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FeII redox couple at a Nafion®-coated Pt electrode
Journal of
Electroanalytical Chemistry Journal of Electroanalytical Chemistry 566 (2004) 269–280 www.elsevier.com/locate/jelechem
EIS study of the FeIII/FeII redox couple at a NafionÒ -coated Pt electrode Jean-Paul Diard, Nicolas Glandut *, Bernard Le Gorrec 1, Claude Montella
2
Ecole Nationale Superieure dÕElectrochimie et dÕElectrom etallurgie de Grenoble, Laboratoire dÕElectrochimie et de Physico-chimie des Materiaux et des Interfaces, UMR 5631 CNRS-INPG-UJF, Domaine Universitaire, BP 75, 38 402 Saint-Martin-dÕHeres Cedex, France Received 26 June 2003; received in revised form 22 October 2003; accepted 8 November 2003
Abstract The redox reaction FeIII + e $ FeII is studied at a NafionÒ -filmed Pt electrode using both steady-state and EIS methods. It is shown that the fitting of the EIS experimental data can be greatly improved by using the new concept of a Ôtheoretical–experimental hybrid modelÕ, i.e., by introducing experimental data obtained at a bare electrode in the theoretical model for a filmed electrode. It is also shown that a combined steady-state and EIS study cannot lead to a separate determination of the three key parameters of the system: the diffusion coefficient in the polymer film (Df ), the thickness of the film (L), and the partition coefficient between the solution and the polymer (c), only the two parameters cDf =L and L2 =Df being obtained from experimental data by a curve fitting procedure. Finally, an exhaustive theoretical EIS study in quiescent solution is also proposed. Ó 2003 Elsevier B.V. All rights reserved. Keywords: NafionÒ -coated electrode; Redox reaction; Electrochemical impedance spectroscopy; Complex non-linear least-squares fitting; Theoretical–experimental hybrid model
1. Introduction The idea to study metallic electrodes coated with NafionÒ , 3 is at the origin of numerous papers written during the 1980s [1–7]. Some of these papers [1,2] dealt with mediated electron transfer, i.e., the oxidation (or reduction) of an electroactive species in solution is due to the presence of an oxidised (or reduced) species in the NafionÒ film, or [3–5] with NafionÒ containing catalysts such as Co3þ /Co2þ complexes or Pt microparticles. The other papers [6,7] dealt with ÔemptyÕ NafionÒ films. The electroactive species in solution diffuses through the film, with a diffusion coefficient (Df ) which is not the same as that in solution (Ds ), and reacts at the electrodejfilm interface.
*
Corresponding author. E-mail address: [email protected] (N. Glandut). 1 Member of the Institut Universitaire de Technologie 1 de Grenoble. 2 Member of PolytechÕ Grenoble. 3 NafionÒ is a registered trademark of E. I. du Pont de Nemours and Company. 0022-0728/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2003.11.038
During the same period, Bard, Saveant and coworkers [8–10] worked on poly(vinyl ferrocene)-coated electrodes, and successfully modeled the mass transport of a species to a rotating disk electrode (RDE) covered with a thin polymer layer, under steady-state conditions. This model is, of course, applicable to the metallic electrodejNafionÒ jsolution system. Due to its applications in proton exchange membrane fuel cells, NafionÒ is again being studied increasingly, and especially when it is deposited on metallic electrodes. Based on the pioneering work of Bard, Saveant and co-workers [8–10], recent papers concerning NafionÒ -coated electrodes can be cited: Zecevic et al. [11] and Maruyama and Abe [12] who studied the oxygen reduction reaction (ORR) at NafionÒ -coated Pt or glassy carbon RDEs, and Mello and Ticianelli [13] and Maruyama et al. [14] who studied the hydrogen oxidation reaction (HOR) at the same kind of electrodes. Nevertheless, an important point must be mentioned concerning steady-state measurements at NafionÒ coated RDEs. In accordance with the model of Bard, Saveant and co-workers [8–10], the diffusion coefficient in the film, Df , the thickness of the film, L, and the
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partition coefficient between the solution and the polymer, c, cannot be separately determined from steadystate measurements, because only the ratio cDf =L is accessible. This means that two parameters must be found first from another method in order to determine the third one. The purpose of this work is to study the redox reaction FeIII + e $ FeII at a NafionÒ -filmed Pt electrode [15,16] using electrochemical impedance spectroscopy (EIS). The measurements are made at the equilibrium potential in quiescent solution (semi-infinite linear diffusion in solution), and under convective diffusion conditions (bounded diffusion approximation in solution). It is shown that, by using the impedance technique, the new accessible parameter is the time constant in the film, sf ¼ L2 =Df , and therefore one parameter, the thickness of the film for example, must be known in order to measure the two others. Hence the problem raised by Bard, Saveant and co-workers [8–10] remains unsolved. Notice that Yu and Yen have recently made a similar remark in their study of Tijporous TiO2 electrodes [17]. Notice also that in 1987, in their theoretical EIS study of a porous layer-coated RDE, Deslouis et al. found [18] that Df and L can be determined separately, but c was not taken into account in their model (which amounts to taking c ¼ 1). Moreover, it is also shown in this paper that the fitting of the experimental data can be greatly improved by using a Ôtheoretical–experimental hybrid modelÕ, i.e., by introducing experimental data obtained at a bare electrode into our theoretical model for a coated electrode. As far as we know, it is the first time that such a fitting procedure is used. This Ôhybrid modelÕ allows us to avoid (but not to solve) the deviations from Warburg behaviour due to natural convection and edge effects when using a fixed electrode [19]. It also allows us to avoid the deviations from the Nernst model (bounded diffusion approximation) when using a rotating disk electrode [20].
2. Experimental The working electrode (WE) consisted in a polycrystalline Pt disk sealed in a Teflon holder, and had a geometric surface area of 3.14 102 cm2 (0.2 cm in diameter). Before the NafionÒ film was deposited and for the experiments on bare Pt, the WE was polished with alumina (3 lm) and rinsed with Millipore water in an ultrasonic bath. The polymer films were spin-coated at 125 rpm for 15 min from a 1.25% NafionÒ solution [21] (Aldrich 5 wt% NafionÒ solution diluted four times with methanol). The weight of the film was 0.1 mg. Knowing the density of recast NafionÒ (1.8 g cm3 [16]) and the effective area (0.2 cm2 ), the thickness of the film could be estimated at 3 lm. The surface of the
Fig. 1. Atomic force microscopy (AFM) roughness profile of the surface of a spin-coated NafionÒ film.
NafionÒ film was characterized by atomic force microscopy (AFM) with a Nanoscope III Digital Instruments system. Fig. 1 shows that the film is very smooth (roughness of 5 nm) and not porous. The WE was then mounted on a EDI 101 type rotating disk electrode (Radiometer Analytical, Villeurbanne, France). Electrochemical measurements were carried out at room temperature (25 °C) in a standard three-electrode cell configuration. An Autolab PGSTAT 30 (Eco Chemie, The Netherlands), which was controlled by GPES 4.9 and FRA 4.9 softwares, was used to perform steady-state and impedance measurements. The WE potential was measured with respect to a SCE reference electrode and a Pt wire was used as a counter electrode. The solution was a 3.9 102 M Fe(ClO4 )3 , 6.3 102 M Fe(ClO4 )2 in 1 M HCl aqueous solution (Aldrich chemical reagents and Millipore water), and was deaerated by bubbling pure Ar gas. Before taking a steady-state jf vs. E curve or an impedance diagram, the electrode was cleaned by cycling the potential at 1 V s1 between the beginning of H2 evolution and the beginning of Cl2 evolution (between )0.25 and +1.1 V vs. SCE) [22,23]. Six electrode rotation rates were used: X ¼ 50, 200, 500, 1000, 2000 and 4000 rpm. For measurements in quiescent solution, the electrode rotation rate was zero.
3. Description of the model The redox reaction FeIII + e $ FeII (formally noted: O + e $ R) was studied on a Pt electrode covered with a thin NafionÒ film. The notation FeIII and FeII is used to denote the formation of complex ions from Fe3þ and Fe2þ in the presence of chloride ions. The thickness of the film is L. In solution (s), mass transport of redox species is governed by diffusion and convection. In the film (f), it is governed only by diffusion. Migration effects can be neglected because supporting electrolytes
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are present in high concentration in both media; HCl 1 M in solution, and –SO3 H 1 M in the film. The sulfonate groups –SO 3 are fixed into the polymeric matrix. Because of the Donnan effect [24], chloride ions are also present in the film. DO;s and DR;s are the diffusion coefficients of oxidized and reduced species, respectively, in the solution. In the polymer phase, the diffusion coefficients are considered as equal for the sake of simplicity DO;f ¼ DR;f ¼ Df :
cO ¼ cR ¼ c:
ð2Þ
There is no diffuse double layer at the polymerjsolution interface, as was recently shown by Wang and Bard [25]. O and R are the constant concentrations of oxidized and reduced species, respectively, in the electrolyte bulk. In Fig. 2 are displayed schematically two steady-state concentration profiles in our system, one at the equilibrium potential (Fig. 2(A)), the other when a species, O or R, is consumed at the metaljpolymer interface (Fig. 2(B)).
0
0
L
∗
O =R
L+δ
film 0
0
x
O∗ = R∗
L
R expðao nÞ O expðar nÞ ; 1 þ cm1 f þ m1R;s expðao nÞ þ cm1 f þ m1O;s expðar nÞ ck 0
ð3Þ where n ¼ nf ðE E0 Þ is the dimensionless potential, E0 the standard potential, k 0 the standard rate constant for electron transfer, n the electron number (n ¼ 1), F the Faraday constant, f ¼ F =ðRT Þ ¼ 38:9 V1 at 25 °C, ao and ar are the symmetry factors for electron transfer in the direction of oxidation and reduction, respectively (ao þ ar ¼ 1), mf ¼ Df =L is the diffusion constant in the polymer film, mO;s ¼ DO;s =dO and mR;s ¼ DR;s =dR are the diffusion–convection constants in solution for the oxi1=3 dized and reduced species, respectively, dO ¼ 1:611DO;s 1=3 m1=6 X1=2 and dR ¼ 1:611DR;s m1=6 X1=2 are the characteristic lengths of diffusion–convection in solution from Levich theory [27], and m is the kinematic viscosity of the solution. The other symbols have been detailed previously. The limiting current densities due to mass transport of the oxidized and reduced species, jdO and jdR , are obtained by setting n ! 1 in Eq. (3) 1 1 jdO ¼ lim jf ¼ nF O þ ð4Þ n!1 cmf mO;s
n!þ1
∗
solution
O,R
Pt
B
jf ¼ nF
jdR ¼ lim jf ¼ nF R
γ O∗ = γ R∗
O,R A
According to the assumptions made in Section 3, the theoretical expression of the steady-state current density (jf ) vs. potential (E) curve for a redox reaction at an ionconducting polymer-coated RDE is [26]
and
solution
film
Pt
4. Steady-state study
ð1Þ
The redox species, O and R, can transfer through the polymerjsolution interface (x ¼ L). It is assumed that this transfer is kinetically reversible (quasi-equilibrium conditions). Hence the partition coefficients cO and cR can be defined as: cO ¼ OðL ; tÞ=OðLþ ; tÞ and cR ¼ RðL ; tÞ=RðLþ ; tÞ, where OðL ; tÞ and RðL ; tÞ are the interfacial concentrations of oxidized and reduced species on the polymer side (x ¼ L ), and OðLþ ; tÞ and RðLþ ; tÞ are the interfacial concentrations on the solution side (x ¼ Lþ ). In what follows, we will assume that
L+δ
x
Fig. 2. Schematic steady-state concentration profiles of redox species for the reaction O + e $ R at a polymer-coated electrode, (A) at the equilibrium potential (E ¼ Eeq , jf ¼ 0), and (B) when a species is consumed (E 6¼ Eeq ; jf 6¼ 0). Bulk concentrations: O ¼ R , and c ¼ 1:5.
271
1 1 þ : cmf mR;s
ð5Þ
Eqs. (3)–(5) clearly show that only the two terms ck o and cmf can be determined from experimental data obtained under steady-state conditions, whereas three important parameters concerning mass transport in the polymer film are present: c, Df and L. That is the reason why c, Df and L cannot be independently determined from steady-state jf vs. E measurements. In Fig. 3 are displayed steady-state jf vs. E experimental curves for the redox reaction FeIII + e $ FeII investigated at a bare Pt RDE (Fig. 3(A)), and at a NafionÒ -coated Pt RDE (Fig. 3(B)), for several electrode rotation rates. The shape of these curves is classical [26,28], and is always the same whatever the value of X. At each X value, mO;s and mR;s can be calculated from the well-known relations: jdO;bare ¼ nFmO;s O and jdR;bare ¼ nFmR;s R , where jdO;bare and jdR;bare are the limiting current densities due to mass transport at a bare electrode (Fig. 3(A)). The diffusion–convection constants, mO;s and mR;s , are then injected in Eqs. (4)
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the well-known expression which applies, for example, to bare Pt electrodes in quiescent solutions [28] 1 ð8Þ MðsÞ ¼ pffiffiffiffiffiffiffi : sDs The concept of the mass-transport function was recently extended to multilayer systems [29–32]. The metaljpolymer filmjelectrolyte system dealt with in this paper is an example. Hence, for a polymer coated-electrode, the characteristic mass-transport function for soluble species at the metal j polymer interface (x ¼ 0) can be written as [29–32] pffiffiffiffiffiffi tanhpffiffiffiffiffiffi sf s cM2 ðsÞ þ m f sf s M1 ðsÞ ¼ ð9Þ pffiffiffiffiffiffi pffiffiffiffiffiffi ; 1 þ cM2 ðsÞmf sf s tanh sf s where M2 is the mass-transport function at the polymer surface (x ¼ Lþ ), sf ¼ L2 =Df the diffusion time constant in the film, and mf ¼ Df =L the diffusion constant in the film. In a quiescent solution, M2 is characteristic of semiinfinite linear diffusion, and is given by the same relationship as in Eq. (8) 1 M2 ðsÞ ¼ pffiffiffiffiffiffiffi : sDs
ð10Þ
Eqs. (9) and (10) lead to
Fig. 3. Experimental steady-state current density vs. potential curves, (A) using a bare Pt RDE, and (B) a NafionÒ -coated Pt RDE, at 25 °C in a 3.9 102 M FeIII + 6.3 102 M FeII + 1 M HCl aqueous solution, with electrode rotation rates (a) 50; (b) 200; (c) 500; (d) 1000; (e) 2000; (f) 4000 rpm.
and (5), and knowing the values of jdO and jdR (Fig. 3(B)), we can obtain the mean value of cmf cmf ¼ 5:5 103 cm s1 :
ð6Þ
5. EIS study in quiescent solution 5.1. Theoretical The characteristic mass-transport function for soluble species at the electrodejelectrolyte interface (x ¼ 0) is defined as [28] MðsÞ ¼
Dcð0; sÞ DJ ð0; sÞ
;
ð7Þ
where Dc and DJ are the Laplace transforms of the concentration and flux variations when the electrode is submitted to a small perturbation in potential or in current, and s is the Laplace complex variable. Due to semi-infinite linear diffusion of soluble species in the electrolyte, the mass-transport function takes on
pffiffiffiffiffiffi tanh sf s c pffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffi m f sf s s Ds M1 ðsÞ ¼ pffiffiffiffiffiffi pffiffiffiffiffiffi : c 1 þ pffiffiffiffiffiffiffiffi mf sf s tanh sf s s Ds
ð11Þ
Using the dimensionless Laplace variable, pffiffiffiffiffiffiffiffiffiffiffiffiffi S ¼ sf s, and the dimensionless parameter K ¼ Ds =Df =c, Eq. (11) can be written in a dimensionless form as pffiffiffi 1 þ K tanh S pffiffiffi : ð12Þ mf M1 ðSÞ ¼ pffiffiffi S ðK þ tanh S Þ The K parameter can be written alternatively as K ¼rf =rs , that is the ratio of the Warburg factor for semi-infinite linear diffusion pffiffiffiffiffi in the polymer film, rf ¼ ð1=O þ 1=R Þ=ðn2 fF c Df Þ, to the Warburg factor for a bare Pt electrode with semi-infinite linear diffusion pffiffiffiffiffiin quiescent solution, rs ¼ ð1=O þ 1=R Þ=ðn2 fF Ds Þ, provided the same diffusion coefficient is assumed for oxidized and reduced species. pffiffiffiffiffiffiffi Setting s ¼ ix and S ¼ iu in Eq. (12), where i ¼ 1, x ¼ 2pf is the angular frequency (rad s1 ), f the frequency (Hz), and u ¼ xsf the dimensionless frequency, the mass-transport function diagrams for several values of K are plotted in Fig. 4(A) using the Nyquist representation Imðmf M1 Þ vs. Reðmf M1 Þ. Figs. 4(B) and (C) also display the Bode plots of mf M1 , i.e., the plots of log jmf M1 j and / vs. log u. First of all, whatever the value of K, there is a common feature between these diagrams: their high frequency part is a )45° phase shift straight line characteristic of semi-infinite linear diffusion in the
J.-P. Diard et al. / Journal of Electroanalytical Chemistry 566 (2004) 269–280
intermediate values of K. In passing, note that our experimental results shown below (Fig. 7) are very close to the diagram plotted in Fig. 4A, c. If it is assumed here that c 1, it can then be concluded that the diffusion coefficient in the polymer phase is smaller than the diffusion coefficient in solution, but has the same order of magnitude (Df K Ds ).
e d c
1
5.2. Bare Pt electrode
b a
0
1 Re (mf M1 )
A
log m f M1
The impedance diagram shown in Fig. 5(A) is measured at a bare Pt electrode in quiescent solution (electrode rotation rate ¼ 0 rpm). The applied dc potential is the equilibrium potential (Eeq ¼ 0:460 V vs. SCE) and
f 4 e d c 2 b
0.6 12 5
a 0 –2
–4
0
4
log u
B
0
− Im Z/(Ω cm2)
−Im (m f M1)
f
0
0 0.3
φ/˚
0.9
−1.6
0 1 0
d
0
6 Re Z/(Ω cm2 )
A
e
12
f –4
0 log u
4
Fig. 4. Dimensionless mass-transport function mf M1 calculated from Eq. (12) and plotted using the Nyquist representation (A), and using the Bode coordinates (B: decimal logarithmp offfiffiffiffiffiffiffiffiffiffiffiffiffi amplitude, and C: phase shift vs. log u). S ¼ iu, u ¼ 2pf sf , and K ¼ Ds =Df =c ¼ 100 (a); 4 (b); 2 (c); 1 (d); 0.3 (e); 0.001 (f).
polymer film. Then, for lower frequencies, three limiting cases should be considered: (i) when K 1 (Figs. 4(a)), the impedance is close to a bounded diffusion impedance due to the polymer layer in series with a Warburg impedance due to diffusion in the electrolyte, these two impedances being well separated in frequency; (ii) when K ¼ 1 (Figs. 4(d)), the polymer-coated electrode can be treated as a bare electrode, and the impedance diagram cannot be distinguished from a Warburg impedance in the whole frequency domain; and (iii) when K 1 (Figs. 4(f)), the shape of the impedance is the shape of a restricted diffusion impedance in a certain frequency range. Other diagrams are plotted in Figs. 4(A–C) for
−1.6
800 − Im Mexp /(s cm− 1)
–90
2
−1
a
–45
43
6
b c
C
273
−1
400
0 21 0 B
0
400 Re Mexp /(s cm− 1 )
800
Fig. 5. (A) Impedance diagram (Nyquist representation) measured at the equilibrium potential (Eeq ¼ 0:460 V/ECS) for a bare Pt electrode in a 3.9 102 M FeIII + 6.3 102 M FeII + 1 M HCl quiescent aqueous solution. Sinusoidal perturbation amplitude dE ¼ 10 mV. Decimal logarithm of the frequency (Hz) is given on the graph. The inset shows the high frequency (HF) part of the diagram. (B) Masstransport experimental data Mexp calculated from the data of A and using Eqs. (13)–(18).
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the resulting dc current is zero: this is the only steady state under such conditions. This diagram can be modeled by the RandlesÕ equivalent circuit shown in Fig. 6 [33], where RX is the ohmic resistance, Rct the chargetransfer resistance, and Cdl the differential double-layer capacitance. The mass-transport impedance, Zmt , is given by Zmt ðsÞ ¼ ZO ðsÞ þ ZR ðsÞ;
ð13Þ
where ZO and ZR are the concentration impedances of the oxidized and reduced species, respectively, and s is the Laplace complex variable. ZO and ZR can be written as [28] ZO ðsÞ ¼ Rct Kr MðsÞ;
ð14Þ
ZR ðsÞ ¼ Rct Ko MðsÞ;
ð15Þ
where Ko and Kr are the rate constants for electron transfer in the direction of oxidation and reduction, respectively. Notice that we assume here that DO;s ¼ DR;s ¼ Ds . At the equilibrium potential, we can write
C dl
RΩ
Rct
Z mt
Fig. 6. RandlesÕ equivalent circuit for the redox reaction FeIII + e $ FeII studied by EIS at a bare Pt electrode or at a NafionÒ filmed Pt electrode, in quiescent solution or under diffusive and convective mass transport conditions.
0.6 12 2 − Im Z/(Ω cm2 )
5 4 3 0 0.3
0.9
6
0
6 Re Z/(Ω cm2 )
ð16Þ
and Rct ¼
n2 fF ða
1 1 ; ¼ nfj0 o Ko R þ ar Kr O Þ
ð17Þ
where j0 is the exchange current density. Combining Eqs. (8), (13)–(17) and using ao þ ar ¼ 1, the masstransport impedance can be expressed as 1 1 1 Zmt ðsÞ ¼ 2 pffiffiffiffiffiffiffi þ ; ð18Þ n fF sDs O R which is the well-known classical Warburg impedance. Its theoretical diagram in the Nyquist representation is a straight line with unity slope (phase shift / ¼ 45°). Nevertheless, the experimental phase shift is greater than 45° (/ 40° in Fig. 5(A)) because of edge effects and natural convection [19]. From Fig. 5(A) and using Eqs. (13)–(17), the experimental mass-transport function Mexp can be calculated at each frequency below 100 Hz (we are not interested here in the high frequency semi-circle due to Rct and Cdl ) as 1 1 1 2 Zmt;exp ðxÞ: ð19Þ Mexp ðxÞ ¼ n fF þ O R The Im Mexp vs. Re Mexp diagram is plotted in Fig. 5(B) using the Nyquist representation. As indicated above, Mexp ðxÞ deviates in the low frequency domain from the theoretical mass-transportpfunction related to semi-infiffiffiffiffiffiffiffiffiffiffi nite diffusion, MðxÞ ¼ 1= ixDs , because of edge effects and natural convection. From the Mexp data and using Mathematica software [34], an interpolating function can be constructed. This approximating function is represented by the solid line in Fig. 5(B). Mexp will be useful in what follows for the fitting of experimental measurements obtained with a NafionÒ -coated Pt electrode (see Section 5.3). 5.3. NafionÒ -coated Pt electrode
−1
0
j0 nF
−1.6
1 0
Ko R ¼ Kr O ¼
12
Fig. 7. Impedance diagram measured at the equilibrium potential (Eeq ¼ 0:460 V/ECS) for a NafionÒ -coated Pt electrode in a 3.9 102 M FeIII + 6.3 102 M FeII + 1 M HCl quiescent aqueous solution. The inset shows the HF part of the diagram.
pffiffiffiffiffiffiffiffiffiffi 5.3.1. Fitting with M2 (x) ¼ 1= ixDs Fig. 7 shows the impedance diagram for a Pt electrode covered with a thin NafionÒ film. The experimental conditions are the same as in Section 5.2. The high frequency part (HF: 100 kHz to 100 Hz) and the low frequency part (LF: below 0.1 Hz) of the two diagrams of Figs. 5(A) and 7 are quite similar. Indeed, a HF semi-circle due to Rct and Cdl , and a LF linear part due to diffusion in solution can be seen (see also the insets). Nevertheless, a difference is noticeable between these two diagrams: in the 100–1 Hz frequency range, the begining of a capacitive arc can be seen for the PtjNafionÒ electrode, whereas this arc is not visible for
J.-P. Diard et al. / Journal of Electroanalytical Chemistry 566 (2004) 269–280
the bare Pt electrode. Considering that the only changing condition between the two experiments is the presence or not of a NafionÒ film, this arc can be attributed to the NafionÒ film. This behaviour is in perfect agreement with the simulation above (see Section 5.1 and Fig. 4A, c). The diagram of Fig. 7 can also be modeled by the RandlesÕ equivalent circuit shown in Fig. 6. Eqs. (13) and (16) are still valid, but Eqs. (14) and (15) must be replaced by the following ZO ðsÞ ¼ Rct Kr M1 ðsÞ;
ð20Þ
ZR ðsÞ ¼ Rct Ko M1 ðsÞ;
ð21Þ
where M1 is given by Eq. (9). The expression of Rct is now a little different from that of Eq. (17)
In Eqs. (25) are given the cmf and sf values plus or minus two standard deviations (2 SD) i.e., there is a 95% chance that the actual values are within the given percentage range (confidence interval).
ð22Þ
At this point, the reader must be careful: the division of Rct for a bare electrode by Rct for a covered electrode, i.e., Eq. (17) divided by Eq. (22), not necessarily lead to the determination of c. Indeed, the exchange current density (j0 ) is, rigorously speaking, different for a bare electrode than for a covered one. In an analogous manner to the above, the combination of Eqs. (11), (13), (16) and (20)-(22) leads to the following equation where s ¼ ix 1 1 1 Zmt ðsÞ ¼ 2 þ n fF O R pffiffiffiffiffiffi tanh sf s 1 pffiffiffiffiffiffiffi þ pffiffiffiffiffiffi cmf sf s sDs : ð23Þ pffiffiffiffiffiffi pffiffiffiffiffiffi 1 1 þ pffiffiffiffiffiffiffi cmf sf s tanh sf s sDs Eq. (23) shows that only two terms can be determined from the experimental data: cmf ¼ cDf =L and sf ¼ L2 = Df , whereas three parameters concerning mass transport in the polymer film are present: c, Df and L. The problem is the same as in the steady-state study made above: Df and L cannot be determined independently by EIS, contrary to what Deslouis et al. [18] have reported. Indeed, in their model, the partition coefficient c was not taken into account, which amounts to taking c ¼ 1. Using Mathematica software [34], and especially its NonlinearRegress least-squares fit function based on the Levenberg–Marquardt algorithm, the experimental results (below 100 Hz) are fitted to the theoretical expression in Eq. (23). Note here that commercial fitting softwares, such as ZView [35,36], cannot be used because Eq. (23) shows that the impedance model cannot be expressed in terms of equivalent circuits. The chosen object-model distance to minimise is the weighted function nd X Zth ðfk ; pÞ Zexp;k 2 1 2 ; v ¼ ð24Þ nd np k¼1 Zexp;k
10
− Im Zmt /(Ω cm2)
1 1 ¼ : n2 fF cðao Ko R þ ar Kr O Þ nf cj0
where p is the list of parameters to be determined (here cmf and sf ), np is the number of parameters (np ¼ 2), Zexp;k is the impedance measured at the fk frequency, nd is the number of data ðfk ; Zexp;k Þ, k ¼ 1; . . . ; nd , and Zth is the calculated impedance at the fk frequency for the p parameter values. The best fit, represented by the solid line and the circles in Fig. 8(A), is obtained for the parameter values: 8 < cmf =ðcm s1 Þ ¼ 6:4 103 4%; ð25Þ s =s ¼ 5:6 102 10%; : f2 v ¼ 5:5 102 :
−1.6
−1
5
0 2 0
1
0
5 Re Zmt /(Ω cm2 )
A
10
10 –1.6 − Im Zmt /(Ω cm2)
Rct ¼
275
–1
5
0 2 0 B
0
1 5 Re Zmt /(Ω cm2 )
10
Fig. 8. (A) Mass-transport impedance data (the same than in Fig. 7 for f 6 100 Hz) fitted with the theoretical expression in Eq. (23) and for Ds ¼ 5 106 cm2 s1 [37]. (B) Same data fitted using the Ôtheoretical– experimental hybrid modelÕ, pffiffiffiffiffiffiffi i.e., by using Mexp shown in Fig. 5(B) instead of M2 ðsÞ ¼ 1= sDs in Eq. (23). In A and B, the small and large dots correspond to the data, and the solid lines and the circles correspond to the result of the fitting (see also Eqs. (25) and (26)).
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5.3.2. Fitting with Mexp Another way can be followed for the fitting of our experimental results: the experimental data Mexp obtained at a bare Pt electrode and including edge effects and natural convection (see Section 5.2 and Fig. 5(B)) can be introduced into Eq. (23) inpffiffiffiffiffiffiffiffiffiffi order to replace the theoretical function M2 ðxÞ ¼ 1= ixDs . This can be done numerically with Mathematica software [34] by using the interpolation function deduced from the Mexp points (Fig. 5(B)). To the best of our knowledge, this is the first time that such a model has been used for the fitting of experimental data. We have called this model a Ôtheoretical–experimental hybrid modelÕ, or Ôhybrid modelÕ. The best fit (see Fig. 8(B)) is obtained for the parameter values: 8 < cmf =ðcm s1 Þ ¼ 7:5 103 3%; ð26Þ s =s ¼ 4:2 102 6%; : f2 v ¼ 2:4 102 : Three observations can be made from these results. First, by eye, it can be seen in Fig. 8 that the result of the fit is better using the hybrid model than the theoretical model. This remark is particularly valid for the low frequencies, where deviation from Warburg behaviour occurs because of edge effects and natural convection, and shows the usefulness of our hybrid model. Next, the difference between the two v2 values confirms our visual observation mathematically. The v2 value in Eqs. (26), when the hybrid model is used, is approximatively better by a factor of two than the v2 value in Eqs. (25), when the theoretical model is used. Finally, although there is a visible difference between Figs. 8(A) and (B), the values of cmf and sf are quite similar in Eqs. (25) and (26). Moreover, the value of cmf obtained by EIS is of the same order as the value of cmf obtained in Eq. (6) from the steady-state polarization curves.
6. EIS study with RDE pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 6.1. Fitting with M4 (x) ¼ tanh ixss =ms ixss What is described in Section 5.3.1 is now applied to experiments with a NafionÒ -coated Pt RDE (rotating disk electrode). The experimental conditions are the same as above: measurements made at the equilibrium potential, identical electrolytic solution, etc. Therefore the electrode becomes a bilayer electrode. Indeed the Nernst layer in solution (d) must be considered now, in addition to the NafionÒ layer deposited on the Pt electrode (Fig. 2). Notice that an exhaustive theoretical study of bilayer electrodes has been made by Diard et al. [31] in the past. Fig. 9 shows the impedance diagrams plotted experimentally for six different electrode rotation rates (X).
These diagrams are all composed of a HF semi-circle due to Rct and Cdl . Two overlapping LF arcs are visible on the diagrams measured for low values of X (Figs. 9(A)–(C)), whereas only one LF arc is visible on the diagrams measured for high values of X (Figs. 9(D)– (F)). Here also, the diagrams of Fig. 9 can be modeled by the RandlesÕ equivalent circuit shown in Fig. 6. The concentration impedances, ZO and ZR , can be written as ZO ðsÞ ¼ Rct Kr M3 ðsÞ;
ð27Þ
ZR ðsÞ ¼ Rct Ko M3 ðsÞ;
ð28Þ
where Rct is the same as in Eq. (22). M3 can be written as M1 was previously written in Eq. (9) pffiffiffiffiffiffi tanh sf s cM4 ðsÞ þ pffiffiffiffiffiffi m f sf s M3 ðsÞ ¼ ð29Þ pffiffiffiffiffiffi pffiffiffiffiffiffi : 1 þ cM4 ðsÞmf sf s tanh sf s The difference between Eqs. (9) and (29) is that the M2 function, characteristic of semi-infinite linear diffusion, has been replaced by M4 which is a bounded diffusion transport function. Its well-known expression, using the Nernst approximation for convective diffusion, is [28] pffiffiffiffiffiffi tanh ss s M4 ðsÞ ¼ ð30Þ pffiffiffiffiffiffi ; ms s s s where ss ¼ d2 =Ds is the diffusion–convection time constant in solution, and ms ¼ Ds =d the diffusion–convection constant in solution. Combining Eqs. (13), (16), (22) and (27)–(30) gives 1 1 1 Zmt ðsÞ ¼ 2 þ n fF O R pffiffiffiffiffiffi pffiffiffiffiffiffi tanh ss s tanh sf s þ pffiffiffiffiffiffi pffiffiffiffiffiffi m s ss s cmf sf s pffiffiffiffiffiffi ; ð31Þ tanh ss s pffiffiffiffiffiffi pffiffiffiffiffiffi 1þ pffiffiffiffiffiffi cmf sf s tanh sf s ms ss s where s ¼ ix. Once again, the problem is that only the two parameters cmf ¼ cDf =L and sf ¼ L2 =Df can be determined from experimental data, whereas three key parameters concerning mass transport in the polymer film are present: c, Df and L. The experimental results shown in Fig. 9 are fitted to the theoretical expression in Eq. (31). The fit is done only for f 6 100 Hz, because we are not interested in the charge-transfer HF semi-circle. The best fits, represented by the solid lines and the circles in Fig. 10, are obtained for the parameter values given in Table 1. In this table are also given the confidence intervals and the v2 values, which are defined as in Section 5.3.1.
6.2. Fitting with Mexp ,RDE A hybrid model can also be used here for the fitting of our experimental results. The procedure is the same as in
– Im Z/(Ω cm2 )
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7
A
−1
277
4
B
−1 0
–2
0 0
0
7
14
3
C
0
0
4
8
3
D
0
0 −1
1 −1
0
0
3
0
6 E
2
0
3
6
2
F
0
0
0
1
1 −1 0
2
4
0
−1 0
2 Re Z/(Ω cm2 )
4
–Im Zmt/(Ω cm2)
Fig. 9. Impedance diagrams measured at the equilibrium potential (Eeq ¼ 0:460 V/ECS) for a NafionÒ -coated Pt electrode in a 3.9 102 M FeIII + 6.3 102 M FeII + 1 M HCl, and for different electrode rotation rate values: (A) 50; (B) 200; (C) 500; (D) 1000; (E) 2000; (F) 4000 rpm.
6
A
−1
4 0
−2
0
2
2 0
0
6
0
12
3
C
0
4
0
3
6
2 0
0
2 0
0
−1 2.5
5
1.5
F
0 2
2 0
E
D
0
−1
2
8
2.5
0
0
B
−1
−1 2
4
0
–1 0
1.5 Re Zmt / (Ω cm2 )
3
Fig. 10. Mass-transport impedance data (the same than in Fig. 9 for f 6 100 Hz) fitted with the theoretical expression in Eq. (31), for different rotation rates (cf. Fig. 9 caption), and for Ds ¼ 5 106 cm2 s1 [37]. The small and large dots correspond to the data, and the solid lines and the circles correspond to the result of the fitting (see also Table 1).
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Table 1 Best fit parameter values (2 SD) determined at different electrode rotation rates from the experimental data of Fig. 9 (f 6 100 Hz) and from Eq. (31). Ds ¼ 5 106 cm2 s1 [37] X (rpm) 50 200 500 1000 2000 4000
103 cmf (cm s1 )
102 sf ðsÞ
9.1 15% 7.8 6% 7.5 4% 7.3 2% 7.0 1% 6.9 1%
2.4 35% 3.5 15% 3.8 9% 4.1 5% 4.4 3% 4.5 4%
Table 2 Best fit parameter values (2 SD) determined at different electrode rotation rates from the experimental data of Fig. 9 (f 6 100 Hz) and from the Ôtheoretical–experimental hybrid modelÕ, i.e., by using pffiffiffiffiffiffi pffiffiffiffiffiffi Mexp;RDE instead of M4 ðsÞ ¼ tanh ss s=ðms ss sÞ in Eq. (31)
102 v2 94.5 18.0 6.3 2.7 1.6 2.4
–Im Zmt/(Ω cm2)
Section 5.3.2. The experimental data Mexp;RDE obtained at a bare Pt rotating disk electrode can be introduced into Eq. (31) in order to replace the theoretical function pffiffiffiffiffiffi pffiffiffiffiffiffi M4 ðsÞ ¼ tanh ss s=ðms ss sÞ. For the sake of clarity, the Mexp; RDE data at each X value are not shown in this paper. The shape of Mexp;RDE in the Nyquist representation is well known: it is the shape of a quarter lemniscate. The results shown in Fig. 9 are fitted by using the hybrid model described above (f 6 100 Hz), and the best fits are displayed in Fig. 11 (see also Table 2). A first remark can be made concerning the v2 values given in Tables 1 and 2, when using the purely theoretical model of Eq. (31), or when using the hybrid model. At each electrode rotation rate, X, the value of 6
103 cmf (cm s1 )
102 sf ðsÞ
102 v2
50 200 500 1000 2000 4000
7.3 3% 7.0 3% 7.3 2% 7.6 1% 7.7 1% 7.8 1%
4.0 5% 4.4 6% 4.1 4% 3.8 3% 3.6 2% 3.5 3%
2.7 3.0 1.6 1.1 0.7 1.3
criterion (24) is always lower for the fitting with the hybrid model than for the fitting with Eq. (31). The visual observation of Figs. 10 and 11 confirms this mathematical result. Such a point shows once again the usefulness of our hybrid model. Next, Table 1 and Fig. 10 show that the fitting with Eq. (31) is particularly poor for the lowest X values, but could be considered as adequate for the highest X values. This can be explained by the fact that the bounded diffusion approximation in solution starts to be valid only for low values of the Nernst layer (d), i.e., for high X values. Then, one can see in Table 2 that the cmf and sf values are very similar whatever the rotation rate: cmf 7:5 103 cm s1 and sf 4:0 102 s. In ad-
A
−1
X (rpm)
4
B
−1 0 −2
0
2
2 0
0
6
0
12
3
C
0
4
2.5
0
0
−1 3
6
2 0
2 0
0
2.5
5
1.5
E
F
0
−1 0
0
−1
2
2 0
D
0
0
2
8
2
4
−1 0
1.5
3
Re Zmt / (Ω cm2 ) Fig. 11. Mass-transport impedance data (the same than in Fig. 9 for f 6 100 Hz) fitted using the Ôtheoretical–experimental hybrid modelÕ, i.e., by pffiffiffiffiffiffi pffiffiffiffiffiffi using Mexp;RDE instead of M4 ðsÞ ¼ tanh ss s=ðms ss sÞ in Eq. (31). Rotation rate values: cf. Fig. 9 caption. The small and large dots correspond to the data, and the solid lines and the circles correspond to the result of the fitting (see also Table 2).
J.-P. Diard et al. / Journal of Electroanalytical Chemistry 566 (2004) 269–280
dition, we find again the values of Eqs. (26) obtained above for a fixed electrode. That means that all the awkward effects coming from mass transport in solution are left out, and that the determination of the parameter values is done under better conditions. To finish, let us try to determine L and c from the Df value recently obtained in the literature from the doubleband microelectrode technique [38]: Df 1:0 106 cm2 s1 , and from Eq. (26) or Table 2. There are now two equations and two unknowns, and one can find: L 2:0 lm, that is not very different from the estimated thickness (L 3 lm, cf. Section 2), and c 1:5.
7. Conclusion In this study of the PtjNafionÒ jFeIII=II system, we dealt with three points. In the first one, a complete theoretical EIS study was made for the redox reaction O + e $ R at a fixed metallic electrode coated with an ion-conducting polymer in contact with a quiescent solution. All the possible impedance diagram shapes were plotted using the Nyquist and Bode representations. In the second point, we showed that a combined steady-state and impedance study of such a system cannot lead to a separate determination of the diffusion coefficient of the electroactive species in the film (Df ) and of the thickness of the film (L), contrary to what Deslouis et al. [18] reported. Indeed, a third parameter, i.e., the partition coefficient between the solution and the polymer phase (c), must be taken into account. The problem of the membrane model raised by Bard, and Saveant and co-workers [8–10] remains unsolved. Hence a third electrochemical technique, or maybe an in situ non-electrochemical technique, which is able to give another combination of Df , L and c as a characteristic parameter, must be used. In the third and last point, the concept of a Ôtheoretical–experimental hybrid modelÕ was introduced for the first time. This Ôhybrid modelÕ consists in introducing the mass-transport experimental data obtained at a bare electrode into our theoretical model for a coated pelecffiffiffiffiffiffiffi trode, instead of the mass-transport function 1= sDs when using a fixed electrode (semi-infinite linear diffusion in solution), or instead of the mass-transport pffiffiffiffiffiffi pffiffiffiffiffiffi function tanh ss s=ðms ss sÞ when using a rotating disk electrode (bounded diffusion approximation in solution). The results showed that the fitting of the EIS experimental data was greatly improved by using the hybrid model because it allows us to avoid (but not to solve) the deviations from the Warburg model (fixed electrode) and from the Nernst model (RDE). One of the outlooks of this work is as follows: we know that the problem of the deviation from the Nernst model in dynamic conditions has been studied [39,40]. The numerical solutions given in [39,40] can also be
279
introduced into Eq. (31) in order to replace the approximative analytical function M4 . It is then possible to compare the results of the fitting when using our hybrid model, and when using the two new models constructed after [39,40]. This will be the aim of a forthcoming paper [41].
Acknowledgements The authors are grateful to Dr. Gregory Berthome (Laboratoire de Thermodynamique et de Physico-Chimie Metallurgiques, Saint-Martin-dÕHeres, France) for performing the AFM measurements. N.G. thanks Prof. J.-Y. Sanchez for his interest in this work, and the French Ministry of the Economy, Finance and Industry for financial support.
References [1] I. Rubinstein, A.J. Bard, J. Am. Chem. Soc. 102 (1980) 6641. [2] I. Rubinstein, A.J. Bard, J. Am. Chem. Soc. 103 (1981) 5007. [3] F.C. Anson, C.-L. Ni, J.-M. Saveant, J. Am. Chem. Soc. 107 (1985) 3442. [4] K. Itaya, H. Takahashi, I. Uchida, J. Electroanal. Chem. 208 (1986) 373. [5] H.-L. Li, J.Q. Chambers, D.T. Hobbs, J. Electroanal. Chem. 256 (1988) 447. [6] S. Gottesfeld, I.D. Raistrick, S. Srinivasan, J. Electrochem. Soc. 134 (1987) 1455. [7] D.R. Lawson, L.D. Whiteley, C.R. Martin, M.N. Szentirmay, J.I. Song, J. Electrochem. Soc. 135 (1988) 2247. [8] C.P. Andrieux, J.M. Dumas-Bouchiat, J.M. Saveant, J. Electroanal. Chem. 131 (1982) 1. [9] J. Leddy, A.J. Bard, J. Electroanal. Chem. 153 (1983) 223. [10] J. Leddy, A.J. Bard, J.T. Maloy, J.M. Saveant, J. Electroanal. Chem. 187 (1985) 205. [11] S.K. Zecevic, J.S. Wainright, M.H. Litt, S.Lj. Gojkovic, R.F. Savinell, J. Electrochem. Soc. 144 (1997) 2973. [12] J. Maruyama, I. Abe, J. Electroanal. Chem. 527 (2002) 65. [13] R.M.Q. Mello, E.A. Ticianelli, Electrochim. Acta 42 (1997) 1031. [14] J. Maruyama, M. Inaba, K. Katakura, Z. Ogumi, Z.-I. Takehara, J. Electroanal. Chem. 447 (1998) 201. [15] J. Ye, K. Doblhofer, Ber. Bunsenges. Phys. Chem. 92 (1988) 1271. [16] J. Langmaier, A. Trojanek, J. Weber, Z. Samec, J. Electroanal. Chem. 469 (1999) 11. [17] G.C. Yu, S.K. Yen, J. Electrochem. Soc. 150 (2003) E190. [18] C. Deslouis, B. Tribollet, M. Duprat, F. Moran, J. Electrochem. Soc. 134 (1987) 2496. [19] J. Navarro-Laboulais, J.J. Garcıa-Jare~ no, F. Vicente, J. Electroanal. Chem. 536 (2002) 11. [20] C. Deslouis, I. Epelboin, M. Keddam, J.C. Lestrade, J. Electroanal. Chem. 28 (1970) 57. [21] J. Wang, S.W. Feldberg, A.J. Bard, J. Phys. Chem. B 106 (2002) 10440. [22] W.J. Blaedel, G.W. Schieffer, J. Electroanal. Chem. 80 (1977) 259. [23] T. Iwasita, W. Schmickler, J. Herrmann, U. Vogel, J. Electrochem. Soc. 130 (1983) 2026. [24] J.-P. Brun, Procedes de separation par membranes, Masson, Paris, 1988. [25] J. Wang, A.J. Bard, J. Am. Chem. Soc. 123 (2001) 498.
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[26] C. Montella, J.-P. Diard, B. Le Gorrec, Exercices de cinetique electrochimique, Tome 1, Regime stationnaire, Hermann, Paris, 2000. [27] V.G. Levich, Physicochemical Hydrodynamics, second ed., Prentice-Hall, Englewood Cliffs, 1962. [28] J.-P. Diard, B. Le Gorrec, C. Montella, Cinetique electrochimique, Hermann, Paris, 1996. [29] J.S. Chen, R. Durand, C. Montella, J. Chim. Phys. 91 (1994) 383. [30] J.S. Chen, J.-P. Diard, R. Durand, C. Montella, J. Electroanal. Chem. 406 (1996) 1. [31] J.-P. Diard, B. Le Gorrec, C. Montella, J. Electroanal. Chem. 471 (1999) 126. [32] J.-P. Diard, B. Le Gorrec, C. Montella, J. Electrochem. Soc. 147 (2000) 296.
[33] M. Sluyters-Rehbach, Pure Appl. Chem. 66 (1994) 1831. [34] S. Wolfram, The Mathematica Book, fourth ed., Wolfram Media/ Cambridge University Press, Cambridge, 1999. [35] ZView, Scribner Associates, Charlottesville, VA, 1993. [36] B.A. Boukamp, Solid State Ionics 20 (1986) 31. [37] R.N. Adams, Electrochemistry at Solid Electrodes, Marcel Dekker Inc., New York, 1969. [38] C. Amatore, C. Sella, L. Thouin, J. Electroanal. Chem. 547 (2003) 151. [39] E. Levart, D. Schuhmann, J. Electroanal. Chem. 28 (1970) 45. [40] B. Tribollet, J. Newman, J. Electrochem. Soc. 130 (1983) 2016. [41] J.-P. Diard, N. Glandut, B. Le Gorrec, C. Montella, in preparation.
Electroanalytical Chemistry Journal of Electroanalytical Chemistry 566 (2004) 269–280 www.elsevier.com/locate/jelechem
EIS study of the FeIII/FeII redox couple at a NafionÒ -coated Pt electrode Jean-Paul Diard, Nicolas Glandut *, Bernard Le Gorrec 1, Claude Montella
2
Ecole Nationale Superieure dÕElectrochimie et dÕElectrom etallurgie de Grenoble, Laboratoire dÕElectrochimie et de Physico-chimie des Materiaux et des Interfaces, UMR 5631 CNRS-INPG-UJF, Domaine Universitaire, BP 75, 38 402 Saint-Martin-dÕHeres Cedex, France Received 26 June 2003; received in revised form 22 October 2003; accepted 8 November 2003
Abstract The redox reaction FeIII + e $ FeII is studied at a NafionÒ -filmed Pt electrode using both steady-state and EIS methods. It is shown that the fitting of the EIS experimental data can be greatly improved by using the new concept of a Ôtheoretical–experimental hybrid modelÕ, i.e., by introducing experimental data obtained at a bare electrode in the theoretical model for a filmed electrode. It is also shown that a combined steady-state and EIS study cannot lead to a separate determination of the three key parameters of the system: the diffusion coefficient in the polymer film (Df ), the thickness of the film (L), and the partition coefficient between the solution and the polymer (c), only the two parameters cDf =L and L2 =Df being obtained from experimental data by a curve fitting procedure. Finally, an exhaustive theoretical EIS study in quiescent solution is also proposed. Ó 2003 Elsevier B.V. All rights reserved. Keywords: NafionÒ -coated electrode; Redox reaction; Electrochemical impedance spectroscopy; Complex non-linear least-squares fitting; Theoretical–experimental hybrid model
1. Introduction The idea to study metallic electrodes coated with NafionÒ , 3 is at the origin of numerous papers written during the 1980s [1–7]. Some of these papers [1,2] dealt with mediated electron transfer, i.e., the oxidation (or reduction) of an electroactive species in solution is due to the presence of an oxidised (or reduced) species in the NafionÒ film, or [3–5] with NafionÒ containing catalysts such as Co3þ /Co2þ complexes or Pt microparticles. The other papers [6,7] dealt with ÔemptyÕ NafionÒ films. The electroactive species in solution diffuses through the film, with a diffusion coefficient (Df ) which is not the same as that in solution (Ds ), and reacts at the electrodejfilm interface.
*
Corresponding author. E-mail address: [email protected] (N. Glandut). 1 Member of the Institut Universitaire de Technologie 1 de Grenoble. 2 Member of PolytechÕ Grenoble. 3 NafionÒ is a registered trademark of E. I. du Pont de Nemours and Company. 0022-0728/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2003.11.038
During the same period, Bard, Saveant and coworkers [8–10] worked on poly(vinyl ferrocene)-coated electrodes, and successfully modeled the mass transport of a species to a rotating disk electrode (RDE) covered with a thin polymer layer, under steady-state conditions. This model is, of course, applicable to the metallic electrodejNafionÒ jsolution system. Due to its applications in proton exchange membrane fuel cells, NafionÒ is again being studied increasingly, and especially when it is deposited on metallic electrodes. Based on the pioneering work of Bard, Saveant and co-workers [8–10], recent papers concerning NafionÒ -coated electrodes can be cited: Zecevic et al. [11] and Maruyama and Abe [12] who studied the oxygen reduction reaction (ORR) at NafionÒ -coated Pt or glassy carbon RDEs, and Mello and Ticianelli [13] and Maruyama et al. [14] who studied the hydrogen oxidation reaction (HOR) at the same kind of electrodes. Nevertheless, an important point must be mentioned concerning steady-state measurements at NafionÒ coated RDEs. In accordance with the model of Bard, Saveant and co-workers [8–10], the diffusion coefficient in the film, Df , the thickness of the film, L, and the
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partition coefficient between the solution and the polymer, c, cannot be separately determined from steadystate measurements, because only the ratio cDf =L is accessible. This means that two parameters must be found first from another method in order to determine the third one. The purpose of this work is to study the redox reaction FeIII + e $ FeII at a NafionÒ -filmed Pt electrode [15,16] using electrochemical impedance spectroscopy (EIS). The measurements are made at the equilibrium potential in quiescent solution (semi-infinite linear diffusion in solution), and under convective diffusion conditions (bounded diffusion approximation in solution). It is shown that, by using the impedance technique, the new accessible parameter is the time constant in the film, sf ¼ L2 =Df , and therefore one parameter, the thickness of the film for example, must be known in order to measure the two others. Hence the problem raised by Bard, Saveant and co-workers [8–10] remains unsolved. Notice that Yu and Yen have recently made a similar remark in their study of Tijporous TiO2 electrodes [17]. Notice also that in 1987, in their theoretical EIS study of a porous layer-coated RDE, Deslouis et al. found [18] that Df and L can be determined separately, but c was not taken into account in their model (which amounts to taking c ¼ 1). Moreover, it is also shown in this paper that the fitting of the experimental data can be greatly improved by using a Ôtheoretical–experimental hybrid modelÕ, i.e., by introducing experimental data obtained at a bare electrode into our theoretical model for a coated electrode. As far as we know, it is the first time that such a fitting procedure is used. This Ôhybrid modelÕ allows us to avoid (but not to solve) the deviations from Warburg behaviour due to natural convection and edge effects when using a fixed electrode [19]. It also allows us to avoid the deviations from the Nernst model (bounded diffusion approximation) when using a rotating disk electrode [20].
2. Experimental The working electrode (WE) consisted in a polycrystalline Pt disk sealed in a Teflon holder, and had a geometric surface area of 3.14 102 cm2 (0.2 cm in diameter). Before the NafionÒ film was deposited and for the experiments on bare Pt, the WE was polished with alumina (3 lm) and rinsed with Millipore water in an ultrasonic bath. The polymer films were spin-coated at 125 rpm for 15 min from a 1.25% NafionÒ solution [21] (Aldrich 5 wt% NafionÒ solution diluted four times with methanol). The weight of the film was 0.1 mg. Knowing the density of recast NafionÒ (1.8 g cm3 [16]) and the effective area (0.2 cm2 ), the thickness of the film could be estimated at 3 lm. The surface of the
Fig. 1. Atomic force microscopy (AFM) roughness profile of the surface of a spin-coated NafionÒ film.
NafionÒ film was characterized by atomic force microscopy (AFM) with a Nanoscope III Digital Instruments system. Fig. 1 shows that the film is very smooth (roughness of 5 nm) and not porous. The WE was then mounted on a EDI 101 type rotating disk electrode (Radiometer Analytical, Villeurbanne, France). Electrochemical measurements were carried out at room temperature (25 °C) in a standard three-electrode cell configuration. An Autolab PGSTAT 30 (Eco Chemie, The Netherlands), which was controlled by GPES 4.9 and FRA 4.9 softwares, was used to perform steady-state and impedance measurements. The WE potential was measured with respect to a SCE reference electrode and a Pt wire was used as a counter electrode. The solution was a 3.9 102 M Fe(ClO4 )3 , 6.3 102 M Fe(ClO4 )2 in 1 M HCl aqueous solution (Aldrich chemical reagents and Millipore water), and was deaerated by bubbling pure Ar gas. Before taking a steady-state jf vs. E curve or an impedance diagram, the electrode was cleaned by cycling the potential at 1 V s1 between the beginning of H2 evolution and the beginning of Cl2 evolution (between )0.25 and +1.1 V vs. SCE) [22,23]. Six electrode rotation rates were used: X ¼ 50, 200, 500, 1000, 2000 and 4000 rpm. For measurements in quiescent solution, the electrode rotation rate was zero.
3. Description of the model The redox reaction FeIII + e $ FeII (formally noted: O + e $ R) was studied on a Pt electrode covered with a thin NafionÒ film. The notation FeIII and FeII is used to denote the formation of complex ions from Fe3þ and Fe2þ in the presence of chloride ions. The thickness of the film is L. In solution (s), mass transport of redox species is governed by diffusion and convection. In the film (f), it is governed only by diffusion. Migration effects can be neglected because supporting electrolytes
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are present in high concentration in both media; HCl 1 M in solution, and –SO3 H 1 M in the film. The sulfonate groups –SO 3 are fixed into the polymeric matrix. Because of the Donnan effect [24], chloride ions are also present in the film. DO;s and DR;s are the diffusion coefficients of oxidized and reduced species, respectively, in the solution. In the polymer phase, the diffusion coefficients are considered as equal for the sake of simplicity DO;f ¼ DR;f ¼ Df :
cO ¼ cR ¼ c:
ð2Þ
There is no diffuse double layer at the polymerjsolution interface, as was recently shown by Wang and Bard [25]. O and R are the constant concentrations of oxidized and reduced species, respectively, in the electrolyte bulk. In Fig. 2 are displayed schematically two steady-state concentration profiles in our system, one at the equilibrium potential (Fig. 2(A)), the other when a species, O or R, is consumed at the metaljpolymer interface (Fig. 2(B)).
0
0
L
∗
O =R
L+δ
film 0
0
x
O∗ = R∗
L
R expðao nÞ O expðar nÞ ; 1 þ cm1 f þ m1R;s expðao nÞ þ cm1 f þ m1O;s expðar nÞ ck 0
ð3Þ where n ¼ nf ðE E0 Þ is the dimensionless potential, E0 the standard potential, k 0 the standard rate constant for electron transfer, n the electron number (n ¼ 1), F the Faraday constant, f ¼ F =ðRT Þ ¼ 38:9 V1 at 25 °C, ao and ar are the symmetry factors for electron transfer in the direction of oxidation and reduction, respectively (ao þ ar ¼ 1), mf ¼ Df =L is the diffusion constant in the polymer film, mO;s ¼ DO;s =dO and mR;s ¼ DR;s =dR are the diffusion–convection constants in solution for the oxi1=3 dized and reduced species, respectively, dO ¼ 1:611DO;s 1=3 m1=6 X1=2 and dR ¼ 1:611DR;s m1=6 X1=2 are the characteristic lengths of diffusion–convection in solution from Levich theory [27], and m is the kinematic viscosity of the solution. The other symbols have been detailed previously. The limiting current densities due to mass transport of the oxidized and reduced species, jdO and jdR , are obtained by setting n ! 1 in Eq. (3) 1 1 jdO ¼ lim jf ¼ nF O þ ð4Þ n!1 cmf mO;s
n!þ1
∗
solution
O,R
Pt
B
jf ¼ nF
jdR ¼ lim jf ¼ nF R
γ O∗ = γ R∗
O,R A
According to the assumptions made in Section 3, the theoretical expression of the steady-state current density (jf ) vs. potential (E) curve for a redox reaction at an ionconducting polymer-coated RDE is [26]
and
solution
film
Pt
4. Steady-state study
ð1Þ
The redox species, O and R, can transfer through the polymerjsolution interface (x ¼ L). It is assumed that this transfer is kinetically reversible (quasi-equilibrium conditions). Hence the partition coefficients cO and cR can be defined as: cO ¼ OðL ; tÞ=OðLþ ; tÞ and cR ¼ RðL ; tÞ=RðLþ ; tÞ, where OðL ; tÞ and RðL ; tÞ are the interfacial concentrations of oxidized and reduced species on the polymer side (x ¼ L ), and OðLþ ; tÞ and RðLþ ; tÞ are the interfacial concentrations on the solution side (x ¼ Lþ ). In what follows, we will assume that
L+δ
x
Fig. 2. Schematic steady-state concentration profiles of redox species for the reaction O + e $ R at a polymer-coated electrode, (A) at the equilibrium potential (E ¼ Eeq , jf ¼ 0), and (B) when a species is consumed (E 6¼ Eeq ; jf 6¼ 0). Bulk concentrations: O ¼ R , and c ¼ 1:5.
271
1 1 þ : cmf mR;s
ð5Þ
Eqs. (3)–(5) clearly show that only the two terms ck o and cmf can be determined from experimental data obtained under steady-state conditions, whereas three important parameters concerning mass transport in the polymer film are present: c, Df and L. That is the reason why c, Df and L cannot be independently determined from steady-state jf vs. E measurements. In Fig. 3 are displayed steady-state jf vs. E experimental curves for the redox reaction FeIII + e $ FeII investigated at a bare Pt RDE (Fig. 3(A)), and at a NafionÒ -coated Pt RDE (Fig. 3(B)), for several electrode rotation rates. The shape of these curves is classical [26,28], and is always the same whatever the value of X. At each X value, mO;s and mR;s can be calculated from the well-known relations: jdO;bare ¼ nFmO;s O and jdR;bare ¼ nFmR;s R , where jdO;bare and jdR;bare are the limiting current densities due to mass transport at a bare electrode (Fig. 3(A)). The diffusion–convection constants, mO;s and mR;s , are then injected in Eqs. (4)
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the well-known expression which applies, for example, to bare Pt electrodes in quiescent solutions [28] 1 ð8Þ MðsÞ ¼ pffiffiffiffiffiffiffi : sDs The concept of the mass-transport function was recently extended to multilayer systems [29–32]. The metaljpolymer filmjelectrolyte system dealt with in this paper is an example. Hence, for a polymer coated-electrode, the characteristic mass-transport function for soluble species at the metal j polymer interface (x ¼ 0) can be written as [29–32] pffiffiffiffiffiffi tanhpffiffiffiffiffiffi sf s cM2 ðsÞ þ m f sf s M1 ðsÞ ¼ ð9Þ pffiffiffiffiffiffi pffiffiffiffiffiffi ; 1 þ cM2 ðsÞmf sf s tanh sf s where M2 is the mass-transport function at the polymer surface (x ¼ Lþ ), sf ¼ L2 =Df the diffusion time constant in the film, and mf ¼ Df =L the diffusion constant in the film. In a quiescent solution, M2 is characteristic of semiinfinite linear diffusion, and is given by the same relationship as in Eq. (8) 1 M2 ðsÞ ¼ pffiffiffiffiffiffiffi : sDs
ð10Þ
Eqs. (9) and (10) lead to
Fig. 3. Experimental steady-state current density vs. potential curves, (A) using a bare Pt RDE, and (B) a NafionÒ -coated Pt RDE, at 25 °C in a 3.9 102 M FeIII + 6.3 102 M FeII + 1 M HCl aqueous solution, with electrode rotation rates (a) 50; (b) 200; (c) 500; (d) 1000; (e) 2000; (f) 4000 rpm.
and (5), and knowing the values of jdO and jdR (Fig. 3(B)), we can obtain the mean value of cmf cmf ¼ 5:5 103 cm s1 :
ð6Þ
5. EIS study in quiescent solution 5.1. Theoretical The characteristic mass-transport function for soluble species at the electrodejelectrolyte interface (x ¼ 0) is defined as [28] MðsÞ ¼
Dcð0; sÞ DJ ð0; sÞ
;
ð7Þ
where Dc and DJ are the Laplace transforms of the concentration and flux variations when the electrode is submitted to a small perturbation in potential or in current, and s is the Laplace complex variable. Due to semi-infinite linear diffusion of soluble species in the electrolyte, the mass-transport function takes on
pffiffiffiffiffiffi tanh sf s c pffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffi m f sf s s Ds M1 ðsÞ ¼ pffiffiffiffiffiffi pffiffiffiffiffiffi : c 1 þ pffiffiffiffiffiffiffiffi mf sf s tanh sf s s Ds
ð11Þ
Using the dimensionless Laplace variable, pffiffiffiffiffiffiffiffiffiffiffiffiffi S ¼ sf s, and the dimensionless parameter K ¼ Ds =Df =c, Eq. (11) can be written in a dimensionless form as pffiffiffi 1 þ K tanh S pffiffiffi : ð12Þ mf M1 ðSÞ ¼ pffiffiffi S ðK þ tanh S Þ The K parameter can be written alternatively as K ¼rf =rs , that is the ratio of the Warburg factor for semi-infinite linear diffusion pffiffiffiffiffi in the polymer film, rf ¼ ð1=O þ 1=R Þ=ðn2 fF c Df Þ, to the Warburg factor for a bare Pt electrode with semi-infinite linear diffusion pffiffiffiffiffiin quiescent solution, rs ¼ ð1=O þ 1=R Þ=ðn2 fF Ds Þ, provided the same diffusion coefficient is assumed for oxidized and reduced species. pffiffiffiffiffiffiffi Setting s ¼ ix and S ¼ iu in Eq. (12), where i ¼ 1, x ¼ 2pf is the angular frequency (rad s1 ), f the frequency (Hz), and u ¼ xsf the dimensionless frequency, the mass-transport function diagrams for several values of K are plotted in Fig. 4(A) using the Nyquist representation Imðmf M1 Þ vs. Reðmf M1 Þ. Figs. 4(B) and (C) also display the Bode plots of mf M1 , i.e., the plots of log jmf M1 j and / vs. log u. First of all, whatever the value of K, there is a common feature between these diagrams: their high frequency part is a )45° phase shift straight line characteristic of semi-infinite linear diffusion in the
J.-P. Diard et al. / Journal of Electroanalytical Chemistry 566 (2004) 269–280
intermediate values of K. In passing, note that our experimental results shown below (Fig. 7) are very close to the diagram plotted in Fig. 4A, c. If it is assumed here that c 1, it can then be concluded that the diffusion coefficient in the polymer phase is smaller than the diffusion coefficient in solution, but has the same order of magnitude (Df K Ds ).
e d c
1
5.2. Bare Pt electrode
b a
0
1 Re (mf M1 )
A
log m f M1
The impedance diagram shown in Fig. 5(A) is measured at a bare Pt electrode in quiescent solution (electrode rotation rate ¼ 0 rpm). The applied dc potential is the equilibrium potential (Eeq ¼ 0:460 V vs. SCE) and
f 4 e d c 2 b
0.6 12 5
a 0 –2
–4
0
4
log u
B
0
− Im Z/(Ω cm2)
−Im (m f M1)
f
0
0 0.3
φ/˚
0.9
−1.6
0 1 0
d
0
6 Re Z/(Ω cm2 )
A
e
12
f –4
0 log u
4
Fig. 4. Dimensionless mass-transport function mf M1 calculated from Eq. (12) and plotted using the Nyquist representation (A), and using the Bode coordinates (B: decimal logarithmp offfiffiffiffiffiffiffiffiffiffiffiffiffi amplitude, and C: phase shift vs. log u). S ¼ iu, u ¼ 2pf sf , and K ¼ Ds =Df =c ¼ 100 (a); 4 (b); 2 (c); 1 (d); 0.3 (e); 0.001 (f).
polymer film. Then, for lower frequencies, three limiting cases should be considered: (i) when K 1 (Figs. 4(a)), the impedance is close to a bounded diffusion impedance due to the polymer layer in series with a Warburg impedance due to diffusion in the electrolyte, these two impedances being well separated in frequency; (ii) when K ¼ 1 (Figs. 4(d)), the polymer-coated electrode can be treated as a bare electrode, and the impedance diagram cannot be distinguished from a Warburg impedance in the whole frequency domain; and (iii) when K 1 (Figs. 4(f)), the shape of the impedance is the shape of a restricted diffusion impedance in a certain frequency range. Other diagrams are plotted in Figs. 4(A–C) for
−1.6
800 − Im Mexp /(s cm− 1)
–90
2
−1
a
–45
43
6
b c
C
273
−1
400
0 21 0 B
0
400 Re Mexp /(s cm− 1 )
800
Fig. 5. (A) Impedance diagram (Nyquist representation) measured at the equilibrium potential (Eeq ¼ 0:460 V/ECS) for a bare Pt electrode in a 3.9 102 M FeIII + 6.3 102 M FeII + 1 M HCl quiescent aqueous solution. Sinusoidal perturbation amplitude dE ¼ 10 mV. Decimal logarithm of the frequency (Hz) is given on the graph. The inset shows the high frequency (HF) part of the diagram. (B) Masstransport experimental data Mexp calculated from the data of A and using Eqs. (13)–(18).
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the resulting dc current is zero: this is the only steady state under such conditions. This diagram can be modeled by the RandlesÕ equivalent circuit shown in Fig. 6 [33], where RX is the ohmic resistance, Rct the chargetransfer resistance, and Cdl the differential double-layer capacitance. The mass-transport impedance, Zmt , is given by Zmt ðsÞ ¼ ZO ðsÞ þ ZR ðsÞ;
ð13Þ
where ZO and ZR are the concentration impedances of the oxidized and reduced species, respectively, and s is the Laplace complex variable. ZO and ZR can be written as [28] ZO ðsÞ ¼ Rct Kr MðsÞ;
ð14Þ
ZR ðsÞ ¼ Rct Ko MðsÞ;
ð15Þ
where Ko and Kr are the rate constants for electron transfer in the direction of oxidation and reduction, respectively. Notice that we assume here that DO;s ¼ DR;s ¼ Ds . At the equilibrium potential, we can write
C dl
RΩ
Rct
Z mt
Fig. 6. RandlesÕ equivalent circuit for the redox reaction FeIII + e $ FeII studied by EIS at a bare Pt electrode or at a NafionÒ filmed Pt electrode, in quiescent solution or under diffusive and convective mass transport conditions.
0.6 12 2 − Im Z/(Ω cm2 )
5 4 3 0 0.3
0.9
6
0
6 Re Z/(Ω cm2 )
ð16Þ
and Rct ¼
n2 fF ða
1 1 ; ¼ nfj0 o Ko R þ ar Kr O Þ
ð17Þ
where j0 is the exchange current density. Combining Eqs. (8), (13)–(17) and using ao þ ar ¼ 1, the masstransport impedance can be expressed as 1 1 1 Zmt ðsÞ ¼ 2 pffiffiffiffiffiffiffi þ ; ð18Þ n fF sDs O R which is the well-known classical Warburg impedance. Its theoretical diagram in the Nyquist representation is a straight line with unity slope (phase shift / ¼ 45°). Nevertheless, the experimental phase shift is greater than 45° (/ 40° in Fig. 5(A)) because of edge effects and natural convection [19]. From Fig. 5(A) and using Eqs. (13)–(17), the experimental mass-transport function Mexp can be calculated at each frequency below 100 Hz (we are not interested here in the high frequency semi-circle due to Rct and Cdl ) as 1 1 1 2 Zmt;exp ðxÞ: ð19Þ Mexp ðxÞ ¼ n fF þ O R The Im Mexp vs. Re Mexp diagram is plotted in Fig. 5(B) using the Nyquist representation. As indicated above, Mexp ðxÞ deviates in the low frequency domain from the theoretical mass-transportpfunction related to semi-infiffiffiffiffiffiffiffiffiffiffi nite diffusion, MðxÞ ¼ 1= ixDs , because of edge effects and natural convection. From the Mexp data and using Mathematica software [34], an interpolating function can be constructed. This approximating function is represented by the solid line in Fig. 5(B). Mexp will be useful in what follows for the fitting of experimental measurements obtained with a NafionÒ -coated Pt electrode (see Section 5.3). 5.3. NafionÒ -coated Pt electrode
−1
0
j0 nF
−1.6
1 0
Ko R ¼ Kr O ¼
12
Fig. 7. Impedance diagram measured at the equilibrium potential (Eeq ¼ 0:460 V/ECS) for a NafionÒ -coated Pt electrode in a 3.9 102 M FeIII + 6.3 102 M FeII + 1 M HCl quiescent aqueous solution. The inset shows the HF part of the diagram.
pffiffiffiffiffiffiffiffiffiffi 5.3.1. Fitting with M2 (x) ¼ 1= ixDs Fig. 7 shows the impedance diagram for a Pt electrode covered with a thin NafionÒ film. The experimental conditions are the same as in Section 5.2. The high frequency part (HF: 100 kHz to 100 Hz) and the low frequency part (LF: below 0.1 Hz) of the two diagrams of Figs. 5(A) and 7 are quite similar. Indeed, a HF semi-circle due to Rct and Cdl , and a LF linear part due to diffusion in solution can be seen (see also the insets). Nevertheless, a difference is noticeable between these two diagrams: in the 100–1 Hz frequency range, the begining of a capacitive arc can be seen for the PtjNafionÒ electrode, whereas this arc is not visible for
J.-P. Diard et al. / Journal of Electroanalytical Chemistry 566 (2004) 269–280
the bare Pt electrode. Considering that the only changing condition between the two experiments is the presence or not of a NafionÒ film, this arc can be attributed to the NafionÒ film. This behaviour is in perfect agreement with the simulation above (see Section 5.1 and Fig. 4A, c). The diagram of Fig. 7 can also be modeled by the RandlesÕ equivalent circuit shown in Fig. 6. Eqs. (13) and (16) are still valid, but Eqs. (14) and (15) must be replaced by the following ZO ðsÞ ¼ Rct Kr M1 ðsÞ;
ð20Þ
ZR ðsÞ ¼ Rct Ko M1 ðsÞ;
ð21Þ
where M1 is given by Eq. (9). The expression of Rct is now a little different from that of Eq. (17)
In Eqs. (25) are given the cmf and sf values plus or minus two standard deviations (2 SD) i.e., there is a 95% chance that the actual values are within the given percentage range (confidence interval).
ð22Þ
At this point, the reader must be careful: the division of Rct for a bare electrode by Rct for a covered electrode, i.e., Eq. (17) divided by Eq. (22), not necessarily lead to the determination of c. Indeed, the exchange current density (j0 ) is, rigorously speaking, different for a bare electrode than for a covered one. In an analogous manner to the above, the combination of Eqs. (11), (13), (16) and (20)-(22) leads to the following equation where s ¼ ix 1 1 1 Zmt ðsÞ ¼ 2 þ n fF O R pffiffiffiffiffiffi tanh sf s 1 pffiffiffiffiffiffiffi þ pffiffiffiffiffiffi cmf sf s sDs : ð23Þ pffiffiffiffiffiffi pffiffiffiffiffiffi 1 1 þ pffiffiffiffiffiffiffi cmf sf s tanh sf s sDs Eq. (23) shows that only two terms can be determined from the experimental data: cmf ¼ cDf =L and sf ¼ L2 = Df , whereas three parameters concerning mass transport in the polymer film are present: c, Df and L. The problem is the same as in the steady-state study made above: Df and L cannot be determined independently by EIS, contrary to what Deslouis et al. [18] have reported. Indeed, in their model, the partition coefficient c was not taken into account, which amounts to taking c ¼ 1. Using Mathematica software [34], and especially its NonlinearRegress least-squares fit function based on the Levenberg–Marquardt algorithm, the experimental results (below 100 Hz) are fitted to the theoretical expression in Eq. (23). Note here that commercial fitting softwares, such as ZView [35,36], cannot be used because Eq. (23) shows that the impedance model cannot be expressed in terms of equivalent circuits. The chosen object-model distance to minimise is the weighted function nd X Zth ðfk ; pÞ Zexp;k 2 1 2 ; v ¼ ð24Þ nd np k¼1 Zexp;k
10
− Im Zmt /(Ω cm2)
1 1 ¼ : n2 fF cðao Ko R þ ar Kr O Þ nf cj0
where p is the list of parameters to be determined (here cmf and sf ), np is the number of parameters (np ¼ 2), Zexp;k is the impedance measured at the fk frequency, nd is the number of data ðfk ; Zexp;k Þ, k ¼ 1; . . . ; nd , and Zth is the calculated impedance at the fk frequency for the p parameter values. The best fit, represented by the solid line and the circles in Fig. 8(A), is obtained for the parameter values: 8 < cmf =ðcm s1 Þ ¼ 6:4 103 4%; ð25Þ s =s ¼ 5:6 102 10%; : f2 v ¼ 5:5 102 :
−1.6
−1
5
0 2 0
1
0
5 Re Zmt /(Ω cm2 )
A
10
10 –1.6 − Im Zmt /(Ω cm2)
Rct ¼
275
–1
5
0 2 0 B
0
1 5 Re Zmt /(Ω cm2 )
10
Fig. 8. (A) Mass-transport impedance data (the same than in Fig. 7 for f 6 100 Hz) fitted with the theoretical expression in Eq. (23) and for Ds ¼ 5 106 cm2 s1 [37]. (B) Same data fitted using the Ôtheoretical– experimental hybrid modelÕ, pffiffiffiffiffiffiffi i.e., by using Mexp shown in Fig. 5(B) instead of M2 ðsÞ ¼ 1= sDs in Eq. (23). In A and B, the small and large dots correspond to the data, and the solid lines and the circles correspond to the result of the fitting (see also Eqs. (25) and (26)).
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5.3.2. Fitting with Mexp Another way can be followed for the fitting of our experimental results: the experimental data Mexp obtained at a bare Pt electrode and including edge effects and natural convection (see Section 5.2 and Fig. 5(B)) can be introduced into Eq. (23) inpffiffiffiffiffiffiffiffiffiffi order to replace the theoretical function M2 ðxÞ ¼ 1= ixDs . This can be done numerically with Mathematica software [34] by using the interpolation function deduced from the Mexp points (Fig. 5(B)). To the best of our knowledge, this is the first time that such a model has been used for the fitting of experimental data. We have called this model a Ôtheoretical–experimental hybrid modelÕ, or Ôhybrid modelÕ. The best fit (see Fig. 8(B)) is obtained for the parameter values: 8 < cmf =ðcm s1 Þ ¼ 7:5 103 3%; ð26Þ s =s ¼ 4:2 102 6%; : f2 v ¼ 2:4 102 : Three observations can be made from these results. First, by eye, it can be seen in Fig. 8 that the result of the fit is better using the hybrid model than the theoretical model. This remark is particularly valid for the low frequencies, where deviation from Warburg behaviour occurs because of edge effects and natural convection, and shows the usefulness of our hybrid model. Next, the difference between the two v2 values confirms our visual observation mathematically. The v2 value in Eqs. (26), when the hybrid model is used, is approximatively better by a factor of two than the v2 value in Eqs. (25), when the theoretical model is used. Finally, although there is a visible difference between Figs. 8(A) and (B), the values of cmf and sf are quite similar in Eqs. (25) and (26). Moreover, the value of cmf obtained by EIS is of the same order as the value of cmf obtained in Eq. (6) from the steady-state polarization curves.
6. EIS study with RDE pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 6.1. Fitting with M4 (x) ¼ tanh ixss =ms ixss What is described in Section 5.3.1 is now applied to experiments with a NafionÒ -coated Pt RDE (rotating disk electrode). The experimental conditions are the same as above: measurements made at the equilibrium potential, identical electrolytic solution, etc. Therefore the electrode becomes a bilayer electrode. Indeed the Nernst layer in solution (d) must be considered now, in addition to the NafionÒ layer deposited on the Pt electrode (Fig. 2). Notice that an exhaustive theoretical study of bilayer electrodes has been made by Diard et al. [31] in the past. Fig. 9 shows the impedance diagrams plotted experimentally for six different electrode rotation rates (X).
These diagrams are all composed of a HF semi-circle due to Rct and Cdl . Two overlapping LF arcs are visible on the diagrams measured for low values of X (Figs. 9(A)–(C)), whereas only one LF arc is visible on the diagrams measured for high values of X (Figs. 9(D)– (F)). Here also, the diagrams of Fig. 9 can be modeled by the RandlesÕ equivalent circuit shown in Fig. 6. The concentration impedances, ZO and ZR , can be written as ZO ðsÞ ¼ Rct Kr M3 ðsÞ;
ð27Þ
ZR ðsÞ ¼ Rct Ko M3 ðsÞ;
ð28Þ
where Rct is the same as in Eq. (22). M3 can be written as M1 was previously written in Eq. (9) pffiffiffiffiffiffi tanh sf s cM4 ðsÞ þ pffiffiffiffiffiffi m f sf s M3 ðsÞ ¼ ð29Þ pffiffiffiffiffiffi pffiffiffiffiffiffi : 1 þ cM4 ðsÞmf sf s tanh sf s The difference between Eqs. (9) and (29) is that the M2 function, characteristic of semi-infinite linear diffusion, has been replaced by M4 which is a bounded diffusion transport function. Its well-known expression, using the Nernst approximation for convective diffusion, is [28] pffiffiffiffiffiffi tanh ss s M4 ðsÞ ¼ ð30Þ pffiffiffiffiffiffi ; ms s s s where ss ¼ d2 =Ds is the diffusion–convection time constant in solution, and ms ¼ Ds =d the diffusion–convection constant in solution. Combining Eqs. (13), (16), (22) and (27)–(30) gives 1 1 1 Zmt ðsÞ ¼ 2 þ n fF O R pffiffiffiffiffiffi pffiffiffiffiffiffi tanh ss s tanh sf s þ pffiffiffiffiffiffi pffiffiffiffiffiffi m s ss s cmf sf s pffiffiffiffiffiffi ; ð31Þ tanh ss s pffiffiffiffiffiffi pffiffiffiffiffiffi 1þ pffiffiffiffiffiffi cmf sf s tanh sf s ms ss s where s ¼ ix. Once again, the problem is that only the two parameters cmf ¼ cDf =L and sf ¼ L2 =Df can be determined from experimental data, whereas three key parameters concerning mass transport in the polymer film are present: c, Df and L. The experimental results shown in Fig. 9 are fitted to the theoretical expression in Eq. (31). The fit is done only for f 6 100 Hz, because we are not interested in the charge-transfer HF semi-circle. The best fits, represented by the solid lines and the circles in Fig. 10, are obtained for the parameter values given in Table 1. In this table are also given the confidence intervals and the v2 values, which are defined as in Section 5.3.1.
6.2. Fitting with Mexp ,RDE A hybrid model can also be used here for the fitting of our experimental results. The procedure is the same as in
– Im Z/(Ω cm2 )
J.-P. Diard et al. / Journal of Electroanalytical Chemistry 566 (2004) 269–280
7
A
−1
277
4
B
−1 0
–2
0 0
0
7
14
3
C
0
0
4
8
3
D
0
0 −1
1 −1
0
0
3
0
6 E
2
0
3
6
2
F
0
0
0
1
1 −1 0
2
4
0
−1 0
2 Re Z/(Ω cm2 )
4
–Im Zmt/(Ω cm2)
Fig. 9. Impedance diagrams measured at the equilibrium potential (Eeq ¼ 0:460 V/ECS) for a NafionÒ -coated Pt electrode in a 3.9 102 M FeIII + 6.3 102 M FeII + 1 M HCl, and for different electrode rotation rate values: (A) 50; (B) 200; (C) 500; (D) 1000; (E) 2000; (F) 4000 rpm.
6
A
−1
4 0
−2
0
2
2 0
0
6
0
12
3
C
0
4
0
3
6
2 0
0
2 0
0
−1 2.5
5
1.5
F
0 2
2 0
E
D
0
−1
2
8
2.5
0
0
B
−1
−1 2
4
0
–1 0
1.5 Re Zmt / (Ω cm2 )
3
Fig. 10. Mass-transport impedance data (the same than in Fig. 9 for f 6 100 Hz) fitted with the theoretical expression in Eq. (31), for different rotation rates (cf. Fig. 9 caption), and for Ds ¼ 5 106 cm2 s1 [37]. The small and large dots correspond to the data, and the solid lines and the circles correspond to the result of the fitting (see also Table 1).
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J.-P. Diard et al. / Journal of Electroanalytical Chemistry 566 (2004) 269–280
Table 1 Best fit parameter values (2 SD) determined at different electrode rotation rates from the experimental data of Fig. 9 (f 6 100 Hz) and from Eq. (31). Ds ¼ 5 106 cm2 s1 [37] X (rpm) 50 200 500 1000 2000 4000
103 cmf (cm s1 )
102 sf ðsÞ
9.1 15% 7.8 6% 7.5 4% 7.3 2% 7.0 1% 6.9 1%
2.4 35% 3.5 15% 3.8 9% 4.1 5% 4.4 3% 4.5 4%
Table 2 Best fit parameter values (2 SD) determined at different electrode rotation rates from the experimental data of Fig. 9 (f 6 100 Hz) and from the Ôtheoretical–experimental hybrid modelÕ, i.e., by using pffiffiffiffiffiffi pffiffiffiffiffiffi Mexp;RDE instead of M4 ðsÞ ¼ tanh ss s=ðms ss sÞ in Eq. (31)
102 v2 94.5 18.0 6.3 2.7 1.6 2.4
–Im Zmt/(Ω cm2)
Section 5.3.2. The experimental data Mexp;RDE obtained at a bare Pt rotating disk electrode can be introduced into Eq. (31) in order to replace the theoretical function pffiffiffiffiffiffi pffiffiffiffiffiffi M4 ðsÞ ¼ tanh ss s=ðms ss sÞ. For the sake of clarity, the Mexp; RDE data at each X value are not shown in this paper. The shape of Mexp;RDE in the Nyquist representation is well known: it is the shape of a quarter lemniscate. The results shown in Fig. 9 are fitted by using the hybrid model described above (f 6 100 Hz), and the best fits are displayed in Fig. 11 (see also Table 2). A first remark can be made concerning the v2 values given in Tables 1 and 2, when using the purely theoretical model of Eq. (31), or when using the hybrid model. At each electrode rotation rate, X, the value of 6
103 cmf (cm s1 )
102 sf ðsÞ
102 v2
50 200 500 1000 2000 4000
7.3 3% 7.0 3% 7.3 2% 7.6 1% 7.7 1% 7.8 1%
4.0 5% 4.4 6% 4.1 4% 3.8 3% 3.6 2% 3.5 3%
2.7 3.0 1.6 1.1 0.7 1.3
criterion (24) is always lower for the fitting with the hybrid model than for the fitting with Eq. (31). The visual observation of Figs. 10 and 11 confirms this mathematical result. Such a point shows once again the usefulness of our hybrid model. Next, Table 1 and Fig. 10 show that the fitting with Eq. (31) is particularly poor for the lowest X values, but could be considered as adequate for the highest X values. This can be explained by the fact that the bounded diffusion approximation in solution starts to be valid only for low values of the Nernst layer (d), i.e., for high X values. Then, one can see in Table 2 that the cmf and sf values are very similar whatever the rotation rate: cmf 7:5 103 cm s1 and sf 4:0 102 s. In ad-
A
−1
X (rpm)
4
B
−1 0 −2
0
2
2 0
0
6
0
12
3
C
0
4
2.5
0
0
−1 3
6
2 0
2 0
0
2.5
5
1.5
E
F
0
−1 0
0
−1
2
2 0
D
0
0
2
8
2
4
−1 0
1.5
3
Re Zmt / (Ω cm2 ) Fig. 11. Mass-transport impedance data (the same than in Fig. 9 for f 6 100 Hz) fitted using the Ôtheoretical–experimental hybrid modelÕ, i.e., by pffiffiffiffiffiffi pffiffiffiffiffiffi using Mexp;RDE instead of M4 ðsÞ ¼ tanh ss s=ðms ss sÞ in Eq. (31). Rotation rate values: cf. Fig. 9 caption. The small and large dots correspond to the data, and the solid lines and the circles correspond to the result of the fitting (see also Table 2).
J.-P. Diard et al. / Journal of Electroanalytical Chemistry 566 (2004) 269–280
dition, we find again the values of Eqs. (26) obtained above for a fixed electrode. That means that all the awkward effects coming from mass transport in solution are left out, and that the determination of the parameter values is done under better conditions. To finish, let us try to determine L and c from the Df value recently obtained in the literature from the doubleband microelectrode technique [38]: Df 1:0 106 cm2 s1 , and from Eq. (26) or Table 2. There are now two equations and two unknowns, and one can find: L 2:0 lm, that is not very different from the estimated thickness (L 3 lm, cf. Section 2), and c 1:5.
7. Conclusion In this study of the PtjNafionÒ jFeIII=II system, we dealt with three points. In the first one, a complete theoretical EIS study was made for the redox reaction O + e $ R at a fixed metallic electrode coated with an ion-conducting polymer in contact with a quiescent solution. All the possible impedance diagram shapes were plotted using the Nyquist and Bode representations. In the second point, we showed that a combined steady-state and impedance study of such a system cannot lead to a separate determination of the diffusion coefficient of the electroactive species in the film (Df ) and of the thickness of the film (L), contrary to what Deslouis et al. [18] reported. Indeed, a third parameter, i.e., the partition coefficient between the solution and the polymer phase (c), must be taken into account. The problem of the membrane model raised by Bard, and Saveant and co-workers [8–10] remains unsolved. Hence a third electrochemical technique, or maybe an in situ non-electrochemical technique, which is able to give another combination of Df , L and c as a characteristic parameter, must be used. In the third and last point, the concept of a Ôtheoretical–experimental hybrid modelÕ was introduced for the first time. This Ôhybrid modelÕ consists in introducing the mass-transport experimental data obtained at a bare electrode into our theoretical model for a coated pelecffiffiffiffiffiffiffi trode, instead of the mass-transport function 1= sDs when using a fixed electrode (semi-infinite linear diffusion in solution), or instead of the mass-transport pffiffiffiffiffiffi pffiffiffiffiffiffi function tanh ss s=ðms ss sÞ when using a rotating disk electrode (bounded diffusion approximation in solution). The results showed that the fitting of the EIS experimental data was greatly improved by using the hybrid model because it allows us to avoid (but not to solve) the deviations from the Warburg model (fixed electrode) and from the Nernst model (RDE). One of the outlooks of this work is as follows: we know that the problem of the deviation from the Nernst model in dynamic conditions has been studied [39,40]. The numerical solutions given in [39,40] can also be
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introduced into Eq. (31) in order to replace the approximative analytical function M4 . It is then possible to compare the results of the fitting when using our hybrid model, and when using the two new models constructed after [39,40]. This will be the aim of a forthcoming paper [41].
Acknowledgements The authors are grateful to Dr. Gregory Berthome (Laboratoire de Thermodynamique et de Physico-Chimie Metallurgiques, Saint-Martin-dÕHeres, France) for performing the AFM measurements. N.G. thanks Prof. J.-Y. Sanchez for his interest in this work, and the French Ministry of the Economy, Finance and Industry for financial support.
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