J. Phys. Chem. 1900, 84, 3330-3338
3336
Kinetic Behavior To Be Expected from Outer-Sphere Redox Catalysts Confined within Polymeric Films on Electrode Surfaces Fred C. Anson Arthur Amos Noyes Laboratory, Divlslon of Chemistry and Chemlcal Ewineering, Caiifornla Institute of Technomy, Pasadena, Calltorn& 9 1125 (Received: July 23, 1980: In Flnal F m : September 19, lQ80)
Some elements of Marcus theory are applied to predict the likely kinetic behavior of redox catalysts held in polymeric films attached to electrode surfaces. It is concluded that the cross reaction between attached catalyst and dissolved substrate will most often be rate limiting. Charge transfer through the attached film may become rate limiting under certain conditions but electron transfer between the electrode and catalysts attached to its surface is predicted not likely to be a rate-limiting process. For catalyst-substrate combinations with cross-reactionrate constanta near the diffusion limit efficient catalysis seems possible even when the equilibrium constant for the cross reaction is much less than unity.
Attention has been directed recently at the acceleration of the rates of electrode processes by means of simple, inorganic, outer-sphere redox catalysts attached to the surfaces of electrodes.ls2 Polymeric ligands: polyelectrol y t e ~ >and ~ various forms of polymerized transition metal ~omplexesl*~*~ have proved to be particularly useful in confining large quantities of potential catalysts within polymeric domains at electrode surfaces. The catalytic responses to be anticipated from such coated electrodes have been discussed in the context of cyclic voltammetry by Saveant and co-workers*who have also presented detailed discussions of the related situation in which the catalysts are present in solution rather than attached to the electrode s ~ r f a c e In . ~ the present work some elementa of Marcus theory1° are applied to the case of attached catalysts in order to reach generalizations about the likely kinetic behavior to be expected from electrodescoated with simple, outer-sphere, redox catalysts. The upper portion of Figure 1 gives a schematic depiction of a cross section near the surface of an electrode coated with a redox catalyst that is to be used in accelerating the oxidation or reduction of a substrate present in the solution. There are three locations, labeled 1,2, and 3 in Figure 1, where the rate of the electron transfer could limit the current. At interface 1 between the electrode and the coating containing the catalyst the current should be given by eq 1, where k, is a heterogeneous electron-transfer
rate constant, CCatis the concentration of catalyst (mol cm-S)present at the interface, n is the number of electrons involved in the electrode reaction of the catalyst, A is the electrode area, and F is the Faraday. For coatings containing I' mol cm-2 of catalyst homogeneously distributed throughout a layer of uniform thickness, d, , C = r / d . For redox catalysts that cycle between two oxidation states the value of k, can be estimated from the rate constant for the homogeneous self-exchangereaction, k,,, by means of the Marcus relationshiplob where Z1 , and Zsolare diffusion-limited collision rates at the electrode surface and in solution, respectively. Thus, eq 1 can be rewritten as t Contribution
No. 6271. QQ22-3654/8Q/2Q84-3336$01 .QO/O
(3)
For homogeneous solutions of small molecules typical values of Ze1and Zwlare lo4 cm s-l and loll M-l s-l, respectively. However, for reactants confined in polymeric matrices at electrode surfaces a smaller value of Zelwould seem appropriate. The region labeled 2 in Figure 1 consists of the coating of thickness d in which the redox catalyst is confined. Current flow through this region under steady-state conditions will be controlled by the rate at which pairs of attached catalyst molecules in oxidized and reduced states are able to exchange eledtrons with each other, a process that will,in turn, be governed by factors such as the rate of diffusional motion of counterions within the coating as well as the rates at which potential reactant pairs are thermally juxtaposed into positions for successful electron t r a n ~ f e r . ~ * ~By J lanalogy J~ with steady-state current flow in thin-layer electrochemical cells13 the limiting steadystate current can be estimated from eq 4, where D is an N
N
(4)
effective diffusion coefficient whose magnitude is determined by some combination of the various motions that are associated with the transfer of charge across region 2. The value of D can often be estimated from current-time or charge-time measurements with catalyst-coated electrodes in the absence of a reactive substrate if the coating is uniform and of known thickness. At interface 3 between the catalyst coating and the solution of substrate, the rate of electron transfer is determined by the rate of the cross reaction between the redox catalyst and substrate2J4
-I-3
- k,,I'Cb (5) nFA where k, is the second-order rate constant governing the cross reaction and Cb is the bulk concentration (moles liter-l) of substrate (stirring is assumed to be adequate to prevent depletion of the substrate concentration at site 3). The lower portion of Figure 1 compares these three estimates of the steady-statecurrents achievable in regions 1,2, and 3 as a function of the self-exchange or cross-reaction rate constants for values of I' and d corresponding to a few monolayers of a polymeric coating containing a 0 1980 Amerlcan Chemical Society
The Journal of Physical Chemistty, Vol. 84, No. 25, 1980 3337
Letters ELECTRODE:
ATTKHED CATALYST LAYER
SOLUTION OF SUBSTRATE
log
hex, kcrs),
M-ls-'
Figure 1. Schematic representation of the interphase between a catalystcoated elecwode and a solution of substrate. The dependence of the current-timitlng proceases at various polnts within the interphase on homogeneous aielfexchsnge or cross-reaction constants is also shown (see text).
T+ LEV
IOZ
10'
IO'
ROTATION RATE, rprn
Figure 2. Calculated current responses for redox catalysts attached to the surface of rotating dlsk electrodes compared to the iimlin Levlch currents. A substrate diffusion coefficient of 6 X loa om2'8 was assumed.
relatively dilute concentration of incorporated redox catalyst. Values of If up to (anorder of magnitude greater can be obtained withi the same value of d by employing polymeric coatings packed more densely with catalyst but the current-limiting steps at each of the three interfaces identified in Figure 1have the same dependence on I? so that conclusions about the relative contributions of each interface to the limitation of the current will not be sensitive to the value of r. Values of d greater than ca. 100 A can result in decreasing accessibility of the bound catalyst to the substrate in solution14which complicates the analysis. The arisumption made here is that all of the catalyst confined at the electrode surface is equally accessible for reaction with substrate molecules transported to the electrode. ,Zdwas taken as 102cm s-l in constructing Figure 1 in recognition of the smaller thermal velocity anticipated for reagents incorporated in polymeric coatings. (Larger values of Zel would lead to the same general conclusions.) Several observahions caul be made on the basis of Figure 1by noting that the line lying lowest at any value of the rate constants will correspond to the current-limiting reaction. Thus, the reaction occurring at interface 3 would appear to be the most likely current-limiting reaction except when catalyst,-substratecombinations exhibiting large cross-reaction rates are utilized under conditions where the diffusional rates within the catalyst layer are quite low, Note, however, th,at, since the current in region 2 diminishes with the square of Q while r presumably increases with the first power of d, limitation of the current by
electron transport processes within the catalyst layer will become more likely with thicker films. It is interesting to note that the electron transfer rate between the electrlode and the catalyst layer at interface 1 is not expected to be current limiting at any accessible steady-statew e n t density. Thus, in the search for better catalysts to be attached to electrodes, it would appear to be sufficient to identify reactants that participate in rapid cross reactions with the substrate to be confident that they will present negligible impediment to current flow at interface 1. There is considerable uncertainty regarding the value of D to be expected within layers of attached catalysts at electrodes but it seems unlikely that values lying far outside the range covered by the dashed lines in Figure 1will be encountered. (Values from to lO-l3 cm2s-l have been estimated for a variety of films and temperatures.) Thus, interface 3 appears most likely to be the seat of the current-limiting process in electrode-catalyst-substrate systems such as that depicted in Figure 1. It therefore seemed wtorthwhile to examine the factors that could affect cross-reaction rates in a little more detail. Rotating disk electrodes provide a particul-arlyconvenient means for establishing steady-state currents at catalyst coated electrode~.~J~J~ The rate of a first-order reaction between catalyst and substrate will be governed by eq 5 so that the current at a catalyst-coated rotating disk electrode can be written as in eq 6, where Imt is the limiting 1 -1= - + 1 (6) 1-t I ~ e v nFAkCJCb current at the rotating disk electrode and I b v is the Levich currenP that would be measured at an uncoated electrode at potentials where the substrate was reduced (or oxidized) as rapidly as it arrived at the electrode. The ratio of the ctitalyst-mediatedcurrent to the Levich current is one measure of the efficiency of a catalyst and this ratio can be calculated from eq 7, where k (cm3mol-l (7)
s-l) is 103kcr,(M-l s-l), reat(mol cmT2)is the quantity of catalyst attached to the electrode, D, is the diffusion coefficient of the substrate, and w is the electrode rotation rate (rpm). (The kinematic viscosity of water was taken as cm2s-l.) Values of Iat/Ibv as great as, say, 0.5, at electrode rotation rates high enough to produce a large substrate flux at the electrode surface, say, lo4rpm, could be regarded as the marks of an adequate catalyst. Thus, it is of interest to determine the values of (kl?,J that would be required to produce such behavior. Figure 2 presents plots of Iat/Ibv for various values of (kr,) which make it evident that the change from a mediocre to a good catalyst encompasseci only about two orders of magnitude in (kI',J. Thus, if a catalyst can be attached to an electrode surface to the extent of mol cm4 it should prove potent if the homogeiieous cross reaction it undergoes with the substrate is governed by a rate constant, k, greater than 5 X 105 M-'s-l (rkr, > 5 X cm 8-9. On the other hand, if k, were 5 >C lo3 M-'s-l or smaller &rat< 5 X lo-" cm s-l), only rather sluggish catalysis would be anticipated. Driving Force and Catalysis Attempted catalysis by means of systems such as that depicted schematically in Figure 1raises a general question: How far apart may the formal potentials for the catalyst and substrate couples be before the catalysis be-
3338
The Journal of Physical Chemistry, Vol. 84,
No. 25, 1980
Letters
trodes to be susceptible to the type of catalysis considered here will probably not be numerous but at graphite electrodes many chromium(II1) and cobalt(II1)ammines appear to be good candidates17and the outer-sphere-catalyzed oxidation of hydrogen peroxide represents another possibility. Nevertheless,it should be recognized that some of the best electrocatalysts employ inner-sphere pathways to achieve their potencyls and their behavior would not be expected to be governed by the same relationships that were used to prepare Figures 1-3. - 4 1 ;I; -0.3
,/' -0.2
/'
I
-0.1
/
/I
0
I
I
I
I
0.1
0.2
0.3
0.4
AEo,volt
Flgure 3. Effect of driving force on the catalytic actlvlty of redox catalysts anchored to electrode surfaces.
comes impractical? Clearly, the catalysis will be more successful the more positive is the difference, AE",between the catalyst and substrate reduction potentials because the desired reaction will be energeticallymore favorable. The question is posed by cases in which AEo is negative, Le., when the desired catalyst-substrate cross reaction would proceed in reverse under standard conditions, One method of inspecting this question is shown in Figure 3 where the effect of changes in PE" on log (kI',J is shown for various combinations of rmtand the rate constants for self-exchange for both the catalyst and the substrate. The Marcus relationshiplo,eq 8, was used to calculate the lines k,, = (kZtk:htKeqfl1J2
(8)
in Figure 3. AE" was calculated as 0.059 log K and f was taken as unity to simplify the calculations.e9l'he plots become steep and nonlinear at the lower left of the figure as the rate of the reverse of the catalyzed reaction approaches the diffusion-limitedrate (kw = 1O1O M-l s-l was assumed). The two heavy horizontal lines in Figure 3 mark the that would produce significant range of values of (kI',) but not diffusion-limitedcatalytic currents in a convenient range of electrode rotation rates (as deduced from Figure 2). For values of rcatin the range 10-10-10-9 mol cme2 catalysts and substrates with self-exchange constants as low as 102-103 M-l s-l should prove useful for PE" 1 0.1 V. Particularly interesting is the prediction that reasonable catalysis should be possible with AEo values as negative as -200 to -250 mV, providing the catalyst and substrate exhibit sufficientlyhigh self-exchangerates. For example, if AE" = -250 mV with kgt = k& = lo7 M-l s-l and rmt = mol cm-2,the limiting current at a catalyst-coated electrode would approach one-half of the convection-limited Levich current at rotation rates as high as lo4 rpm. On the basis of Figure 3 it would appear that the proper balancing of driving force, concentration of attached catalyst, and self-exchange rates should allow a wide range of potential catalysts to be considered for any substrate whose electrooxidation (or -reduction) it is desired to catalyze. The few experimental results presently available to test the predictions resulting from this s t ~ d y ~ have i~~@ supported the basic notions. However, additional experimentation is clearly called for. It should be emphasized again that the analysis presented is strictly applicable only to simple, outer-sphere redox catalysis because it resta on the portion of Marcus theory derived for such cases. Examples of outer-sphere substrates that react slowly enough at clean, bare elec-
Conclusions By appealing to the Marcus correlations of rate constants for electronic self-exchange with those for homogeneous cross sections and for heterogeneous reaction at electrode surfaces, some aspects of the expected catalytic behavior of outer-sphere redox catalysts attached to the surfaces of electrodes have been exposed. In many cases it appears that the catalyzed currents will be limited by the rates of the cross reaction between catalyst and substrate before the rate of electron transfer between the electrode and the attached catalyst, or through the attached layer containing the catalyst, begin to limit the current. Significant catalysis appears to be possible even when the reversible potential of the catalyst is up to 200 mV more negative than that of the substrate so long as the self-exchange rates of the two reactants are sufficiently large. Some general guidelines for choosing among outer-sphere reactants for use as redox catalysts to be attached to electrode surfaces are one result of the analysis outlined here. Acknowledgment. This work was supported by the National Science Foundation and the US. Army Research Office. Numerous discussions with Dr. Kiyotaka Shigehara were helpful and enlightening. Much of the material in this paper was presented at the US.-France Symposium on Modified Electrodes (Bendor, France, June, 1980) where helpful criticisms were offered by several of those attending, especially Professors J. M. Saveant and A. J. Bard.
Raferences and Notes A. 6. Bocarsly, E. 0. Walton, and M. S. Wrlghton, J . Am. Chem. Soc., 102, 3390 (1980); N. S . Lewlg, A. B. Bocarsly, and M. S. Wrlghton, J . Phys. Chem., 842 2033 (1980). N. Oyama and F. C. Anson, Anal. Chem., 52, I192 (1980). N. Cyama and F. C. Anson, J. Am. Chem. Soc.,101,3450 (1979). N. Oyama and F. C. Anson, J. Ektrochem. Soc., 127,640 (1980). N. Oyama, T. Shimomura, K. Sh ehara, and F. C. Anson, J . Ekctroanal. chem., j ~ ~ ~ ~ p & R. J. Nowak, F. A. Schultr, M. Umafia, R. Lam, and R. W. Murray, AM/. Chem., 52, 315 (1980). H. 6. Abruiia, P. Denisevlch, T. J. Meyer, and R. W. Murray, J. Am. Chem. Soc., In press. C. P. Andrleux and J. M. Saveant, J. Nectroanal. Chem., 93, 163 (1978). C. P. Andrleux, J. M. Dumas-Bouchiat,and J. M. Saveant, J. Ekctroanal. Chem., 87, 39, 55 (1978); 88, 43 (1978). (a) R. A. Marcus, Discuss. Faraday Soc., 29, 21 (1960); (b) J . Chem. Phys., 43, 679 (1965). F. B. Kaufman and E. M. Engler, J. Am. Chem. Soc.,101, 547 (1979). P. Daum, J. R. Lenh@xlLD. Rollson, and R. W. Mway, J. Am. chem. Soc., 102, 4649 (1980). A. T. Hubbard and F. C. Anson In "Electroanalytical Chemistry", Vol. 4, A. J. Bard, Ed., Marcel Dekker, New York, 1970. K. Shlgehara, N. Oyama, and F. C. Anson, Inorg. Chem., In press. V. G. Levlch, "PhyslcochemlcalHydrodynamics", PrentlcaHall, Englewood Cllffs, NJ, 1962. K. Shlgehara and F. C. Anson, experiments In progress. M. J. Weaver and F. C. Anson, Inorg. Chem., 15, 1871 (1976); T. L. Satterberg and M. J. Weaver, J. Phys. Chem., 82, 1764 (1978). H. Gerlscher, MI.Bur. Stand(U.S.),Spec. Publ., No. 455 (1976).
