the cyclic voltammogram response. In a finite differences simulation kf and kb have fixed values at each potential step, so the potential sweep can be simulated by solving eq 5 assuming kf and kb are independent of time. Under those conditions the incremental change in xO for a potential step from E1 to E2 of length ∆t is
∆xO )
[
kb
(kb + kf)
]
- xO,i (1 - exp[-(kb + kf)∆t])
(13)
where xO,i is again the initial mole fraction of oxidized adsorbate, but at the beginning of each potential step E1, and kf and kb are the rate constants calculated at the final potential E2. The value of xO at the end of the potential step becomes the initial value in the next step. The (16) Laviron, E. J. Electroanal. Chem. 1979, 101, 19-28. (17) Schmickler, W. Interfacial Electrochemistry; Oxford University Press: Oxford, U.K., 1996.
Cyclic Voltammetric Response of Redox Adsorbates
Figure 2. Plots of peak potential versus log(|ν|/k°) for several values of the reorganization energy λ. Solid lines from bottom to top: λ ) 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 2.00 eV. Calculations were performed using eqs 2 and 4 with kmax ) 3.2 × 105 s-1 and T ) 273 K. Dashed line shows a calculation based on ButlerVolmer kinetics with k° ) 1 s-1.
Figure 3. Plots of peak height versus log(|ν|/k°) for several values of the reorganization energy λ. Solid lines from right to left: λ ) 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 2.00 eV. Calculations were performed using eqs 2 and 4 with kmax ) 3.2 × 105 s-1 and T ) 273 K. Dashed line shows a calculation based on ButlerVolmer kinetics with k° ) 1 s-1.
dimensionless current is calculated by substituting ∆xO/ ∆t into eq 7 with ν ) (E2 - E1)/∆t. Equations 2 and 13 were solved numerically using programs (notebooks) written in mathematica 3.0 (Wolfram Research, Champaign, IL) and run on a Apple Power Macintosh 9500/150 computer. The mathematica notebooks are available electronically from the author. Results and Discussion Figures 2-4 show plots of peak potential, peak height, and peak width at half peak height versus log(|υ|/k°) for several values of the reorganization energy. The separation of the lines in Figures 2-4 can be explained by noting that in order to do calculations with varying λ and constant kmax the value of k° for each value of λ differs as indicated in Table 1. Unlike the Butler-Volmer case (dashed line), where the plots of peak potential versus log(|υ|/k°) are linear in the irreversible region, the plots calculated from GMLD kinetics are initially Butler-Volmer like when FEp , NAλ and then display upward curvature before curving back when FEp > NAλ. This was also a feature of plots calculated
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Figure 4. Plots of peak width at half peak height versus log(|ν|/k°) for several values of the reorganization energy λ. Solid lines from bottom to top: λ ) 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 2.00 eV. Calculations were performed using eqs 2 and 4 with kmax ) 3.2 × 105 s-1 and T ) 273 K. Dashed line shows a calculation based on Butler-Volmer kinetics with k° ) 1 s-1.
previously in which both kmax and λ were varied simultaneously.11,12 At sweep rates sufficiently rapid to produce FEp > NAλ, the peak height rapidly decreases with increasing sweep rate (Figure 3). From the trend shown in Figure 2 it might be expected that the peak potential would asymptotically approach a constant value for very large sweep rates, but this has not been explored, since Figure 3 shows that at these sweep rates the peak height becomes so small (it approaches zero) that the presence of a voltammetric peak would not be apparent experimentally. This contrasts with the Butler-Volmer model in which a constant peak height is predicted for an irreversible reaction regardless of the sweep rate. In practice cyclic voltammogram measurements of peak potential are not normally made over as wide a range of sweep rates as shown in Figure 2 due to experimental complications, most notably the effect of uncompensated resistance. Focusing on the λ ) 1.0 eV plot we can see that if the range of sweep rates were reduced to 2-3 decades the plot may appear linear but give rise to a slope greater than that predicted on the basis of Butler-Volmer kinetics, as shown for example in Figure 2 of ref 10 and Figure 9 of ref 11. Attempts to fit the Butler-Volmer model to this result would lead to the sum of the anodic and cathodic transfer coefficients being *1. Figure 4 shows how the peak width at half peak height varies with increasing sweep rate. As the sweep rate is increased, the cyclic voltammogram peak becomes more sigmoidal in character and consequently the peak width at half peak height increases rapidly at very high sweep rates. Some of the lines in Figure 4 have a smaller range than others due to an inability to measure the peak width at half peak height with accuracy as the peak potentials approach the limit of the potential range used in this study. A comparison between cyclic voltammograms calculated numerically by eqs 2 and 13 and those calculated using the approximate analytical solution, eqs 10 and 11 is shown in Figure 5. When FEp f NAλ, the cyclic voltammograms calculated from eq 11 are in excellent agreement with those calculated numerically. Even for smaller values of FEp the peak sizes and shapes calculated by the two methods agree well; however, the cyclic voltammograms calculated by the analytical method are offset slightly along the potential axis. The difference between the peak potentials of the analytical approximation and the numerical method depends on the ratio FEp/NAλ and seems
5162 Langmuir, Vol. 15, No. 15, 1999
Honeychurch
Figure 5. (A) Calculated cyclic voltammograms for λ ) 1.00 eV and log(|ν|/k°) ) 1, 3, and 4: black lines, finite difference simulation using eqs 2 and 13; dashed lines, analytical approximation using eq 11. (B) Calculated cyclic voltammograms for λ ) 0.25 eV and log(|ν|/k°) ) 3, 5, and 6. black lines, finite difference simulation using eqs 2 and 13; dashed lines, analytical approximation using eq 11.
Figure 6. (A) Simulated cyclic voltammograms at constant kmax. T ) 273 K, kmax ) 3.2 × 105 s-1, log(|υ|) ) 3, λ (eV) ) (right to left) 0.25, 0.50, 0.75, 1.00, 1.25, and 1.50. (B) Simulated cyclic voltammograms at constant k°. T ) 273 K, k° ) 1 s-1, log(|ν|) ) 1, λ (eV) ) (bottom to top) 0.25, 0.50, 0.75, 1.00, 1.25, and 1.50. (C) Simulated cyclic voltammograms with variable kmax. T ) 273 K, λ ) 1 eV, log(|ν|) ) 3, kmax (s-1) ) (right to left) 3.2 × 107, 3.2 × 106, 3.2 × 105, 3.2 × 104, 3.2 × 103, and 3.2 × 102.
to follow the empirical relationship cathodic/anodic offset (mV) ) -20 ( 16.7(FEp/NAλ) for 0.25 < λ < 1.50 eV. The advantage of using the analytical approximation with Mathematica is that calculations were performed greater than an order of magnitude faster than the numerical solution. The analytical approximation can also be readily performed using a scientific spreadsheet/graphing program. The numerical solution requires that kf and kb be initially calculated at fixed (in this case 1 mV) intervals and then called when {ψ, η} pairs are calculated with eq 13. Once a list of kf and kb values had been tabulated and stored their was little difference in the time required to simulate a cyclic voltammogram by the analytical and numerical methods. In Figure 6 cyclic voltammograms were simulated using eq 11 in order to demonstrate the effect of λ and kmax on the size, shape, and position of the cyclic voltammograms under irreversible conditions. Figure 6a shows some simulated cyclic voltammograms at constant kmax. ButlerVolmer cyclic voltammograms retain their shape in the irreversible region independent of the value of the dimensionless rate constant, RTk°/nF|ν|, and shift cathodically as its value increases. The GMLD cyclic voltammograms do shift cathodically as λ is increased, that is, increasing irreversibility, but their shape continually changes. The cyclic voltammograms become broader and flatter with increasing λ. The cyclic voltammograms do not become Butler-Volmer like with increasing λ; in fact, they become less Butler-Volmer like. Figure 6b shows simulated cyclic voltammograms at constant k° which are consistent with those shown in previous studies.11,12 At constant k° there is little variation
in the peak potential with variation of λ, a point noted previously.12 The impression one gets from Figure 6b is that because the peaks become broader and flatter with decreasing λ, the reaction is becoming more irreversible as λ is decreased. Alternatively, the impression gained is that the cyclic voltammograms are becoming more ButlerVolmer like with increasing λ, just as Tafel plots become more Butler-Volmer like with increasing λ. As shown in Figure 6a, the reaction is becoming more irreversible (the peaks shift cathodically) as λ is increased, as expected, and the cyclic voltammograms become less and less Butler-Volmer like with increasing λ. The departure from Butler-Volmer behavior with increasing λ is the opposite of what is observed from Tafel plots. The difference in the two sets of simulations (Figure 6b and c) arises from the need to exponentially increase kmax as λ is increased in order to maintain k° at a constant value. The exponential increase in kmax apparently has a greater influence on peak shape leading to a more Butler-Volmer like appearance of the cyclic voltammogram (Figure 6b). Figure 6a shows simulated cyclic voltammograms with variable kmax and constant λ and log(|ν|). The behavior is analogous to that observed in Figure 6c. In both cases the reaction becomes more irreversible with decreasing kmax or increasing λ, producing a shift in the cyclic voltammograms along the potential axis and increasing peak width at half peak height, as indicated by the data in Figures 2 and 4. In the discussions above we have disagreed with the approach taken previously in which generalized predictions of the effect of λ on cyclic voltammogram shape and position have been made using a constant k° because the
Cyclic Voltammetric Response of Redox Adsorbates
simulations also reflect the effects of a second variable, kmax. On the other hand, if simulated data are being matched to experimental data, kmax and λ must be determined from k°, since this is usually the experimentally obtained parameter. This seems reasonable if done over a narrow range of λ and with an upper limit of kmax such that the condition of nonadiabaticity is maintained rather than the case where one is making generalized predictions of electrochemical responses for wide ranging values of λ. In this respect the procedure described below to determine λ and kmax is similar to that used previously.11,12,18 Initially k° can be determined experimentally from small peak separations in the low sweep rate region using Laviron’s working curve.16 In this region, there is little difference in the cyclic voltammogram peak potentials predicted by Butler-Volmer and GMLD kinetics. By determining k° in this region, however, the peak separation should be measured from the peak potentials in the reversible region rather than measuring the absolute peak separation.19 Following the determination of k°, λ can be estimated from eq 14, which was obtained from nonlinear regression of the simulated data and is valid over the range 0.25 < λ < 1.50 eV and -3 < log(|ν|/k°) < 8 and |Ep - Er| > 0.1 V
λ ) (T/1000){2.968 - 0.6581 log(|ν|/k°) + 5.300Ep 0.2986Ep log(|ν|/k°)} (r2 ) 0.9762) (14) where λ is in electronvolts and Ep in volts. Having obtained λ and k°, kmax can be calculated from eq 4 and the cyclic (18) (a) Methods for measuring λ and kmax by diffusional cyclic voltammetry have been described in detail by Miller. (b) Miller, C. J. Heterogeneous Electron Transfer Kinetics at Metallic Electrodes. In Physical Electrochemistry Principles, Methods, and Applications; Rubenstein, I., Ed.; Marcel Dekker: New York, 1995; pp 27-80. (19) (a) Honeychurch, M. J. Langmuir 1998, 14, 6291-6296. (b) Assumption iv of ref 19a and refs 14 and 15 make reference to desorption being rapid. Since these references deal only with strongly adsorbed and chemisorbed molecules, rapid desorption is clearly in conflict with the model discussed; therefore, this statement should be disregarded.
Langmuir, Vol. 15, No. 15, 1999 5163
voltammogram can then be simulated using eqs 2 and 13 or eqs 10-12. The present treatment considers that the redox adsorbates behave “ideally” whereas there is a well-documented list of factors which lead to nonideal behavior that would need to be included in this model for it to be considered a “complete” model for cyclic voltammogram of redox adsorbates.14,15 In particular much recent work has focused on the effect of the interfacial potential distribution (IPD) on the cyclic voltammograms of redox adsorbates.19-27 IPD or other effects are diagnosed from the nonideal response under reversible conditions. We are currently attempting to incorporate IPD effects into a GMLD kinetic model, but for the time being we make some qualitative observations. The effect of the IPD is to reduce the overpotential experienced by the adsorbate so that the “true” overpotential is smaller than the experimentally measured overpotential. By including the “true” overpotential in eqs 2 and 13 or eqs 10 and 11, the cyclic voltammogram response in the presence of IPD effects could be simulated; however, the numerical simulation is complicated by the fact that the “true” overpotential is dependent on xO, and therefore, the solution requires an iteration within an iteration, as with the Butler-Volmer case.19 Since the “true” overpotential, is lower than the measured overpotential it would seem that in the presence of IPD effects the Tafel plots and cyclic voltammograms should appear Butler-Volmer like for a slightly wider potential range and therefore the onset of curvature from plots of peak potential versus log(|ν|/k°) should be extended over a slightly wider potential sweep range. LA990169U (20) Smith, C. P.; White, H. S. Anal. Chem. 1992, 64, 2398-2405. (21) Smith, C. P.; White, H. S. Langmuir 1993, 9, 1-3. (22) Fawcett, W. R. J. Electroanal. Chem. 1994, 378, 117-124. (23) Fawcett, W. R.; Fedurco, M.; Kovacova, Z. Langmuir 1994, 10, 2403-2408. (24) Andreu, R.; Fawcett, W. R. J. Phys. Chem. 1994, 98, 1275312758. (25) Andreu, R.; Calvente, J. J.; Fawcett, W. R.; Molero, M. J. Phys. Chem. B 1997, 101, 2884-2894. (26) Andreu, R.; Calvente, J. J.; Fawcett, W. R.; Molero, M. Langmuir 1997, 13, 5189-5196. (27) Honeychurch, M. J. J. Electroanal. Chem. 1998, 445, 63-69.
