J. Phys. Chem. 1988, 92, 1011-1013
1011
Electron Hopping between Localized Sites. Coupling with Electroinactive Counterion Transport Jean-Michel Savdant Laboratoire d’Electrochimie MolPculaire de I’UniversitP Paris 7 , Unit2 AssociPe au CNRS No. 438, 2 place Jussieu, 75251 Paris Cedex 05, France (Received: October 28, 1987) Electron hopping between fixed sites is necessarily accompanied by electroinactive counterion displacement for maintaining electroneutrality. The various coupling modes of the two movements are discussed taking into account ion pairing between the fixed redox ions and the mobile electroinactive counterions. A square-scheme model and a concerted electron transfer-ion displacement model are developed allowing one to understand how the electroinactive counterion displacement may (partially or totally) control the rate charge transport in the context of an electron-hopping mechanism. The recent development of redox polymer electrode coatings’ has attracted active attention to the mechanism and kinetics of charge transport by electron hopping between fixed sites.’qz To maintain electroneutrality, the propagation of electrons is accompanied by the displacement of mobile electroinactive counterions. The question thus arises of the possibility of a mixed kinetic control of charge transport by electron hopping between redox sites and displacement of electroinactive counterions. Although this has been repeatedly invokedZ for explaining the variation of the charge transport rate with the electroinactive counterion, no satisfactory description of the physical nature of this mixed control has been given so far. The successive steps in approaching the problem have been the following. Ignoring ion-ion interactions (or, more precisely, assuming that they do not depend upon the redox state on the film), including ion pairing, the propagation of electrons under a chemical potential gradient by means of electron hopping has been shown to be formally equivalent to the “diffusion”of the immobile redox centers with the same diffusion coefficient for the ox and red states3 DE = kECoEb2 (kE,bimolecular isotopic electron transfer rate constant; CE, total concentration of redox sites; 6 , mean distance between two adjacent redox sites). Charge transport in many redox polymers has been analyzed along these lines.’q2 The effect of an electric field on the rate of electron hopping has been analyzed later on, under the same starting assumptions, coupling between the electron and electroinactive counterion movements being viewed as resulting solely from electroneutrality. This is formally equivalent to a “migration” of the immobile redox centers under the electric field.& However, the usual form of the Nernst-Planck law, applicable for physical migration of ions, is not valid in the present case but should be replaced by a second-order expression resulting from the bimolecular nature of electron hopping.4a For an A ne @ B reaction, the electron flux is then given by (x, distance; a, potential; C,, C,, concentrations):
+
( I ) For an overview, see Murray, R. W. In Electroanalytical Chemistry; Bard, A. J., Ed.; Dekker: New York, 1984; pp 191-368. ( 2 ) (a) Based on a large body of previous experimental data by various workers, a general discussion of the factors controlling charge transport in redox polymer films is given in ref 1, pp 334-339; see also ref 2b-h. (b) Buttry, D. A.; Anson, F. C. J. Am. Chem. SOC.1983, 105,685. (c) Majda, M.;Faulkner, L. R., J. Electroanal. Chem. 1984, 169, 77. (d) Ibid. 1984, 169,97. (e) Elliott, C.; Redepenning, J. G. Ibid. 1984, 181, 137. (f) Chen, X.;He, P.; Faulkner, L. P. Ibid. 1987, 222, 223. (g) Jernigan, J. C.; Murray, R. W. J . Phys. Chem. 1987, 91,2030. (h) Jernigan, J. C.; Murray, R. W. J. Am. Chem. SOC.1987, 109, 1138. ( 3 ) (a) Andrieux, C. P.; Saveant, J. M. J. Electroanal. Chem. 1980,111, 377. (b) Laviron, E. J. Electroanal. Chem. 1980, 112, 1. ( 4 ) (a) Savtant, J. M. J. Electroanal. Chem. 1986, 201, 211; 1987, 227, 299. (b) Savtant, J. M. J . Electroanal. Chem., in press. (c) Andrieux, C. P.; SavCant, J. M. J. Phys. Chem. submitted for publication.
0022-3654/88/2092-1011$01.50/0
SCHEME I
kC
It z-
+
----*
+
Z-
+
-
+
It Z‘
+
The s t e a d y - ~ t a t eand ~ ~ transient“ current responses may then be significantly different from those deriving from a classical Nernst-Planck treatment of migration. Common to both cases is however the fact that the steady-state response is independent of the electroinactive counterions’ mobility.” The latter are indeed macroscopically immobile in the sense that diffusion and migration exactly balance each other. Potential step transients do depend upon the mobility of the electroinactive counterion but not in the expected manner: the smaller the mobility the larger the response.“ Models based on simple electroneutrality coupling of the electrons and ions movement ignoring activity and ion-pairing effects thus do not fit the concept of a (partial or total) rate control of the charge transport by the electroinactive counterion displacement. Ion pairing, and possibly higher ion aggregation, is in fact expected to occur extensively in redox polymer coatings in view of the large ionic concentrations present and of the hydrophobicity of the polymer backbone.s The question of the influence of ion ( 5 ) (a) Eisenberg, A. Macromolecules 1970, 3, 147. (b) Eisenberg, A.;
King, M. Ion-Containing Polymers; Academic: New York, 1977. (c) Komozovski, R. A,; Mauritz, K. A. In Perfluorinated Ionomer Membranes; Eisenberg, A., Yeager, H. L., Eds.;American Chemical Society: Washington, DC, 1982; ACS Symp. Ser. No.180, pp 113-138 and references cited therein. (d) In polymers where the redox centers are electrostatically bound to a polymeric polyelectrolyte, strong ion pairing is the very basis of their immobilization.
0 1988 American Chemical Society
1012 The Journal of Physical Chemistry, Vol. 92, No. 5, 1988
pairing between the localized redox centers and the mobile electroinactive counterions on the dynamics of charge transport has recently been addressed6 in the framework of the mechanism depicted in the lower half (starting from A Z and going downward) of Scheme I, where the ion pairs (AZ) are immobile and do not participate in the charge transport, while electron hopping involves the free immobile ions (A’) and where the electroinactive counterions (Z-) are physically displaceable. Under these conditions not only transient responses but also steady-state responses depend upon the nature of the counterion: the current response rapidly decreases as the ion-pairing equilibrium constant increases? When, in this context, the charge-transport rate tends to become low as a consequense of extensive ion pairing, the question arises of a relay by another charge-transport mechanism involving the direct participation of the ion-paired redox centers and thus a more intimate coupling of electron hopping and electroinactive counterion displacement than considered before. The discussion of this point is the aim of the present note. The ion-paired immobile redox centers may participate in the charge transport (upper part of Scheme I) according to the following square scheme mechanism AZ,
Letters
Figure 1. Potential energy profiles.
developed for reactions in which electron transfer is concerted with the dissociation of a covalent bond.* In the present case, whatever the exact form of the potential energy curves, they are symmetrical around the middle of the two reactants when they are at their reacting distance as shown in Figure 1. The free energies of the system in states AZ1 Bz and B, AZz are respectively
+
Kl
1 2
Gp
BZj-
+
A;
i -
B1
+
AZ2
k-I
(the subscripts 1 and 2 symbolize the location of the corresponding species in two adjacent sites) where electron hopping occurs along the vertical reactions and electroinactive counterion displacement along the horizontal reactions. AI+,BZC, BZl-, Az+are produced by strongly uphill reactions and can therefore be considered as obeying the steady-state assumption. Furthermore, the A I f BZC and BZ,- A2+ states of the system have the same free energy. Thus, k l / k - , = kE/k-E = K . Under these conditions, it immediately appears that d[AZl]/dt = d[Bz]/dt = -d[AZ,]/dt = -d[Bl]/dt = kC([AZZl P I 1 - [AZII P21)
+
+
+
GR = GoR+ R(r) + C - ( x I - x , , ] ) ~
k-I + B2 e A; + BZ,
kx
1
IPotential Energy
=
+ P(r) + C -Kl( x ,
cop
1 2
- x1,2)2
The second terms represent the dissociation of the A Z ion pair. Since the two potential energy curves are symmetrical around 6/2 R(6-r) = P ( r ) The third terms represent the outer- and inner-sphere reorganization (the Kl’s are the force constants, assuming as usual that the antisymmetric components of the inner-sphere force constants are negligible’). Thus at the transition state ( r = r’; x , = x l * ) R ( r * )- P ( r * )
+ C KlT ( X ~- *X J
- ( x l *- x , , ~ ) ’= O
I
the activation free energy being for this zero driving force reaction: Kl Kl AGO*= R ( r * ) + z ~ ( x -~x,,])’ * = P ( r * ) + z y ( x 1 *- x , , ~ ) ’
with (l/kC) = (l/k-I) + (l/k-E) Charge transport is thus formally equivalent to a diffusion of the immobile redox centers, A Z and B, having the same diffusion coefficient? DC = [ ( k - ~ k - ~ ) / (+k k-I)]cE -~ 62
I
Minimizing the activation free energy leads to r* = 6 / 2 and thus to
(2)
The above relationships are typical of a mixed kinetic control of and electroinactive charge transport by electron hopping ((kE) and of how one of them will precounterion displacement (k1) dominate over the other (LE >> k-,, control by counterion displacement; kI>> k+ control by electron hopping). The above-described model assumes that the two elementary steps of the charge-transport process are sequential, involving the passage of the system through energetically quite unfavorable states (Al+ BZz-) or (BZ1- A2+). This is the reason why a process where electron hopping and counterion displacement are concerted rather than sequential (as represented by the dotted arrows in the above scheme) may prove energetically more favorable. This can be modeled as followed by a modification of Marcus theory for outer-sphere electron transfers’ that would take into account as accompanying electron transfer not only solvent reorganization and changes in bond lengths and angles but also the dissociation of an ion pair. A similar theory has been recently
+
I
+
(6) SavQnt, J. M. J . Phys. Chem., in press. (7) (a) Marcus, R. A. Annu. Reu. Phys. Chem. 1964, IS, 155. (b) Marcus, R. A. J . Chem. Phys. 1965,43, 679. (c) Waisman, E.; Worry, G.; Marcus, R. A. J. Electroanal. Chem. 1977,82,9. (d) Marcus, R. A. Faraday Discuss. Chem. Soc. 1982, 74, 7. (e) Marcus, R. A,; Sutin, N. Biochim. Biophys. Acta 1985, 811, 265.
The first term represents the contribution of the electroinactive counterion displacement and the second, the contribution of solvent and inner-sphere reorganizations to electron hopping. Although the model should be improved to permit a precise quantitative confrontation with experimental data (analysis of preexponential factors, estimation of the ion-pair dissociation energetics), the above relationship allows one to understand how the electroinactive counterion displacement may (partially or totally) control the rate of charge transport in the context of an electron-hopping mechanism. We have taken the example of a positively charged ion-paired redox ion in its oxidized form and of a mobile negative electroinactive counterion. The model is immediately transposable to the reverse case of a negatively charged ion-paired redox ion in its oxidized form and of a mobile positive electroinactive counterion. Transport of electrons under a chemical potential gradient will thus obey the Fick laws with a diffusion coefficient deriving either from eq 2 or from the activation free energy defined in eq 3. As (8) Saveant, J. M . J . Am. Chem. SOC.1987, 109, 6788.
J . Phys. Chem. 1988, 92, 1013-1016 aAZ -at
regards electron transport in an electric field the same second-order rate law as previously established for simple electron hopping will apply.9 The overall rate law in terms of fluxes is thus the same as is eq 1 where A is replaced by AZ and DE by Dc as derived from eq 2 or 3. In electrochemical applications where a chemical potential gradient is created by the reduction of AZ or the oxidation of B at an electrode surface, the electroinactive counterion Z- must be transported through the redox polymer film (from the electrode toward the solution in the first case and vice versa in the second). It follows that the charge-transport mechanism then follows the overall foregoing reaction scheme where electron hopping between A+ and B sites and physical displacement of Z - coexist with electron-ion hopping between AZ and B sites. The rate laws are thus given by the following relationships (by analogy with previous treatments6)
1013
-
aA _ at
introducing three diffusion coefficients: Dc = kCAX2CoE, DE = kEhXZcE,and QI = kIAx2(the definitions of the various symbols are shown in Scheme I).
(9) Since the potential energy curves are symmetrical and in the case the electrical potential variation between two adjacent redox sites are small, the electron-ion transfer coefficient (i.e., the symmetry factor) can be considered as close to 0.5 as required by eq 1.4‘ For large spatial variations of the electrical potential the linearization implied in eq 1 may not remain valid and the full exponential variation of the electron-ion transfer kinetics with potential may have to be considered.
Acknowledgment. Discussions with C. P. Andrieux (UniversitE Paris 7) on the matter of the present paper were, as always, very helpful.
Preparation of Ultrafine Amorphous Fe,, C, Alloy Particles on a Carbon Support Jacques van Wonterghem and Steen Marup* Laboratory of Applied Physics II, Technical University of Denmark, DK- 2800 Lyngby. Denmark (Received: October 29, 1987)
Iron pentacarbonyl has been thermally decomposed on a carbon support at 353 K. By use of Mossbauer spectroscopy it is shown that the reaction leads to formation of ultrafine amorphous Fq8C22 alloy particles with a mean diameter of about 3.9 nm. The particles exhibit a superparamagnetic behavior at 80 K.
Introduction Thermal decomposition of iron carbonyls on various supports has been studied e~tensively.l-~The aim of most of the work has been to form ultrafine metallic particles which may have interesting catalytic properties. Recently, we have studied the thermal decomposition of iron pentacarbonyl in organic liquids with a view to prepare ferrofluids containing metallic particles with a large It was found that the decom(1) Phillips, J.; Clausen, B. S.;Dumesic, J. A. J. Phys. Chem. 1980, 84, 1814. (2) Phillips, J.; Dumesic, J. A. Appl. Surf. Sci. 1981, 7 , 215. (3) Lazar, K.;Matusek, K.;Mink, J.; D o h , S.;Guczi, L.; Vizi-Orosz, A.; Marko, L.; Reiff, W. M. J. Catal. 1984, 87, 163. (4) Schay, Z.; Lazar, K.;Mink, J.; Guczi, L. J . Catal. 1984, 87, 179. ( 5 ) Rojas, D.;Bussiere, P.;Dalmon, J. A.; Choplin, A.; Basset, J. M.; Olivier, D. Surf. Sei. 1985, 156, 516. (6) Trautwein, A. X.;Bill, E.;Bl&, R.; Doppler, G.; Seel, F.; Klein, R.; Gonser, U.Surf. Sci. 1985, 156, 140. (7) Doppler, G.; Bill, E.;Gonser, U.; Seel, F.;Trautwein, A. X . Hyperfine Interact. 1986, 29, 1307. (8) van Wonterghem, J.; Merup, S.;Charles, S. W.; Wells, S.;Villadsen, J. Phys. Rev.Lett. 1985, 55, 410. (9) van Wonterghem, J.; Msrup, S.; Charles, S. W.; Wells, S.; Villadsen, J. Hyperfine Interact. 1986, 27, 333. (10) Msrup, S.;Christensen, B. R.;van Wonterghem, J.; Madsen, M. B.; Charles, S . W.; Wells, S . J. Magn. Magn. Mater. 1987, 67, 249. (1 1) van Wonterghem, J.; Msrup, S.; Charles, S.W.; Wells, S.J. Colloid Interface Sci., in press.
0022-3654/88/2092-lO13$01 S O / O
position of the iron pentacarbonyl led to formation of amorphous Fel,CX alloy particles. We have also shown that amorphous particles of Fe-B, Fe-Co-B, and Fe-Ni-B alloys can be formed by chemical reactions at a low temperature.12 Amorphous alloys are normally prepared as thin ribbons or films by the liquid quench technique or by vapor deposition. In order to avoid crystallization during the preparation, the material must be cooled rapidly to a temperature below the glass transition temperature. However, when an alloy is formed by a chemical method at a low temperature, the material may be amorphous if the chemical reaction takes place below the glass transition temperature of the alloy.8~12 In this Letter we show that ultrafine amorphous Fel& particles can be prepared on a carbon support by thermal decomposition of Fe(CO),. Such supported particles are of great interest because they can be used for studies of surface and catalytic properties of amorphous alloys.
Experimental Section The sample was prepared in a sample holder consisting of a copper tube closed in both ends with heat resistant plastic film (Kapton). A hole was drilled in the copper tube through which the sample material could be injected. Before impregnation, the ~
~
~~~
~~~
~~
~
(12) van Wonterghem, J.; Msrup, S.; Koch, C. J. W.; Charles, S.W.; Wells, S.Nature (London) 1986, 322, 622.
0 1988 American Chemical Society
