Anal. Chem. 1994,66, 3164-3172
This Research Contribution is in Commemoration of the Life and Science of
I. M. Kolthoff (1894- 1993).
Voltammetry of Redox-Active Groups Irreversibly Adsorbed onto Electrodes. Treatment Using the Marcus Relation between Rate and Overpotential Kara Weber and Stephen E. Creager' Department of Chemistry, Indiana University, Bloomington, Indiana 4 7405
A treatment of linear sweep voltammetry for redox-activegroups irreversibly immobilized on electrodes is presented with use of the Marcus theory of electrode kinetics to relate rate constants to overpotential. The present treatment extends an earlier treatment of the same problem (Laviron, E. J. Electrosnal. Chem. 1979,101, 19) that used the Butler-Volmer theory to relate rate constants to overpotential. The behavior predicted in the present treatment matches that of the earlier treatment for very high reorganization energies; however, for reorgdzation energies below about 2.0 eV, voltammograms are predicted to be broader and peak potentials are in most cases predicted to shift further from E O f than in the earlier treatment. These effects are most pronounced at high overpotentials and at high sweep rates. Voltammetric data acquired over a wide range of sweep rates for ferrocene oxidation/reduction in selfassembled monolayers of N-( 15-mercaptopentadecy1)ferrocenecarboxamide coadsorbed with 16-mercaptohexadecanol onto gold were analyzed using the present treatment. Predictions of broader voltammograms and greater shifts in peak potential with increasing sweep rate were fully realized in the experimental data. A protocol based on fitting peak potential vs log (sweep rate) data to predictions from theory is suggested as the preferred means of analyzing voltammetric data. Fitting was accomplished using a simplex algorithm with the heterogeneous electron-transfer self-exchange rate constant, km and the reorganization energy, A, as fitting parameters. Values of ko = 7.0 s-l and X = 0.87 eV, and ko = 6.6 s-l and X = 0.87 eV, were obtained from fits to two independent voltammetric data sets for two separate ferrocene-containing monolayers. These ko and X values are in excellent agreement with those in a recent report (Chidsey, C. E. D.Science 1991,251,919) on a related system studied by potential-step amperometry. An important research problem in modern electrochemistry is the study of long-range electron transfer, specifically that between a metal electrode and a redox-active group held at a fixed distance from the electrode surface. This seemingly simple process has relevance to the flow of electrons in biological superstructures such as redox proteins and photosynthetic reaction It is also likely to be a primary underpinning of the future science and technology of molecular electronic^.^,^ In this regard, molecular self-assembly chem~~
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(1) McLendon, G. ACC.Chem. Res. 1988,21, 160. (2) Williams, R. J. P. Mol. Phys. 1989,68, 1. (3) Moser, C. C.; Keske, J. M.; Warncke, K.; Farid, R. S.;Dutton, P. L. Nature 1992,355, 796.
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Anaiytical Chemistry, Voi. 66,No. 19, October 1, 1994
istry6 has been widely used to create structures that enforce a specific distance of closest approach for an electroactive group to an ele~trode.~ Voltammetric methods in particular have been used to study the rate and mechanism of electron transfer across such blocked interfaces.s-24 Recent work from Miller and c o - w ~ r k e r has s ~ demonstrated ~~~ that, with proper treatment of mass-transfer effects, one can derive the potential dependence of the interfacial electron-transfer rate from voltammetric data on freely diffusing redox-active probe molecules. Accounting for mass transfer can be problematic, however, particularly if the blocked interface contains defect sites at which electron transfer occurs preferentially.*JO Several research groups have noted the advantages of attaching a redox-active group to an electrode through a specific molecular bridging m0iety.*~-3~Advantages of this (4) Brecque, M. L. Mosaic 1989,20, 16. ( 5 ) Mirkin, C. A.; Ratner, M. A. Annu. Reu. Phys. Chem 1992,43, 719. (6) Ulman, A. An Introduction to Ultrathin Organic films. From LangmuirElodgett to Self-Assembly.; Academic Press, Inc.: San Diego, CA, 1991. (7) Molecular Design of Electrode Surfaces; Murray, R. W., Ed.; Techniques of Chemistry Series; John Wiley & Sons, Inc.: New York, 1992; Vol. 22. (8) Finklea, H. 0.;Avery, S.;Lynch, M.; Furtsch, T. Langmuir 1987,3, 409. (9) Finklea, H. 0.;Snider, D. A.; Fedyk, J. Langmuir 1990,6 , 371. (10) Finklea, H. 0.; Snider, D. A.; Fedyk, J.; Sabatani, E.; Gafni, Y.; Rubinstein, I. Langmuir 1993,9, 3660. (1 1) Sabatani, E.; Rubinstein, I.; Maoz, R.; Sagiv, J. J. Electroanal. Chem. 1987, 219, 365. (12) Bilewicz, R.; Majda, M. J. Am. Chem. SOC.1991, 113, 5464. (13) Chailapakul, 0.;Crooks, R. M. Langmuir 1993,9, 884. (14) Becka, A. M.; Miller, C. J. J. Phys. Chem. 1993,97, 6233. (15) Becka, A. M.; Miller, C. J. J. Phys. Chem. 1992,96, 2657. (16) Miller, C.; Gratzel, M. J . Phys. Chem. 1991,95, 5225. (17) Miller, C.; Cuendet, P.; Gratzel, M. J . Phys. Chem. 1991,95, 877. (18) Porter, M. D.; Bright, T. E.; Allara, D. L.; Chidsey, C. E. D. J . Am. Chem. SOC.1987,109, 3559. (19) Chidsey, C. E. D.; Loiacono, D. N. Longmuir 1990,6 , 682. (20) Creager, S.E.; Hockett, L. A.; Rowe, G. K. Langmuir 1992,8 , 854. (21) Groat, K. A.; Creager, S.E. Longmuir 1993,9, 3668. (22) Demoz, A.; Harrison, D. J. Longmuir 1993,9, 1046. (23) Wang, J.; Wu, H.; Angnes, L. Anal. Chem. 1993,65, 1893. (24) Malem, F.; Mandler, D. Anal. Chem. 1993,65, 37. (25) Acevedo, D.; Abruna, H. D. J. Phys. Chem. 1991,95, 9590. (26) BundingLee, K. A. Langmuir 1990,6 , 709. (27) Chidsey, C. E. D. Science 1991,251, 919. (28) Collard, D. M.; Fox, M. A. Langmuir 1991,7, 1192. (29) Collinson, M.; Bowden, E. F.; Tarlov, M. J. Langmuir 1992,8, 1247. (30) Rowe, G. K.; Creager, S . E . Longmuir 1991,7 , 2307. (31) Delong, H. C.; Buttry, D. A. Langmuir 1992,8, 2491. (32) Finklea, H. 0.;Ravenscroft, M. S.;Snider, D. A. Longmuir 1993,9, 223. (33) Redepenning, J.; Tunison, H. M.; Finklea, H. 0. Langmuir 1993,9, 1404. (34) Black, A. J.; Wooster, T. T.; Geiger, W. E.; Paddon-Row, M. N. J. Am. Chem. SOC.1993,115, 7924. (35) Hickman, J. J.; Laibinis, P. E.; Auerbach, D. I.; Zou, C. F.; Gardner, T. J.; Whitesides, G. M.; Wrighton, M. S.Langmuir 1992,8, 357. (36) Katz, E.; Itzhak, N.; Willner, I. Langmuir 1993,9, 1392. (37) Curtin,L.S.;Peck,S.R.;Tender,L. M.;Murray,R. W.;Rowe,G.K.;Creager, S.E. Anal. Chem. 1993,65, 386. (38) Obeng, Y. S.; Laing, M. E.; Friedli, A. C.; Yang, H. C.; Wang, D.; Thulstrup, E. W.; Bard, A. J.; Michl. J. J. Am. Chem. SOC.1992, 114, 9943.
0003-2700/94/0366-3164$04.50/0
0 1994 American Chemical Society
strategy include the following: (i) mass transfer is eliminated as a factor affecting electrochemical currents; (ii) every redoxactive group is connected to the electrode via the same bridging group; and (iii) the local chemical microenvironment surrounding the redox-active groups can be controlled. Voltammetric methods have also been used to analyze redox kinetics in such structures. The usual formalism is that of Lavi1-011,~~ which uses the classic Butler-Volmer model of electrochemical kinetics41to predict how electron-transfer rate constants vary with applied potential. Laviron showed that one could extract values for the heterogeneous self-exchange rate constant, ko, and the electrochemical transfer coefficient, a,for irreversibly adsoibed redox-active groups from the dependence of the voltammetric peak potentials on sweep rate. This formalism has been widely used to obtain kinetics information from voltammetric data on irreversibly adsorbed redox-active groups. Unfortunately, it is based on a model (Butler-Volmer) that is too simplistic to adequately describe the physical reaiity of such systems. In this paper, we present a new treatment of the problem in which we use the modern theory of electrode kinetics developed principally by Marcus to relate electron-transfer rate constants to applied potential. We present voltammetric data for oxidation/reduction of immobilized ferrocene groups in a self-assembled monolayer on a gold electrode and show that the data can be explained by invoking a Marcus-type treatment of the electron-transfer kinetics. The system that we have studied is similar to one studied recently by Chidsey using potential-step amperome t r ~ . ~Our ’ conclusions parallel thoseof Chidsey but are based entirely on voltammetric data. Advantages of voltammetry over amperometry in this research include the following: (i) it is often easier to implement; (ii) it is less demanding on potentiostatic instrumentation; (iii) it is less prone to data analysis errors related to subjective fitting of portions of the data to a linear function; and (iv) it is less sensitive to mild dispersions in rate constants caused by heterogeneity in monolayer structure.
EXPERIMENTAL SECTION Electrodes and Materials. Gold electrodes were prepared by sealing 1.0-mm-diameter gold wires in epoxy (EPON 825 (Shell) cured with 1,3-~henylenediamine)~’ and were skid polished with alumina and etched in 1:3:4concentrated H N 0 3 / concentrated HCl/water for 1 min prior to coating with a self-assembled monolayer.20 Coating was accomplished by immersing the freshly etched electrode in an ethanol solution of 0.25 mM N-( 15-mercaptopentadecy1)ferrocenecarboxamide (Fc-CONH-C15-SH) and 0.75 mM 16-mercaptohexadecanol (HO-C16-SH) for at least 12 h, followed by rinsing with ethanol and soaking in a second ethanol solution of 1.0 mM HOC1&H for at least 48 h. The double-coating procedure, which was first suggested by Chid~ey,~’ is intended to reduce the fraction of ferrocene adsorbates at defect sites in the monolayer. Failure to include the second coating usually resulted in broader, more poorly defined voltammetric waves. Electrodes were rinsed with ethanol and water one final time prior to use (39) Uosaki, K.; Sato, Y.; Kita, H. fungmuir 1991, 7 , 1510. (40) Laviron, E. J . Elecfroanal. Chem. 1979, 101, 19. (41) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; John Wiley & Sons: New York, 1980.
in electrochemical experiments. Total ferrocenecoverage was always in the range of (5-10) X lo-” mol cm-2 as determined by integration of the charge under the oxidation peak from a slow-scan (0.1 V s-l) cyclic voltammogram. F c - C O N H - C I ~ S Hwas prepared from ferrocene and 16bromohexadecanoic acid as previously de~cribed.~2The product was purified using flash chromatography (hexanes/ ethyl acetate) on silica gel. 16-Mercaptohexadecanol was prepared from 16-hydroxyhexadecanoic acid (Aldrich) by conversion of the alcohol to the bromide by refluxing in HBr/ acetic acid,43 followed by reduction of the acid to an alcohol using 1.0 M borane in THF ( A l d r i ~ h )and ~ ~ subsequent conversion of bromide to thiol by treatment with thi0urea.~5 The product was purified by recrystallization from diethyl ether. Both products were characterized by TLC and IH NMR. Water was purified via a Barnstead Nanopure system. Perchloric acid (Mallinckrodt) and absolute ethanol (Midwest Grain) were reagent grade and were used as received. Cell and Instruments. Electrochemical experiments were performed using a single-compartment cell that was continuously purged with water-saturated nitrogen. The cell was housed in a Faraday cage to reduce stray electronic noise. Voltammetry was referenced to a silver quasi-reference electrode (1-mm-diameter Ag wire encased in the epoxy next to the working electrode). The auxiliary electrode was a l-cmz Pt foil attached to Pt wire at each corner and held approximately parallel to the surface of the working electrode at a distance of approximately 1 mm. Cyclic voltammetry was performed using a low-current, rapid-response potentiostat of local design and construction (EIS-969, Indiana University electronics shop) with a microprocessor-controlled analog function generator (KrohnHite Model 5900C) to generate potential sweeps and a digital oscilloscope (Tektronix Model TDS-320) to digitize the data. Instrument control was accomplished using a 16-MHz 80386SX-based microcomputer with programs written in BASIC in the Labwindows (Version 2.2, National Instruments) programming environment. CalculationsandSimulations. Calculation of rate constants, simulation of voltammograms, and fits to data were accomplished using programs written in QuickBASIC 4.5 and run on a 33-MHz 80486-based microcomputer. Details of the simulations are given in the main text. A typical execution time for calculating a set of rate constants using the Marcus expressions with a pair of input ko and X values, simulating a set of voltammograms, and calculating a peak potential vs log (sweep rate) curve was 30 s. A typical time for fitting a peak potential vs log (sweep rate) data set to a unique pair of ko and X values using the simplex algorithm was 3 h.
RESULTS AND DISCUSSION Cyclic Voltammetry of Fc-CONH-C&H/HO-CI~-SH. The major focus of this work is application of the Marcus (42) Creager, S. E.; Rowe, G. K. J . Elecfroanal. Chem. 1994, 370, 203. (43) Chidsey, C. E. D.; Bertozzi, C. R.; Putvinski, T. M.; Mujsce, A. M. J. Am. Chem. Soc. 1990, 112, 4301. (44) Ywn, N. M.; Pak, C. S.; Brown, H. C.; Krishnamurthy, S.; Stocky, T. P. J . Org. Chem. 1973, 38, 2786. (45) Cossar, B. C.; Fournier, J. 0.;Fields, D. L.; Reynolds, D. D. J. Org. Chem. 1962, 27, 93.
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treatment of electrode kinetics in place of the conventional Butler-Volmer treatment for the specific case of cyclic voltammetry of irreversibly adsorbed redox-active groups. We begin by presenting some cyclic voltammetric data for ferrocene moieties immobilized in a self-assembled monolayer. These data suggest that the advanced treatment is indeed required. Self-assembled monolayers of FC-CONH-CIS-SH coadsorbed with HO-C&H have been particularly wellbehaved in our hands and are considered exclusively in the present work. We hypothesize that the polar nature of the terminal hydroxy group on the adsorbed alkanethiolates and of the amide group on the ferrocene creates a local microenvironment around ferrocene sites that is similar to that around ferrocene in bulk water46 and is approximately equivalent from site to site. It is important that the immobilized redox-active groups (ferrocenes) all react with the same intrinsic rate constant; otherwise, thevoltammograms become difficult to interpret. Figure 1 shows a series of cyclic voltammograms for a FC-CONH-C~&H/HO-C~~-SH monolayer acquired at sweep rates between 0.1 and 1000 V s-l. The full width at halfmaximum of the oxidative wave at the slowest sweep rate studied (0.1 V s-I, Figure 1A)is approximately 95 mV, which is close to the ideal value of 90.6 mV expected for Nernstian behavior of an ensemble of identical, independent sites at room t e m p e r a t ~ r e .As ~ ~the sweep rate increases, the peak potentials shift symmetrically in positive and negative directions about E O ’ , the waves become broader, and the peak currents (after normalization to sweep rate) decrease. This is the behavior expected of a system undergoing a transition from reversible to quasi-reversible to irreversible electron transfer. Using Laviron’s formalism, it is possible to estimate ko and a for these immobilized ferrocenes from the magnitude of the peak splitting and the dependence of peak potential on sweep rate.40 One aspect of Laviron’s formalism that is particularly useful for analyzing data is the prediction that, at sweep rates for which the peak splitting is greater than 200 mV, the anodic and cathodic peak potentials should both scale linearly with log (sweep rate). The slopes of the anodic and cathodic branches of a plot of peak potential vs log (sweep rate) can be used to estimate a. Observation of such a linear scaling of peak potential with log (sweep rate) is in fact diagnostic of the validity of the Butler-Volmer model. Figure 2 presents a plot of peak potential vs log (sweep rate) for the ferrocene system considered in this study. The symbols are from experimental data, and the lines were calculated using Laviron’s formalism with k, = 10 s-*, a = 0.5, and T = 298 K. The symmetry between the two branches of the plot strongly suggests that the transfer coefficient is 0.5 for this system, in agreement with an abundanceof data on both soluble and immobilized ferrocene^.^^^^^-^^ The data are not well fit by the lines calculated using Laviron’s formalism, primarily due to a nonlinear dependence of peak potential on log (sweep (46) Rowe, G . K.; Creager, S.E. J . Pfiys. Cfiem. 1994, 98, 5000. (47) Ahang, X.; Leddy, J.; Bard, A. J. J . Am. Cfiem. SOC.1985, 107, 3719. (48) Baranski, A. S.;Winkler, K.;Fawcett, W. R. J . Electroanal.Cfiem. 1991,313. 367. (49) Bond, A. M.; Henderson, E. L. E.; Mann, D. R.; Mann, T. F.; Thormann, W.; Zoski, C. G. Anal. Cfiem. 1988,60, 1878. (SO) Gennett, T.; Milner, D. F.; Weaver, M. J . J . Pfiys. Cfiem. 1985, 89, 2787. (51) Wipf,D. 0.;Kristensen,E.W.; Deakin, M. R.; Wightman,R. M. Anal. Cfiem. 1988, 60, 306.
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-0.5
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rate) and a divergence of the anodic and cathodic peak potentials at rapid sweep rates. The uncompensated solution resistance in our cell was estimated to be less than 25 0 from double-layer charging transients in response to a smallamplitude potential steps. The peak current in the voltammogram at 1000 V s-I is approximately 300 PA, yielding a
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electron transfer (electron transfer with weak electronic coupling between reactants), the following expressions can be used for calculating reductive and oxidative rate constants as a function of overpotential for electron transfer between a redox-active molecule and a metal electrode?
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Sweep Rate (VIS) Figure 2. Plot of ( E b k - E O ’ ) vs log (sweep rate) for the mixed monolayer in Figure 1 in aqueous 1.0 M HCIO, at room temperature (circles). E o ‘ was taken as +0.18 V vs Ag wire, which is the average of the anodic and cathodic peak potentials for a slow-scan cyclic - E o ’ )vs log (sweep rate) voltammogram. Caiculation of the (&k curves was performed using the Laviron formalism with a = 0.5, k, = 10 s-’,and T = 298 K (line).
maximum ohmic loss at the peak of less than 7 mV. Ohmic losses are therefore small enough in these data that they may be ignored. The nonlinear dependence of peak potential on sweep rate causes rate constants calculated using the Laviron formalism tovary depending on the choice of sweep rate. By fitting data over only a limited range of sweep rates, it is possible to estimate values for a and k, for that specific range; unfortunately, the values for a are always either much higher or much lower than 0.5 (for estimates from the anodic and cathodic branches, respectively), and values for both a and k, still depend on the sweep rate range. Modifications of the Butler-Volmer theory that incorporate potential-dependent a values have been proposed to alleviate these problem^;^^,^^?^^ however, they are really only approximations to the more rigorous modern theoretical treatments of electrode kinetics. We now present our implementation of those treatments for interpreting voltammetric data for irreversibly adsorbed redox-active groups. The Marcus Treatment of Electrochemical Kinetics. Deviations of the data from the predictions of Laviron’s formalism can be understood by noting that the Butler-Volmer theory on which the Laviron treatment is based is a very simple, approximate model that on close examination is not really correct. It considers the reaction surface to be linear rather than parabolic (Le., potential energy varies linearly rather than parabolically along the reaction coordinate), and it ignores contributions to the rate from states in the electrode at any potential other than the applied potential (Le., the Fermi level). Both of these restrictions are easily removed, and methods for doing so were outlined in the late 1950s and early 1960s when theoretical treatments of electron transfer were undergoing For the specific case of nonadiabatic rapid (52) Weaver, M. J.; Anson, F. C . J . Phys. Chem. 1976, 80, 1861. (53) Carrigan, D. A,; Evans, D. H. J. Electroanal. Chem. 1980, 106, 287. (54) Marcus, R. A. J. Chem. Phys. 1956, 24,966. (55) Marcus, R. A. J. Chem. Phys. 1965, 43,679. (56) Levich, V. G. In Physical Chemistry; An Aduanced Treatise; Eyring, H., Henderson, D., Jost, W., Eds.;Academic Press: New York, 1970; Vol. 9B; pp 985.
The term p is the density of electronic states in the metal electrode, ILI is the electronic coupling matrix element between electronic states in the electrode and the redox-active molecule, X is the reorganization energy, e is the energy of a given state in the electrode, e~ is the Fermi level of the electrode (Le,, the applied potential), and 7 is the overpotential (Le., the applied potential relative to the formal potential). These expressions have been developed by several authors, and the interested reader is referred to the original literature for a more thorough discussion of their origin. They are formally identical to expressions given by Chidsey in his study of electron-transfer kinetics in similar self-assembled monolayers using potential step methods.27 We have written computer programs that calculate k,, and krd by evaluating the integrals in eqs 1A and 1B numerically by use of the trapezoidal rule. Input parameters are the self-exchange rate constant, k,, the reorganization energy, A, and the temperature. Rate constants were typically calculated every 5 mV, and integrations were performed over a range of f 9 V about the Fermi level at a resolution of 20 mV. Small variations in the integration resolution had very little effect on the final curves. The group of constant terms outside the integrals (all of which are assumed here to be potential independent) was evaluated for a given set of input parameters by first performing the integration at zero overpotential, where krd = ko, = k,. This value was then used to calculate rate constants at any desired overpotential by reevaluating the expressions at that overpotential. This protocol eliminates the need to supply specific values for p and ILJand is particularly convenient for fitting data for which preliminary estimates of k, and X are available. Figure 3 presents a series of calculated log (krd + kox)vs overpotential curves for a specific value of k, ( 5 s-l) and for several values of the reorganization energy. The curves are similar in form to those published recently by Chidsey” and also by Finklea and c o - w ~ r k e r s in ~ ~related , ~ ~ work on a different system. If the reorganization energy is made extremely large (e.g.,25 eV), then thecalculated rate constant sum does indeed increase almost exactly logarithmically with overpotential, as predicted in the simple Butler-Volmer model. This can be understood by noting that the parabolic reaction (57) Finklea, H. 0.;Hanshew, D. D. J. Am. Chem. SOC.1992, 114, 3173.
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have therefore resorted to finite difference simulation methods to learn how the voltammetry should be affected by the use of modern theories to relate electron-transfer rate constants to potential. Our implementation of the finite difference simulation method considers the voltammetric scan as a series of discrete small-amplitude potential steps over fixed time intervals whose duration depends on sweep rate. The current at each interval is proportional to the amount of electroactive material that is oxidized/reduced in response to each potential step. It is conventional to present current in dimensionless form as the fractional degree of oxidation per unit of dimensionless potential (i.e., potential normalized by the factor RT/F).The expression for calculating dimensionless current is therefore as follows:
Figure 3. Dependence of log (krd kox)on overpotential for several values of the reorganization energy. Curves were calculated by using the Marcus model; see text for details. Simulation parameters: T = 298 K, ko = 5 s-l, potential range -2.0 to 2.0 V, and Increment 0.005 V.
coordinate assumed in the Marcus treatment is closely approximated by a linear function when the applied potential is much less than the reorganization energy. As the reorganization energy decreases, however, the calculated curves begin to roll off at high overpotential, until a pair of plateau regions is predicted for which the sum of the reductive and oxidative rate constants is nearly independent of potential. This occurs when the overpotential is much larger than the reorganization energy and reflects the fact that the entire distribution of redox-active molecules with states available to transfer electrons (unoccupied states for reduction, occupied for oxidation) is bracketed exclusively by states in the electrode that are also available to transfer electrons (occupied states for reduction, unoccupied for oxidation). At this point, increasing the driving force by applying a greater overpotential does not make any new states available for electron transfer, nor does it increase the probability of electron transfer between any two states; therefore, the rate does not change. This is the electrochemical analog of the Marcus inverted region; the rate does not decrease with increased driving force, as in molecular donor-acceptor electron transfer, rather it becomes independent of driving force. It is significant that the reorganization energies for which these effects are important include those of most common ‘outer-sphere redox reactions; these effects should therefore be the rule rather than the exception and should be commonly observed in experiments involving oxidation/reduction at large overpotentials. After inspecting Figure 3, it is clear why the peak potentials in Figure 2 diverge; Butler-Volmer theory predicts that rate constants will continue to rise exponentially with applied potential, whereas the present theory predicts that they will rise much more slowly. Slower increases in rate constants translate into slower rates and peak potentials that are further removed from E O’. Simulation of Cyclic Voltammetric Behavior. It proved possible in Laviron’s treatment to derive analytical expressions for the exact shape of the cyclic voltammetric waves. In the present treatment, with its more complex dependence of the rate constants on potential, that did not prove possible. We 3108
AnalyticalChemistty, Vol. 66,No. 19, October 1, 1994
where Af is the change in the fractional degree of oxidation during a given interval, AE is the potential increment for each interval, is the fractional degree of oxidation in a given interval before the potential step is applied, fmrg is a “target” fractional degree of oxidation calculated by using the Nernst equation at the step potential for a given interval, At is the time interval over which the potential is applied (At = A E / v , where v the sweep rate), and krd and koxare the reductive and oxidative electron-transfer rate constants at the step potential for a given interval. Voltammograms were calculated iteratively, starting with a value Ofjnit for the first interval calculated by application of the Nernst equation at the initial potential and then constantly updatingJnit by the value of Af calculated at each interval by using eq 2. Figure 4 presents the results of several such simulations performed using rate constants calculated by using the Marcus treatment of electrode kinetics. A fixed value of 5 s-l was assumed for k, in these calculations, and three representative values of the reorganization energy were selected to illustrate the effect of using the Marcus treatment. Each set of simulated voltammograms includes sweep rates between 0.1 and 10 000 V s-l, and includes the range of sweep rates in the data set in Figure 1. The first set of simulatedvoltammograms (Figure 4A) calculated using a reorganization energy of 25 eV behaves as expected from Laviron’s treatment based on Butler-Volmer theory. At low sweep rate thevoltammetry appears reversible; however, as sweep rate increases, the peaks become broadened and skewed as electron transfer becomes rate-limiting, eventually assuming a fixed shape with a peak potential that depends linearly on log (sweep rate). When a reorganization energy of 1.O eV is used (Figure 4B), the same general trends are observed but the peaks do not assume a fixed form at high sweep rates; rather, they continue to broaden and shift to progressively greater peak potentials as sweep rate increases. When a very low reorganization energy of 0.2 eV is used, the effect is even more pronounced; at high sweep rates, the voltammograms have become so broad and shifted that the peaks are difficult to discern. In a real experiment with background current included, it would be difficult to reliably
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E - E" (V) Figure 4. Simulated linear-sweep voltammograms for a surfaceconflnedredox-actlve species. Simulationsused the Marcusrelationship between rateand overpotential,and current was calculated as described in the text with T = 298 and k, = 5 s-l.Reorganization energies were (A) = 25 eV, (6) = 1.0 eV, and (C) = 0.2 eV. Voltammograms are plotted as dimensionless current vs overpotentialat sweep rates of 0.1, 1.0, 10.0, 100.0, 1000.0, and 10 000.0 V 5-l (left to right).
analyze such a voltammogram. The differences among the simulated voltammograms are fairly modest when the sweep rate is low; however, they become striking at higher sweep rates where the voltammograms appear at higher overpotentials. (Compare, for example, the three voltammograms at lOOOV s-l.) This behaviorisanticipatedfrom therateconstant calculations at different overpotentials; rate constants depend only weakly on the choice of reorganization energy at low overpotentials; however, at higher overpotentials, the choice of reorganization energy has a greater effect. It is instructive to compare the simulated voltammograms in Figure 4 with the measured voltammograms in Figure 1. On close inspection, the measured voltammograms do indeed appear broader and the currents lower than expected at very high sweep rates (consider, for example, Figure 1D and E). We take this as strong evidence that predictions from Marcus theory of a leveling of rate at increased overpotentials are indeed realized in our data. Dependence of Peak Potential and Peak Current on Sweep Rate. We now demonstrate how the divergence of peak potential with log (sweep rate) can be used to derive values for k, and X from a set of voltammograms acquired at different sweep rates. Figure 5A presents a set of calculated curves of peak potential vs log (scan ratelk,). Each curve is calculated for a different value of the reorganization energy and includes
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Scan Ratelko Flgure 5. (A) Effect of reorganization energy on calculated plots of peak potentlal (relative to E O f ) VI log (sweep ratelk,). (6)Effect of reorganizationenergy on calculated plotsof dimensionlesspeak current vs log (sweep ratelk). In both cases, calculations were performed using the Marcus treatment at a temperature of 298 K. Each curve consists of 55 points, each of which was taken from a single simulated voltammogram.
55 separate points corresponding to specific ratios of sweep rate to k,. It is evident that the Butler-Volmer model describes the behavior well at high reorganization energies but that the curves begin to diverge at lower reorganization energies, exactly as observed in the data in Figure 2. At very low reorganization energies, the curves begin to fold back on themselves, predicting peak potentials that are unusually low. Inspection of the simulated voltammograms in Figure 4C reveals that this folding over occurs in concert with the extreme flattening of the voltammetric waves, which in turn occurs when the overpotential is such that the forward rate constant is nearly invariant with overpotential. While this is certainly a very real and interesting phenomenon, it would probably be difficult to detect experimentally, due to the small magnitude of the currents relative to the background and the extreme breadth of the waves. Figure 5B illustrates how the dimensionless peak current is predicted to vary with sweep rate and with reorganization energy. At high reorganization energy, the peak current drops initially but then assumes a value that is independent of sweep rate over a wide range. At lower reorganization energies, the dependence of peak current on sweep rate is more complex. The general trend is that peak currents at a given sweep rate always decrease as reorganization energy decreases. This, of AnalyticalChemisW, Vol. 66, No. 19, October I, 1994
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course, reflects the broadening of the voltammetric waves at lower reorganization energy, as illustrated in Figure 4. In principle, one could compare either experimental peak potential or peak current data with the results of calculations such as those in Figure 5 to estimate the rate constant and reorganization energy for a given system. We have chosen to fit peak potentials rather than the peak currents, primarily because of the relative ease in selecting values for peak potentials from experimental data compared with the difficulties encountered in performing reliable background subtraction and current normalization. The problem with using peak currents is illustrated in the voltammogram in Figure 1A; the background current in this voltammogram is clearly rising at positive overpotentials, rendering proper selection of the baseline subjective. On the other hand, the peak potential is fairly easy to discern despite the rising background and should remain relatively unaffected by sloping backgrounds as long as the slope is fairly modest. Another factor in our decision to fit peak potentials rather than peak currents is that even fairly minor nonidealities in the voltammetry (for example, effects due to a distribution of formal potentials and/or electron-transfer rate constants, interaction energies among redox-active moieties in the monolayer, and double-layer effects) can have dramatic effects on the voltammetric wave shape and therefore on the peak current.58d0 The specific subject of heterogeneity in the distribution of rate constants will be taken up later in the paper; however, for now let it suffice to say that including some heterogeneity in the distribution of rate constants affects peak width and peak current much more than it affects peak potential. Having elected to fit data on peak potentials, we must next decide how to properly fit the data to our model to obtain the desired rate constants and reorganization energies. One possible procedure is to prepare a series of working curves of peak potential vs log (sweep ratelk,) at different overpotentials, then by sliding thedata set along the log (sweep ratelk,) axis, estimate X by selecting the curve that fits the data best and then estimate k, from the position of the data set on the log (sweep ratelk,) axis. This cumbersome and subjective procedure would be difficult to implement and would yield only very approximate estimates of X and k,. A much better procedure is to minimize the difference (actually the sum of the squares of the differences) between a calculated curve and the data set. We selected a simplex-based method for performing this least-squares minimization, primarily because the simplex method is very robust and does not require explicit evaluation of derivatives of the function being fit (in our case a very complex function) .61 Simplex-based methods have been suggested for general application in nonlinear curve fitting primarily for these reasons.62 The algorithm we used is based on the simplex algorithm originally proposed by Nelder and Mead63and is available as a prepackaged program in BASIC written for general use in nonlinear curve fitting.64 Obviously, it was necessary to incorporate our code for calculating rate (58) Kano, K.; Uno, B. Anal. Chem. 1993, 65, 1088. (59) Brown, A. P.;Anson, F. C. Anal. Chem. 1977, 49, 1589. (60) Smith, C. P.; White, H. S. Anal. Chem. 1992, 64, 2398. (61) Gans,P. DaraFirringinrheChemicalSciencesby theMerhodofLaastSquares; John Wiley & Sons, Ltd.: Chichester, England, 1992. (62) Phillips, G . R.; Eyring, E. M. Anal. Chem. 1988, 60, 738. (63) Nelder, J. A.; Mead, R. Comput. J. 1965, 8, 308.
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constants, voltammograms, and peak potential vs log (sweep rate) curves into the prepackaged program. The modified program uses two fitting parameters, k, and A, to minimize the sum of the squares of the differences between the calculated peak potential and the measured peak potential at each sweep rate for which a point was measured. Figure 6 presents the results of one such fit of a data set for a Fc-CONH-C~S-SH/HO-C&H monolayer on gold at room temperature in a solution of 1.O M perchloric acid. The solid line corresponds to values of k, = 7.0 s-1 and X = 0.87 eV obtained from the fitting routine. We also analyzed a second data set for an identical but independently prepared monolayer and obtained values of k, = 6.6 s-’ and X = 0.87 eV; hence, the behavior is reproducible. To first order, the position of the curve on the abscissa determines k, and the curvature at high sweep rate determines A. The dashed lines in Figure 6 correspond to curves calculated with the same k, but different X values; they are included to illustrate the sensitivity of the fit to A. Reorganization energies from these fits are in excellent agreement with the value of 0.85 eV reported by Chidsey2’ from a fit to data from a series of potential step experiments. They are also in good agreement with an estimate from theory of 0.95eV for the outer-sphere (64) Ebert, K.; Ederer, H.; Isenhour, T. L.Computer Applications in Chemistry; An Introductionfor PC Users; VCH Publishers: New York, 1989.
reorganization energy for ferrocene, obtained using Marcus’ expressionSSwith a ferrocene radius of 3.8 R,50 a ferroceneto-electrode distance of 22 A (1.3 A per linkage in the chain linking the ferrocene group to the electrode),** and the tabulated optical and dielectric properties of water at 298 K.b5 The rate constants obtained from our fits are slightly higher than Chidsey’s value of 2.5 s-l; this could be due to the fact that in our system there are 18 atoms in the chain linking the ferrocene to the electrode (1 sulfur atom, 15 methylene carbons, and 1 amide linkage) whereas in Chidsey’s system there were 19 (1 sulfur, 16 methylenes, and 1 ester linkage). It will be interesting to see how the rates vary with chain length and with the specific structures of the adsorbates. Effect of a Heterogeneous Distribution of Rate Constants. A central issue in this research is the importance of a heterogeneous distribution of electron-transfer rate constants. Both the Butler-Volmer theory and the present theory assume that the intrinsic electron-transfer rate constant for each immobilized molecule is the same at any given potential. A distribution of rate constants, caused for example by a distribution of E O’s or reorganization energies (perhaps reflecting a range of local microenvironments) or of preexponential factors (perhaps reflecting a range of electrontransfer distances), would have a serious effect on the measured response in any electrochemical experiment, including voltammetry. The double-soaking protocol used in this work is in fact intended to mitigate the effects of this kind of heterogeneity. We have not fully solved this difficult problem, which ultimately reduces to one of eliminating defects in the monolayers. We have, however, considered via simulation the effect that a distribution of rate constants would have on thevoltammetric behavior, and we present the results of those simulations below. Figure 7 illustrates the effect of assuming a particular distribution of rate constants on the voltammetric behavior simulated using the Butler-Volmer model. The voltammograms in Figure 7A were simulated by assuming a single k, value of 5.0 s-’ and an CY value of 0.5. The distribution of rate constants used to calculate the voltammograms in Figure 7B is completely arbitrary and assumes that sites are equally distributed into five independent populations having characteristic k, values of 1.25, 2.5, 5.0, 10, and 25 s-l. This distribution of rate constants clearly results in calculated voltammograms that are noticeably broadened relative to those calculated by assuming a single rate constant. This is not unexpected; each voltammogram in Figure 7B is simply the sum of five individual voltammograms, each of which is shaped ideally (Le., as in Figure 7A) but with a different characteristic peak potential at a given sweep rate. If one had only a single voltammogram to interpret, then it would be very difficult to tell the difference between broadening caused by a distribution of rate constants and broadening caused by more fundamentally interesting phenomena such as the Marcus-type effects illustrated in Figure 4. If, however, one examines the dependence of peak potential on sweep rate, the situation is much better; peak potential still scales linearly with log (sweep rate) in Figure 7B despite the broad range of rate constants used in the simulation. Thus, (65) Gordon, A. J.; Ford, R. A. The Chemist’s Companion; John Wiley & Sons: New York, 1972.
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E - Eo (V) Figure 7. Simulated linear-sweep voltammograms for a surfaceconfined redox-acttve species calculated using the Butler-Voimer relationship betweenrate and overpotentialwith (A) a single rate constant ko = 5.0 s-l and (B) with a dispersion of rate constants ko = 1.25,2.5, 5.0, 10.0, and 25.0 s-l.Contributions from each rate constant are equally weighted. Current was calculated as described in the text with T = 298 K. Voltammograms are plotted as dlmensloniess current vs overpotential at sweep rates of 0.1, 1.0, 10.0, 100.0, 1000.0, and 10 000.0 V s-’ (from left to right).
a divergence in peak potential with log (sweep rate) such as that observed in Figure 2 is very strong evidence of Marcustype effects and is not consistent with a simple distribution of rates causing peak broadening. This exercise highlights a potential pitfall in overinterpreting the magnitude of the peak current and/or the wave shape of a voltammogram to obtain reorganization energies. Such methods depend fairly heavily on the assumption of a homogeneous site distribution and can easily lead to erroneous conclusions. Finally, a note of caution is in order regarding asymmetry between the anodic and cathodic peaks in cyclic voltammograms of immobilized redox-active groups. The voltammograms in Figure 1and the data in Figure 2 are quite symmetric about E O’, suggesting that fits to data at either positive or negative overpotentials would yield similar results. This will not always be the case, however, and when the peak potentials do not shift symmetrically about E O’, it would seem that one must then decide which branch to fit. In fact, neither branch of such a data set should be fit using the method presented here, since it assumes that such asymmetry does not exist. Asymmetry in the voltammetry is ultimately related to asymmetry in the reaction coordinate describing electron transfer. The Butler-Volmer theory accounts for such asymmetry via the electrochemical transfer coefficient; however, theversion of Marcus theory presented here assumes that the reaction coordinate is symmetric and that the transfer coefficient at zero overpotential (cY,,=o) is 0.5. Hupp and Weaver have discussed such asymmetry in the context of two distinct reorganization energies, Xr and A,, for the forward and backward electron-transfer reactions.66 The energies Xf and Xr can be quite different, particularly when there is a Analytical Chemistry, Vol. 66,No. 19, October 1, 1994
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large inner-sphere component to the overall activation energy or when there are strong, specific interactions of the redoxactive group with the solvent. Accounting for asymmetry in the voltammetry will therefore require that the theory be further developed to account for asymmetry in the reaction coordinate.
ACKNOWLEDGMENT Financial support of this work from the National Science Foundation (Grant CHE-9216361) and from the U S . (66) Hupp, J. T.; Weaver, M. J. J . Phys. Chem. 1984, 88, 6128. (67) Nahir, T. M.; Clark, R. A.; Bowden, E. F. Anal. Chem. 1994, 66, 2595.
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Department of Education (for a National Needs Fellowship to K.W.) is gratefully acknowledged. Scientific Parentage of the Author. S . E. Creager, Ph.D. under R. W. Murray, Ph.D. under R. C. Bowers, Ph.D. under I. M. Kolthoff.
Note Added in Proof. Recent work by Nahir and coworkers6' invokes the Marcus theory to describe the specific case of irreversible electron transfer involving immobilized redox-active groups on electrodes. Received for review April 15, 1994. Accepted June 21, 1994.' e Abstract
published in Advance ACS Absfrucfs, August 1, 1994.
