19364
J. Phys. Chem. B 2006, 110, 19364-19368
Thermodynamic Aspects of Ion Intercalation in KhFek[Fe(CN)6]l‚mH2O Compounds: Application to the Everit’s Salt/Prussian Blue Transition D. Gime´ nez-Romero,† P. R. Bueno,‡ J. J. Garcı´a-Jaren˜ o,§ C. Gabrielli,*,† H. Perrot,† and F. Vicente§ UPR 15 du CNRS, Laboratoire Interfaces et Syste` mes Electrochimiques, UniVersite´ Pierre et Marie Curie, 4 place Jussieu, 75252 Paris, France, Instituto de Quı´mica, Departamento de Fı´sico-Quı´mica, UniVersidade Estadual Paulista, P.O. Box 355, 14801-907, Araraquara, Sa˜ o Paulo, Brazil, and Departament de Quı´mica Fı´sica, UniVersitat de Vale` ncia, C/Dr Moliner, 50, 46100, Burjassot, Vale` ncia, Spain ReceiVed: March 13, 2006; In Final Form: July 10, 2006
The K+ reversible processes for ion exchange in KhFek[Fe(CN)6]l‚mH2O host compounds (Prussian Blue) were thermodynamically analyzed. A thermodynamic approach was established and developed based on the consideration of a lattice-gas model where the electronic contribution to the chemical potential is neglected and the ion-host interaction is not considered. The occupation fraction of the intercalation process was calculated from the kinetic parameters obtained through ac-electrogravimetry in a previous paper. In this way, the mass potential transfer function introduces a new way to evaluate the thermodynamic aspect of intercalation. Finally, based on the thermodynamic approach, the energy used to put each K+ ion into the host material was calculated. The values were shown to be in good agreement with the values obtained through transient techniques, for example, cyclic voltammetry. As a result, this agreement between theory and experimental data validates the thermodynamic approach considered here, and for the first time, the thermodynamic aspects of insertion were considered for mixed valence materials.
Introduction It has been previously demonstrated that the behavior of the insertion reaction mechanism in transition metal hexacyanoferrates of the general formula KhFek[Fe(CN)6]l‚mH2O (Prussian Blue, denoted as PB) is composed of mainly three different charge modes involving three different ionic sites inside the crystalline structure.1 The processes involve three different guests that are exchanged with the host material (KhFek[Fe(CN)6]l‚mH2O): H3O+, K+, and H+. The dominant species that is exchanged with the host depends on the applied potential, but the most pronounced process is related to the K+ ion exchange due to the kinetic aspects involved in the intercalation. Another important feature concerning KhFek[Fe(CN)6]l‚mH2O host materials that has been outlined is the presence of two distinct structural environments inside the crystalline structure of KhFek[Fe(CN)6]l‚mH2O compounds. These two distinct environments have been ascribed to be the PB structure itself, KhFek[Fe(CN)6]l, and its hydrated part, mH2O. The PB atomic structure sustains in its interstitial sites, and into [Fe2+(CN)6]4- vacancies, two different water crystalline structures, according to a structural analysis.2 The water crystalline structure existing inside of the interstitial canals appears to be most important for K+ and H3O+ ion exchange. The kinetic analyses are also in agreement with this approach. On the other hand, the structural water, which is part of the coordination shell of Fe3+ and fills empty nitrogen sites of the Fe(CN)64- vacancies, is related to H+ exchange.1 The crystalline structure of water existing in the host is very important for ionic * To whom correspondence should be addressed. E-mail:
[email protected]. † Universite ´ Pierre et Marie Curie. ‡ Universidade Estadual Paulista. § Universitat de Vale ` ncia.
exchanges occurring in this compound, as already investigated by previously detailed ac-electrogravimetric studies.3 The part of the crystalline water hydrating the interstitial sites of the PB structure influences the K+ dynamic movement during electronic compensation, that is, K+ exchange is related to the amount of water present in these channels. Furthermore, the water molecules hydrating interstitial sites are responsible for H3O+ exchange with the aqueous electrolyte environment during the electronic compensation. On the other hand, the channels formed by the Fe(CN)64- vacancies hydrated by water are responsible for H+ exchange.1 The ac-electrogravimetry technique is a relatively new technique that allows more information to be obtained on the role of different ions during an electrochemical process, as well as on the kinetic parameters of this process.4 In general, the ac-electrogravimetry technique consists of coupling electrochemical impedance spectroscopy (EIS) with a fast response quartz crystal microbalance used in ac mode.5-7 This technique has been already used for studying a great amount of electrochemical processes,8-11 including a full kinetic separation of the different exchanged processes existing in KhFek[Fe(CN)6]l‚ mH2O host materials in a KCl electrolyte environment.3 The ac-electrogravimetry technique consists of obtaining simultaneously the response in current (∆I/∆E(ω), electrical transfer function) and mass (∆m/∆E(ω), mass potential transfer function) to a sinusoidal potential perturbation with a small amplitude. The advantage in combining such transfer functions is the possibility of the full separation of the electrochemical processes which involve concomitantly the mass and electronic changes. Considering the intercalation properties of KhFek[Fe(CN)6]l‚ mH2O compounds and the full separation of the three exchanged processes involved, according to an approach considered
10.1021/jp061534a CCC: $33.50 © 2006 American Chemical Society Published on Web 09/06/2006
Ion Intercalation in KhFek[Fe(CN)6]l‚mH2O
J. Phys. Chem. B, Vol. 110, No. 39, 2006 19365
elsewhere,3 the thermodynamic aspects of K+ guest intercalation into a KhFek[Fe(CN)6]l‚mH2O host structure in 0.25 M KCl electrolytes were developed. The electrochemical process involved in the H3O+ exchange was not considered because it is not a reversible process as those related to the K+ guests. This is important because the intercalation term is mostly related to a reversible insertion of guest atoms or molecules into solid hosts, provided the host structure is not significantly altered during the process.12 For the thermodynamic development of these processes, the most important aspect to be considered is the fact that the concentration of the guest can be changed. Therefore, there is a change of the Gibbs free energy with ηi, in which ηi is the number density of intercalated atoms in a given i kind of guest site. In this paper, a lattice-gas model was adapted to our own problem, where the simple mean-field expression for the latticegas model was rewritten by taking into account mass-transfer function aspects. Accordingly, the main goal of this work was to develop the thermodynamic formalism in the light of mass and electronic aspect of intercalation. Specifically, the thermodynamic aspects of K+ intercalation into KhFek[Fe(CN)6]l‚mH2O specific sites were fully calculated and interpreted. Thermodynamics of Insertion in KhFek[Fe(CN)6]l‚mH2O Compounds In general, the kinetics of insertion and extraction of ions in the host lattice is largely dominated by their transport. However, in this work, the kinetics of the processes that occur in the solid state during the electrochemical intercalation or deintercalation of a small ion is derived under the assumption that the film is very thin, so the transport can be considered very fast. In this way, the concentration of the intercalated ions and the chemical potential are both homogeneous in the working electrode (host). Therefore, the complexity of the transport is not considered to seek for another kind of effect, which is assumed to take place in homogeneous conditions. Then, this approach is applied here for very thin films of KhFek[Fe(CN)6]l‚mH2O compounds. Usually, the composition of the intercalation electrodes is described in terms of an occupancy fraction, that is, y ) ηi/Ny, where ηi is the number density of the intercalated ions for a given kind i of guest in an insertion/deinsertion process and Ny is the total number density of one type of site. For instance, one can consider the number density of one type of site or one type of host atom (i.e., K+ in KhFek[Fe(CN)6]l‚mH2O) as h ) ηh/Nh. The electrical potential E is related to the chemical potential of the exchanged atoms due to the reduction processes, µred, and to the chemical potential of the exchanged atoms due to the oxidation processes, µox, by
E)-
1 1 (µ - µox) ) - µ ze red ze
capacitance depends on the ions and electrons contribution to the potential. Indeed, the chemical potential µ of the intercalated atoms, which is determined by the potential (eq 1), can be separated into two contributions: one from the intercalated ions and another one from the charge-compensating electrons that enter the host structure. The electron contribution corresponds to the chemical potential of the electrons, that is, the position of the Fermi level in the electronic band structure of the host material. If one assumes that the incorporation of the metal ion does not affect the electronic band structure (the rigid-band model), then the physicochemical equations can be simplified. In this work, the electronic contribution to the chemical potential µ will be neglected and only the contribution of the intercalated ions to the intercalation capacitance will be considered. The electronic contribution is neglected due to the small thickness of the KhFek[Fe(CN)6]l‚mH2O host films used in these experiments (∼0.1 µm) allied to the highly electronic conductivity presented by such compounds.1,13 For these host thin films, the voltammetric peaks are very narrow and reversible,1 especially for the process involving the K+ intercalation. Here, it is also assumed that the energetic band is able to maintain its shape enough to keep constant the Fermi level so that the compound can be considered to possess a rigid-band electronic structure (the Fermi level is shifted upward as much as necessary to fill the former vacant sites with electrons compensating the ionic charge). This could be also inferred from the high electrochemical reversibility of such a compound.1 In KhFek[Fe(CN)6]l‚mH2O host thin films, a simple insertion reaction mechanism, in which ion guests go directly from the solution to the possible sites into the structure, can be represented by means of the following reaction scheme kred
βi + zie- {\ } ηi k
The oxidation and reduction kinetic processes are accompanied by an insertion/desinsertion process to reach the electroneutrality of the host film. βi ) Ny - ηi is the number density of sites available for the cation intercalation or the number density of intercalated anions, depending on the intercalation process which takes place (it is related to the number density of oxidized metallic atomic species in the electronic counterpart), and ηi is the number density of intercalated cations or of sites available for anion intercalation (which from the electronic counterpart is related to the reduced metallic atomic species) inside the host. Accordingly, the theoretical faradaic transfer functions of this reaction mechanism were calculated previously3 as
CV )
(1)
In this equation, e stands for the positive elementary charge, z is the number of electrons passed through the external circuit, and µ is the chemical potential of the working electrode (i.e., of the host thin film). If the guest has the charge ze (i.e., z ) 1 for K+), then the z ions are intercalated in the host from the solution whereas z electrons passed from the electrode to the host to reach the electroneutrality of the host. The electrochemical system is in equilibrium at the beginning of the electrochemical experiment (steady-state conditions for electroactive films), and therefore, the current intensity in the steady state is zero. It is possible to define the intercalation capacitance by deriving the total charge as a function of E. The intercalation
(1)
ox
dmV dE
)
1
∑Ci Vfilm n
∑
dmi
Vfilm i)1 dV
-Gi
n
)
ziF ∑ jω + K i)1
∑ i)1
-Gi
zi
n
)
i
xi
(2) i
MWi
jω + Ki
(3)
in which dmV is the variation of mass per volume unit of the electrode and CV is the total volumetric capacitance which is also known as the chemical capacitance. dmi and Ci are, respectively, the mass variation and the volumetric equilibrium intercalation capacitance of each i insertion/deinsertion process. F is the Faraday constant (96 500 C/mol) and xi is the charge of the exchanged species, MWi represents the molecular weight of the charged species involved in the ith faradaic process, and i is assumed to be -1 when there is an insertion/deinsertion
19366 J. Phys. Chem. B, Vol. 110, No. 39, 2006
Gime´nez-Romero et al.
process of cationic species (the guest is positively charged) occurring in the host and +1 when there is an insertion/desertion process of anionic species (the guest species is negatively charged). Therefore, the sign indicates the direction of the ionic or cationic movement during the insertion/deinsertion process for ions. Finally, Gi and Ki are the kinetic parameters of each i insertion/deinsertion process.3 Gi is related to the transfer rate of the ith insertion/deinsertion process, whereas the Ki parameter is related to the kinetic rate constants (oxidation and reduction) of the faradaic process.3 By combining eq 2 to E ) -µ/e (eq 1 in which z ) 1), it is possible to obtain thermodynamic information for the system. Thus, the equilibrium intercalation capacitance of each i insertion/deinsertion process is related to the derivative of the potential-composition curve, E(y), which basically reflects the reciprocal of ∂µ/∂y, as follows
Ci )
dQi dηi ) -zFVfilm dE dE
(4)
d(yNy) dy dy Ci ) izFVfilm ) izFVfilmNy ) -izeFVfilmNy dE dE dµ (5) where Qi is the exchanged charge due to the ith insertion/ deinsertion process, Vfilm is the volume of the electroactive film, and Ny is the number density of total available sites for intercalation. This last parameter is constant during the acelectrogravimetry measurement since the KhFek[Fe(CN)6]l‚mH2O electroactive thin host does not change during the measurement. The most important aspect of the thermodynamics of insertion is that the concentration of the guest can change, as stated in the Introduction section. Consequently, the changes in the Gibbs free energy G⊥ can be related to the number of intercalated guest atoms. Note that the symbol ⊥ was used to differentiate between the kinetic parameter, G, defined in eq 2, and G⊥ as the Gibbs free energy. The thermodynamic quantity describing these changes is the chemical potential, defined as µ ) ∂G⊥/∂ηi. The actual form of E(y) depends on the specific thermodynamic properties of the intercalation system, and a model has to be used to describe it. The lattice-gas model stands as a widely used approach to describe such properties.12,14,15 It uses the statistical mechanics, and it takes into account a lattice in which each site has two states. If the sites in the host are considered as “full” or “empty”, a lattice-gas model can be derived for the intercalation compounds.12 If the states are “spin up” and “spin down”, an Ising model can be defined for a magnetic system, for instance. Thus, this model can be also used to explain the variation of the magnetic properties of the KhFek[Fe(CN)6]l‚ mH2O hosts with respect to the applied potential. A difficulty in applying these models to intercalation compounds occurs when the dissociation of the intercalated atom into ions and electrons has to be detailed. The chemical potential can be written as a sum of the contributions of the ions and electrons, that is, µ ) µi + µe for a single electronic and ionic charge-transfer process (µe is usually called the Fermi energy). However, for thin KhFek[Fe(CN)6]l‚mH2O host films, it is possible to consider the electronic term contribution to the chemical potential changes as negligible, so that µe is considered not to vary significantly with respect to the perturbation potential. Thus, it is possible to infer that ∂µ ) ∂µi or, in other words, the chemical potential varies just due to the changes on the ionic part with respect to the potential perturbation. In highly electronic conductors, such as the KhFek[Fe(CN)6]l‚mH2O
compounds, considering the variation of µe with respect to the potential perturbation as negligible is very plausible. By assuming that there is no interaction between the ions and host or ions with themselves and still that the sites are all energetically equivalent, the sites can be considered to be occupied at random. Therefore, the entropy of the ions randomly distributed over a fraction y of Ny available sites can be given by12
Si ) iNykB[y ln y + (1 - y)ln(1 - y)]
(6)
where kB is the Boltzmann constant. Then the partial derivative of the entropy is
∂Si/∂ηi ) ikB ln(y/(1 - y))
(7)
Assuming the energy, Λ0, as the Gibbs free energy change due to put an isolated ion and electron into the lattice (∆G⊥ per exchanged particle), then
µ ) Λ0 - ikBT ln[y/(1 - y)]
(8)
This equation ignores interactions between guest atoms. To consider interactions, we suppose gkBT is supposed to be the total interaction energy that a given ion would feel if all the other sites were full. When only a y fraction of the sites is occupied, it costs an extra energy, gkBTy, to add another ion to the lattice, so that it is possible to rewrite eq 8 to consider the interaction contribution, and µ becomes
µ ) Λ0 - ikBT ln[y/(1 - y)] - gikBTy
(9)
where g can vary between 4 and -4 and values outside this range indicate that a phase change has taken place in the host.15 Accordingly, the equilibrium capacitance can be calculated from eqs 4 and 9
Ci ) -izeFVfilmNy
(
)
-1 -i dy -izeFVfilmNy - giy ) dµ kBT y(1 - y) (10)
Thus, in the steady state (when ω f 0), the occupancy fraction of the host, that is, yj, for each insertion/deinsertion process can be calculated considering eqs 2 and 10
(
)
(
)
Ciωf0 )
-izeFVfilmNy -i - giyj kBT yj(1 - yj)
-
Gi - ieNy -i ) - giyj Ki kBT yj(1 - yj)
-1
) -ziVfilmF
Gi Ki (11)
-1
(12)
Therefore, from eq 3, the mass-transfer function at low frequency is equal to
dmV
ωf0
dE
∑ i)1
- ieNy
zi
n
)
i
xi
MWi
kBT
(
-i
yj(1 - yj)
)
-1
- giyj
(13)
Note that the -Gi/Ki term in eq 12 represents the ionic isotherm of intercalation.3 So, if the chemical potential is related to the cell potential from eq 1, then the energy to put an isolated ion and electron into the lattice, Λ0, can be calculated. It can be carried out by following the evolution of the chemical potential with respect to the occupancy fraction of the host at equilibrium (eq 8), which can be calculated from the measure-
Ion Intercalation in KhFek[Fe(CN)6]l‚mH2O
J. Phys. Chem. B, Vol. 110, No. 39, 2006 19367
Figure 1. Thermodynamic aspects of the PB host. The curve of eE vs yj for the charge-transfer mechanism involving K+ exchange. The yj values were calculated from eq 12 and by means of the kinetics parameters of this insertion/desinsertion mechanism calculated proposed in the previous paper from the simulation of the ac-electrogravimetry spectra.3 The continuous line is the simulation of the yj parameters using eq 14 (eE ) 0.423-0.037 ln(yj/1 - yj), R2 ) 0.97). ac-Electrogravimetry experiments were carried out in 0.25 M KCl at T ) 298 K, pH ) 2.23. Finally, all potentials in this figure refer to the normal hydrogen electrode (NHE).
ment of the cell potential at equilibrium with respect to the charge passed between the electrodes. An equation that correlates the cell potential to the occupancy fraction can be obtained by combining eqs 1 and 9
eE ) -Λ0 + ikBT ln[yj/(1 - yj)] + gikBTyj
(14)
As it is possible to observe in eq 12, the calculus of the parameter yj must be made by means of an iterative method, since the parameter g cannot be directly determined. Therefore, the parameter yj must be first inferred by ignoring the interaction between guests (g f 0). Subsequently, this parameter is recalculated using the parameter g calculated by means of the simulation of experimental data from eq 14. This interactive procedure finishes when the parameters calculated from eq 14 and that used to calculate the parameter yj are the same. Accordingly and considering the reaction mechanism of the KhFek[Fe(CN)6]l‚mH2O host developed elsewhere,1,3 Figure 1 shows the representation of eq 14 for the K+ cations insertion/ desinsertion mechanism in the electron transfer of KhFek[Fe(CN)6]l‚mH2O compounds (electron transfer known as Everit’s Salt (ES) T Prussian Blue (PB)), where the yj parameter of this plot has been calculated from eq 12, by using the kinetics parameters (Gi, Ki) obtained in the previous paper from the analysis of the ac-electrogravimetry spectra.3 Consequently and as both kinetic parameters take into account the ion hydration process,3 the energy of this dehydration process is also considered in the calculus of the necessary energy to insert each ion into the host. Before proceeding with the application of the theory to experimental data, it is important to comment that the probability of the sites occupation in the model considered here is given by y ) ηi/Ny ) F(Λ0,µ) which, in agreement with the kinetic model proposed in the previous paper,3 turns into
k0red ηi y) ) 0 0 Ny k k
(15)
ox red
in which k0ox and k0red are the kinetic rate constants at steady state for the cationic deintercalation and intercalation, respec-
Figure 2. Thermodynamic aspects of the PB host. The curve of eE vs yj for the charge-transfer mechanism involving K+ exchange. The yj values were calculated from eq 12 and by means of the kinetics parameters of this insertion/desinsertion mechanism calculated in the previous paper from the simulation of the ac-electrogravimetry spectra.3 The continuous line is the simulation of the yj parameters using eq 14 (eE ) 0.45 - kBT ln(yj/1 - yj) - 2kBTyj). ac-Electrogravimetry experiments were carried out in 0.25 M KCl at T ) 298 K, pH ) 2.23. Finally, all potentials in this figure refer to the normal hydrogen electrode (NHE).
tively. These kinetic rate constants present an exponential potential dependence such as
kox ) khox exp(boxE h)
(16)
kred ) khred exp(bredE h)
(17)
In the above equations, E h is the steady-state potential and hkox and khred are the preexponential factors of the kinetic rate constant for the cationic deintercalation and intercalation processes, respectively. bi is the Tafel coefficient of the kinetic rate constant. This coefficient is related to the symmetry of the energetic patterns for deintercalation (box) and intercalation (bred). It must be pointed out that these parameters are related to the transfer coefficients according to the Butler-Volmer kinetic dependence established in the previous paper.3 As it can be observed in Figure 1, eq 14 can be fitted to experimental data even if the interaction parameter is not considered (g f 0). Accordingly, in this particular case, the value of the slope is equal to the parameter kBT (4.12 × 10-21 J, i.e., ∼0.0257 eV at 298 K). However, the experimental value obtained is about 5.91 × 10-21 J, that is, ∼0.0369 eV. As it can be noted, there are some differences between theoretical and experimental data. These differences are due to the noconsideration of the interaction term. If interaction is considered, then a more accurate fitting is observed. From this fitting, it is possible to calculate the Gibbs free change, Λ0, due to put an isolated potassium ion and an electron into the lattice. The Gibbs free energy change obtained experimentally is about -7.21 × 10-20 J (i.e., ∼ -0.450 eV). To consider a mole of guest exchanged to the host, the Avogadro number is considered so that the value obtained per mole is about -43 kJ‚mol-1, which is in very good agreement with the value of -41.8 kJ‚mol-1 calculated from the cyclic voltammetry procedure.16 It is important to stress that the same thermodynamic parameter was obtained by means of two different methodologies which validates the theory developed here concerning the thermodynamics of intercalation in KhFek[Fe(CN)6]l‚mH2O compounds. Furthermore, it is also important to comment on the total interaction energy that an intercalated K+ ion would experience
19368 J. Phys. Chem. B, Vol. 110, No. 39, 2006 if all the other sites were full. This energy is, in this case, repulsive and its value is about 2kBT J. This repulsive energy begins to become effective at more anodic potentials than 0.2 V with respect to the saturated calomel reference electrode, which corresponds to the voltammetric peak.1 Accordingly, the hypotheses made to obtain eq 12 and above all and the kinetic model developed in the previous paper3 of the involved process considered here, that is, K+ exchange, are validated according to the thermodynamic approach developed in this work. The validation of eqs 13 and 14 confirms that the electronic contributions to the chemical potential can be neglected; it is confirmed that the KFe3+Fe2+(CN)6‚mH2O structure possesses a high electronic conductivity, as it has been shown by means of the evolution of the impedance spectra with respect to temperature.13 Finally, eq 14 can be also used to study the other reversible process in the electron-transfer ES T PB, related to the H+ intercalation process. The problem with this intercalation process is that the total repulsive interaction energy that an intercalated H+ ion would experience if all the other sites were full is equal to 150 kBT J (calculated form the fitting of eq 14 where yj is calculated by means of the kinetic parameters calculated in the previous paper3). This energy is too high, and therefore, this process must have special particularities, which will be studied carefully in depth in a future analysis. Conclusion A thermodynamic formalism for the ion intercalation in the PB host compound was developed and correlated to the kinetic parameters. This validates the kinetic model as the values obtained are in good agreement with that obtained by transient techniques. The fact that the simple model, like the lattice-gas model used in this work, can be applied so well to K+ ion intercalation supports the idea that there is no interaction between the guest and the host material at more anodic potentials than 0.20 V with respect to the saturated calomel reference electrode, at least for low concentrations of K+. The approach used in this work offers an opportunity to investigate, in the future, how ions are distributed and how they could interact within the host material at higher K+ concentrations. The ionic movement controls the electronic counterpart and then the
Gime´nez-Romero et al. approach is able to provide further essential understanding to the intercalation processes in mixed valence compounds such as Prussian Blue and derivatives and correlate electrochemical changes to physical properties such as optical and magnetic properties. Acknowledgment. This work has been supported by FEDERCICyT Project CTQ 2004-08026/BQV. P.R.B. acknowledges the Sa˜o Paulo state research funding institution (FAPESP) under Project No. 02/06693-3. D.G.-R. acknowledges a Fellowship from the Generalitat Valenciana, Postdoctoral Program. J.J.G.J. acknowledges their position to the Ramon y Cajal Program (Spanish Ministry Office of Education and Science). We appreciate the very useful discussions with Nuria PastorNavarro. References and Notes (1) Gime´nez-Romero, D.; Bueno, P. R.; Gabrielli, C.; Perrot, H.; Garcia-Jaren˜o, J. J.; Vicente, F. J. Phys. Chem. B 2006, 110, 2715. (2) Herren, F.; Fischer, P.; Ludi, A.; Halg, W. Inorg. Chem. 1980, 19, 956. (3) Gime´nez-Romero, D.; Bueno, P. R.; Gabrielli, C.; Perrot, H.; Garcia-Jaren˜o, J. J.; Vicente, F. J. Phys. Chem. B, 19352. (4) Benito, D.; Gabrielli, C.; Garcia-Jaren˜o, J. J.; Keddam, M.; Perrot, H.; Vicente, F. Electrochem. Commun. 2002, 4, 613. (5) Bourkane, S.; Gabrielli, C.; Keddam, M. Electrochim. Acta 1989, 34, 1081. (6) Cordoba-Torresi, S.; Gabrielli, C.; Keddam, M.; Takenouti, H.; Torrese, R. J. Electroanal. Chem. 1990, 290, 269. (7) Bourkane, S.; Gabrielli, C.; Keddam, M. Electrochim. Acta 1993, 38, 1023. (8) Yang, H.; Kwak, J. J. Phys. Chem. B 1997, 101, 774. (9) Gabrielli, C.; Garcia-Jaren˜o, J. J.; Keddam, M.; Perrot, H.; Vicente, F. J. Phys. Chem. B 2002, 106, 3192. (10) Oh, I.; Lee, H.; Yang, H.; Kwak, J. Electrochem. Commun. 2001, 3, 274. (11) Gabrielli, C.; Keddam, M.; Nadi, N.; Perrot, H. Electrochim. Acta 1999, 44, 2095. (12) McKinnon, W. R.; Haering, R. R. In Modern Aspects of Electrochemistry; White, R E., Bockris, J. O. M., Conway, B. E., Eds.; Plenum Press: New York, 1983; Vol. 15. (13) Garcia-Jaren˜o, J. J.; Sanamatias, A.; Navarro-Laboulais, J.; Benito, D.; Vicente, F. Electrochem. Acta 1998, 43, 235. (14) McKinnon, W. R. In Solid State Electrochemistry; Bruce, P. G., Ed.; Cambridge University Press: Cambridge, U.K., 1995. (15) Mattson, M. S. Phys. ReV. B 1998, 58, 11015. (16) Garcia-.Jaren˜o, J. J.; Sanmatias, A.; Navarro-Laboulais, J.; Vicente, F. Electrochim. Acta 1998, 44, 395.
