Kinetic Aspects of Ion Exchange in KhFek[Fe(CN)6]lÂ·mH2O

19352

J. Phys. Chem. B 2006, 110, 19352-19363

Kinetic Aspects of Ion Exchange in KhFek[Fe(CN)6]l‚mH2O Compounds: A Combined Electrical and Mass Transfer Functions Approach D. Gime´ nez-Romero,† P. R. Bueno,‡ C. Gabrielli,*,† J. J. Garcıá-Jaren˜ o,§ 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 present paper quantifies and develops the kinetic aspects involved in the mechanism of interplay between electron and ions presented elsewhere1 for KhFek[Fe(CN)6]l‚mH2O (Prussian Blue) host materials. Accordingly, there are three different electrochemical processes involved in the PB host materials: H3O+, K+, and H+ insertion/extraction mechanisms which here were fully kinetically studied by means of the use of combined electronic and mass transfer functions as a tool to separate all the processes. The use of combined electronic and mass transfer functions was very important to validate and confirm the proposed mechanism. This mechanism allows the electrochemical and chemical processes involved in the KhFek[Fe(CN)6]l‚mH2O host and Prussian Blue derivatives to be understood. In addition, a formalism was also developed to consider superficial oxygen reduction. From the analysis of the kinetic processes involved in the model, it was possible to demonstrate that the processes associated with K+ and H+ exchanges are reversible whereas the H3O+ insertion process was shown not to present a reversible pattern. This irreversible pattern is very peculiar and was shown to be related to the catalytic proton reduction reaction. Furthermore, from the model, it was possible to calculate the number density of available sites for each intercalation/deintercalation processes and infer that they are very similar for K+ and H+. Hence, the high prominence of the K+ exchange observed in the voltammetric responses has a kinetic origin and is not related to the amount of sites available for intercalation/ deintercalation of the ions.

Introduction In a previous paper, it was shown that the insertion reaction mechanisms in transition metal hexacyanoferrates of the general formula KhFek[Fe(CN)6]l‚mH2O (Prussian Blue, PB) involve mainly three different charge modes related to three different ionic sites inside of the crystalline structure1 in aqueous media. These charge modes are related to the exchange of H3O+, K+, and H+ guest ions with the KhFek[Fe(CN)6]l‚mH2O host structure. The predominance of one or another exchange process, between the host and the electrolyte (solution), was shown to depend on the applied potential. It was also demonstrated that the presence of such processes are related to two distinct structural environments existing inside the crystalline structure of the KhFek[Fe(CN)6]l‚mH2O compounds. These two different environments are the PB structure itself, represented by KhFek[Fe(CN)6]l, and its hydrated part, represented by mH2O.1 Accordingly, it was demonstrated that the PB atomic structure sustains in its interstitial sites and into [Fe2+(CN)6]4- vacancies two different crystalline water structures.1,2 Apparently, the water structure existing inside of the interstitial sites appears to be the most important for the K+ and H3O+ ions exchange movement. On the other hand, the structural water which is part of the coordination shell of Fe3+ and fills the empty nitrogen * To whom correspondence should be addressed. E-mail: [email protected]. † Universite ´ Pierre et Marie Curie. ‡ Universidade Estadual Paulista. § Universitat de Vale ` ncia.

sites of the Fe(CN)64- vacancies appears to be responsible for the H+ exchange.1 Therefore, the water crystalline structure into the host is very important, and in one way or another, it influences the exchange processes of K+, H+, and H3O+.1 The main goal of the present paper is to develop the kinetic aspects involved in the previous picture of the insertion/ deinsertion processes and separate all of them by using electrical and mass charge transfer functions. Once separated, the processes were kinetically modeled and interpreted according to the general structural model proposed1 and shortly described in the above paragraphs. In developing the kinetic model, important information arises from the analysis of the results as will be demonstrated here. All of them are carefully discussed here. Furthermore, the superficial oxygen reduction occurring in parallel to the intercalation reactions was included in the formalism. This superficial reaction is separated to the other bulk intercalation reactions. Mass and Electronic Exchanges There are several methodologies or strategies used in the literature to deal with ionic and solvent molecules exchange in electroactive host thin films. For instance, it is possible to record current and mass changes during potential sweeps by using an ammeter and a quartz crystal microbalance3-6 or to measure the electrochemical impedance of the coated electrode7-10 or even to measure the beam deflection (mirage effect).11-16 However, all these strategies or techniques are not sufficiently

10.1021/jp061533i CCC: $33.50 © 2006 American Chemical Society Published on Web 09/06/2006

Ion Exchange in KhFek[Fe(CN)6]l‚mH2O Compounds efficient and selective concerning the separation of the different kinetic contributions due to the different species involved in the electrode charge compensation.17 Moreover, these techniques solely provide information on the global response of the electrode/film/electrolyte interface. As an example, cyclic voltammetry measures global current values, and in such approach, it is not possible to discriminate between the anion ingress or cation expulsion for the charge compensation occurring in the film during a redox process. In addition, if neutral species are involved, then their movements are hardly perceptible. To face this problem, during the last years, a new approach has been developed which allows more information about the ionic insertion/deinsertion to be obtained during an electrochemical reaction that involves more than two ionic species.21 This new approach is based on a combination of two different transfer functions, an electrical transfer function (electrochemical impedance spectroscopy) and a mass potential transfer function (ac-electrogravimetry). The approach is summarized as the coupling of electrochemical impedance spectroscopy (EIS) with a fast response electrochemical quartz crystal microbalance (EQCM). The EQCM technique gives access to the mass change on the working electrode22-24 during a sinusoidal electrochemical potential perturbation. This technique has been already used to investigate processes occurring in oxide materials, for example, WO3,25,26 to describe iron behavior in acidic medium,27 processes occurring in polypyrrole28-31 and polyaniline polymer films32-36 as well as in other conducting film materials.17,36-38 Specifically, the combined transfer function approach allows the sinusoidal current response of a potential perturbation (electrochemical impedance function, Z) to be simultaneously measured with the sinusoidal mass response (electrogravimetric transfer function, dm/dE). The main advantage of this approach is to allow the separation between fast and slow ionic coupled electrochemical processes by changing the frequency of the potential perturbation. Another important advantage of the use of the combined mass and electronic change approach is the possibility to calculate the ratio between the mass change and electrical charge variation (F(dm/dq) function) during an experiment. This ratio allows information on the molecular weight of the species being exchanged between film and electrode/electrolyte during an electrochemical process17 to be extracted by means

J. Phys. Chem. B, Vol. 110, No. 39, 2006 19353 Theoretical Description of Transfer Functions for Intercalation KhFek[Fe(CN)6]l‚mH2O compounds are treated as intercalation compounds according to the solid-state mechanism presented previously.1 An intercalation or insertion compound is always a solid material made of host atoms and guest ions. The host atoms provide a lattice or framework, and the guest ions occupy sites in the host lattice. Two properties distinguish intercalation compounds from other solids: the guest can be added to the host or removed from it, so the concentration of guests can change. These two properties are exploited when insertion/ deinsertion reactions occur in the KhFek[Fe(CN)6]l‚mH2O compound in aqueous KCl media. It is widely admitted that the kinetics of insertion and extraction is largely dominated by the transport of inserted ions in the guest lattice. However, the kinetic processes that occur in the solid state during the electrochemical interfacial transfer can be considered as the rate-determining step if the film is very thin (the estimated film thickness is around 90 nm and it was corroborated by a theoretical estimation1). By this way, both the concentration of the intercalated ions and the chemical potential are practically homogeneous in the working electrode, at any modulation frequency. Therefore, we get rid of the complexity of the transport to seek for kinetic effects, which take place in homogeneous conditions. In this framework, the main interest lies in the analysis of the film capacitance; the manifestation of internal kinetic processes consists on a frequency-dependent capacitance, which will be analyzed further as in the following. Accordingly, the composition of intercalation electrodes is usually described in terms of an occupancy fraction, that is, y ) ηi/Ny, where ηi is the number density of intercalated atoms in a given kind i of the guest sites during an insertion/deinsertion process and Ny is the number density of one type of site (it is proportional to the size of the host for that kind of site), that is, a y type site. For instance, one can write 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 η ) ηk+/Nk+). The potential E of the working electrode 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)zi dm (ω) ) Σi MWiνi(ω) ( F dq xi mass changes due to uncharged species (1) where xi is the charge of the exchanged species, i is assumed to be -1 when there is a cationic insertion/deinsertion process (the guest is positively charged) occurring in the host and +1 when there is an insertion/deinsertion process of anionic species (the guest species is negatively charged). Indeed, the sign indicates the direction of the anionic or cationic movement during the ionic insertion/deinsertion process, F is the Faraday constant (96500 C/mol), dq the electric charge that crosses the film, dm is the mass change of the film, MWi is the molecular weight of the charged species involved in the ith faradaic process, zi is the number of electrons involved in the faradaic process. and νi represents a percentage of the electrical charge balanced by the participation of the ion involved in the global electrochemical process (a percentage which depends on the pulsation (ω) (ω ) 2πf where f is the frequency)).

1 1 (µ - µox) ) - µ ze red ze

(2)

given that the electrochemical system is in equilibrium at the beginning of the electrochemical experiment (steady-state conditions for electroactive films) and, therefore, the current intensity at steady state is zero. In this equation, e stands for the positive elementary charge (1.602 × 10-19 C), z is in this case the valence of the exchanged ion between the solution and the film (i.e., z ) 1 for K+. Note that K+ can be inserted just into sites of h type in the KhFek[Fe(CN)6]l‚mH2O compound), and µ is the chemical potential of the working electrode. Deriving the total charge as a function of E, it is possible to define the intercalation capacitance. The contribution to the intercalation capacitance is related to the ion and to the electron contribution to the chemical potential. In fact, the chemical potential µ of the intercalated atoms, which is determined by the potential (eq 2), can be separated into two contributions: one from the intercalated ions and another one from the chargecompensating electrons that enter the host structure. The electron contribution corresponds to its own chemical potential, that is,

19354 J. Phys. Chem. B, Vol. 110, No. 39, 2006

Gimeńez-Romero et al.

Figure 1. Curves of current vs potential during a linear potential scan at a PB film in the ES T PB electron transfer. Experimental conditions were 0.25 M KCl, pH ) 2.23, and at a scan rate 10 mV/s.

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. This simplification has also been used previously for this kind of film in the literature.17 In this work, the electronic contribution to the chemical potential µ was neglected, and the attention was mainly focused on the contributions of the intercalated ions to the intercalation capacitance. The electronic contribution to the chemical potential was neglected mainly because of the high electronic conductivity of the KhFek[Fe(CN)6]l‚mH2O compound during the experiments. It was also assumed that the band is able to maintain its shape enough to keep the Fermi level constant and also that the material is considered to possess a rigid-band electronic structure due to the almost metallic conductivity of the KhFek[Fe(CN)6]l‚mH2O compound1,18 and the reversibility of this kind of film (Figure 1). Therefore, in KhFek[Fe(CN)6]l‚mH2O host thin films, a simple insertion reaction mechanism of ion guest directly from the solution to possible sites into the structure can be represented by means of the following reaction scheme -

kred

βi + zie {\ } ηi k

are considered to be governed, as a function of potential, by a Butler-Volmer dependence. Therefore, the kinetic constant of trapping and releasing are considered to possess a potential dependence. All of these aspects will be discussed in a further work. For the calculation of the theoretical faradaic capacitance function of this kinetic reaction mechanism, it is important to emphasize that the partial derivative of βi with respect to the applied potential (dβi/dE) is calculated by considering the insertion and deinsertion as elementary steps. In other words, the ion insertion produces at each steady state a uniform concentration of guest ions in the intercalation film, and thus, the chemical potential is considered homogeneous. This means that there are paths available for the intercalated ions to fill the host structure uniformly and the number density of total available sites for intercalation, Ny, is constant. Furthermore, it will be assumed that the insertion and deinsertion kinetic constants follow a Butler-Volmer dependence with regard to the applied potential,40 so that

ki ) khi exp(biE)

(4)

In addition, the potential E is sinusoidally perturbed by ∆E during an ac-electrogravimetry coupled to EIS by means of the following expression41

E)E h + ∆E ) E h + ∆E exp(jωt)

(5)

dE ) jω∆E dt

(6)

in which E h is the steady-state potential or the potential and ∆E is an oscillating potential perturbation of small amplitude around a stationary value E h , during the steady-state experiment, ∆E is the amplitude of the variation of the electrochemical potential, and j is the square root of -1. Thus, the kinetic constants can be written as

h + ∆E)) ) ki0 exp(bi∆E) ki ) ki exp(bi(E

(7)

(3)

ox

where the oxidation and reduction kinetic processes are accompanied by an insertion/desinsertion process to reach the electroneutrality of the film. ηi is the number density of intercalated cations or of free host sites available for anion intercalation, which is related to the reduced metallic atomic species in the host. βi ) Ny - ηi is the number density of the free host sites available for cation intercalation or of intercalated anions, depending on the intercalation process which takes place; it is also related to the number density of oxidized metallic atomic species in the electronic counterpart of the host. It is important to comment that the model to be described here can be also interpreted according to the approach of ionic trapping by assuming fast transport.19,20 However, here it is considered particularly as a case in which the concentration of ion charge in the “fast states” follows instantly the applied potential and the total capacitance is considerably larger than the “fast states” capacitance. In this particular situation, the “fast states” network can be viewed as a relay from the solution to the traps. In other words, in the context described in refs 19 and 20, the ηi values described here can be viewed as the trap states and βi as the release states or the “fast states”. Furthermore, there are important differences that reside on the fact that, in our present model, the dynamics between the ηi and βi states

For sufficiently small amplitude potential perturbations, βi and ηi can be represented in terms of complex functions. Thus, ηi can be described as

ηi ) ηi + ∆ηi ) η j i + ∆ηi exp(jωt)

(8a)

dηi ) jω∆ηi dt

(8b)

where η j i is the concentration of intercalated cations or of sites available for anions intercalation at the polarization potential E h . Note that η j i is constant for a particular electrolyte concentration and at a steady-state potential, as argued in a previous paper.1 As a result, taking into account the kinetic laws for this mechanism, eq 3, the variation of ηi with time can be calculated from the following expression

dηi ) jω∆ηi ) kred/iβi - kox/iηi dt

(9)

As already commented, Ny is considered as a constant during the measurements, and therefore, after differentiation of eq 9, it is possible to consider that

Ion Exchange in KhFek[Fe(CN)6]l‚mH2O Compounds

jω

J. Phys. Chem. B, Vol. 110, No. 39, 2006 19355

d∆ηi dηi ) jω ) dE dE

CV )

dηi dηi - box/ikox/iηi - kox/i (10) bred/ikred/iβi - kred/i dE dE Now, considering that ηi follows the variation of the potential according to the equilibrium conditions previously stated and that the perturbation is neglected, one has

dηi bred/ikred/i0βi - box/ikox/i0ηi ) dE jω + (k 0 + k 0)

(11)

dηi Gi ) dE jω + Ki

(12)

red/i

ox/i

where Gi ) bred/ikred/i0β h i - box/ikox/i0η j i and Ki ) kred/i0 + kox/i0. The equilibrium intercalation capacitance can be easily defined from the Faraday law given that the initial current density at steady-state is zero (steady-state conditions for electroactive films), such as

∆qi ) - ziVfilmF∆ηi

-ziFVfilm

where Vfilm is the volume of the electroactive film. At the same time, the mass transfer function can be easily deduced by considering the molecular weight of the guest species exchanged during the process. Thus, this transfer function presents, for species i, the following dependence with respect to the perturbation frequency

∆mi ) -iVfilm

zi MWi∆ηi xi

(15)

zi dηi zi Gi dmi ) -iVfilm MWi ) -iVfilm MWi (16) dE xi dE xi jω + Ki dmi and Ci are, respectively, the mass variation of each i insertion/deinsertion process and the equilibrium intercalation capacitance of each i insertion/deinsertion process. For an irreversible insertion or deinsertion ionic mechanism, Gi obeys the limit imposed by either kred/i0/kox/i0 f 0 or kred/i0/kox/i0 f ∞ and can be written as Gi ) bred/ikred/i0β h i or Gi ) -box/ikox/i0η j i, respectively, because kox/i0 . kred/i0 or kox/i0 , kred/i0. It is also possible to consider different independent insertion/ deinsertion mechanisms similar to that considered in eq 3. As a result, the total electric and mass transfer functions can be calculated as the sum of individually theoretical transfer functions of each insertion/deinsertion process. To normalize this value, the total faradaic impedance is divided by the electroactive host volume. In this way, the calculated impedance of the electroactive host is an impedance value per volume of the host. Therefore, the global mass and electrical impedance functions correspond, respectively, to

dmV dE

)

1

n

zi

i

ziF ∑ jω + K i)1

(18) i

GH3O+ G H+ GK + CV ) + + F jω + KH+ jω + KH3O+ jω + KK+

(20)

where the subscript H+ indicates the exchange reaction mechanism which involves the proton (MWH+ ) 1 g‚mol-1), the subscript H3O+ indicates the exchange reaction mechanism which involves the hydrated protons (MWH3O+ ) 19 g‚mol-1), and the subscript K+ indicates the exchange reaction mechanism which involves the potassium cations (MWK+ ) 39 g‚mol-1). Finally, as three quantities are unknown, which correspond to the three ion responses to ∆E, and that only two quantities, Z(ω) ) dE/dI (ω) and dm/dE (ω) are measured, it is not possible to separate the three unknown quantities. However, it is possible to eliminate the contribution of one species from the total mass transfer function. The response of one cation (or that cation considered numbered as one, for instance) can be eliminated from the global electrogravimetric transfer function, dm/dE (ω), by considering33,34 the partial electrogravimetric transfer function dmV23/dE

dmV23 dmV expe ) dE dE

1

z1 MW1 x1 CV expe z1F

(21)

By using eqs 17 and 18, dmV23/dE is equal to

dmV23 dE

n

)

∑ i)1

(

zi i

xi

MWi - 1

z1 x1

)

MW1

-Gi jω + Ki

(22)

Similarly, other contributions can be eliminated in the mass change function. From the plots of the partial electrogravimentric transfer functions, the number of species involved in the charge compensation process can be evaluated.35 The elimination of one species over three (or more), as described here, is close to the method used by Bruckenstein and Hillman as a diagnostic for species interference.3,43,44 Consideration of Parallel Superficial Reactions

-Gi

∑ i x MWi jω + K

Vfilm i)1

)

GH3O+ GH+ G K+ dmV )1 + 19 + 39 (19) dE jω + KH+ jω + KH3O+ jω + KK+ -

Gi (14) jω + Ki

Vfilm

-Gi

n

in which dmV is the mass variation per volume of the electrode and CV is the total volumetric capacitance which is also known as chemical capacitance. These theoretical transfer functions are similar to the functions proposed in the literature.17 All of them were adjusted to consider the host as possessing an occupancy fraction for ions.17 In this way, the model considered here has a different physicohemical regard concerning the interpretation of the Gi and Ki kinetic parameters. On the other hand, the dehydration effect is considered indirectly in these models. It is indirectly included in the kinetic parameters (transfer parameters). For instance, the G parameter (exchange rate) is lower when the dehydration process is more important.45 According to the ES T PB reaction mechanism considered elsewhere,1 the partial transfer functions can be described as

(13)

dqi d∆ηi dηi Ci ) ) -ziFVfilm ) -ziFVfilm ) dE dE dE

∑ Ci

(17) i

Superficial parallel reactions occurring during insertion/ deinsertion processes can be also modeled by the previous



approach. For instance, the theoretical electrical transfer function for an irreversible superficial reaction, in which the reactants and products are in the electrolyte, can be easily considered. For an irreversible oxygen reduction occurring at the surface of the host material, the electrical transfer functions can be written as

jω

dCirrev dCirrev ) birrervkirrevCirrev + kirrev ) dE dE birrevkirrevCirrev ) Girrev (23) Ci ) -zVsolutionF

dCirrev Girrev ) -ziVsolutionF dE jω

Equation 24 is deduced under the assumption that the amount of reactive species is constant, considering they are an excess in solution. Note that, in this case, Cirrev is, naturally, the concentration of reactant species in the electrolyte. kirrev is the kinetic constant for this irreversible reaction. For this situation, eq 19 does not change because the irreversible reaction is occurring on the surface so that it does not contribute to the mass change.1 However, eq 20 must be rewritten by considering the electronic contribution of the parallel irreversible reaction occurring on the electrode surface and then

-

GH 3 O + G O2 GK + GH + CV + + + ) (25) F jω + KH+ jω + KH3O+ jω + KK+ jω

where the subscript O2 indicates the molecular oxygen reduction catalyzed by the PB host surface.42 Note that eq 25 took into account the irreversible oxygen reaction that is commonly observed to occur on the surface of the PB host.1 The kinetic model developed here tries to describe completely all the processes involved on PB host material. However, special attention will be given to the processes related to the ionic exchange. Experimental Section The electrochemical experiments were carried out in a typical three-electrode cell, where the working electrode was one gold electrode, of about 25 mm2, of the quartz crystal of a microbalance (Matel-Fordahl, France). A platinum plate was the counter electrode and a saturated calomel electrode (SCE) was used as the reference electrode. To use the fast EQCM in ac mode, the modified working electrode was polarized at a chosen potential and a sinusoidal small potential perturbation was superimposed.33 The microbalance frequency change corresponding to the mass response of the modified working electrode was detected by means of a special homemade frequency/potential converter.33 The resulting signal was simultaneously sent with the current response of the electrode to a four-channel frequency response analyzer (Solartron 1254), which allowed the electrogravimetric transfer function to be simultaneously obtained with the electrochemical impedance. EQCM was calibrated by means of a galvanostatic copper electrodeposition, which gave an experimental Sauerbrey constant equal to 16.3 × 107 Hz‚g-1‚cm2. This calibration procedure has been commonly used.17,31,34,45 PB films were galvanostatically deposited on the gold electrode of the quartz crystal.46 FeCl3 (A. R., R. P., NORMAPUR), K3Fe(CN)6 (A. R., R. P., NORMAPUR), and HCl (A. R., R. P. NORMAPUR) were used for the synthesis

Figure 2. Dependence of faradaic electric impedance (Figure 2a) and faradaic mass impedance (Figure 2b) spectra on the polarization potential. Experimental conditions were 0.25 M KCl, pH ) 2.23: 0, the experimental spectra at 0.10 V; 4, at 0.20 V; ×, at 0.30 V; O, at 0.40 V; and ), at 0.50 V. Continuous lines are the simulated spectra from the values of Table 1. Both simulations (electrical and mass impedance) are performed by means of the same kinetic parameters and considering a nonideality factor of 0.9 (see text for more explanation on this). The noncompensated resistance and the double layer capacitance were eliminated from the experimental spectra.

of the PB films. The galvanostatic experiments were carried out by means of an AUTOLAB potentiostat-galvanostat (PGSTAT100). The gold electrode was immersed into a 0.02 M K3Fe(CN)6, 0.02 M FeCl3, and 0.01 M HCl solution and a controlled cathodic current of 40 µA cm-2 was applied during 150 s to obtain the electrodeposited PB films. After that, the PB films were stabilized before the impedance measurements by means of cyclic voltammetry around the ES T PB transition [0.60, -0.20] V in 0.25 M KCl (A. R., R. P. NORMAPUR), pH ) 2.23. These PB films were sufficiently thin to allow the use of the Sauerbrey equation47 without limitation coming from the viscoelastic properties of the film.48,49 Other nonideal contributions which could cause frequency shifts, like roughness, are not expected for the thin films used in these experiments.50 The frequency response measurements were fulfilled in the frequency interval from 0.01 to 6 × 104 Hz. These experiments were initially stabilized during 60 s at the steady-state potential. All these electrochemical experiments were carried out in 0.25 M KCl, pH ) 2.23, and in the potential range between 0.50 and -0.2 V. Results and Discussion Transfer Function Response Analysis. Figure 2 shows the complex capacitance plane (Figure 2a) and the mass potential transfer function (Figure 2b) plotted in the complex plane of the ES T PB electron-transfer process. Thus, it can be observed


Figure 3. Dependence of a real part value at the low-frequency limit of a F(dm/dq) function on the polarization potential. Experimental conditions were the same as those indicated on the label of Figure 2.

that the magnitude of the complex capacitance spectra increases when the potential becomes more cathodic up to 0.20 V and it decreases when the potential becomes more anodic up to 0.20 V. Such potential is very specific because the oxidation and reduction current intensities on the voltammograms are maximum, thus a small variation in the potential causes a great change in the current intensity (see, Figure 1), and therefore, a great change in the admittance response of the system is obtained in the potential range around this value. The mass potential transfer function (Figure 2b) is characterized by a negative value of the real part. This feature indicates that, in these experimental conditions, the species exchanged are mainly cations, according to the theory presented in the introductory section. Furthermore, as the electron transfer is associated with a cationic matter flux to maintain the system electroneutrality,1 it is foreseen that the mass complex diagrams should present similarities in shape to that found in the complex capacitance diagrams (Figure 2). Consequently, it is observed that the maximum potential to the matter flux is the same potential found to the maximum current intensity, that is, 0.20 V. According to what was stated in eq 1, the ratio between the electrical and mass changes (F(dm/dq) function) allows the molecular weight of the species exchanged during the electrontransfer process to be estimated. The evolution of the real part value at the low-frequency limit of the mass transfer function is shown in Figure 3 with respect to the steady-state potential. The value of the low-frequency limit of the real part goes from about -5 g‚mol-1 at 0.50 V up to about -40 g‚mol-1 at 0.20 V. This is in agreement with the values found in ref 17in a more limited potential range. The ions present in the electrolyte are the potassium cations, the protons, and the chloride anions in the 0.25 M KCl, pH 2.23, solution. However, as it was demonstrated elsewhere,1 the anions do not take part in the compensation process during the ES T PB electron transfer. The value of the low-frequency limit of the real part of the mass transfer function is about -40 g‚mol-1 at 0.20 and 0.00 V so that the ES T PB electron transfer involves mainly a potassium cation exchange at these steadystate potentials (MWK+ ) 39 g‚mol-1). Accordingly, the exchange of the potassium ion takes place inside the solid phase without its solvation shell. This result corroborates the structural localization of the potassium ions inside the hydrated PB crystalline structure, as argued in a previous paper.1 Furthermore, from Figure 3, it is also possible to observe a slight decrease of the value of the low-frequency limit of the

J. Phys. Chem. B, Vol. 110, No. 39, 2006 19357 real part of the mass transfer function at about 0.10 V. This behavior could be attributed to the fact that at steady-state potential the potassium ions must be inserted simultaneously with other ionic movement or exchange. As it was experimentally demonstrated in a previous paper,1 indeed, the other processes involved are related to H+ and H3O+ exchange processes. It is important to mention the existence of a molecular oxygen reduction catalyzed by the PB films occurring as a secondary reaction, as stated in the Introduction section. However, as already commented, this reaction does not influence the mass change function because it is occurring on the surface. This reaction influences only the capacitance function. Regarding the parallel oxygen reaction, it could still be noted that at most cathodic potentials, a region where this secondary reaction domain, the F(dm/dq) function decreases in agreement with the voltammetric response.1 Although this trend can also be observed at more anodic potentials, the decrease at these anodic potentials can be attributed to another cause, that is, it is related to the proton exchanged process. The catalysis of the molecular oxygen reduction is not very important at these potentials.42 It could be evidenced that the decrease of the lowfrequency limit of the real part of the F(dm/dq) function cannot be associated with the nonfaradaic currents because, at anodic potentials, the voltammogram pattern (see Figure 1) presents only an oxidative current intensity (it is known that nonfaradaic currents lead to the presence of both oxidative and reductive currents, negative and positive polarization of the host during the voltammetric scans). To look deeply into the reaction mechanism of the ES T PB electron transfer, the ac-electrogravimetry in 0.25 M KCl was simulated in Figure 2 (continuous line) to be compared with the experimental results. The simulated curves were obtained according to the models represented by eqs 19 and 25. It is important to emphasize that, for the simulation of the experimental response, the nonfaradaic components were not considered in the calculations. The nonfaradaic impedance response has mainly two origins: (i) the charge distribution in the host/electrolyte interface related to potential differences existing between the electrode and electrolyte (i.e., double layer capacitance effects, Cdl) and (ii) the ohmic drop between the electrolyte and working electrode contact (i.e., noncompensated resistance, Rnc). Therefore, such a nonfaradaic impedance contribution can be simulated by means of a double layer capacitance, parallel to the faradaic impedance and in series with the noncompensated resistance.51 By considering these two components, the faradaic impedance, Zfaradaic(ω), can be calculated from the experimental global response, Zexperimental(ω)

Zfaradaic(ω) )

(

1 - jωCdl Z experimental(ω) - Rnc

)

-1

(26)

It is important to stress that the nonfaradaic components are evaluated from the experiment results at high frequency so that it does not influence the low-frequency faradaic impedance. Furthermore, as the double layer capacitance is constituted exclusively by heterogeneous processes on the surface of the working electrode, this component does not contribute to the experimental mass transfer function. Nevertheless, the contribution of the noncompensated resistance must be considered in the calculus of the mass change due to the faradaic processes.17 Furthermore, in the simulations observed in Figures 2 and 4, some dispersion on the frequency was considered. In other words, a constant phase element (CPE)-like behavior was introduced in the theoretical functions (see eqs 26 and 27). This



Figure 4. Complex capacitance curve of the partial electrogravimentric transfer function for ES T PB electron transfer. Experimental conditions were 0.25 M KCl, pH ) 2.23, and at 0.20 V. The continuous lines are the spectra simulated from the parameters of Table 1.

modification does not cause any change in the physical meaning of the kinetic parameters. Therefore, eqs 17 and 18 can be rewritten as

dmV dE

zi

n

)

i MWi ∑ x i)1 i

n

CV )

ziF ∑ i)1

-Gi (jω)0.9 + Ki

(27)

- Gi (jω)0.9 + Ki

(28)

It is important to emphasize that the CPE constant was considered as a constant and not significantly different from unity (i.e., it was considered as equal to 0.9 in all the simulations). A CPE-like behavior is generally assumed to be due to some nonuniformity of the electrode surface, and it is frequently observed in intercalation systems. For frequency mass change measurements, it was also observed and similar causes can be evoked. Such a behavior can also be attributed to the fact that the quartz microbalance has a radial dependence on the mass sensitivity which could affect the mass transfer function frequency by generating some frequency dispersion effects. Furthermore, the nonuniformity of the host thickness may also be a source of the frequency dispersion of the mass transfer function. Indeed, the thickness distribution has already been shown to influence the impedance response in that way.52,53 During the simulations, it was noted that the CPE behavior introduced in the model just causes a minimal error to the kinetic parameter values obtained from the modeling procedure. Pure capacitive behavior can also be used with a minimal variation to the kinetic parameter values extracted from the model.

From Figures 2 and 4, it can be observed that the electrical and mass frequency responses, simultaneously simulated, are both in good agreement with the experimental data (see Figure 4). Besides, the simultaneous adjustment of the experimental data to both electrical and mass theoretical transfer functions in a large frequency range favors the minimization of the errors. However, the most important conclusion brought by this study is that the adjustment of the theoretical functions to the experimental data validate the mechanism for interplay between electron and ionic fluxes previously proposed,1 which is reflected here in eqs 20 and 21. The proposed mechanism is totally in agreement with the experimental data, and now, the theoretical functions can be exploited to study separately the kinetic parameters of each exchange process. Interpretation of the Kinetic Parameters. The simulations of the ac-electrogravimetry data shown in Figures 2 and 3 and discussed in the previous section will now be used to obtain the kinetic parameters of the ES T PB electron-transfer process. The interpretation and the physicochemical meaning of the kinetic parameters lead to new insights concerning the electrochemical reactions involved in PB host material. The validation of the model and of the approach used here is reinforced by the analysis of the noncompensated impedance data obtained from high frequency, according to the procedure presented in the previous section. Hence, according to Table 1, when normalized to the working electrode geometrical area, it is verified that the noncompensated resistance value is constant whatever the applied potential, as is logical to assume. On the other hand, it can be observed that the double layer capacitance presents a maximum value at a similar potential where the complex capacitance spectra presents its maximum value, that



TABLE 1: Parameters Calculated from the Simulations of the Electrical and Mass Impedance Spectra at Each Polarization Potential of the ES T PB Electron-Transfer Mechanism in 0.25 M KCl and pH ) 2.23a E V

Rnc Ω cm2

Cdl 106F cm-2

KH3O+ s-1

-GH3O+ 104 mol s-1 V-1 cm-3

KH+ 10-2 s-1

-GH+ 104 mol s-1 V-1 cm-3

KK+ s-1

-GK+ 102 mol s-1 V-1 cm-3

-GO2 104 mol s-1 V-1 cm-3

0.50 0.40 0.30 0.25 0.20 0.15 0.10 0.05

6.7 6.7 6.8 6.8 6.8 6.7 6.8 6.8

4 5 5 12 12 29 14 7

0.5 0.9 1.5 2.0 3.0 4.0 5.0 9.0

1.4 26 41 61 120 150 200 260

18 13 4.5 3.0 2.3 1.9 1.4 1.0

1.2 1.2 1.0 1.1 1.1 0.9 0.8 0.6

140 8.0 5.4 3.4 3.2 5.0 6.0

1.4 4.1 5.1 4.3 3.4 2.5 2.0

1 2 2

a Figures 2 and 4 show the plot of these values as a function of the potential. R nc is the noncompensated resistance and Cdl the double layer capacitance.

is, 0.15 V. This behavior is attributed to the fact that at such a potential the maximal current intensity occurs so that a maximum amount of ionic species is expected to exist in the double layer region. Through the use of the approach of eq 25, it was possible to study the insertion/deinsertion mechanism proposed, even when a parallel heterogeneous irreversible reaction occurs. As a result, from Table 1, it is possible to infer that the oxygen reduction reaction occurs up to 0.20 V42 and increases when the potential becomes more cathodic. The other three intercalation processes occur at all polarization potentials, although the H+ and H3O+ exchanges are easily evidenced in the potential interval between [0.30, 0.50] V. Table 1 also shows the evolution of the kinetic parameter Ki of ES T PB electron transfer as a function of the potential. For reversible electrochemical processes, this kinetic parameter can be considered as the sum of two exponential trends related to the Butler-Volmer reaction rate constants (K ) kred/i0 + kox/i0). For nonreversible processes such as a parallel reaction, this parameter is equal to only one kinetic constant which is obtained directly from the dependence of Ki as a function of the polarization potential. However, when the reaction is reversible, the Ki value is practically equal to the reduction (cationic insertion) kinetic constants at cathodic potentials and to the oxidation (cationic deinsertion) kinetic constants at anodic potentials. The behavior of Ki as a function of potential naturally depends on the domination of one or the other rate constants. Finally, the reaction rate constants can be obtained from the dependence of Ki on the polarization potential. The rate constants of insertion and deinsertion calculated from the Ki values for the three cationic species involved on the interplay mechanism, that is, K+, H+, and H3O+ 1, are, therefore, presented in Figure 5. The good correlation between the rate constants and the exponential trends validates the ButlerVolmer dependence of these kinetic parameters proposed in the model formalism. On one hand, apparently, it is possible to infer that, for ES T PB electron-transfer processes, the ionic exchange process that can be considered as reversible in this potential range is only that related to K+. The explanation is due to the fact that this is the only process in which both direct (insertion) and reverse (deinsertion) constants were obtained. On the other hand, in the other two cases related to H+ and H3O+, either the process is reversible and one of the two constants, direct (insertion) or reverse (deinsertion), is very large compared with the other in the range of potential considered or the process is irreversible. More information on the kinetics can be obtained by performing the analysis of the Gi parameters, which for the ES T PB electron-transfer process is considered as a function of the potential. They are all presented in Table 1. For instance, if one anal-

yzes the GK+ value, that is, the Gi kinetic parameter associated with the K+ exchange, a peak is observed as a function of potential (Figure 6). This behavior is expected from eq 11. Therefore, the maximum observed is related to the fact that the GK+ parameter is the product of two opposed trends: the trends of the kinetic constants which decrease (increase) and the number density of intercalated K+ ions which increases (decreases). Nonetheless, the other electrochemical processes do not present any peaks. The explanation for this fact is similar to that already used to evaluate the evolution of Ki as a function of potential. For the intercalation reversible processes, the evolution of the Gi kinetic parameters with respect to the potential can be simulated by considering the kinetic laws. Therefore, from eq 9, it is possible to deduce the βi dependence with respect to the polarization potential

βi )

kox/i0Ny kox/i0 + kred/i0

(29)

Accordingly, the dependence of the Gi kinetic parameters with respect to the potential in an intercalation reversible reaction depends exclusively on the insertion and deinsertion kinetic constants of the ith faradaic process and on the number density of the total available sites for intercalation Ny (which depends exclusively on the host characteristics). Therefore, at steadyh i ) kox/i0η j i, Gi (combining eqs 11, 12, and state where kred/i0β 29) can be turned on

Gi ) (bred - box)kred/i

0

kox/i0Ny

kox/i0 + kred/i0

(30)

Figure 6 shows the simulations of the Gi kinetic parameters with respect to the applied potential or potential. All of the simulations are in agreement with the experimental data. Gi or Ki values can be calculated from the sum of the two kinetic constants involved in each process (Ki ) kred/i0 + kox/i0). Both parameters are in good agreement with the experimental results and reinforce the theory behind the transfer functions and ionic exchange model proposed. As a result, from these simulations, it is possible to fully characterize the PB electrochemical processes, since all kinetic parameters of the reversible PB electrochemical processes are calculated from this study (see Figures 5 and 6). It is possible to say that the K+ and H+ exchange involves reversible reactions whereas the H3O+ exchange involves an irreversible reaction, since the GH3O+ kinetic parameter cannot be simulated from eq 30. The experimental value of the total electroactive sites (Nt) ) involved in the ES T PB electron-transfer process is Nexp t 4.5 mmol‚cm-3 obtained from the voltammetric charge (by



Figure 5. Change of the kinetic parameters Ki (Table 1) vs the polarization potential. The continuous lines are the simulations of the reaction rate constants: (a) kox/K+0 ) 0.014 e23E s-1, R ) 0.975, kred/K+0 ) 8.6 e-6E s-1, R ) -0.972; (b) kox/H+0 ) 0.007 e6.7E s-1, R ) 0.990; and (c) kred/H3O+0 ) 10 e-6.1E s-1, R ) 0.994, in which E is the applied potential.

means of the Faraday laws by considering one electron for each electronic transfer). At the same time, the theoretical value obtained by considering the NK+ and NH+ available sites for insertion is NK+ + NH+ ) 1.6 + 2.1 ) 3.7 mmol‚cm-3. Note that it is impossible to obtain directly the value for NH3O+ from GH3O+ because it was shown that it is a nonreversible process (and in this case it is not possible to calculate Ny for the process, since this can be associated with a reversible process). Nonetheless, note that NK+ + NH+ and Nexp are very close and, as a t result, it is possible to affirm that the accuracy involved in the calculations of the kinetic parameters is very high. Another important conclusion to be made here is that the values of NK+ + NH+ do not differ greatly. This is very interesting information that arises from the analysis and can lead to the conclusion that the high prominence of K+ exchange on the voltammetric pattern is due to a kinetic origin (because the rate constants of the K+ insertion/deinsertion process are larger than those for the H+ insertion/deinsertion process) and it is not related to the amount of available sites for intercalation/ deintercalation. As the value of the NH3O+ parameter cannot be calculated for the H3O+ exchange process, the simulation for the GH3O+ kinetic parameter could not be performed as well, according to the pattern presented in Figure 6 for this process. The irreversibility found for the H3O+ exchange process is in agreement with the fact that the cathodic (reduction) charge was systematically found higher than the anodic (oxidation) charge during the voltammetric cyclic experiments (Figure 1). This irreversible process is in accordance with the PB catalytic reaction of the proton reduction postulated by Abe et al.55 Accordingly and as

both reactions are irreversible, it is plausible to propose this reaction to explain the irreversibility of the hydrated proton reduction, as well as the fact that the cathodic (reduction) charge is systematically found higher than the anodic (oxidation) charge during the voltammetric cyclic experiments. From Figure 6, it is also possible to infer that the K+ and H+ exchange PB reversible electrochemical processes present different characteristic potentials: the electrochemical process which involves the K+ exchange has a potential peak at 0.25 V, whereas the process which involves the H+ exchange has a potential at 0.31 V. These potentials found from the model are in agreement with the values, found in the literature, extracted from voltammetric experiments,54 which corroborate the values of the kinetic parameters obtained here. The ratio between the Gi and Ki kinetic parameters for each separated exchange process corresponds to the derivative of ηi with respect to the polarization potential in the steady-state condition (eq 12, i.e., when ω f 0). Accordingly, Figure 7a shows the plot of the -Gi/Ki quantity for each exchange process involved in PB host material. In other words, it corresponds to the dη j i/dE function (see eq 12). Accordingly, Figure 7b shows the isotherm laws for the three ions involved in the PB electrochemistry1 obtained by integration of Figure 7a. Figure 7a allows the number density of the exchanged species to be separately extracted for each electrochemical process from the ac-electrogravimetric experiments. It is important to stress that the simulations from the kinetic parameters obtained from Figure 6 are in agreement with the results of Figure 7a. From Figure 7, it is also possible to confirm that the H+ and H3O+ exchange or movement are more



Figure 6. Curve of the kinetic parameters Gi (Table 1) vs the polarization potential. The continuous lines are the simulations of these kinetic parameters. The factors of these simulations, eq 30, are as follows: NK+ ) 0.0016 mol‚cm-3, kox/K+0 ) 0.016 e22E s-1, kred/K+0 ) 15 e-10E s-1, NH+ ) 0.0021 mol‚cm-3, kox/H+0 ) 0.003 e8E s-1, and kred/H+0 ) 0.01 e-1E s-1, in which E is the applied potential.

Figure 8. Schematic representation of the chemical reaction coupled to H3O+ exchange. Note that the apparent global reaction is the maintenance of the ES form of the host material. The proposed mechanism is based on the fact that the ES reacts chemically to give molecular hydrogen and simultaneously the PB formed due to this reaction is electrochemically reduced to ES. In the chemical part, it is not possible to know if H3O+ is the reaction that gives rise to H2 + H2O or if H+ is the reaction which gives the H2 molecule directly.

Figure 7. (a) Curve of the ratio -Gi/Ki (Table 1) vs the polarization potential. (b) Curve of the insertion with respect to the potential. The continuous lines are the simulations from the kinetic parameters of Figure 6.

pronounced at high anodic potential, whereas at higher cathodic potential the K+ exchange dominates.

From Figure 7b, it is also possible to evidence that ηH3O+ has practically a linear dependence in this potential range. Such dependence implies a constant rate of the process which is generally related to a chemical reaction, in this case, a chemical reaction occurring inside the PB host material. The first question that arises here is how to explain such an apparently contradictory result. First, the H3O+ movement is most likely the same as the one proposed by Abe et al.,55 the catalytic reduction of the proton. This reaction is, therefore, a chemical catalytic reaction of proton reduction that occurs in parallel to the movement of H3O+ due to the charge-transfer reaction. Figure 8 helps to better understand the proposal made here based on the evaluation of the kinetic parameter of the mechanism. The mechanism says that there is a chemical reaction in which ES is converted to PB by means of the following reaction


2H+ (or H3O+) + 2e- f H2 (+ 2H2O)


(31)

in which e- is used to convert ES into PB. However, during the application of the potential, there is an H3O+ movement occurring in parallel to the chemical reaction in which PB is converted into ES. The apparent global reaction is the maintenance of the ES form of the host material so that it is possible to propose that the ES reacts chemically to give molecular hydrogen, and simultaneously, the PB formed due to this reaction is electrochemically reduced to ES. Beside this, another important consequence that comes from the analysis of the kinetic parameters is that both kinetic reduction rate constants for K+ and H3O+ insertions are similar (Figure 5). In other words, the insertion (reduction) kinetic constant of this process is similar to that occurring during potassium (reduction) insertion. Therefore, it is possible to say that the hydrated proton reduction takes place in the K+ interstitial sites. The schematic representation of the reaction mechanism involving proton reduction reaction of Figure 8 is useful in understanding the nonstability observed in the ES compound when exposed to air. The ES compounds when exposed in ambient air without polarization are spontaneously converted into PB form and such irreversible change can be explained by the chemical reaction scheme of Figure 8. Furthermore, this reaction can explain the fact that the voltammetric reduction charge is always bigger than the voltammetric oxidation charge (Figure 1). Conclusion The different electrochemical processes which take place during the Prussian Blue ionic insertion/deinsertion mechanism were fully separated by using the most recent frequency response electrochemical technique such as ac-electrogravimetry. The interplay between ionic and electronic fluxes was kinetically quantified and analyzed. The kinetic analysis leads to the conclusion that K+ and H+ exchange processes are both reversible. The irreversible H3O+ exchange process leads to the conclusion that the H3O+ movement occurs parallel to a chemical reaction in which PB is converted into ES. The approach given here is in accordance with the PB catalytic reaction of proton reduction postulated by Abe et al.55 However, the kinetic analysis made here gives more information on this catalytic proton reduction, showing that it is part of a mechanism involving the H3O+ exchange. Therefore, the apparent global reaction is the maintenance of the ES form of the host material. The proposed mechanism is based on the fact that ES reacts chemically to give molecular hydrogen and simultaneously the PB formed due to this reaction is electrochemically reduced to ES. Finally, the analysis of the kinetic parameters indicates that the calculated number density of available sites for K+ and H+ exchange processes are very similar so that the high prominence of K+ exchange observed in the voltammetric patterns is due to a kinetic origin and is not related to the amount of sites available for intercalation/deintercalation. As a conclusion, the model can now be used to foresee and quantify trends besides optimizing the applications of PB host materials as a function of electrochemical potentials. It could be exploited especially in regards to amperometric biosensors applications. Acknowledgment. This work has been supported by FEDERCICyT Project CTQ 2004-08026/BQV. D.G.-R. acknowledges a Fellowship from the Generalitat Valenciana, Postdoctoral Program. P.R.B. acknowledges the Saõ Paulo state research

funding institution (FAPESP) under Project No. 02/06693-3. J.J.G.-J. acknowledges their position to the Ramon y Cajal Program (Spanish Ministry of Education and Science). We appreciate the very useful discussions with Nuria PastorNavarro. Symbol Legend bi ) exponential factor of the kinetic constant of the ith faradaic process Ci ) capacitance function of the ith faradaic process CV ) total volumetric capacitance function Cirrev ) concentration of reactant species in the electrolyte e ) positive elementary charge e- ) electron E ) potential E h ) polarization potential to reach the steady state ∆E ) oscillating potential perturbation ∆E ) amplitude of the potential perturbation EQCM ) Quartz Crystal Microbalance EIS ) Electrochemical Impedance Spectroscopy ES ) Everit’s Salt F ) Faraday constant F(dm/dq) ) ratio between the change of mass and the electrical charge passed during an electrogravimetry experiment j ) square root of -1 ki ) kinetic constant of the ith faradaic process khi ) preexponential factor of the kinetic constant of the ith faradaic process ki0 ) kinetic constant in the steady state of the ith faradaic process KhFek[Fe(CN)6]l‚mH2O ) Prussian Blue mi ) mass deposited on the working electrode due to the ith faradaic process mV ) mass deposited on the working electrode per volume of the electrochemical film mV23 ) mass deposited on the working electrode per volume of the electrochemical film where the n°1 response is eliminated MWi ) molecular weight of the charged species involved in the ith faradaic process Ni ) total number density of intercalated atoms in a given i kind of guest PB ) Prussian Blue qi ) electrical charge due to the ith faradaic process t ) time V ) volume xi ) charge of the exchanged species y ) occupancy fraction Z ) impedance function zi ) number of electrons involved in the faradaic process i ) -1 when there is an insertion/deinsertion process of cationic species occurring in the host and +1 when there is an insertion process of anionic species βi ) number density of the free host sites available for cation intercalation or of intercalated anions ηi ) number density of intercalated cations or of free host sites available for anion intercalation η j i ) number density of intercalated cations or of free host sites available for anion intercalation in the steady state ∆ηi ) variation of the number density of intercalated cations or of free host sites available for anion intercalation during the ac experiment ∆ηi ) amplitude of variation of the number density of intercalated cations or of free host sites available for anion intercalation during the ac experiment

Ion Exchange in KhFek[Fe(CN)6]l‚mH2O Compounds µi ) chemical potential of the exchanged atoms due to the i processes νi ) percentage of the electrical charge balanced by the participation of the ion involved in the global electrochemical process ω ) Frequency of the potential perturbation in an ac experiment References and Notes (1) Gimeńez-Romero, D.; Bueno, P. R.; Gabrielli, C.; Perrot, H.; Garcia-Jarenõ, 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) Bruckenstein, S.; Hillman, A. R. J. Phys. Chem. 1988, 92, 4837. (4) Orata, D.; Buttry, D. A. J. Am. Chem. Soc. 1987, 109, 3574. (5) Torresi, R. M.; Cordoba-Torresi, S. I.; Gabrielli, C.; Keddam, M.; Takenouti, H. Synth. Met. 1993, 61, 291. (6) Naoi, K.; Lien, M.; Smyrl, W. H. J. Electrochem. Soc. 1991, 138, 440. (7) Maiai, G.; Torresi, R. M.; Ticianelli, E. A.; Nart, F. C. J. Phys. Chem. 1996, 100, 15910. (8) Deslouis, C.; Musiani, M.; Tribollet, B. J. Phys. Chem. 1996, 100, 8994. (9) Deslouis, C.; Musiani, M.; Tribollet, B.; Vorotyntsev, M. A. J. Electrochem. Soc. 1995, 142, 1902. (10) Glarum, S. H.; Marshall, J. H. J. Electrochem. Soc. 1987, 134, 142. (11) Mathias, M. F.; Haas, O. J. Phys. Chem. 1992, 96, 3174. (12) Matencio, T.; Vieil, E. Synth. Met. 1991, 44, 349. (13) El Rhazi, M.; Lopez, C.; Deslouis, C.; Musiani, M. M.; Tribollet, B.; Vieil, E. Synth. Met. 1996, 78, 59. (14) Barbero, C.; Miras, M. C.; Kotz, R.; Haas, O. J. Electroanal. Chem. 1997, 437, 191. (15) Henderson, M. J.; Hillman, A. R.; Vieil, E. J. Phys. Chem. B 1999, 103, 8899. (16) Henderson, M. J.; Hillman, A. R.; Vieil, E. J. Electroanal. Chem. 1998, 454, 1. (17) Gabrielli, C.; Garcia-Jarenõ, J. J.; Keddam, M.; Perrot, H.; Vicente, F. J. Phys. Chem. B. 2002, 106, 3182. (18) Garcia-Jarenõ, J. J.; Sanmatias, A.; Navarro-Laboulais, J.; Vicente, F. Electrochem. Acta 1998, 43, 235. (19) Bisquert, J. Electrochim. Acta 2002, 47, 2435. (20) Diard, J. P.; Montella, C. J. Electroanal. Chem. 2003, 557, 19. (21) Benito, D.; Gabrielli, C.; Garcia-Jarenõ, J. J.; Keddam, M.; Perrot, H.; Vicente, F. Electrochem. Commun. 2002, 4, 613. (22) Bourkane, S.; Gabrielli, C.; Keddam, M. Electrochim. Acta 1989, 34, 1081. (23) Cordoba-Torresi, S.; Gabrielli, C.; Keddam, M.; Takenouti, H.; Torresi, R. J. Electroanal. Chem. 1990, 290, 261. (24) Bourkane, S.; Gabrielli, C.; Huet, F.; Keddam, M. Electrochim. Acta 1993, 38, 1023, 1827. (25) Gabrielli, C.; Keddam, M.; Perrot, H.; Torresi, R. J. Electroanal. Chem. 1994, 378, 85.

J. Phys. Chem. B, Vol. 110, No. 39, 2006 19363 (26) Bohnke, O.; Vuillemin, B.; Gabrielli, C.; Keddam, M.; Perrot, H.; Takenouti, H.; Torresi, R. Electrochim. Acta 1995, 40, 2755. (27) Gabrielli, C.; Keddam, M.; Minouflet, F.; Perrot, H. Electrochim. Acta 1996, 41, 1217. (28) Yang, H.; Kwak, J. J. Phys. Chem. B 1997, 101, 774. (29) Yang, H.; Kwak, J. J. Phys. Chem. B 1997, 101, 4656. (30) Yang, H.; Kwak, J. J. Phys. Chem. B 1998, 102, 1982. (31) Gabrielli, C.; Garcia-Jarenõ, J. J.; Keddam, M.; Perrot, H.; Vicente, F. J. Phys. Chem. B 2002, 106, 3192. (32) Gabrielli, C.; Garcia-Jarenõ, J. J.; Perrot, H. Electrochim. Acta 2001, 46, 4095. (33) Gabrielli, C.; Keddam, M.; Nadi, N.; Perrot, H. Electrochim. Acta 1999, 44, 2095. (34) Gabrielli, C.; Keddam, M.; Nadi, N.; Perrot, H. J. Electroanal. Chem. 2000, 485, 101. (35) Gabrielli, C.; Keddam, M.; Perrot, H.; Pham, M. C.; Torresi, R. Electrochim. Acta 1999, 44, 4217. (36) Gabrielli, C.; Garcia-Jarenõ, J. J.; Perrot, H. ACH-Models Chem. 2000, 137, 269. (37) Garcia-Jarenõ, J. J.; Sanmatias, A.; Vicente, F.; Gabrielli, C.; Keddam, M.; Perrot, H. Electrochim. Acta 2000, 45, 3765. (38) Oh, I.; Lee, H.; Yang, H.; Kwak, J. Electrochem. Commun. 2001, 3 274. (39) Jin, W.; Toutianoush, A.; Pyrasch, M.; Schnepf, J.; Gottschalk, H.; Rammensee, W.; Tieke, B. J. Phys. Chem. 2003, 107, 12062. (40) Bortels, L.; Van den Bossche, B.; Deconinck, J.; Vandeputte, S.; Hubin, A. J. Electroanal. Chem. 1997, 429, 139. (41) Bard, A. J.; Faulkner, L. R. Electrochemical Methods Fundamentals and Applications, 2nd ed.; John Wiley & Sons: New York, 2001; pp 370371. (42) Itaya, K.; Shoji, N.; Uchida, I. J. Am. Chem. Soc. 1984, 106, 3423. (43) Bruckenstein, S.; Hillman, A. R. J. Phys. Chem. 1991, 95, 10748. (44) Hillman, A. R.; Loveday, D. C.; Swann, M. J.; Brukenstein, S.; Wilde, C. P. J. Chem. Soc., Faraday Trans. 1991, 87, 2047. (45) Garcia-Jarenõ, J. J.; Gimeńez-Romero, D.; Vicente, F.; Gabrielli, C.; Keddam, M.; Perrot, H. J. Phys. Chem. B 2003, 107, 11321. (46) Itaya, K.; Ataka, T.; Toshima, S. J. Am. Chem. Soc. 1982, 104, 4767. (47) Sauerbrey, G. Z. Phys. 1959, 155, 206. (48) Brukenstein, S.; Brzezinska, K.; Hillman, A. R. Electrochim. Acta 2000, 45, 3801. (49) Garcia-Jarenõ, J. J.; Gabrielli, C.; Perrot, H. Electrochem. Commun. 2000, 2, 195. (50) Gileady, E.; Tsionsky, V. J. Electrochem. Soc. 2000, 147, 567. (51) Prokopowiz, A.; Opallo, M. Solid State Ionics 2003, 157, 209. (52) Gabrielli, C.; Takenouti, F.; Haas, O.; Tsukada, A. J. Electroanal. Chem. 1991, 302, 59. (53) Mathias, M. F.; Haas, I. J. Phys. Chem. 1992, 96, 3175. (54) Garcia-Jarenõ, J. J.; Navarro-Laboulais, J.; Vicente, F. Electrochim. Acta 1997, 42, 1473. (55) Abe, T.; Taguchi, F.; Tokita, S.; Kaneko, M. J. Mol. Catal. A: Chem. 1997, 126, L89.

Kinetic Aspects of Ion Exchange in KhFek[Fe(CN)6]lÂ·mH2O

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