J. Phys. Chem. 1994,98, 13396-13402
13396
Heterogeneous Electron-Transfer Dynamics of Decamethylferrocene from 130 to 181 K John N. Richardson, Justin Harvey2 and Royce W. Murray* Kenan Laboratories of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290 Received: July 25, 1994; In Final Form: October 1, 1994@
Kinetic parameters for heterogeneous electron transfer are reported for decamethylferrocene in two lowtemperature solvent systems: butyronitrile and 2: 1 (v:v) ethyl chloridehutyronitrile. The (MesC&Fe+'O rate to c d s over the temperatures investigated. Temperature-dependent constant ranges from 4 x heterogeneous electron-transfer rate constants used for the determination of activation parameters were obtained from comparison of experimental cyclic voltammograms to those produced via digital simulation employing Marcus-DOS kinetic theory. The activation parameters are similar in the two solvents (AG* = 0.27 and 0.26 eV) and are somewhat larger than predicted for an outer-sphere reorganizational barrier from dielectric continuum theory (AG* = 0.22 and 0.21 eV). Available literature values for the inner sphere barrier for (Me+25)2Fe+'O and the thermal activation of solvent dipole longitudinal relaxation times are compatible with the experimental barriers containing small contributions from these sources. This work represents the lowest temperature electrochemical kinetic study to date employing a diffusing redox species in liquid solution.
Among recent research efforts1-' aimed at an improved understanding of electrochemical processes at ultralow temperatures has been the design' of a low-temperature solvent system that produced microband voltammetry in a liquid electrolyte solution at temperatures down to 88 K.' The liquid electrolyte' is a mixture of a relatively polar liquid, butyronitrile, with a low-viscosity component, ethyl chloride, containing dissolved Bu4NPF6 supporting electrolytes. This mixed solvent has been applied in low-temperature investigations of superconducting interfaces2 and of electroactive alkanethiol monolayer~.~,~ This paper describes the electron-transfer dynamics of the metallocene decamethylferrocene, (MesC&Fe, at Au electrodes in the ethyl chloridehutyronitrile (EtCWrCN) solvent mixture over a temperature range of 130-175 K. Measurements were also made in butyronitrile over a higher temperature interval, 162-181 K. The results illustrate the effectiveness of thermally quenching the dynamics of an ordinarily very facile electrontransfer c o ~ p l (k, e ~= ~ 3.5 ~ and 0.4 c d s and kexhomo= 3.5 x lo7 M-' s-', at room temperature in acetone, benzonitrile, and CH3CN, respectively), by a factor of as much as lo6, enabling kinetic measurements on modest time scales by conventional linear sweep voltammetry.10 In these measurements, the rates become sufficiently slow, and the applied overpotentials ( U p ) sufficiently large, that analysis of the electrode kinetics is made with the full Marcus relation" more accurately than with the traditionally employed Butler-Volmer equation.'O Activation parameters obtained from the temperature dependency of the heterogeneous electron-transfer rate constants are compared to contemporary predictions. There are no previous observations of homogeneous or heterogeneous electron-transfer rates in metallocene solutions at reduced temperatures comparable to those employed here.
Experimental Section Chemicals. Butyronitrile (Aldrich, 99+%), decamethylferrocene (Strem), and tetra-n-butylammonium hexafluorophosphate (Fluka, puriss, Bu4NPF6) were used as received. Ethyl Present address: Department of Chemistry, University of Melbourne, Parkville, Victoria 305 1, Australia. @Abstractpublished in Advance ACS Abstracts, November 15, 1994.
0022-365419412098-13396$04.50/0
chloride (Linde) was condensed and stored in sealed vacuum transfer pipets at room temperature. Electrode Preparation and Electrochemical Measurements. Working and reference electrodes were 2.0- and 0.5mm diameter gold disks (Johnson Matthey Electronics, 99.999%), respectively, encapsulated in a cylindrical epoxy assembly (Shell Epon 828, m-phenylenediamine curing agent, cured overnight at 70 "C). The disk electrodes were polished before each use with 1-pm alumina and 1-pm diamond paste (Buehler), followed by extensive rinsing with water (Bamstead Nanopure, =- 18 MQI cm2) and sonication. The polished workingheference electrode assembly was fitted into a slotted stainless steel sleeve whose bottom served as a coplanar auxiliary electrode spaced ca. 0.50 mm from the working electrode. This assembly was placed in an aluminum container with the desired quantities of butyronitrile solvent, decamethylferrocene (0.5 mM final concentration), and B u m F 6 supporting electrolyte (0.075 M final concentration) which was sealed and subjected to one freeze-pumpthaw cycle. For experiments in the mixed solvent, ethyl chloride was vacuum transferred into the cell solution to a final 2: 1 EtCV PrCN volume ratio. The A1 cell bottom was bolted directly to the cold finger of a Janis helium refrigerator cryostat interfaced to a Lakeshore Cryogenics 320 autotuning temperature controller. Equilibration at the initial target temperature was achieved within 4 h; changes of 10 "C or less required ca. 30 min for temperature equilibration. Cyclic voltammetric current-potential data and potential step chronoamperometric current-time data were acquired with a locally constructed potentiostat and stored digitally. A 20-Hz low-pass filter was employed in recording slow potential scan cyclic voltammograms; potential step current-time transients were filtered at 2 kHz. Diffusion coefficients of decamethylferrocene were determined from current-time transients plotted according to the Cottrell equation,'* using positive-going potential steps large enough (typically 0.6 V) to produce a fully diffusion-controlled reaction and to swamp effects of uncompensated resistance. Transients for background current correction were obtained with identical potential steps in an adjacent featureless potential region. The uncompensated resistance RWC in the mixed solvent system and the cell described, measured by ac impedance with a Schlumberger Solartron Model 1255 frequency response 0 1994 American Chemical Society
Electron-Transfer Dynamics of Decamethylferrocene
TABLE 1: b E p and k" for Decamethylferrocenein 2:l (v:v) EtCYPrCN as a Function of Potential Sweep Rate T,K v, mV/s AEp, mV k"," c d s RLJNC,~ k P RwcIck P 130 10 212 5.8 x 69 65 20 261 4.3 x 325 3.1 x 50 352 2.9 x 74 378 2.4 x 102 140 10 142 1.3 x 39 39 20 178 1.1 x 10-5 50 222 9.3 x 10-6 74 239 9.0 x 102 256 8.1 x IOw6 150 10 98 9.9 x 10-5 11 12 115 1.1 x 20 50 159 8.4 x 74 173 8.4 x 102 198 6.8 x 160 20 85 3.3 x 10-4 5.9 8.4 40 io0 3.3 x 10-4 102 127 3.3 x 10-4 153 151 2.8 x 203 166 2.5 x From comparison to A E p values calculated by using Marcus-DOS theory, assuming I = 0.15 eV (see text). bMeasured by using ac impedance spectroscopy. Estimated from the equation for feedback circuit. analyzer and Model 1286 potentiostat, is R ~ =c 69 and 2.8 kS2 at 130 and 175 K, respectively. The consequences of these substantial resistances are mitigated by the relatively small voltammetric currents (typically < A), the low decamethylferrocene concentration, and small diffusion coefficient (D = 6 x lo-* cm2/s at 130 K) in the cold solvent. The iRmc effects were compensated electronically, using "positive feedback compensation". l3 The observations in Table 1 are examples of cyclic voltammetric data taken with positive feedback compensation over a 130- 160 K temperature range. Invariance of rate constants as the potential sweep rate (and thus AEp) increases is a classical criterion for the absence of IRmc effects. Table 1 shows that rate constants evaluated from AE, using simulations based on Marcus theory (assuming L = 0.15 eV; see later discussion) typically vary by 2-fold or less over a decade of sweep rates. Also, in a voltammogram taken at 50 mV/s at 130 K and exhibiting AE, = 325 mV, the decrease in A E p produced with positive feedback compensation, 19 mV, was close to the iRmcrelated AEp value, 16.5 mV, calculated by comparing digitally simulated voltammograms based on assuming RIJNC= 0 and 69 kS2. The digital simulations were performed on an 80 x 86 based computer using the program "MCDELUX' written in QuickBasic V4.5 using the fast implicit finite differences method as described by Rudolph14aand the Marcus theory based kinetic approach described by C h i d ~ e y . ' ~ ~
Results and Discussion Analysis of Cyclic Voltammograms for Heterogeneous Electron-Transfer Rate Constants. Examples of voltammograms at the extremes of temperature explored, 130 and 175 K, in the mixed solvent system PrCN/EtCl are shown in Figure 1, A(-) and B(-), respectively. The voltammogram at 175 K is quasi-reversible (AEp = 71 mV), whereas that at 130 K is nearly completely irreversible ( U p = 326 mV). Observations at intermediate temperatures show that AE, increases smoothly with decreasing temperature and with increasing potential sweep rate and that peak currents are proportional to the square root of the potential sweep rate, behavior characteristic of a diffusion-
J. Phys. Chem., Vol. 98,No. 50, 1994 13397
I*
0.1 O**
3 c1
st o
a -0.1
:
-0.2
0.8
0
-0.2
0.1 -0.1 Potential 01) vs. Au QRE
-0.3
0.6
0.2
0.4
2.0
B
-2.0
I
I 0.3
Figure 1. Positive feedback corrected cyclic voltammograms (bold curves) of decamethyl ferrocene (0.5 mM) in 2:l (v:v) ethyl chloride/ butyronitrile with 0.075 M B W F 6 at 130 K (panel A) and 175 K (panel B), at a 2-mm-diameter Au disk electrode and potential sweep rate of 50 mV/s. Open circles are the best-fit results of the Marcus and 1.4 x cds theory digital simulation, taking k" = 3.1 x at 130 and 175 K, respectively, and assuming I S M= 0.15 eV. Curve - - - in panel A was simulated by Butler-Volmer theory for k" = 5.7 x lo-' c d s and a = 0.5 at 130 K. 0.3 1
EE u3
I
O
-0.1 -o,2
i
-0.34 1 .o
~
0.4
'
~
0.2
,
I
~
0
-0.2
-0.4
B
-1.0 1
I 0.3
0.1 -0.1 Potential (V) vs. Au QRE
-0.3
Figure 2. Positive feedback corrected cyclic voltammograms (bold curves) of decamethylferrocene (0.5 mM) in butyronitrile with 0.075 M BUPF6 at 162 K (panel A) and 181 K (panel B), at a 2-mmdiameter Au disk electrode and potential sweep rate of 50 mV/s. Open circles are the best-fit results of the Marcus theory digital simulation, and 9.4 x c d s at 162 and 181 K, taking k" = 1.3 x respectively, and assuming &M = 0.15 eV. controlled reaction with a slow, thermally activated heterogeneous electron-transfer rate. Figure 2 (-) shows analogous voltammograms in butyronitrile solvent over a smaller temperature range. The voltammograms, nearly reversible at 181
Richardson et al.
13398 J. Phys. Chem., Vol. 98, No. 50, 1994 K (Figure 2B, AEp = 63 mV), are quasi-reversible at 162 K (Figure 2A, A& = 109 mV), which is near the freezing point of the solution (ca. 160 K). The Nicholson and Shain methodlo was in a preliminary analysis applied to the dependency of experimental AE, on potential sweep rate, producing rate constants which were at the lower temperatures somewhat variable with sweep rate (from 10 to 100 mV) but which gave linear Arrhenius plots (ln[k,] vs l / Q over the above temperature ranges. At the lower temperatures, however, the voltammetric waveshapes are at all sweep rates much broader than voltammograms digitally simulated by using standard Butler-Volmer kinetic theory. Figure 1A shows the huge difference between the simulated Butler-Volmer voltammetric waveshape (- -, thin dashes) in the mixed solvent and the experimental waveshape (- -, thick dashes) at 130 K. The substantial broadening of the experimental cyclic voltammetry (relative to simulated voltammograms) gradually diminishes as the temperature is increased from 130 to 170 K. At the highest temperatures (Figures lB, 2A,B), the shapes of the Butler-Volmer theoretical and experimental voltammograms can be made to agree. The behavior at the lower temperatures (Figure 1A) reveals a complication in using the Butler-Volmerbased Nicholson and Shainlo analysis to obtain k" values there for the (Me~C5)2Fe+'~ couple. There are at least two possible explanations for the broadening of voltammograms like Figure 1A (-): ( a ) inappropriateness of Butler-Volmer theory for these kinetic circumstances and (b) a heterogeneity of chemical state. Both of these effects may be important, as next discussed. The standard Butler-Volmer model12 for the relation between the electron-transfer rate and the electrode potential is a limiting case of contemporary Marcus" theory, for reaction free energy (which measured as Ep - EO' , or [E,,o, - E,,d]/2 = AEp/2, is overpotential) much less than the reorganizational energy banier (A = 4AG*). It appears that this assumption may break down at the large overpotentials required to drive the decamethylferrocene electron-transfer reaction at the lower temperatures, requiring the full Marcus relationship" for analysis of the electrode kinetics. This relation predicts that voltammetric waves are broadened for such a circumstance. Marcus theory has been applied by C h i d ~ e yto ' ~explain ~ the rate-overpotential results in the potential step experiments on chemisorbed ferrocene alkanethiol monolayers; his analysis of the electrode kinetics included accounting for the continuum of electronic states in the metal electrode. We have presented analogous theory for potential sweep experiments with immobilized redox monolayers3a and here extend it to the cyclic voltammetry of diffusing redox solutes. The behavior of cyclic voltammograms digitally simulated based on an integration of the Marcus relation over a range of energies about the Fermi level of the metal electrode is illustrated in Figure 3, A and B for 130 and 175 K, respectively. The calculations use parameters appropriate for the Figure 1A (thick dashes) and 1B (thick dashes) voltammograms i.e., AEp = 326 and 7 1 mV, respectively, and illustrate at each temperature the effect of reorganizational energy (A = 0.15-0.85 eV). At the higher temperature (Figure 3B), the simulated current-potential curves are nearly identical over this range of A since reaction overpotential (e.g., AEp/2 = 0.035 V) in Figure 3B is much smaller than even the smallest reorganization energy represented. These Marcus theory simulated current-potential curves are indistinguishable from those predicted using standard ButlerVolmer kinetics. At the lower temperatures, in Figure 3A, AEd2 is larger (0.163 V), and for values of A of cu. 0.4 eV and less, the calculated peak currents become smaller and the
0.4
1
I
A
A
-0.4 0.4 1
I
1
-0.4 0.6
0.4
0.2
0
-0.2 -0.4
-0.6
Overpotential (v)
Figure 3. Digitally simulated voltammogramsbased on Marcus-DOS theory, for the conditions of Figure 1, illustrating the effect of 1on the shape of the current vs potential curve: (panel A) T = 130 K and A E p = 325 mV; 1 s =~0.15 eV and ~ S M = 0.006 c d s (0);&.M = 0.4 eV and k s =~ 0.0029 c d s (- -); ASM = 0.85 eV and k s =~ 0.0018 c d s (-); (panel B) T = 175 K and A E p = 70 mV; 1 s =~0.15 eV and k s =~ 0.65 c d s (-); ISM= 0.85 eV and k s =~ 0.70 c d s (+).
voltammogram becomes substantially broader than in ButlerVolmer behavior (which is approximated by the curve for A = 0.85 eV). The broadening of the theoretical voltammetry can be qualitatively understood by recognizing that as reaction free energy (i.e., AEp/2) approaches A, one accesses a kinetic regime where heterogeneous electron-transfer rates become less than exponentially dependent on, and ultimately independent of, ~verpotential.'~ Examination of the legend in Figure 3A shows that over a nearly 10-fold range (0.15-0.85 eV) of reorganizational energy A, the heterogeneous electron-transfer rate constant k" needed to produce a AE, of 325 mV in the voltammograms simulated at 130 K varies by only 4-fold. At 175 K, the same variation in A corresponds to only a 9% change in k". In general, we find that the shapes of the cyclic voltammetric peaks are strongly dependent on the ratio of A and reaction overpotential, while the electron-transfer rate constant k" is the stronger detenninant of reaction overpotential AE,. That is, electron-transfer rate constants assessed solely by fitting theoretical simulations to experimental AE, values are only slightly dependent on the assumed value of reorganizational energy A. These characteristics were also found in the calculation^^^ of cyclic voltammograms of surface-immobilized redox species such as chemisorbed ferrocene alkanethiols. Even though ilcould, in principle, be determined from shape fitting of voltammetric waves, we regard (vide infra) the temperature dependency of k" as a more reliable way to evaluate the electron-transfer activation parameters. In the following, rate constants are evaluated from experimental hE, values based on the extremes of large and small reorganizational energies used in Figure 3, with the purpose of revealing the minor extent to which the katemperature dependence analysis depends on the choice of a reorganizational energy value in interpretingthe low-temperature voltammetry. To evaluate standard heterogeneous electron-transfer rate constants (k")from experimental U p values, voltammograms were calculated (as in Figure 3) for incremented values of the
J. Phys. Chem., Vol. 98, No. 50, 1994 13399
Electron-Transfer Dynamics of Decamethylferrocene
TABLE 2: Heterogeneous Electron-Transfer Rate Constants for Decamethylferrocenein Butyronitrile and in 2:l (v:v) Ethyl Chloride/Butyronitrile lo%", c d s T,K PCN" WNb EtCyRCNC EtCyRCNb 0.40 0.58 130 135 0.98 1.3 2.8 3.0 140 5.8 6.0 145 10.4 9.9 150 23 20 155 34 33 160 162 13 13 165 19 19 59 54 169 32 32 173 44 44 175 138 128 177 68 70 181 94 94 "Assumes Isim= 0.87 eV. bAssumesIsh= 0.15 eV. 'Assumes I s h = 0.84 eV. dimensionless rate constant
kdim,
kdim= k"(-)RT vDF
112
where R is the gas constant, T is the temperature, v is the potential scan rate, D is the diffusion coefficient of (Me&&Fe, and F is the Faraday constant. The finite difference digital simulation employed the Marcus relation,l' integrating it over the continuum of electronic states about the Fermi level in the metal electrode, as had been done14 for surface-immobilized redox species. Adsorption and double-layer effects are neglected. Working curve plots of peak potential separation (AE,) in simulated voltammograms were prepared as a function of k b and for different temperatures and values of I . As in Figure 3, working curves prepared for selected values of I were similar. The working curves were used with experimental AE, values to estimate k"; calculations of complete voltammograms for that k" were made for more detailed comparisons such as in Figures 1 (0)and 2 (0). The results for k" summarized in Table 2 are for the slowest potential sweep rates employed, to minimize any vestiges of iRmc effects. The small reorganizational barrier energy, 0.15 eV, gave the best (but certainly not perfect) fit of mixed solvent voltammograms at the lowest temperatures (e.g., Figure 1A). The larger reorganization energies chosen, 0.87 eV in butyronitrile and 0.84 eV in EtCl/PrCN, are theoretical values calculated from the Marcus dielectric continuum expression, using extrapolated low-temperature dielectric values, vide infra. In butyronitrile, -Table 2 shows that there is no difference in the obtained k" value for the extremes of assumed I . In the EtCl/PrCN solvent, differences in k" (Table 2) are minor, consistent with the above discussion. A second possible source of voltammetric peak broadening is a loss of proportionality between activity and total concentrations of (MesC5)zFe and (MesC&Fe+ species at the lowest temperatures, producing the appearance of a dispersion in redox potentials (E"' - dispersion). We can only speculate as to a mechanism for this; an example would be a dispersion in the microscopic environments of the (MesC&Fe and (MesC&Fe+ species that is kinetically sluggish on the voltammetric time scale. The sluggish chemical process could involve strong { (MesC&Fe+,PF6-} ion pairing or ion multiplet aggregation. Differing populations of (Me5Cs)zFe and (MesC&Fe+ states might further exhibit different rate constants or diffusion coefficients. The behavior of the data requires that E"'-
0.4
0.6
0.4
0 -0.2 Overpotential
0.2
-0.4
-0.6
Figure 4. Digital simulations illustrating the effect of a distribution of E"' values about a central value: (-) no distribution; simulation parameters were ksm = 0.01 c d s , ,Ism = 0.7 eV, T = 130 K, E"' = 0.0 V; AEP = 240 mV; (- -) sum of five equally weighted voltammograms employing the same parameters but with E"' = -0.06, -0.03, 0.0, 0.03, and 0.06 V; the resulting AEP = 300 mV.
dispersive effects become important only at the lowest temperatures, where band broadening persists at all potential sweep rates but vanishes at higher temperatures. (Additionally,doublelayer effects on electrode kinetics can broaden voltammetry.) Figure 4 shows a simple example of E"'-dispersive peak broadening generated by summing simulated voltammograms of five equal populations having E"' spaced by 30 mV. While the match of this particular example to a Marcus-simulated voltammogram (corresponding approximately to the experiment of Figure 1A) is not very good, we are confident that a good match could be produced by sufficient manipulation of other involved parameters. Lacking explicit experimental details, perfecting this exercise has little appeal, but the point is clear that E"' and/or kinetic-dispersive voltammetric peak broadening may comprise in total or part the source of the voltammetric peak broadening seen below 170 K (as in Figure 1A (-)). That is, the peak broadening is not necessarily due to an usually small value of I . Temperature Dependence of Electron-Transfer Rate Constants. For both homogeneous and heterogeneous electrontransfer processes, the standard rate constant is related to activation free energy by
k" = Kp~,,v,, exp
(-zET) -
where Kp is the equilibrium constant for precursor complex formation, ~~1 is the electronic transmission coefficient (assumed to be unity, e.g., (MesC5),Fe+/O is adiabatic), and v n is the nuclear frequency factor. 15-18 Given the previous literature on metallocenes,8J9we assume, for now at least, that the activation energy, AG*ET,is predominately an outer-sphere reorganizational barrier, (i.e., reorganization of solvent dipoles). The dependency of the reaction rate on temperature is a wellknown method to determine the activation energy, AG*, using the slope of a plot of h [ k ] vs T'.(We assume here that AH* x AG*.) This approach has been used for both heterogeneous and homogeneous electron-transfer r e a c t i ~ n s . ~ , ~ J ~ - * ~ Arrhenius plots for the data of Table 2 (Figure 5) are linear for both solvent systems, irrespective of whether (Table 2) a large or small reorganizational energy was assumed in the experimental analysis (vide supra). The resulting activation barriers, AG*Ex~(Table 3, footnotes f-h), are again rather insensitive to which I was used in evaluating k" from the voltammetry. Also, AG*Ex~is nearly the same in the two solvents. Recall that,' lacking significant activation entropy, reorganizational and activation barrier energies are related by 2 = ~AG*Ex~.
Richardson et al.
13400 J. Phys. Chem., Vol. 98, No. 50, 1994 I
\
-14 J
0.005
0.006
1K
0.007 (K-’)
1 0.008
Figure 5. Arrhenius plots of observed rate constants for decamethylfemene in butyronitde (0)and 2:l (v:v) ethyl chloridehutyronitde (0). (0.075 M) supporting electrolyte.
There are limited previous activation data for decamethylferrocene and no data in either of the two solvents employed here. In acetone, the activation enthalpy is reported8 as 0.20 eV, somewhat lower than that for butyronitrile in Table 3. Extrapolation of the Figure 5 Arrhenius plots to room temperature produces k029g and preexponential (A) values (Table 3) that can be compared with the results8 for k298 of 3.5 and 0.4 cm/s in acetone and benzonitrile, respectively, and for A of 8 x lo3 cm/s in acetone. The Table 3 results are reasonably close to these previous, room-temperature data. The Table 3 barrier results for AG*ED can also be compared with outer-sphere reorganizational energies calculated from the Marcus relation”
where e is the electronic charge, N is Avogadro’s number, co is the permittivity of free space, copand eSare the optical and static dielectric constants respectively, a is the reactant molecular radius, and d is the distance between the reacting molecule and the electrode surface (the 1/2d term for image effects is ignored8). To calculate AG*os, it is necessary to account for the temperature dependencies of cop and cs. Russel and J a e n i ~ k e ?for ~ example, noted a 3% variation in A G * E over ~ a 60 “C temperature range for pyrazine in DMF. Values of and €2’ for butyronitrile and ethyl chloride exhibit mild, linear temperature dependencies from 201 to 333 K; for example, es for ethyl chloride changes (linearly) by 26% over a 72 K temperature range. Since these changes were small, we extrapolated the literature data for copand E , to the experimental temperature range employed here. These calculations predict a minor temperature dependence for AG*os; predicted for butyronitrile are 0.217 eV at 162 K and 0.218 eV at 181 K, corresponding to an average activation energy AG*os = 0.22 eV. We assume that the ethyl chloride/butyronitrile mixture behaves as a continuum with “average” dielectric properties estimated by a mole-fraction-weighted sum of dielectric properties of the individual solvent components, giving from the individual cop= 2.108 and 2.119 and es = 40.98 and 20.71 at 130 K and cop= 2.054 and 2.051 and cs = 36.51 and 18.08 at 175 K for F K N and EtC1, respectively, cop= 2.12 and cS = 26.59 at 130 K and cop= 2.05 and E~ = 23.42 at 175 K. The estimates of AG*os = 0.206 and 0.211 eV at 130 and 175 K, respectively, again indicate a minor temperature dependence of AG*os. Examining Table 3 shows that the measured activation barriers A G * m are in both solvents near but slightly larger than the energies AG*os estimatedz8 from eq 3. Lowtemperature electron-transfer kinetics is largely uncharted territory, so even an approximate correspondence between theory
and new ultra low-temperature data is interesting and strongly indicates that the voltammetric wave broadening described (Figure 1A) should be ascribed to a dispersion of chemical state and not to unusually small reorganizational barriers. We have observed3 analogous wave broadening in the low-temperature voltammetry of self-assembled ferrocene monolayers in the PrCN/EtCl solvent; these monolayers also give activation barriers slightly larger than (but closer to) eq 3 values and display a dispersion in their kinetic reactivities. In what follows, we consider the possible origins of the ca. 0.05 (hO.01)eV difference between A G * m and AG*os, which could be (a) an inner-sphere term A G * I ~and/or (b) a solvent dynamics contribution. Weaver et aLZ9have estimated that AG*1s = ca. 0.5 kcaYmol or 0.02 eV for the (Me5C5)2Fe+Io couple, which is about one-half of the AG*Ex~vs AG*os difference in Table 3. We recall at this point the voltammetric wave broadening at lower temperatures. If this broadening arises from a chemical state dispersion, the value of A G * I ~may be enhanced as an additional consequence. Therefore, the combination of an approximately doubled (but still minor) innersphere barrier29 with the calculated AG*os could provide a satisfactory accounting for the experimental observations. Considering a possible solvent dynamics effect, it is now recognized that, to an extent dependent on the reaction adiabaticity,*Othe dynamics of solvent dipole reorientation influence electron-transfer rates through the preexponential nuclear barrier crossing frequency, v,. Slow solvent repolarization (described by the longitudinal relaxation time, T L ) provokes repeated unfruitful crossings of the transition barrier that depress the net electron transfer rate.8,24 Solvent dynamics effects have been studied as a function of s ~ l v e n t ? redox ~ ~ , ~species,* ~ solution visc0sity,3~-~~ and, in a few cases, a function of t e m p e r a t ~ r e . ~ ~ ~ ~ ~ ~ Solvent dynamics become temperature dependent by way of thermal activation of solvent dipole reorientations. Hypothesizing that the temperature dependence ( h G * ~ m of ) the diffusion coefficient (D)of decamethylferrocene may provide a crude analogy to that of the solvent relaxation parameter TL, we connect A G * D to ~ the nuclear frequency factor, v,, which has inverse proportionality to r ~ , ~ ~ (4) by the equations
tL =
E)% kT
v = G
(7)
where a is the solvent molecular radius, 17 is the viscosity, T D is the Debye relaxation time, E , is the high-frequency dielectric constant, and r is the hydrodynamic radius of the diffusing species. Combining these equations gives an equation which includes a temperature-dependent v,
Electron-Transfer Dynamics of Decamethylferrocene
J. Phys. Chem., Vol. 98, No. 50, 1994 13401
TABLE 3: Activation Parameters for Decamethylferrocenein Butyronitrile and 2:l (v:v) Ethyl ChlorideButyronitriIe solvent AG*os/ eV AG*=,b eV A G * D ~ :eV AG*= - AG*Dm,d eV AEW; x cds k"298~,~ cds PrCN
0.22
EtCYPrCN
0.21
0.27f 0.278 0.24 f 0.01f 0.26 f O.Olh
0.20 0.13
0.07f 0.078 0.1If 0.13h
4.5v 2.328 0.84f 3.99
0.81f 0.809 0.88f
lXh Theoretical value, eq 3, for mixed solvent based on cop = 2.05 and c8 = 23.42estimated by extrapolation of literature data33,36to 175 K, a = 3.8 x m for (Me5Cs)~Fe.~From Figure 5. From Figure 6.*Equation 8, see text. e Preexponential term of eq 2 from Figure 5 interceptsfl = 0.15eV assumed in analysis of k". 8 1 = 0.87eV assumed in analysis of k". = 0.84eV assumed in analysis of k". -12
The main points that can be made from the present study are I as follows: (a) The electrochemical kinetic and activation data
-20 0.005
0.006
0.007
0.008
1rr (K')
Figure 6. Arrhenius plots of diffusional data (Table 4)for decamethylferrocene in butyronitrile (0)and 2:l (v:v) ethyl chloridehutyronitrile (0).Bu$\TPF6(0.075M) supporting electrolyte. TABLE 4: Diffusion Coefficients of Decamethylferrocenein Butyronitrile and 2:l Ethyl ChlorideButyronitrile 10'0, cmVs 130 135 140 145 150 155 160 162 165 169 173 175 177 181
0.61 0.97 1.4 1.9 2.9 4.4 5.7 12 17 28 42
6.8 14
71 111
where AD^ is the diffusion preexponential factor. This equation shows that AG*Ew determined as in Figure 5 consists of a summation of diffusion and electron-transfer barrier energies. A G * D was ~ evaluated from chronoamperometrically determined diffusion coefficients presented in Table 4 and plotted in Figure 6. D is larger and the diffusion activation barrier smaller in the mixed solvent (Table 3), a result of the lower viscosity of its ethyl chloride component. Table 3 shows that the A G * w - A G * Ddifference ~ is quite small and much smaller than the AG*os barriers calculated from eq 3. This disrepancy suggests that either (a) the (MesC5)2Fe+'O reaction is not fully adiabatic and that a partial, not full, solvent dynamics correction should be applied (i.e., that ZL appears30 in eq 4 as Z L - ~and a FY 0.25 and 0.38 for PrCN and PrCN/ EtCl, respectively) or (b) the thermal activation A G * D of ~ (Me5Cs)zFe diffusion is larger than that for self-diffusion of the solvent components themselves. The latter interpretation is supported by estimations8*28c of the activation enthalpy AH*, of ZL of ca. 0.09,0.05, and 0.06 eV for butyronitrile, acetonitrile, and butyronitrile, respectively, that are much smaller than the A G * D results ~ in Table 3. It appears that the use of the (Me5Cs)zFe diffusion activation A G * Dproduces ~ a substantial overestimation of the solvent component self-diffusion activation.
demonstrate the efficacy of thermal quenching of a nominally facile electron-transferreaction. At the same time, undetermined characteristics of the low-temperature solvent (chemical association phenomena) and electrode interface (double layer structure) limit the degree of interpretation. Further experimental evidence will be required to understand the extent to which ultra low-temperature electron transfers offer properties that are not simple extrapolations of room-temperature behavior. (b) The experimental electron-transfer barrier AG*Ex~is somewhat larger than the calculated outer-sphere reorganizational energies AG*os. The difference may arise from an innersphere barrier AG*Is that has been enhanced by chemical interaction effects as signaled by the broadened voltammetric waves. If the AG*Ew vs AG*os difference is due entirely to an AG*,s term, of magnitude ca. 0.05 eV, solvent dynamics effects are probably unimportant. Alternatively, the AG*EXP vs AG*os difference could be comprised of a smaller (ca. 0.02 eV29) AG*Is term and a partial solvent dynamics correction. We are able to raise but not distinguish between these possibilities. It is worth noting that an analogous (but smaller) difference has been observed3 between AG*Ex~and AG*os in studies of self-assembled ferrocene monolayers, where solvent dynamics effects for reasons of nonadiabaticity are not expected to be important. (c) An attempt to derive a solvent repolarization barrier by analogy to that for (MesC5)zFe diffusion suggests that the latter substantially overestimates the probable temperature dependence of ZL. (d) The heterogeneous electron-transfer rates vary exponentially with temperature and show no obvious sign of an approach to temperature insensitivity as would occur with a vibronic tunneling mechanism. This effect, if present, could produce a diminished activation barrier. (e) A Marcus theory based modification of the classical theory for cyclic voltammetric waveshapes has been presented that is useful for circumstances where the electrochemical overpotential is not negligible with respect to the electron-transfer reorganizational energy barrier. Digital simulations show that significant deviations from classical Butler-Volmer kinetic behavior occur at low temperatures when the overpotential exceeds about 40% of the reorganizational energy.
Acknowledgment. This research was supported in part by grants from the Office of Naval Research and the National Science Foundation and a Commonwealth Postgrad Research Award (Australia) to J. H. We gratefully acknowledge helpful discussions and advice from Dr. S. Feldberg, Brookhaven National Labs, and Prof. A. Bond, La Trobe University, Australia. The digital simulation program was prepared by J. H. while at Brookhaven. References and Notes (1) chin^. S.: McDevitt. J. T.: Peck. S. R.: Murrav. R. W. J. Elec;rochem."Soc. 1991,138, 2308. (2) Peck, S. R.; Curtin, L. S.; McDevitt, J. T.; Murray, R. W.; Collman, J. P.; Little, W. A,; Zetterer, H. M.; Duan,H. M.; Dong, C.; Hermann, A. M. J. Am. Chem. SOC. 1992,114, 6771.
Richardson et al.
13402 J. Phys. Chem., Vol. 98, No. 50, 1994 (3) (a) Tender, L. M.; Carter, M. T.; Murray, R. W. Anal. Chem., in press. (b) Richardson, J. N.; Peck, S. R.; Curtin, L. S.; Tender, L. M.; Tenill, R. H.; Carter, M. T.; Murray, R. W.; Rowe, G. K.; Creager, S. E. Electrochim. Acta, in press. (c) Richardson, J. N.; Rowe, G. K.; Carter, M. T.; Tender, L. M.; Curtin, L. S.; Peck, S. R.; Murray, R. W. J . Phys. Chem., in press. (d) Carter, M. T.; Rowe, G.K.; Richardson, J. N.; Tender, L. M.; Terrill, R. H.; Murray, R. W. J. Am. Chem. Soc., submitted. (4) Curtin, L. S.;Peck, S. R.; Tender, L. M.; Murray, R. W.; Rowe, G. K.;Creager, S. E. Anal. Chem. 1993, 65,386. ( 5 ) Green, S.J.; Rosseinsky, D. R.; Toohey, M. J. J. Am. Chem. Soc. 1992, 114, 9702. (6) O’Connell. K. M.: Evans. D. H. J. A m Chem. SOC.1983,105, 1473. (7) Nagaoka, T.; Okkaki, S. J. Phys. Chem. 1985, 89, 2340. (8) Gennet, T.; Milner, D. F.; Weaver, M. J. J. Phys. Chem. 1985.89, 2787. (9) McManis, G. E.; Nielson, R. M.; Gochev, A.; Weaver, M. J. J. Am. Chem. Soc. 1989, 111, 5533. (10) Nicholson, R. S. Anal. Chem. 1965, 37, 1351. (11) Marcus, R. A. J. Chem. Phys. 1965, 43, 679. (12) Bard, A. J.; Faulkner, L. R. ElectrochemiculMethods; Wiley: New York, 1980. (13) This resistance compensation methodI2 depends on continuously applying to the working electrode a supplementary potential proportional to the instantaneouscell current. In practice, the proportionality constant is increased until the circuit becomes electronically unstable, then backing off slightly. The induced change in voltammetric A E p is near the point of electronic instability relatively insensitive to the proportionality constant. We have found that positive feedback compensation is relatively straightforward to use at the slow times, small currents, and low rate constants characteristic of the low-tempxature-regime. Also,calculations from a circuit equation of the amount of resistance compensation being applied are generally in good agreement (Table 1) with R ~ values c measured using ac imwdance, indicating nearly full Compensation was Dossible. (14) (a) Rudolph, MrJ. EleEtroanal. dhem. 1991,3li, 13. (b) Chidsey, C. E. D. Science 1991, 251, 919. (15) Maroncelli, M.; MacInnis, J.; Fleming, G. R. Science. 1989, 243, 1674. (16) Weaver, M. J.; McManis, G. E. Ace. Chem. Res. 1990, 23, 295. (17) Sutin, N. Prog. Inorg. Chem. 1983, 30, 441. (18) Hupp, J. T.; Weaver, M. J. J. Electroanal. Chem. 1983, 152, 1. (19) Nielson, R. M.; McManis, G.E.; Golovin, M. N.; Weaver, M. J. J. Phys. Chem. 1988, 92, 3441. (20) Grampp, G.;Kapturkiewicz, A.; Jaenicke, W. Ber. Bunsenges. Phys. Chem. 1990,94,439.
(21) Yang, E. S.; Chan, M.; Wahl, A. C. J. Phys. C h m . 1980,84,3094. (22) Kirchner, K.; Dang, S.; Stebler, M.; Dogden, H. W.; Wherland, S.; Hunt, J. P. Inorg. Chem. 1989, 28, 3604. (23) Bernhard, P.; Helm, L.; Ludi, A,; Merbach, A. J. Am. Chem.Soc. 1985, 107, 312. (24) Weaver, M. J. Chem. Rev. 1992, 92, 463. (25) Russel, C.; Jaenicke, W. Electrochim. Acta 1982, 27, 1745. (26) Riddick, J. A.; Bunder, W. B.; Sahano, T. K. Organic Solvents-Physical Properties and Methods of Purification; John Wiley and Sons: New York, 1986. (27) Madelung, O., Ed. Landolt-Bomstein: Numerical Data and Functional Relationships in Science and Technology: Vol6, Static Dielectric Constants of Pure Liquids and Binary Liquid Mixtures; Springer-Verlag: New York, 1991. (28) (a) Disagreement between A G * m results and calculations from eq 3 has been seen before?~28b*28e and seeking improvements, Fawcett and OpalloZMhave applied the mean spherical approximation (MSA) to account for the effects of electrolyte ions on solvent repolarization energetics. In butyronitrile, a barrier of 0.158 eV (smaller than that obtained from eq 3) is predicted,2Mfrom the expression
AG*,,,
=
-(x)[(Ay 32m,
(L)(L)]
1 - E,,, r, - 1 - E, ra i6,
where r, is the molecular radius and 6,, the MSA polarization parameter, m for F‘ICN.~~ (b) Nielson, R. M.; Weaver, M. J. J. is 0.83 x Electroanal. Chem. 1989, 260, 15. (c) Fawcett, W. R. Chem. Phys. Lens. 1992,174, 167. (d) Fawcett, W. R.; Opallo, M. J. Electroanal. Chem. 1993, 349, 273. (29) Reference 9, footnote 47. (30) (a) Bixon, M.; Jortner, J. Chem. Phys. 1993,176,467. (b) Phelps, D.; Weaver, M. J. J. Am. Chem. SOC. 1992, 96, 7187. (31) Farmer, J. K.; Gennett, T.; Weaver, M. J. J. Electroanal. Chem. 1985, 191, 357. (32) Zhang, X . ; Leddy, J.; Bard, A. J. J. Am. Chem. Soc. 1985, 107, 3719. (33) Zhang, X.; Yang, H.; Bard, A. J. J. Am. Chem. SOC. 1987, 109, 1916. (34) Zhang, H.; Murray, R. W. J. Am. Chem. SOC. 1993, 115, 2335. (35) Baranski, A.; Winkler, K.; Fawcett, W. R. J. Electroanal. Chem. 1991, 313, 367. (36) Calef, D. F.; Wolynes, P. G. J. Phys. Chem. 1983, 87, 3387.
