J. Phys. Chem. 1983, 87, 214-219
214
ARTICLES Self-Exchange Reactions at Redox Polymer Electrodes. A Kinetic Model and Theory for Stationary Voltammetrlc Techniques Fred C. Anson,'
Jean-Mlchel Saveant,*$ and Klyotaka Shlgeharati
Arthur Amos Noyes Laboratorles, Dlvision of Chemisby and Chemical Engineering, Caiifornle Institute of Technology, Pasadena, California 9 1125, and Laboratoire d'Electrochimie, Universite de Paris 7, 7525 1 Paris, Cedex 05, France (Received: September 24, 1982)
A theoretical analysis of the charge transfer kinetics for systems in which the same redox couple is attached to a polymeric coating on an electrode surface and is dissolved in the supporting electrolyte solution is presented for the case of voltammetry at a rotating disk electrode. Two types of exchange reactions between the bound and unbound reactants present in the polymer film are identified electron-exchangeand place-exchange. The variations of the limiting disk currents with the speed of rotation of the electrode give rise to linear Koutecky-Levich plots in all circumstances. As a result, the separation of the effects on the current produced by the diffusion of the reactants in the solution from those associated with kinetic phenomena occurring inside the coating is straightforward. A closed-form expression is derived for the limiting current at the rotating disk electrode in terms of three characteristic currents that measure the diffusion rate of the unbound reactant, the rate of diffusion-like transport of charge along the polymer chains by the attached reactant, and the rate of the exchange reactions. Expressions are also given for the steady-state concentration profiles that develop in the coating. Procedures are given for using the derived expressions to determine rate constants of exchange reactions that proceed within the polymeric coatings and several limiting situations are described that may be exploited to facilitate the analysis of data.
The kinetics of electrochemical reactions mediated by redox polymer films have been considered in a number of studies in recent years.'-" Initially, attention was focused on the electron-exchange reaction between the substrate dissolved in solution and the active form of the mediator incorporated in the polymer film as the current-limiting factor (after mass transfer of the substrate from the bulk of the solution to the film-solution interface had been accounted for). Well-known theories of stationary voltammetric techniques12J3 were adapted for this purpose.2~6~8J1J4J5 The corresponding theory for the case of linear sweep voltammetry has also been worked out' with explicit consideration of the diffusion of the substrate from the film-solution interface to the electrode surface as a possible rate-limiting factor. An expression for the thickness of the reaction layer for the exchange reaction in this context was also derived.' Somewhat later, the diffusion-like transport of charge between the electrodefilm and film-solution interfaces was also taken into account as a possible current-controlling It is now well recognized that these three factors (i.e., the kinetics of electron-exchange reactions, substrate diffusion, and charge transport) may act simultaneously to determine the currents observed at electrodes coated with redox polym e r ~ . ~ 'However, ~ ~ ~ , the ~ ~only , ~ existing ~ theory that accounts rigorously for the interplay of the three currentlimiting factors along with the diffusion of the substrate in the solution4J0is limited to the case of an irreversible cross-exchange reaction: 'Arthur Amos Noyes Laboratories.
* Laboratoire d'Electrochimie.
Present address: Department of Polymer Chemistry, Waseda University, 3-4-1Ohkubo, Shinjuku-ku, Tokyo 160, Japan. # Contribution No. 6676.
P+e+Q
Q + A +P
+ product
(2)
where P and Q are the oxidized and reduced half, respectively, of the redox couple present in the polymer coating and A is the substrate that reacts with the reduced form of the redox couple in an irreversible cross-reaction. In other recent approaches to the p r ~ b l e m ~ ,qualitative '-~ or semiquantitative estimates of the role of the various rate-limiting factors have been offered but no general equations were derived that are reliably applicable to all the situations likely to be encountered in steady-state experiments with rotating disk electrodes. For the case of cyclic voltammetry, a direct digital simulation approach (1)Andrieux, C. P.; Saveant, J. M. J.Electroanal. Chem. 1978,93,163. (2)Oyama, N.;Anson, F. C. Anal. Chem. 1980,52,1192. (3) Daum, P.; Lenhard, J. R.; Rolison, D.; Murray, R. W. J . Am. Chem. Soc. 1980,102,4649. (4) Andrieux, C. P.; Dumas-Bouchiat, J. M.; Saveant, J. M. J . Electroanal. Chem. 1980,114,159. (5)Anson, F. C. J. Phys. Chem. 1980,84, 3336. (6) Shigehara, K.;Oyama, N.; Anson, F. C. Inorg. Chem. 1981,20,518. (7) Daum, P.; Murray, R. W. J. Phys. Chem. 1981,85,389. (8)Rocklin, R. D.; Murray, R. W. J . Phys. Chem. 1981,85, 2104. London 1981,302,237. (9)Murray, R. W.Philos. Trans. R. SOC. (10)Andrieux, C. P.; Dumas-Bouchiat, J. M.; Saveant, J. M. J . E k c troanal. Chem. 1982,131, 1. (11)Rubinstein, I.; Bard, A. J. J . Am. Chem. SOC.1981,103,5007. (12)Levich. V.G.'Phvsicochemical Hvdrodvnamics": Prentice-Hall: - Englewood Cliffs, NJ, 1962. (13)Koutecky, J.; Levich, V. G. Zh.Fiz. Khim. 1956,32,1565. (14)Albery, W.J.; Foulds, A. W.; Hall, K. J.; Hillman, A. R. J . Electrochem. SOC. 1980,127, 654. Albery, W.J.; Bowen, W. R.; Fischer, F. S.; Foulds, A. W.; Hall, K. J.; Hillman, A. R.; Egdall, R. G.; Orchard, A. F. J . Electroanal. Chem. 1980,107,37. (15)Andrieux, C.P.; Dumas-Bouchiat, J. M.; Saveant, J. M. J . Electroanal. Chem. 1981,123,171.
0022-3654/83/2087-0214$01.50/0@ 1983 American Chemical Society
The Journal of Physical Chemistry, Vol. 87, No. 2, 1983 215
Self-Exchange Reactions at Redox Polymer Electrodes
for the calculation of current-potential curves has been developed.ll Recently the theory for the case of an irreversible cross-reaction proceeding at a coated rotating disk electrode4J0was reformulated in terms of four characteristic currents corresponding to the four possible rate-limiting phenomena.16 This allowed most of the heavy algebra in the earlier treatments to be eliminated and produced more transparent final equations that are easier to use. In this report the same approach is applied to the case of a reversible “self-exchange” reaction in which the substrate redox couple in solution and the attached redox couple are the same.6 The analysis is given for a reduction reaction but transposition to an oxidation reaction is straightforward. The reaction scheme consists of electrode reaction 1followed by chemical reaction 3, where A and B represent
A+Q+B+P
to the polymer reaction 3 is accompanied by the movement of electroinactive ions from the neighboring solution phase onto or off of the polyelectrolyte or polymer chain to maintain electroneutrality. As with covalently attached reactants, true self-exchange, (i.e., K,, = l),requires that the A/B and p/&couples have the same formal potentials, so that the complete set of ions involved in reaction 3 (including the counterions) must experience small or similar interactions with their environment when bound to the polyelectrolyte or polymer chains and when dissolved in the solution surrounding the chains. It should also be recognized that there are exchange reactions that do not involve the exchange of electrons. Instead, a physical exchange of positions by pairs of attached and unattached reactants may occur, either at the same oxidation level, as in reactions 4 and 5
(3)
the oxidized and reduced forms of the redox couple in solution and P and Q represent the same two species when they are attached to the polymeric coating. The self-exchange reaction (reaction 3) is assumed to have an equilibrium constant, K,,, of unity. By measuring the formal potentials of the redox couples in the attached and unattached sites, one can determine whether this condition is fulfilled. The simple reaction scheme comprising reactions 1 and 3 is likely to be observed with redox centers that are covalently attached to the polymeric coating. For example, electrodes coated with p~ly(vinylferrocene)~J~-~~ or poly( p - n i t r o ~ t y r e n e ) ~would ~ - ~ ~ be anticipated to reduce ethylferrocene or p-nitroethylbenzene, respectively, according to this scheme. In such cases equality of the formal potentials for the A/B and P/Q couple (leading to K,, = 1)implies that the attachment of A and B to the coating results in small or similar alterations in the interactions (solvation,ion pairing, etc.) that they experience with their surroundings. If the formal potentials of the bound and unbound reactant couple are not equal, the situation becomes equivalent to one in which the two reactants undergo a cross-reaction. The steady-state electrochemical responses to be expected in that case have been previously presented for irreversible cross-reactions1°and an analysis of reversible cross-reactions will soon become availableeZ7 With systems involving metal ion complexes that are electrostatically bound to polyelectrolyte chains in the electrode c ~ a t i n g ~ vor~coordinated J ~ , ~ ~ to ligands attached (16)Andrieux, C.P.; Saveant, J. M. J.Electroanal. Chem. 1982,134, 163. (17)Merz, A.; Bard, A. J. J. Am. Chem. Soc. 1978,100,3222. (18)Peerce, P. J.; Bard, A. J. J. Electroanal. Chem. 1980,108,121. (19)Peerce, P. J.; Bard, A. J. J. Electroanal. Chem. 1980,112,97. (20)Peerce, P. J.; Bard, A. J. J. Electround Chem. 1980,114,89. (21)Dautartas, M.; Evans, J. F. J.Electroanal. Chem. 1980,103,301. (22)D a w , P.; Murray, R. W. J. Phys. Chem. 1981,85,389. (23)Rolison, D. R.; Umana, M.; Burgmayer, P.; Murray, R. W. Inorg. Chem. 1981,20,2996. (24)Kerr. J. B.: Miller. L. L. J. Electround. Chem. 1979. 101. 263. (25)Kerr; J. B.iMiller,L. L.; Van de Mark,M.R. J.Am. h e m : SOC. 1980,102,3383. (26)Andrieus, C. P.;Saveant, J. M. J.Electround Chem. 1980,111, 377. (27)Andrieux, C.P.;Saveant, J. M. J. Electroanal. Chem. 1982,142, 1. (28)Oyama, N.;Sato, K.; Matsuda, N. J. Electroanal. Chem. 1980, 115,149. (29)Facci, J.; Murray, R. W. J. Electround Chem. 1981,126,339. (30)Facci, J.; Murray, R. W. J. Phys. Chem. 1981,85,2870. (31)Rubinstein, I.; Bard, A. J. J. Am. Chem. SOC.1980,102,6041. (32)Buttry, D.; Anson, F. C. J. Electroanal. Chem. 1981,130,333. (33)Bruce, J. A.; Wrighton, M. S. J. Am. Chem. SOC.1982,104,75.
or at different oxidation levels, as in reaction 3 which may involve both place-exchange and electron-exchange components. Both types of physical exchange involve the same counterion displacements to maintain electroneutrality separately in the polymer and solution phases. The assumption that K,, = 1 is equivalent to assuming that the attachment equilibrium constants for reactions 4 and 5 are the same for both A and B. This may often not be the case because A and B usually bear different charges but in this analysis we will pursue the consequences of this assumption. In what follows we examine the particular case of ionic reactants that are bound electrostatically to oppositely charged sites in polyelectrolyte coatings. However, extension to reactants that are bound by labile coordination to ligand sites on uncharged polymers would be straightforward. We will regard reactions 4 and 5 as simple first-order reactions in both directions. This implies that the quantities of reactants bound electrostatically to the polyelectrolyte chains are far enough below their saturation values to correspond, at least approximately, to a Henry’s law type of binding isotherm. For more highly loaded coatings the problem could still be treated by employing an appropriate isotherm but this additional complication has been avoided in this first approach to the problem.
Formal Statement of the Kinetic Problem and Its Solution The problem is to calculate the steady-state disk current (at the plateau of the wave) that will result when the reaction scheme is that given by the combination of reactions 1 and 3-5. The experimental variables are the following: CA,CB, Cp, and CQare the concentrations of the indicated species; x is the distance from the electrode surface; D is the diffusion coefficient of A and B in solution; and 6 is the thickness of the diffusion layer12formed at the surface (34)Mortimer, R. J.; Anson, F. C. J. Electround. Chem. 1982, 138, 325. (35)Oyama, N.;Anson, F. C. J. Am. Chem. SOC.1979,101, 3650. (36)Oyama, N.;Anson, F. C. J. Electrochem. SOC.1980,127,640. (37)Scott, N.S.;Oyama, N.; Anson, F. C. J. Electroanal. Chem. 1980, 110,303. (38)Shigehara, K.; Oyama, N.; Anson, F. C. J.Am. Chem. SOC.1981, 103,2552. (39)Haas, 0.; Vos, J. G. J.Electroanal. Chem. 1980,113,139. Kruus, M.; Vos, J. G. J.Am. Chem. SOC.1981,193,1318. (40)Haas, 0.; (41)Abruna, H. D.;Denisevich, P.; Umana, M.; Meyer, T. J.; Murray, R. W. J. Am. Chem. SOC.1981,103,1.
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Anson et al.
The Journal of Physical Chemistry, Vol. 87,No. 2, 1983
of the rotated electrode commencing at the film-solution Ds(dCA/dx),=*_ = D(dCA/dX),=*+ (13) interface (6 = 4.98D1/3u1/6w-1/2; 6 in centimeters, D and u (CA),=*-= K(CA)~=O+ in cm2s-l, w in rpm). The diffusion coefficients of A and B in the film are assumed to have the same value, Ds, as Ds(dCB/dx),=*_ = D(dCB/dx),=*+ (14) are the "diffusion coefficients" of P and Q, DE. The latter (CB)~=*= K(CB),=@+ coefficient characterizes a diffusion-like transport of charge along the polyelectrolyte chains that may involve both The limiting current density is given by electron hopping and ion displacement. The ion disilim = J'[Ds(dCA/dx),=o + D~(dCp/dx)x=o] (15) placement may involve solution-like diffusion or ion hopthat are covalently attached to the coatings, ping from site to site as has been discussed r e ~ e n t l y . ~ ~For ~ ~reactants ~ kl = k2 = 0 and linear combination and integration of eq K is the coefficient that governs the partitioning of A and B between the bulk solution and the solution occupying 6 9 leads to CA + CB = KCAOand Cp + C, = Cpo inside the film. The same is true, although less obvious, for electhe volume between the polyelectrolyte chains inside the trostatically or coordinatively attached reactants where k1 coating. If CAois the concentration of A in the bulk of the and k 2 may be different from zero. This assertion is solution, KCAOwill be its initial equilibrium concentration demonstrated in Appendix I. in the film. The initial equilibrium concentration of P in It is convenient at this stage to introduce four characthe film, Cpo, is rPo/9, where rPis the total quantity of teristic current densities that are measures of the four reactant bound to the polyelectrolyte chains and 9 is the possible rate-limiting factors (the symbolism used in the film thickness. Writing Cpo = Fpo/9 involves the apfollowing is essentially the same as that in the previous proximation that the properties of the coating are uniform publications1°J6with the exception that we now use i to in the direction perpendicular to the electrode surface. In represent the current density instead of the current inthe case of electrostatic binding of ions to swollen polyetensity): lectrolytes the actual concentration of the reactant in the vicinity of the fixed charge sites, Cp*, may be larger than A i = FCA'D/6 (16) Cpo: Cpo/Cp* y where y is the fraction of the total volume of the film occupied by the polyelectrolyte moleiA is the solution Levich current density12that corresponds cules themselves. However, the assumption of a uniform to the limiting current that would be obtained at a bare macroscopic coating composition allows Cpoto be obtained electrode. by averaging the actual reactant concentrations over the is = FCAo~Ds/9 (17) whole area of a section of the film parallel to the electrode surface. is is the current density arising from the diffusion of the Since we have assumed (vide supra) that the reductions unattached reactant through the coating. of A and P take place a t potentials that do not differ (18) iE = FCpoDE/9 = Frp0DE/@' significantly, only a single wave that encompasses both reductions is to be expected. We are thus seeking the iE is the current arising from the diffusion-like transport limiting current density, ih, on the plateau of the current of the bound reactant along the polymer chains. potential curve that corresponds to the sum of both reik = F(kCpo kl)KcAo@ = F(kKCA' k2)cpo@= ductions. If we use the Nernst diffusion layer approxiF(~KCA'+ k,)rpo(19) mation, the stationary concentrations of the four reactants in the film and in the solution obey the following set of ik is the current that measures the rate of the exchange equations and boundary conditions: reactions. In the film, i.e., for 0 < x < 9 We also introduce the following dimensionless variables: Ds(d2CA/dX2)- k1CA + k&p - ~ ( C A C-QC ~ c p = ) 0 (6) y = x/9 a = CA/KCAO b = CB/KCB' k & ~ + ~ ( C A C Q CBCP) = 0 Ds(d2CB/dX2)- k1cB p = Cp/cp' q = CQ/Cpo (7) A dimensionless formulation of the problem can then be DE(d2Cp/dX2)+ klc~ - kzCp + k(cAcQ - C ~ c p =) 0 written involving two differential equations and two sets of boundary conditions that describe the phenomena (8) taking place inside the film DE(d2CQ/dX2)+ k 1 C ~- k & ~- ~ ( C A C-QCBCP)= 0 (9) d2a/dy2- ( i k / i s ) ( a- p) = 0 (20) In the solution, i.e., for 9 < x < 9 + 6 d2p/dy2 + (ik/iE)(a- p ) = 0 (21) cp = CQ = 0 y=o a=O p=o (22) dCA/dx = [CA' - (CA),=*+]/~ (10) y =1 dp/dy = 0 1 - u = (is/iA)(da/dy) (23) dCB/dx = (Cdx=*+/6
-
+
At the electrode surface, Le., for x = 0 CA = cp = 0 dCA/dX + dCB/dX = 0 dCp/dx + dCQ/dx = 0 A t the film-solution interface, i.e., for x = CP (dCp/ dx = *- = (dCQ/ dx ), =*.= 0
+
The dimensionless current on the plateau is given by ilim/iA = (1 - a),=1 = (is/iA)(da/dy),=l (24) (11)
(12)
(42) Anson, F. C.; Saveant, J. M.; Shigehara, K. J . Am. Chem. SOC., in press.
Integration of this system of equations (Appendix 11)leads to the following expression for the limiting current: l/ilim = l / i * + 1/iF (25) with
The Journal of Physical Chemistry, Vol. 87, No. 2, 1983 217
Self-Exchange Reactions at Redox Polymer Electrodes
The corresponding concentration profiles are given by a=
( :: y+,x
sinh
[ (; + 4)'2y]
+7 LE ik ) I 2
cash
i
f 1s k
+7 1E ik )'2]))/ log (
+:
E
)
Flgure 1. Variation of the ratio of the "film current", i,, to the "penetration current", is, with the two "competition parameters", ik/is i k / i E and iE/is (see text) as calculated from eq 26.
+
0
sinh
tanh
0
"R + S"
b=l-a q = l - p (29) The departure of the exchange reaction from equilibrium is measured in dimensionless form by a - p which can be calculated from eq 27 and 28. Film Current and Concentration Profiles A first important point that follows from eq 25 and 26 is that Koutecky-Levich plots6J2-'*constructed from the variations of the limiting current with the electrode rotation rate are linear in all circumstances. This contrasts with the case of cross-exchange reactions where nonlinear Koutecky-Levich plots may r e s ~ l t . ' ~ JThe ~ ~ ~"film ~ current", iF, derived from the intercept of the KouteckyLevich plots, reflects the competition between the three rate-limiting processes in the film as measured by the three characteristic currents, ik, is, and iE. While the expression for iF,eq 26, involves these three currents, the competition between the rate-limiting processes actually depends upon two current ratios: ik/iS + ik/iEand iE/iS. Important limiting situations are found when these two competition parameters take extreme values. This can be seen in Figure 1, which shows how the ratio iF/iS varies as a function of the two competition parameters. In Figures 2 and 3 are shown the corresponding concentration profiles and the departure of the exchange reaction from equilibrium inside the film. The letters R, S, and E are used, as they were in the case of irreversible10J6 and reversible2Icross-reactions, to designate which of the possible rate-limiting phenomena control the film current under the various circumstances. R designates control by the rate of the self-exchange reaction; S designates control by the diffusion of A through the film; and E designates control by the diffusion-like transport of P along the
"
R t 5 + E''
I
R+E'
3
"SR"
' ' ( S +E I R '
I
E9
'p
3
'E :
"S + E " a)
I
10
's
I
S
0
Figure 2. Dlmensioniess concentration profiles of A (a = C A / ~ C A o ) and Q (9 = C o / C $ ) inside the fllm for varlous values of the two competition parameters i E / i sand ( / k / i s / k / / ~ ) " ' . The abscissa in each diagram Is the dlmensbnless distance from the electrode surface: xla. The curves were calculated for i s / i A = 0.25.
+
-
polymer chains within the film.1°J6 When ik/is + &/iE 0, i.e., when the exchange reactions are slow compared to the diffusion of A and the transport of P, the limiting situation designated by R is obtained whatever the value of the second parameter, iE/iS (Figure 1). In this case iF = is; that is, the film current is the same as it would be for the reduction of A by its penetration through an inert film with the same diffusion coefficient, Ds.CQ remains equal to Cpo (Le., q = 1in Figure 2) and Cp = 0 throughout the film, indicating that the contributions from the exchange reactions are kinetically negligible. (Nevertheless, we retain the designation R because, as the concentration profiles in Figure 2 make clear, the situation is closest to that identified in previous work as R and differs substantially from that labeled S.) The departure from equilibrium of the exchange reaction is then maximal at all points in the film (Figure 3, case R). In other words,
218
Anson et al.
The Journal of Physical Chemistty, Vol. 87, No. 2, 1983
parent the larger the ratio of iE to is. When ik/iS+ ik/iE becomes large enough, a mutual compensation of the exchange reactions and the two diffusional processes ensues, producing exponentially decaying concentration profiles in a thin reaction layer adjacent to the film-solution interface (Figure 2, cases SR, S E). The exchange reactions remain in equilibrium throughout the film except for this thin reaction layer (Figure 3). In the expression for iF (eq 26) this situation corresponds to the term involving the hyperbolic tangent becoming equal to 1. If, at the same time, iE >> is, the expression for iFassumes the simple form given in eq 30 which corresponds to the oblique straight line labeled SR in Figure 1.
+
7 -
iF = (ikiS)1/2
(30)
When the penetration of the solution reactant into film is very small and/or its reaction rate with the attached reactant is very large, the layer within which the reaction occurs tends toward a single monolayer. Under these conditions the film current density is given by eq 31 l/iF = l/iE l/ik (31)
+
with ik = F(kMCAo
Flgure 3. Departure of the exchange reaction from equilibrium as expressed by a - p as a function of the distance from the electrode surface for various values of the two competition parameters, and ( i k / i s h/iE)l’z. The ordinate in each diagram is a - p calculated from eq 27 and 28. The abscissa is the dimensionless distance from the electrode surface: x / @ . The curves were calculated for i s / i A = 0.25.
+
there is no coupling between the diffusion of A through the solution within the coating and the transport of P along the polyelectrolyte chains. In this case, under steady-state conditions, no charge is propagated across the coating by P because it cannot be regenerated by the electron- or place-exchange reactions within the time scale of the experiment. Conversely, when ik/iS ik/iE m, the exchange reactions are so fast compared to the two diffusional processes that coupling between the diffusion of A and the transport of P is complete. The two diffusional processes thus jointly limit the rate, giving rise to an S + E situation (Figure 3, case S + E). The film current is then simply the sum of the two diffusional currents: iF = iE + is. If iE
