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Kinetics of zinc anodic dissolution from the EIS characteristic points
Electrochemistry Communications 5 (2003) 722–727 www.elsevier.com/locate/elecom
Kinetics of zinc anodic dissolution from the EIS characteristic points D. Gimenez-Romero, J.J. Garcıa-Jare~ no, F. Vicente
*
Departament de Quımica Fısica, Universitat de Val encia, C. Dr. Moliner 50, 46100, Burjassot, Val encia, Spain Received 29 May 2003; received in revised form 13 June 2003; accepted 16 June 2003 Published online: 15 July 2003
Abstract A possible faradaic impedance function for the complex mechanism of metals electrodissolution across two consecutive electrotransferences has been developed in this work. The analysis of this function provides some characteristics points from which it is possible to calculate kinetic parameters of these processes. The dependence of these parameters on the potential has been studied in the case of Zn. These ones have been interpreted in terms of changes in the controlling stages of the overall rate of reaction. Ó 2003 Elsevier B.V. All rights reserved. Keywords: EIS simulation; Zinc anodic dissolution; Kinetic constants and electron transfer
1. Introduction The research for complex heterogeneous electrochemical processes often requires transient techniques. Depending on the technique used, the response of the transient is analyzed in either the time or the frequency domain. Electrochemical impedance spectroscopy (EIS) is an example of frequencies response analysis which has proved to be useful in both applied and fundamental electrochemistry [1,2], as well as in other disciplines [3]. The power of the technique is its ability to distinguish among processes with different time constants at the interface [4]. EIS is now a common and useful technique for studying metals dissolution reactions [5–12]. One of its advantages is the possibility of studying the relaxation of adsorbed intermediates. The impedance spectra of the dissolution reactions of metals usually show one or several relaxations (capacitive or inductive), which have been assigned to adsorbed intermediates participating in the reaction [13].
*
Corresponding author. Tel.: +34-963543022; fax: +34-963544564. E-mail address: [email protected] (F. Vicente).
1388-2481/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1388-2481(03)00150-4
The impedance model construction is the final aim of an experimental study [14] and should correspond to the properties of processes taking place in the object under study. Therefore, this one is usually very complicated, even for simple mechanism with little elementary steps involved. The parametrical identification can be made by using these impedance models and gives important information about the influence of the physical parameters on the reaction mechanism. The identification procedures are in principle numerical computer techniques and, therefore, sophisticated software is necessary for this calculation [15]. The aim of this work is the study of the faradaic mechanism composed of two consecutive monoelectronic transferences, since the dissolution of numerous metals takes place by means of this kinetic model [16– 20]. Nevertheless, the simultaneous two electron transferences can be done in other experimental conditions [21,22]. Relevant information on the mechanism of these electrochemical processes is provided by the parametrical identification of the electrochemical impedance model. The theoretical results of this work are applied to the experimental study of the zinc anodic dissolution reaction in deaerated sulphate medium, since the
D. Gimenez-Romero et al. / Electrochemistry Communications 5 (2003) 722–727
postulated mechanism in the bibliography for this process at acid pH corresponds to two consecutive monoelectronic transferences [16,23].
2. Theory The electrochemical dissolution of zinc and other metals is described in the bibliography by means of the following reaction mechanism [16–20] k1
Mðh0 Þ ! MðIÞðh1 Þ þ 1e k2
MðIÞðh1 Þ ! MðIIÞðh2 Þ þ 1e k4
MðIIÞðh2 Þ ! M2þ ðh2 Þ
ðiiÞ ðiiiÞ
where the participation of other species is omitted. The use of k1 , k2 , k4 is a consequence on the analogy with Cachet and Wiart [24] model for the zinc anodic dissolution which considers also a self-catalytic process characterized by a kinetic constant k3 . In these experimental conditions, the influence of the self-catalytic process on the EIS has proved to be negligible [27]. The theoretical impedance function of these electrochemical systems can be deduced through the following hypothesis: 1. The initial surface concentration of active centres is constant in an EIS experiment. 0
h ¼ h1 þ h2 þ h0 ;
ð1Þ
where hi is the surface concentration of i species and h0 is the initial surface concentration of active centres. 2. The monoelectronic transferences take place following kinetics of first order: dh1 ¼ k1 h0 k2 h1 ; dt
ð2Þ
dh2 ¼ k2 h1 k4 h2 ; ð3Þ dt where ki is the kinetic constant of the i transference. 3. The kinetic constants of the electronic transferences have a Butler–Volmer dependence on the applied potential [25]. ki ¼ ki0 ebi DE ;
ð4Þ
where DE is the applied potential variation in the EIS experiment, ki0 is the pre-exponential factor of the kinetic constants and bi is the exponential factor of these constants. 4. The diffusion rate follows a first order law with respect to the surface concentration of divalent zinc. 5. The applied potential during a EIS experiment can be emulated by means of the following expression [26]: E ¼ E þ DEejxt ;
dE ¼ jxDE dt;
ð5Þ
ð6Þ
where x is the frequency signal, t is the time, E is the stabilization potential, DE is the amplitude of the potential perturbation and j is the square root of )1. 6. The potential variations in this experiment are very small and, therefore, can be depreciated so that the differentials are considered as increases, Eq. (7), and the kinetic constants can be developed in power series, Eq. (8):
ðiÞ
723
dhi Dhi ¼ ; dE DE
ð7Þ
ki ¼ ki0 ð1 þ bi DE þ . . .Þ:
ð8Þ
7. Finally, the steady state is also considered and, at the same time, the bi are considered to be similar. Subsequently, the theoretical faradaic impedance function for these systems is [27] FA
dE ðx2 þ RÞðS þ T x2 Þ þ Y /x2 ¼ 2 di x2 /2 þ ðT x2 þ SÞ þj
ðTY þ /Þx3 þ xðSY /RÞ x2 /2 þ ðT x2 þ SÞ2
;
ð9Þ
where: R ¼ ðk20 þ k10 Þk4 þ k10 k20 ;
ð10Þ
S ¼ ððk20 þ k10 Þk4 þ k10 k20 Þðk10 b1 h0 þ k20 b2 h1 Þ 2ðk20 Þ2 k10 b2 h1 ;
ð11Þ
T ¼ ðk10 b1 h0 þ k20 b2 h1 Þ;
ð12Þ
/ ¼ ðk10 þ k20 þ k4 Þðk10 b1 h0 þ k20 b2 h1 Þ k10 k20 b2 h1 ;
ð13Þ
Y ¼ ðk10 þ k20 þ k4 Þ:
ð14Þ
It can be proved that the inductive loop appears when the second electronic transference (k2 ) is slower than the diffusive process (k4 ) and it is determined by means of kinetic parameters of all faradaic processes. However, the capacitive loop at high frequencies exclusively depends on the kinetic parameters of the first electronic transference.
3. Experimental The electrochemical experiments were carried out by means of a typical three electrodes cell where a platinum plate was the counter electrode and a AgjAgCljKClsat the reference electrode. The working electrode was a steel shooping plate supplied by GALESA, 0.5 cm2 of surface and 100–200 lm of zinc thickness. The working solution was composed of H3 BO3 (R.P. Normapurâ ) 0.32 M, NH4 Cl (p.a. Panreac) 0.26 M and Na2 SO4 (GEHE & Co. A.G. Dresden) 1.33 M, pH ¼ 4.4
724
D. Gimenez-Romero et al. / Electrochemistry Communications 5 (2003) 722–727
[28]. The cell was thermostatized at 24.4 °C by means of a HETO DENMARK bath and bubbled with Ar (AIR LIQUIDE) for 5 min before the electrochemical experiments take place. The EIS experiments were carried out by means of a potentiostat–galvanostat PAR 273A and the frequency analyser was used with the lock-in-amplifier PAR 5210. The impedance measurements were made in the frequency range from 0.05 to 104 Hz, with signal amplitude of 5 mV rms. 4. Results and discussion Fig. 1 shows an example of experimental impedance spectra under kinetic control for an electrochemical system with the previous reaction model. In this case, the faradaic impedance function presents five characteristic points: 1. The high frequencies limit of impedance characterized by Zimag ¼ 0. Resistive behaviour. 2. The low frequencies limit of impedance in which Zimag ¼ 0. 3. The intercept point of impedance function and the Zreal axis on the Nyquist plot. Characterized by Zimag ¼ 0. 4. The point in which the imaginary part of the impedance is maximum.
5. The point in which the imaginary part of the impedance is minimum. The impedance of these systems in the first characteristic point (initial point of the capacitive loop) presents the following value: x!1 ¼ 0; FAZimag x!1 FAZreal ¼
ð15Þ
k10 b1 h0
1 : þ k20 b2 h1
ð16Þ
As can be seen, this value only depends on the rate of the processes that involve electrons transfer. At the low frequencies limit: x!0 FAZimag ¼ 0; x!0 FAZreal ¼
ð17Þ 1
k10 b1 h0
þ ððk4
k20 Þ=ðk4
þ k20 ÞÞk20 b2 h1
:
ð18Þ
The faradaic impedance value depends on all the parameters of the faradaic processes that take place on the electrode, Eq. (18). The third point is characterized by the fact that the imaginary part of the faradaic impedance is zero. At this point, it is obtained that: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xZimag ¼0 ¼ k10 ðk4 k20 Þ; ð19Þ Z
imag FAZreal
¼0
¼
1 k10 b1 h0
ð20Þ
:
The following characteristic points are calculated by differentiating the faradaic impedance function with respect to the frequency and equating to zero, since these points are the maximum and minimum values of this function. That way, the impedance and frequency values where the imaginary part of impedance reaches maximum values are: xZimag maximum ¼ ðk4 þ k20 Þ þ ðk4 k20 Þ Z
imag FAZimag
maximum
¼ 2k10 b1 h0
Z
imag FAZreal
1 ¼ 2 Fig. 1. Dependence of electrical impedance spectra on the stabilization potential. Experimental conditions were H3 BO3 0.32 M, Na2 SO4 1.32 M, NH4 Cl 0.26 M and T ¼ 297:5 0:1 K. The working solution was deaerated by bubbling Ar for 5 min. (h) are the experimental EIS values at )1.000 V, (s) at )0.975 V, (D) at )0.950 V and (x) at )0.925 V, with regard to the AgjAgCljKClsat electrode. Continuous lines are the simulated spectra from the kinetic parameters obtained by means of the singular points (Table 2). The uncompensated resistance is 3.4 X cm2 and the double layer capacitance is 6.8 105 F cm2 .
1
k10 b1 h0 k20 b2 h1
k20 b2 h1 ; k10 b1 h0
k4 þk20 k4 k20
ð21Þ ;
ð22Þ
þ1
maximum
! 1 1 þ : k10 b1 h0 k10 b1 h0 þ k20 b2 h1 ððk4 k20 Þ=ðk4 þ k20 ÞÞ ð23Þ
And, in the minimum, these values are: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /2 þ 2TS k10 k20 ;k40 k10 Zimag minimum ; ¼ x T k30 !0 1 þ ðk20 b2 h1 =k10 b1 h0 Þ ð24Þ
D. Gimenez-Romero et al. / Electrochemistry Communications 5 (2003) 722–727 Z
imag FAZimag
minimum
Zimag minimum FAZreal
¼ 1 ¼ 2
1
; 1 þ ðk10 b1 h0 =k20 b2 h1 Þ ! 1 1 þ : k10 b1 h0 k10 b1 h0 þ k20 b2 h1
2k10 b1 h0
ð25Þ
h1 ¼
h2 ¼
Table 2 Kinetic parameters of the zinc anodic dissolution in deaerated sulphate medium E (V)
k10 ðs1 Þ
k20 ðs1 Þ
k4 ðs1 Þ
h0 ðlmol m2 Þ
)0.900 )0.925 )0.950 )0.975 )1.000
10236 9108 5724 3702 2150
13.9 13.2 9.0 5.0 3.9
19.4 21.0 24.6 20.5 7.4
638 697 692 617 340
ð26Þ
These equations allow the easy parametrical identification of the impedance model. The non-linear equations system used in this identification procedure is composed of Eqs. (18)–(20), (24) and (25). The other deduced expressions are not used in this calculation due to the uncertainly in its experimental measurement. Besides, in this equations system, the initial rates of the faradaic processes are also considered equal, since the faradaic system is at the steady state at the beginning of the EIS experiments. Then, the initial surface concentrations present the following relations: h0 ¼
725
k10 k20
k20 k4 h0 ; þ k10 k4 þ k20 k4
ð27Þ
k10 k20
k10 k4 h0 ; þ k10 k4 þ k20 k4
ð28Þ
k10 k20
k10 k20 h0 : þ k10 k4 þ k20 k4
ð29Þ
Fig. 1 shows the experimental impedance data of zinc anodic dissolution in deaerated sulphated medium and Table 1 collects values for the measured characteristic points. In the calculation of these points, the contributions to the electrical impedance of the uncompensated resistance and the double layer capacitance should be considered. These values are 3.4 X cm2 for the uncompensated resistance and 6.8 105 F cm2 for the double layer capacitance and are evaluated from the high frequencies response. Thus, considering this and using the value of the characteristic points, the parametrical identification of this system is made in Table 2. This identification allows the simulation of the EIS experimental spectra (Fig. 1) by means of Eq. (9). As can be observed in Fig. 1, the EIS spectra simulation from the characteristic points agrees with the experimental data. And, therefore, hy-
These parameters are calculated by means of the characteristic points of Table 1.
pothesis made for the calculation of these characteristic points are validated. Fig. 2 shows the evolution with respect to the stabilization potential of the pre-exponential factors of the kinetic constants of the zinc anodic dissolution process. That way, the parameter bi can be calculated by means of the fitting of these data to an exponential law, since these factors have also a Butler–Volmer dependence on the stabilization potential ki0 ¼ ki0 ebi E ;
ð30Þ
where ki0 is a constant. Thus, both electronic transference coefficients, ai , are calculated and their values are 0.60 0.01 and 0.65 0.01, respectively. Conversely, the diffusion constant, k4 , does not vary with the stabilization potential. Diffusion corresponds to a process dependent on the concentration gradient but not dependent on the stabilization potential. The discrepancy in the k4 value at )1.00 V can be due to the hydrogen evolution reaction which takes place simultaneously with the dissolution process at this potential. This fact can cause that the number of active centers for the zinc electrodissolution is smaller since there is always this competitive reaction. Then an apparent smaller k4 value is obtained. On the other hand, Fig. 2 also shows that, at potentials more cathodic than )0.88 V, the rate of the overall reaction is controlled by the second monoelectronic transference while, at potentials more anodic than )0.88 V, it is the diffusion of Zn2þ species which controls the overall rate.
Table 1 Evolution with respect to the stabilization potential of characteristic points of the impedance function of the zinc anodic dissolution in deaerated sulphated medium Z
E (V)
x!0 Zreal ðXÞ
xZimag ¼0 ðrad s1 Þ
imag Zreal ðXÞ
)0.900 )0.925 )0.950 )0.975 )1.000
2.2 1.9 2.0 3.4 11.7
236.7 266.5 299.4 239.9 86.1
2.5 2.3 2.8 5.2 14.8
¼0
Z
xZimag minimum ðrad s1 Þ
imag Zimag ðXÞ
5472.7 4869.5 3060.5 1979.2 1149.8
)1.0 )1.0 )1.2 )2.0 )6.2
minimum
Z
imag Zreal ðXÞ
1.3 1.2 1.9 3.0 9.4
minimum
Z
xZimag maximum ðrad s1 Þ
imag Zimag ðXÞ
36.7 34.5 31.5 28.8 15.6
0.3 0.2 0.3 0.7 1.5
maximum
Z
imag Zreal ðXÞ
maximum
2.4 2.1 2.5 5.1 13.5
The impedance dates showed in this table are corrected with the values of the non-compensated resistance and the double layer capacitance. The non-compensated resistance value is 3.4 X cm2 and the double layer capacitance value is 6.8 105 F cm2 .
726
D. Gimenez-Romero et al. / Electrochemistry Communications 5 (2003) 722–727
Fig. 2. Curve of the pre-exponential factors of the kinetic constants, ki0 , vs the applied potential, E. (h) are the values of ln k10 ; (s), ln k20 ; and (D), ln k4 . The units of kinetic constants are s1 .
Thus, it is possible to obtain a good estimation of h0 , the initial surface concentration of active centres, by means of the parameter bi . Table 2 shows the dependence of this parameter on the potential. This value does not vary significantly on the applied potential except at very cathodic potentials (E ¼ 1:00 V) where h0 reaches smaller values. A possible explanation is that hydrogen species associated to the hydrogen evolution can be adsorbed on the metallic surface and therefore these centres keep inactive for the zinc faradaic processes. In order to corroborate that this methodology is appropriate for the study of this system, the dependence of characteristic points on the potential has been simulated. Thus, for example, the simulation of the evolution Zimag minimum of Zreal according to the stabilization potential is shown in Fig. 3. This simulation is carried out by means of the data interpolated of the Fig. 2 and the theoretical Zimag minimum expression of Zreal , Eq. (26). And, as commented, this simulation agrees with the experimental data.
5. Conclusion The study of the electrochemical systems with two consecutive monoelectronic transferences by means of the characteristics points allows the parametrical identification from EIS without a fitting procedure, but only by measuring some quantities on the NyquistÕs plot. Parameters obtained by this procedure can be used themselves as well as be introduced in a numerical fitting as a good first approach. From these parameters, it is
Fig. 3. Simulation of the evolution with respect to the stabilization Zimag minimum potential of the Zreal characteristic point. (h) are the values of this characteristic point (Table 1) and the continuous line is its simulation.
possible to simulate the whole impedance spectra with good results. The characteristics points study also allows to establish the kinetic behaviour of the electrochemical systems with respect to the experimental variables. Thus, the dependence of the kinetic magnitudes of the zinc anodic dissolution on the stabilization potential has been established. These dependences are fully consistent and allow to discern between two potential zones: a zone in which the rate of the overall reaction is controlled by the second electronic transference and another zone where the diffusion of Zn(II) is the steady state.
Acknowledgements This work has been partially supported by CICyTMAT/2000-0100-P4. D. Gimenez-Romero acknowledges a Fellowship from the Generalitat Valenciana, Program FPI. J.J. Garcıa-Jare~ no acknowledges the financial support by the ‘‘Ramon y Cajal’’ program (Ministerio de Ciencia y Tecnologia-Spanish Ministry of Science and Technology).
References [1] F. Vicente, A. Roig, J.J. Garcıa Jare~ no, A. Sanmatıas. Procesos electr odicos del Nafi on y del Azul de Prusia/Nafion sobre xido de Indio-Esta~ electrodo transparente o no: Un modelo de electrodos multicapa. Ed. Moliner 40, Burjassot, 2001. [2] O.E. Barcia, O.R. Mattos, Electrochim. Acta 35 (1990) 1601.
D. Gimenez-Romero et al. / Electrochemistry Communications 5 (2003) 722–727 [3] J.R. Macdonald, Impedance Spectroscopy, Wiley, New York, 1987. [4] E. Ahlberg, H. Anderson, Acta Chem. Scand. 46 (1992) 1. [5] E. Ahlberg, H. Anderson, Acta Chem. Scand. 46 (1992) 15. [6] M. Keddam, O.R. Mattos, H. Takenouti, J. Electrochem. Soc. 128 (1981) 266. [7] L. Fedrizzi, L. Ciaghi, P. Bonora, R. Fratesi, G. Roventi, J. Appl. Electrochem. 22 (1992) 247. [8] L. Bai, B.E. Conway, J. Electrochem. Soc. 138 (1991) 2897. [9] L. Bai, B.E. Conway, Electrochim. Acta 38 (1993) 1803. [10] J.A.L. Dobbelaar, J.H.W. de Wit, J. Electrochem. Soc. 139 (1992) 716. [11] B.H. Erne, D. Vanmaekelbergn, J. Electrochem. Soc. 144 (1997) 3385. [12] M.R. Barbosa, J.A. Bastos, J.J. Garcıa-Jare~ no, F. Vicente, Electrochim. Acta 44 (1998) 957. [13] A. Ahlberg, H. Anderson, Acta Chem. Scand. 47 (1993) 1162. [14] Z. Stoynov, Electrochim. Acta 35 (1990) 1493. [15] J.R. Macdonald, Solid State Ion. 58 (1992) 97. [16] R. Wiart, Electrochim. Acta 35 (1990) 1587.
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[17] J.P. Diard, J.M. Le Canut, B. Le Gorrec, C. Montella, Electrochim. Acta 43 (1998) 2469. [18] M.A.V. Devanathan, Electrochim. Acta 17 (1972) 1683. [19] C. Sanz, J. Lıopez, F. Vicente, An. Quim. 88 (1992) 530. [20] M. Itagaki, H. Nakazawa, K. Watanabe, K. Noda, Corros. Sci. 39 (1997) 901. [21] E. Gileadi, J. Electroanal. Chem. 532 (2002) 181. [22] M.A. Nu~ nez-Flores, A. Ruano, F. Vicente, An. Quim. 84 (1988) 289. [23] L. Koene, M. Sluyters-Rehbach, J.H. Sluyters, J. Electroanal. Chem. 402 (1996) 57. [24] C. Cachet, R. Wiart, J. Electroanal. Chem. 111 (1980) 235. [25] L. Bortels, B. Van den Bossche, J. Deconinck, S. Vandeputte, A. Hubin, J. Electroanal. Chem. 429 (1997) 139. [26] A.J. Bard, L.R. Faulkner, in: Electrochemical Methods Fundamentals and Applications, second ed., Wiley, New York, 2001, pp. 370–371. [27] D. Gimenez-Romero, J.J. Garcıa-Jare~ no, F. Vicente, J. Electroanal. Chem. (2003) in press. [28] T. Ohtsuka, A. Komori, Electrochim. Acta 43 (1998) 3269.
Kinetics of zinc anodic dissolution from the EIS characteristic points D. Gimenez-Romero, J.J. Garcıa-Jare~ no, F. Vicente
*
Departament de Quımica Fısica, Universitat de Val encia, C. Dr. Moliner 50, 46100, Burjassot, Val encia, Spain Received 29 May 2003; received in revised form 13 June 2003; accepted 16 June 2003 Published online: 15 July 2003
Abstract A possible faradaic impedance function for the complex mechanism of metals electrodissolution across two consecutive electrotransferences has been developed in this work. The analysis of this function provides some characteristics points from which it is possible to calculate kinetic parameters of these processes. The dependence of these parameters on the potential has been studied in the case of Zn. These ones have been interpreted in terms of changes in the controlling stages of the overall rate of reaction. Ó 2003 Elsevier B.V. All rights reserved. Keywords: EIS simulation; Zinc anodic dissolution; Kinetic constants and electron transfer
1. Introduction The research for complex heterogeneous electrochemical processes often requires transient techniques. Depending on the technique used, the response of the transient is analyzed in either the time or the frequency domain. Electrochemical impedance spectroscopy (EIS) is an example of frequencies response analysis which has proved to be useful in both applied and fundamental electrochemistry [1,2], as well as in other disciplines [3]. The power of the technique is its ability to distinguish among processes with different time constants at the interface [4]. EIS is now a common and useful technique for studying metals dissolution reactions [5–12]. One of its advantages is the possibility of studying the relaxation of adsorbed intermediates. The impedance spectra of the dissolution reactions of metals usually show one or several relaxations (capacitive or inductive), which have been assigned to adsorbed intermediates participating in the reaction [13].
*
Corresponding author. Tel.: +34-963543022; fax: +34-963544564. E-mail address: [email protected] (F. Vicente).
1388-2481/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1388-2481(03)00150-4
The impedance model construction is the final aim of an experimental study [14] and should correspond to the properties of processes taking place in the object under study. Therefore, this one is usually very complicated, even for simple mechanism with little elementary steps involved. The parametrical identification can be made by using these impedance models and gives important information about the influence of the physical parameters on the reaction mechanism. The identification procedures are in principle numerical computer techniques and, therefore, sophisticated software is necessary for this calculation [15]. The aim of this work is the study of the faradaic mechanism composed of two consecutive monoelectronic transferences, since the dissolution of numerous metals takes place by means of this kinetic model [16– 20]. Nevertheless, the simultaneous two electron transferences can be done in other experimental conditions [21,22]. Relevant information on the mechanism of these electrochemical processes is provided by the parametrical identification of the electrochemical impedance model. The theoretical results of this work are applied to the experimental study of the zinc anodic dissolution reaction in deaerated sulphate medium, since the
D. Gimenez-Romero et al. / Electrochemistry Communications 5 (2003) 722–727
postulated mechanism in the bibliography for this process at acid pH corresponds to two consecutive monoelectronic transferences [16,23].
2. Theory The electrochemical dissolution of zinc and other metals is described in the bibliography by means of the following reaction mechanism [16–20] k1
Mðh0 Þ ! MðIÞðh1 Þ þ 1e k2
MðIÞðh1 Þ ! MðIIÞðh2 Þ þ 1e k4
MðIIÞðh2 Þ ! M2þ ðh2 Þ
ðiiÞ ðiiiÞ
where the participation of other species is omitted. The use of k1 , k2 , k4 is a consequence on the analogy with Cachet and Wiart [24] model for the zinc anodic dissolution which considers also a self-catalytic process characterized by a kinetic constant k3 . In these experimental conditions, the influence of the self-catalytic process on the EIS has proved to be negligible [27]. The theoretical impedance function of these electrochemical systems can be deduced through the following hypothesis: 1. The initial surface concentration of active centres is constant in an EIS experiment. 0
h ¼ h1 þ h2 þ h0 ;
ð1Þ
where hi is the surface concentration of i species and h0 is the initial surface concentration of active centres. 2. The monoelectronic transferences take place following kinetics of first order: dh1 ¼ k1 h0 k2 h1 ; dt
ð2Þ
dh2 ¼ k2 h1 k4 h2 ; ð3Þ dt where ki is the kinetic constant of the i transference. 3. The kinetic constants of the electronic transferences have a Butler–Volmer dependence on the applied potential [25]. ki ¼ ki0 ebi DE ;
ð4Þ
where DE is the applied potential variation in the EIS experiment, ki0 is the pre-exponential factor of the kinetic constants and bi is the exponential factor of these constants. 4. The diffusion rate follows a first order law with respect to the surface concentration of divalent zinc. 5. The applied potential during a EIS experiment can be emulated by means of the following expression [26]: E ¼ E þ DEejxt ;
dE ¼ jxDE dt;
ð5Þ
ð6Þ
where x is the frequency signal, t is the time, E is the stabilization potential, DE is the amplitude of the potential perturbation and j is the square root of )1. 6. The potential variations in this experiment are very small and, therefore, can be depreciated so that the differentials are considered as increases, Eq. (7), and the kinetic constants can be developed in power series, Eq. (8):
ðiÞ
723
dhi Dhi ¼ ; dE DE
ð7Þ
ki ¼ ki0 ð1 þ bi DE þ . . .Þ:
ð8Þ
7. Finally, the steady state is also considered and, at the same time, the bi are considered to be similar. Subsequently, the theoretical faradaic impedance function for these systems is [27] FA
dE ðx2 þ RÞðS þ T x2 Þ þ Y /x2 ¼ 2 di x2 /2 þ ðT x2 þ SÞ þj
ðTY þ /Þx3 þ xðSY /RÞ x2 /2 þ ðT x2 þ SÞ2
;
ð9Þ
where: R ¼ ðk20 þ k10 Þk4 þ k10 k20 ;
ð10Þ
S ¼ ððk20 þ k10 Þk4 þ k10 k20 Þðk10 b1 h0 þ k20 b2 h1 Þ 2ðk20 Þ2 k10 b2 h1 ;
ð11Þ
T ¼ ðk10 b1 h0 þ k20 b2 h1 Þ;
ð12Þ
/ ¼ ðk10 þ k20 þ k4 Þðk10 b1 h0 þ k20 b2 h1 Þ k10 k20 b2 h1 ;
ð13Þ
Y ¼ ðk10 þ k20 þ k4 Þ:
ð14Þ
It can be proved that the inductive loop appears when the second electronic transference (k2 ) is slower than the diffusive process (k4 ) and it is determined by means of kinetic parameters of all faradaic processes. However, the capacitive loop at high frequencies exclusively depends on the kinetic parameters of the first electronic transference.
3. Experimental The electrochemical experiments were carried out by means of a typical three electrodes cell where a platinum plate was the counter electrode and a AgjAgCljKClsat the reference electrode. The working electrode was a steel shooping plate supplied by GALESA, 0.5 cm2 of surface and 100–200 lm of zinc thickness. The working solution was composed of H3 BO3 (R.P. Normapurâ ) 0.32 M, NH4 Cl (p.a. Panreac) 0.26 M and Na2 SO4 (GEHE & Co. A.G. Dresden) 1.33 M, pH ¼ 4.4
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D. Gimenez-Romero et al. / Electrochemistry Communications 5 (2003) 722–727
[28]. The cell was thermostatized at 24.4 °C by means of a HETO DENMARK bath and bubbled with Ar (AIR LIQUIDE) for 5 min before the electrochemical experiments take place. The EIS experiments were carried out by means of a potentiostat–galvanostat PAR 273A and the frequency analyser was used with the lock-in-amplifier PAR 5210. The impedance measurements were made in the frequency range from 0.05 to 104 Hz, with signal amplitude of 5 mV rms. 4. Results and discussion Fig. 1 shows an example of experimental impedance spectra under kinetic control for an electrochemical system with the previous reaction model. In this case, the faradaic impedance function presents five characteristic points: 1. The high frequencies limit of impedance characterized by Zimag ¼ 0. Resistive behaviour. 2. The low frequencies limit of impedance in which Zimag ¼ 0. 3. The intercept point of impedance function and the Zreal axis on the Nyquist plot. Characterized by Zimag ¼ 0. 4. The point in which the imaginary part of the impedance is maximum.
5. The point in which the imaginary part of the impedance is minimum. The impedance of these systems in the first characteristic point (initial point of the capacitive loop) presents the following value: x!1 ¼ 0; FAZimag x!1 FAZreal ¼
ð15Þ
k10 b1 h0
1 : þ k20 b2 h1
ð16Þ
As can be seen, this value only depends on the rate of the processes that involve electrons transfer. At the low frequencies limit: x!0 FAZimag ¼ 0; x!0 FAZreal ¼
ð17Þ 1
k10 b1 h0
þ ððk4
k20 Þ=ðk4
þ k20 ÞÞk20 b2 h1
:
ð18Þ
The faradaic impedance value depends on all the parameters of the faradaic processes that take place on the electrode, Eq. (18). The third point is characterized by the fact that the imaginary part of the faradaic impedance is zero. At this point, it is obtained that: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xZimag ¼0 ¼ k10 ðk4 k20 Þ; ð19Þ Z
imag FAZreal
¼0
¼
1 k10 b1 h0
ð20Þ
:
The following characteristic points are calculated by differentiating the faradaic impedance function with respect to the frequency and equating to zero, since these points are the maximum and minimum values of this function. That way, the impedance and frequency values where the imaginary part of impedance reaches maximum values are: xZimag maximum ¼ ðk4 þ k20 Þ þ ðk4 k20 Þ Z
imag FAZimag
maximum
¼ 2k10 b1 h0
Z
imag FAZreal
1 ¼ 2 Fig. 1. Dependence of electrical impedance spectra on the stabilization potential. Experimental conditions were H3 BO3 0.32 M, Na2 SO4 1.32 M, NH4 Cl 0.26 M and T ¼ 297:5 0:1 K. The working solution was deaerated by bubbling Ar for 5 min. (h) are the experimental EIS values at )1.000 V, (s) at )0.975 V, (D) at )0.950 V and (x) at )0.925 V, with regard to the AgjAgCljKClsat electrode. Continuous lines are the simulated spectra from the kinetic parameters obtained by means of the singular points (Table 2). The uncompensated resistance is 3.4 X cm2 and the double layer capacitance is 6.8 105 F cm2 .
1
k10 b1 h0 k20 b2 h1
k20 b2 h1 ; k10 b1 h0
k4 þk20 k4 k20
ð21Þ ;
ð22Þ
þ1
maximum
! 1 1 þ : k10 b1 h0 k10 b1 h0 þ k20 b2 h1 ððk4 k20 Þ=ðk4 þ k20 ÞÞ ð23Þ
And, in the minimum, these values are: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /2 þ 2TS k10 k20 ;k40 k10 Zimag minimum ; ¼ x T k30 !0 1 þ ðk20 b2 h1 =k10 b1 h0 Þ ð24Þ
D. Gimenez-Romero et al. / Electrochemistry Communications 5 (2003) 722–727 Z
imag FAZimag
minimum
Zimag minimum FAZreal
¼ 1 ¼ 2
1
; 1 þ ðk10 b1 h0 =k20 b2 h1 Þ ! 1 1 þ : k10 b1 h0 k10 b1 h0 þ k20 b2 h1
2k10 b1 h0
ð25Þ
h1 ¼
h2 ¼
Table 2 Kinetic parameters of the zinc anodic dissolution in deaerated sulphate medium E (V)
k10 ðs1 Þ
k20 ðs1 Þ
k4 ðs1 Þ
h0 ðlmol m2 Þ
)0.900 )0.925 )0.950 )0.975 )1.000
10236 9108 5724 3702 2150
13.9 13.2 9.0 5.0 3.9
19.4 21.0 24.6 20.5 7.4
638 697 692 617 340
ð26Þ
These equations allow the easy parametrical identification of the impedance model. The non-linear equations system used in this identification procedure is composed of Eqs. (18)–(20), (24) and (25). The other deduced expressions are not used in this calculation due to the uncertainly in its experimental measurement. Besides, in this equations system, the initial rates of the faradaic processes are also considered equal, since the faradaic system is at the steady state at the beginning of the EIS experiments. Then, the initial surface concentrations present the following relations: h0 ¼
725
k10 k20
k20 k4 h0 ; þ k10 k4 þ k20 k4
ð27Þ
k10 k20
k10 k4 h0 ; þ k10 k4 þ k20 k4
ð28Þ
k10 k20
k10 k20 h0 : þ k10 k4 þ k20 k4
ð29Þ
Fig. 1 shows the experimental impedance data of zinc anodic dissolution in deaerated sulphated medium and Table 1 collects values for the measured characteristic points. In the calculation of these points, the contributions to the electrical impedance of the uncompensated resistance and the double layer capacitance should be considered. These values are 3.4 X cm2 for the uncompensated resistance and 6.8 105 F cm2 for the double layer capacitance and are evaluated from the high frequencies response. Thus, considering this and using the value of the characteristic points, the parametrical identification of this system is made in Table 2. This identification allows the simulation of the EIS experimental spectra (Fig. 1) by means of Eq. (9). As can be observed in Fig. 1, the EIS spectra simulation from the characteristic points agrees with the experimental data. And, therefore, hy-
These parameters are calculated by means of the characteristic points of Table 1.
pothesis made for the calculation of these characteristic points are validated. Fig. 2 shows the evolution with respect to the stabilization potential of the pre-exponential factors of the kinetic constants of the zinc anodic dissolution process. That way, the parameter bi can be calculated by means of the fitting of these data to an exponential law, since these factors have also a Butler–Volmer dependence on the stabilization potential ki0 ¼ ki0 ebi E ;
ð30Þ
where ki0 is a constant. Thus, both electronic transference coefficients, ai , are calculated and their values are 0.60 0.01 and 0.65 0.01, respectively. Conversely, the diffusion constant, k4 , does not vary with the stabilization potential. Diffusion corresponds to a process dependent on the concentration gradient but not dependent on the stabilization potential. The discrepancy in the k4 value at )1.00 V can be due to the hydrogen evolution reaction which takes place simultaneously with the dissolution process at this potential. This fact can cause that the number of active centers for the zinc electrodissolution is smaller since there is always this competitive reaction. Then an apparent smaller k4 value is obtained. On the other hand, Fig. 2 also shows that, at potentials more cathodic than )0.88 V, the rate of the overall reaction is controlled by the second monoelectronic transference while, at potentials more anodic than )0.88 V, it is the diffusion of Zn2þ species which controls the overall rate.
Table 1 Evolution with respect to the stabilization potential of characteristic points of the impedance function of the zinc anodic dissolution in deaerated sulphated medium Z
E (V)
x!0 Zreal ðXÞ
xZimag ¼0 ðrad s1 Þ
imag Zreal ðXÞ
)0.900 )0.925 )0.950 )0.975 )1.000
2.2 1.9 2.0 3.4 11.7
236.7 266.5 299.4 239.9 86.1
2.5 2.3 2.8 5.2 14.8
¼0
Z
xZimag minimum ðrad s1 Þ
imag Zimag ðXÞ
5472.7 4869.5 3060.5 1979.2 1149.8
)1.0 )1.0 )1.2 )2.0 )6.2
minimum
Z
imag Zreal ðXÞ
1.3 1.2 1.9 3.0 9.4
minimum
Z
xZimag maximum ðrad s1 Þ
imag Zimag ðXÞ
36.7 34.5 31.5 28.8 15.6
0.3 0.2 0.3 0.7 1.5
maximum
Z
imag Zreal ðXÞ
maximum
2.4 2.1 2.5 5.1 13.5
The impedance dates showed in this table are corrected with the values of the non-compensated resistance and the double layer capacitance. The non-compensated resistance value is 3.4 X cm2 and the double layer capacitance value is 6.8 105 F cm2 .
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D. Gimenez-Romero et al. / Electrochemistry Communications 5 (2003) 722–727
Fig. 2. Curve of the pre-exponential factors of the kinetic constants, ki0 , vs the applied potential, E. (h) are the values of ln k10 ; (s), ln k20 ; and (D), ln k4 . The units of kinetic constants are s1 .
Thus, it is possible to obtain a good estimation of h0 , the initial surface concentration of active centres, by means of the parameter bi . Table 2 shows the dependence of this parameter on the potential. This value does not vary significantly on the applied potential except at very cathodic potentials (E ¼ 1:00 V) where h0 reaches smaller values. A possible explanation is that hydrogen species associated to the hydrogen evolution can be adsorbed on the metallic surface and therefore these centres keep inactive for the zinc faradaic processes. In order to corroborate that this methodology is appropriate for the study of this system, the dependence of characteristic points on the potential has been simulated. Thus, for example, the simulation of the evolution Zimag minimum of Zreal according to the stabilization potential is shown in Fig. 3. This simulation is carried out by means of the data interpolated of the Fig. 2 and the theoretical Zimag minimum expression of Zreal , Eq. (26). And, as commented, this simulation agrees with the experimental data.
5. Conclusion The study of the electrochemical systems with two consecutive monoelectronic transferences by means of the characteristics points allows the parametrical identification from EIS without a fitting procedure, but only by measuring some quantities on the NyquistÕs plot. Parameters obtained by this procedure can be used themselves as well as be introduced in a numerical fitting as a good first approach. From these parameters, it is
Fig. 3. Simulation of the evolution with respect to the stabilization Zimag minimum potential of the Zreal characteristic point. (h) are the values of this characteristic point (Table 1) and the continuous line is its simulation.
possible to simulate the whole impedance spectra with good results. The characteristics points study also allows to establish the kinetic behaviour of the electrochemical systems with respect to the experimental variables. Thus, the dependence of the kinetic magnitudes of the zinc anodic dissolution on the stabilization potential has been established. These dependences are fully consistent and allow to discern between two potential zones: a zone in which the rate of the overall reaction is controlled by the second electronic transference and another zone where the diffusion of Zn(II) is the steady state.
Acknowledgements This work has been partially supported by CICyTMAT/2000-0100-P4. D. Gimenez-Romero acknowledges a Fellowship from the Generalitat Valenciana, Program FPI. J.J. Garcıa-Jare~ no acknowledges the financial support by the ‘‘Ramon y Cajal’’ program (Ministerio de Ciencia y Tecnologia-Spanish Ministry of Science and Technology).
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