Anal. Chem. 1994,66, 2595-2598
Linear-Sweep Voltammetry of Irreversible Electron Transfer in Surface-Confined Species Using the Marcus Theory Tal M. Nahlr, Rose A. Clark, and Edmond F. Bowden’ Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695-8204
A theoretical analysis suggests a relationship between an observed reaction rate constant and a differential rate constant, which is related to the Marcus theory of electron transfer. Consequently, assumptions from the Marcus theory are used to derive an expression for the current as a function of overpotential during linear-sweep voltammetry in the case of kinetically irreversible electron transfer between an electrode and a surface-confined redox species. A method for the extraction of kinetic parameters from experimentalcurrent vs potential data is also described. The application of this procedure is demonstrated in the electron transfer between gold electrodes and strongly adsorbed cytochrome c through a self-assembled alkanethiolate monolayer. The measurement of rates of charge transfer in electrochemical systems is a topic of much interest. In the past few decades, several reports have proposed that the accurate determination of kinetic parameters should consider the implications of the principles of electron-transfer reactions outlined by Marcus’ and the electrochemical kinetic theory proposed by Levich et a1.2 These studies suggest that a Tafel behavior may not be observed in all irreversible reactions and that the transfer coefficient, a,could depend on the electrode overpotential.3 Recently, the Marcus theory has been applied successfully in the kinetic analysis of electron transfer between gold electrodes and redox species through alkanethiol monolayer^.^ The results included semilogarithmic plots of rate constants vs overpotential which exhibited significant curvature, and an “inverted” region.4e In this work, the analysis of irreversible linear-sweep voltammograms of surface-confined redox molecules using the Marcus theory is presented. One of the advantages of choosing such a system is the relatively simple mathematics that are needed for the description of the physical and chemical processes (compared to the analysis of systems with diffusing Initially, a simple scheme redox species, for for the extraction of rate constants is shown, and the interpretation of kinetic parameters is discussed. Then, an (1) (a) Marcus, R. A. J. Chem. Phys. 1956,24,966-978. (b) Marcus, R. A. Can. J. Chem. 1959,37,155-163. (c) Marcus, R.A. Electrochim. Acta 1968,13, 995-1 004.
(2) Bockris, J. OM.; Khan, S . U. M. Quantum Elecrrochemisrry; Plenum Press: New York, 1979; Chapters 7, 13. (3) (a) Parsons, R.; Passeron, E. J . ElectroaMl. Chem. 1966, 12, 524-529. (b) Weaver, M. J.;Anson,F.C.J.Phys.Chem. 1976,80,1861-1866. (c) Savtant, J.-M.;Tessier,D. Faraday Discuss. Chem.Soc. 1982,74,57-74. (d) Corrigan, D. A.; Evans, D. H. J. Electroanal. Chem. 1980,106,287-304. (4) (a)Chidsey,C.E.D.Science1991,251,919-922. (b)Finklca,H.O.;Hanshew, D. D,J. Am. Chem.Soc. 1992,114,3173-3181. (c) Miller,C.;Cuendet,P.; Griltzel, M. J . Phys. Chem. 1991, 95, 877-886. (d) Miller, C.; Griltrel, M. J . Phys. Chem. 1991,95,5225-5233. (e) Bccka, A. M.; Miller, C. J. Phys. Chem. 1992. 96, 2657-2668. 0003-2700/94/0366-2595$04.50/0 0 1994 American Chemlcal Society
example showing the effects of Marcus theory on electrontransfer kinetics between a gold electrode and strongly adsorbed cytochrome c through an alkanethiol monolayers is presented. Finally, a complete current vs overpotential relationship is derived.
EXPER I MENTAL SECT1ON Evaporated gold mirror electrodes (1000-AAu on 50-A Ti on glass) were purchased from Evaporated Metal Films (Ithaca, NY). They were heated in concentrated nitric acid until boiling and rinsed in water at room temperature before a thiol monolayer was adsorbed on the gold by soaking the electrodes in an ethanol solutionof HS(CH2)lsCOOH. Details about the reagents, instrumentation, and procedure for the cytochrome c system were provided in ref 5b. Voltammetric data were collected every 1 mV. Mathematical solutions were obtained, in part, using the computer program Mathematica (Wolfram Research, Inc., Champaign, IL). Throughout this work, currents during oxidation are positive. RESULTS AND DISCUSSION Procedure for the Extraction of Rate Constants. The rate constants of electron transfer reactions involving surfaceconfined redox species have been commonly measured using chron~amperometry.~~,~ A disadvantage of that method is the need to apply a potential step at individual overpotentials. Next, a method to extract rate constants from linear-sweep voltammograms is developed. In a kinetically irreversible single-electron oxidation of a surface-confined species R,6 the current is i = FAkI’,
(1)
where F is the faraday, A is the area of the electrode, k is the reaction rate constant, and rR is the surface concentration of the reactant R (the reduced species). It is assumed that there is no contribution to the current from diffusing species. Equation 1 may also be written in a differential form: dik = FA dI’, In addition, the current is related to the rate of consumption of reduced species with respect to time: (3)
Given a linear potential sweep rate, v = dq/dt, where q is ( 5 ) (a) Tarlov, M. J.; Bowden, E. F. J . Am. Chem. SOC.1991,113, 1847-1849. (b) Song, S.; Clark, R. A.; Bowden, E. F.;Tarlov, M. J. J. Phys. Chem. 1993. 97, 6564-6572. (6) (a) Laviron, E. J. Electroanal. Chem. 1914.52, 355-393. (b) Laviron, E. J . Electroanal. Chem. 1979, 101, 19-28. (c) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, Fundamenrals and Applications; John Wiley & Sons: New York, 1980; pp 523-525.
Analytical Chemistty, Vol. 66,No. 75,August 7, 7994 2595
a
Flgure 1. Illustrationof the method for extracting rate constantsfrom linear-sweepvoltammograms. The experimental current curve /from a to bis integrated(with respectto time) to yield the chargecorresponding to the shaded region. The magnitude of this area is subtracted from the total peak area (or total charge of adsorbed species) to yield the ratio ilk at b. The current i is then divided by this quantity, ilk, to give the total reaction rate constant at b (currents during oxidation are positive).
i dq = -FAv drR
(4) Dividing eq 4 by eq 2 to eliminate the surface concentration and rearranging yields (5)
Consider the current obtained during a linear potential sweep from a very negative overpotential a to some overpotential b, such as that designated by i in Figure 1. After rearrangement, eq 5 is written in an integral form corresponding to the overpotential limits:
The left side is the total charge passed up to q = b (represented by the shaded area in Figure l), which is obtained by integration of experimental voltammograms. If the right side of eq 6 is replaced with the integral of the right side of eq 2, then the integral value is FArR at q = a less FArR at q = b. Since all the redox species is in the reduced form at q = a, the quantity i l k at q = a is FArtota1 (where rtotal is the sum of the concentrations of both forms of the redox couple, which is found from the charge corresponding to a total electrochemical conversion of the redox sites). Therefore, i l k at q = b is FArtotalless the quantity on the left side of eq 6. The rate constant at any overpotential b can then be found by dividing the measured i by this i l k . Interpretation of Kinetic Results. According to the Marcus theory, the calculation of kinetic parameters in an electrontransfer reaction is greatly simplified when the work required to bring the reactants together and to separate the products can be ignored. Under such conditions, the expression for the activation free energy from the fully classical form of Marcus theory is
(A+ AGO)^ (7) 4x where X is the reorganization energy and AGO is the reaction free energy. Examples are electron transfers in intramolecular reactions' and between surface-confinedredox species and an e l e ~ t r o d e Note . ~ ~ that ~ ~ eq ~ ~7 ~implies ~ that the magnitude of AG* =
2596
Therefore, according to the Marcus theory, the largest value of kM for a single-electron-transfer reaction is observed at q = &X/F,where is for oxidation and - is for reduction. As Gurney initially pointed out through quantum mechanics, the analysis of an electron transfer from an electrode to an oxidized solution species has to include the possibility that electrons at all energy levels in the electrode may transfer (i.e., the reactant energy surfaces are represented by many parabola^).^ He concluded, however, that practically all of the reacting electrons in his investigations of electrolysis of acids came from the Fermi level (a similar observation was also made by Marcuslc). Several recent reports also considered thecontribution of lower energy electrons to the total reaction and presented results which were interpreted in conjunction with the Marcus These investigations made several assumptions about the electron-transfer system: (i) the identical redox sites are at a certain fixed distance from the electrode; (ii) the density of electron states in the electrode is approximately constant in the contributing energy levels; (iii) no electron transfer out of the electrode (i.e., reduction) is allowed from energy levels above the Fermi level, since the electron population there is in~ignificant~~ (a similar approach applies to oxidation, except that the electrons cannot enter the populated energy levels in the electrode below the Fermi level). Under these circumstances,a rate constant for a singleelectron reduction was suggested in refs 2 and 4:
+
overpotential, the last equation becomes
i i d 7 =-vd- k
the rate constant as a function of the overpotential is symmetrical about the reorganization energy. It also assumes that there are no other contributions which influence the variation of the rate constants with overpotential. Although this may be an idealized model, our present work is restricted to such conditions. The free energy of a single-electron-transferreaction at an electrode is related to the overpotential according to AGO = &Fq, where + is for reduction and - is for oxidation (Le., a negative overpotential favors reduction; q = 0 corresponds to AGO = 0). Using the usual Arrhenius exponential dependence, k a: exp(-AG*/RT), the rate constant corresponding to a single overpotential q (Le., the reactant energy surface is represented by a single parabola) is
Analytical Chemistry, Vol. 66, No. 15, August 1, 1994
(9) where k9,s is a constant. A comparison between eqs 8 and 9 suggests that there are two types of rate constants, which are related by an integral. The total rate constant, k, given by eq 9, is the observed (measured) quantity. The differential rate constant, kd, given by dkldq, is equal to the integrand in eq 9. Thus, it is linearly proportional to the rate constant at a single overpotential, kM, from the Marcus theory (eq 8). An important kinetic parameter is the transfer coefficient that corresponds to a particular overpotential, a9,given by
=L+Fll
2 4x for a single electron transfer. Note that this is not a from ~~~~
~
(7) Closs, G. L.; Miller, J. R. Science 1988, 240, 440-447. ( 8 ) Curtin, L. S.;Peck,S. R.;Tender, L. M.; Murray, R. W.; Rowe,G. K.; Creager, S. E. Anal. Chem. 1993,65, 386392. (9) Gurney, R. W. Proc. R. SOC.London 1931, A134, 137-154.
600 lDol
-
~~~~~~
0 -0.2
0.2
0
0.4
#
Butler-Volmer theory (see the Appendix for derivation). A more general expression, which we suggest in this work, replaces '/z with a constant: a,,= a,,o +
The example in the next section shows how the magnitudes of both a and a,,can be extracted from experimental data. Analysisof StronglyAdsorbedCytochrome c. An important component in the binding of cytochrome c to carboxyterminated alkanethiol self-assembled monolayers is most likely the electrostatic attraction between positively charged amine sites on lysine groups in the protein and negatively charged carboxylate sites on the film (at pH 7). This type of ion pairing would be similar to that observed in the interaction of alkylamineslOa and polylysinelob with selfassembled alkanethiolate monolayers. Cyclic voltammetry of these systems shows a redox potential consistent with the adsorbed protein being in a native configuration.' In this work, the faradaic current is assumed to be solely due to electron transfer between the gold electrode and adsorbed cytochrome c through the self-assembled monolayer, as previously des~ribed.~J The cytochrome c system does not exhibit ideal voltammetry, as is most evident at fast (- 1 V/s) cathodic sweeps, which show a largecurrent "shoulder", and in peak broadening at reversible electron-transfer c0nditions.5~ Nevertheless, irreversibleanodic sweepsyielded what seem to begood current vs potential plots (after background subtraction; see Figure 2). The dependence of k and kd (or dk/dq) on overpotential is found using the procedure outlined earlier, and by finitedifference approximation, respectively, from the experimental data. The electrochemical transfer coefficient for irreversible oxidation is usually calculated from the rate constants at each overpotential according to
k - In k,)
+1
0.3
0.2
0.4
Overpotential [VI
0.6
Overpotential [VI Figure 2. Experimental(dashed lines)and theoretical (solid line) iinearsweep voltammograms for the adsorbed cytochrome c system. The theoretical result was obtained from eq 14 with Y = 0.5 Vls, A = 0.32 cm*, rW= 15.2 pmoi/cm*, X = 0.28 eV, a,,,o= 0.59, and k,,,o= 2.87 V-I s-I.
a = - =(ln Fq
0.1
7t
I I
/
0.1
0
"inverted region
0.2
0.3
\{
1
0.4
Overpotential [VI
Figure 3. Experimental (dots) and theoretical (line) rate-constant dependence on overpotential (vs experimentally determined formal potential .F' = -0.040 V vs a Ag-AgCI reference electrode): (a) total rate constant, k, and (b) differential rate constant, k,,. The kinetic parameters for the theoretical curves are the same as for Figure 2.
Vyieldeda=O.51+0.71q(251 points,R2=0.999). Equation 10 shows that X can be calculated if the dependence of a,,on q is known. a,, is computed from kd using an expressionsimilar to eq 11, but for q-subscripted variables (this is consistent with the definition of a,,in the Appendix). The result is a,, = 0.59 0.89q (25 1 points, R 2 = 0.993), and a reorganization energy value of 0.28 eV is calculated. Figure 3 presents the results for the total and differential rate constants. The lines are theoretical results which were calculated using the extracted values of A, a,,,~, and k,,o. Both k and kd were calculated from the integral and integrand, respectively, in eq 12. The peak overpotential at 0.23 V in Figure 3b corresponds to the magnitude of the reorganization energy only if a,,,~= '/z (eq A3). The main purpose of the foregoing discussion has been to demonstrate the application of the voltammetric analysis to a real system. However, because the adsorbed cytochrome c example is not fully ideal, the value given above for X should not be considered definitive. For example, the effect of anomalous peak broadening due to a dispersed formal potentialsb must be minimized or accounted for in a more careful fashion. Nevertheless, the preliminary value obtained for X here is comparable to that in previous reports.sbJ2 Theoretical Current-Voltage Relationship. Using an expression similar to eq 9 and a,,from eq lob, the total reaction rate constant, k, can be evaluated for a single electron-transfer oxidation:
+
(1 1)
Regression analysis of a vs q using experimental current overpotential data (Figure 2) from 0.05 (to avoid a possible contribution from quasi-reversible current near q = 0) to 0.3 (10) (a) Sun, L.; Crooks, R. M.; R i m , A. J. Longmuir 1993, 9, 1775-1780. (b) Jordan, C. E.:Frey, 8.L.; Kornguth, S.; Corn, R.M., submitted for publication to Lungmuir. (11) Rccves. J. H.; Song, S.; Bowdcn, E. F. Anal. Chem. 1993.65, 683688.
k ( 4c ~ ) " 2 e p ~ k+~erf[c'/'($ l go,,
+ q ) ] ) (12)
where b = -(I - a,,,o)F/RT,c = F2/4ART, k,,,0 = k,, exp(12) (a) Churg, A. K.;Wciss, R. M.; Warshcl, A.; Takano, T. J. J. Phys. Chem. 1983, 87, 1683-1694. (b) Dixon,D.W.; Hong, X.; Wahlcr, S. E.; Mauk, A. G.; Sishta, B. P.J . Am. Chem. Soc. 1990,112,1082-1088. (c) Andrew, S. M.; Thomasson, K. A.; Northrup, S. H. J . Am. Chem. Soc. 1993, 115,
55 16-5521.
Ana&-ticalChemlstry, Vol. 66, No. 15, August 1, 1994
2587
curve and comparing the result with the experimentally determined total surface coverage of redox species.
0.4
0.2
0
ACKNOWLEDGMENT We thank the National Science Foundation for supporting this work (CHE-9307257). R.A.C. thanks the US.Department of Education for an Area of National Needs Fellowship in Electronic Materials. We also acknowledge Professor Robert M. Corn of The University of Wisconsin-Madison for providing us with a preliminary copy of ref lob.
0.6
Overpotential [VI
Flgure 4. Theoretical Irreversible oxidation linear-sweep voltammograms at various reorganization energies: 0.2 (..e); 0.5 (- -); 1.0 (- -): and m (-) eV. The ratio of k,,,oto v Is 1 V-*, a,,,o= 0.5, and T = 295 K. Ail peak areas are normalized.
-
(-X/4RT) if a,,,o = l / 2 , and q’ is an integration variable. This expression is inserted into eq 5 to give a differential equation of i as a function of q:
k.i‘;..eg/4c
(~ ) l ~ ’ e - c ( W + d z
4c
+ 11 + erf[c1/’(b/2c
1
1 where a = -k,o/v.
ti.,
+ q)])’
APPEND I X The traditional analysis of kinetics of electrode reactions introduces a transfer coefficient, a.13 For example, the rate constant of an irreversible cathodic reaction is
where ko is the rate constant at q = 0 and n is the number of electrons transferred. The derivative of the log of k, with respect to overpotential is
= di(d
(13) + erf[c1/’(b/2c + q ) ] do
The solution is
Also, the derivative of the log of the rate constant from Marcus theory with respect to overpotential (see eq 8) is
-=-(-+ dlnk~
)
1 nFq nF da 2 2X R T If the contribution to k, is solely from a single energy level corresponding to q,13 such as the Fermi level, the last two expressions may be equated to give
aa aq
nF 1 2 ~ + i and the solution of this differential equation is a+q--=-
where C1 is a constant. Figure 4 shows a series of linear-sweep voltammograms for several values of reorganization energy (all peaks have the same area). The most noticeable effects are the substantial peak broadening and the decrease in the magnitude of peak currents as X decreases. The peak with largest current is from Laviron’s work: which provides the relationship between current and overpotential for irreversible electron-transfer kinetics in adsorbed systems such as discussed in this work, except that the transfer coefficient is assumed to be constant (this is equivalent to setting X at co in eq 10). From eq 8 and the integral in eq 12, the total reaction rate constant, k,in this case is related to the rate constant from the Marcus theory through a constant multiplier (since al)is constant). The electrons in the electrode may then be considered to be at a single overpotential, q, which is equivalent to assuming that all participating electrons are at the Fermi level. Finally,a comparison between experimental and theoretical (eq 14) current-overpotential plots for the adsorbed cytochrome c system is shown in Figure 2. The constant C1 was evaluated numerically by integrating the theoretical current (1 3) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, Fundamentals and Applications; John Wilcy & Sons: New York, 1980; Chapter 3.
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Analyticel Chemistry, Vol. 66, No. 15, August 1, 1994
1 nF cnst 2 4X q Assuming that a has a finite value as q 0, cnst must be 0. This results in eq loa, and an identical result is also derived for oxidation. Although two ways of defining CY are found in the literature (4X is sometimes replaced with 2X), the expression used in this work was also recommended for a more convenient way to determine the reorganization It is crucial to note that this derivation shows that the transfer coefficient here (eq A5) is really a,,, since one cannot assume a priori that electrons with energies significantly different from a single (Fermi) level do not participate in the reaction. Thus, the observed coefficient a (as defined in the main text body) is related to the Marcus theory only through the integration of rate constants over all contributing energy levels. Tobeconsistent with the Marcus theory, theevaluation of X must be from a,,.Nevertheless, the above expression was applied to the analysis of total rate constants, k, or k,and a.3 a=-+-+-
Received for review January 3, 1994. 1994.’
-
Accepted April 21,
.Abstract published in Aduance ACS Abstracts. June 15, 1994.
