4584
J. Phys. Chem. B 2005, 109, 4584-4592
Graphical Analysis of Electrochemical Impedance Spectroscopy of Two Consecutive Irreversible Electron Transfers. 1. Theoretical Study of the Anodic Dissolution of Metals J. J. Garcı´a-Jaren˜ o,† D. Gime´ nez-Romero,† M. Keddam,‡ and F. Vicente*,† Departament de Quı´mica Fı´sica, UniVersitat de Vale` ncia, C/Dr Moliner, 50, 46100, Burjassot, Vale` ncia, Spain, and UPR 15 CNRS, Laboratoire des Interfaces et Syste` mes Electrochimiques (LISE), UniVersite´ P. et M. Curie, 4 place Jussieu, F-75252 Paris Cedex 05, France ReceiVed: July 20, 2004; In Final Form: September 28, 2004
A general function for the faradaic impedance associated to a two consecutive single electron transfer mechanism followed by a irreversible first-order step has been obtained on the basis of formal kinetics. Kinetic parameters associated to this reaction mechanism can be obtained from the different kind of plots of the impedance function: Nyquist, Cole-Cole, and different Bode plots. A strategy for obtaining all parameters is suggested and analyzed on a detailed flowchart.
Introduction The anodic dissolution of many metals to the divalent oxidation state takes place by means of similar mechanisms consisting of a two consecutive single-electron transfers followed by a heterogeneous irreversible process, which implies the mass transfer from the surface to the solution.1-7 However, impedance spectra of different shapes are reported depending on the metal and a given metal in experimental conditions. This is perfectly consistent with the theoretical prediction of impedance models based on the relaxation of the surface coverage by adsorbed intermediates. Considering this, the theoretical analysis of this reaction mechanism is of outstanding importance for getting a deeper insight into electrochemical processes such as electromachining, discharge at the negative electrode of primary and secondary batteries, metal corrosion, and corrosion protection including that provided by inhibitors and coatings. For such intricate heterogeneous electrochemical processes, involving several overlapping steps, steady-state measurements are unable to yield a detailed kinetic description; transient techniques are required. Over the last three decades, ac electrochemical impedance spectroscopy (EIS) has proved to be the best suitable in both applied and fundamental electrochemistry8-12 as well as in other disciplines.13,14 The technique is potential selective and allows the separation of processes with different relaxation time constants by sweeping the frequency of the signal perturbation. For metal dissolution reactions,15-23 the impedance spectra usually show one or several relaxations (capacitive or inductive), which have been assigned to adsorbed intermediates participating in the reaction mechanism.24 Furthermore, for reactions where the rate is controlled by the diffusion of soluble species, the impedance spectra show the well-known Warburg or Warburg-Nernst impedance.24 In the literature, different strategies have been carried out for the analysis of impedance results. It is very common to analyze impedance results in terms of equivalent circuits.25-28 * Author to whom correspondence should be addressed. E-mail:
[email protected]. † Universitat de Vale ` ncia. ‡ Universite ´ P. et M. Curie.
The values of the electrical components of these circuits are tentatively associated with the physical/chemical properties of the metal-electrolyte interface.29 Hence, in a way, the evolution of these elements displays the influence of the metal and reaction medium on the electrodic mechanism. Anothersand in the present trends considered as more relevantsmodeling is the derivation of an analytical or, when not possible, numerical function of the frequency. The starting point is a set of equations depicting the physicochemical processes at the interface. Such models may contain ingredients such as electron transfers, mass transport, 2D and 3D phases nucleation, and growth. That way, Cao30,31 developed a methodology for the mathematical modeling of the faradaic impedance of irreversible electrode reactions based on the state variable theory that has been successfully applied to the analysis of impedance data of several systems. Diard et al.32 proposed a method based on the positions of the impedance poles and zeros on the p-plane for the analysis of impedance data instead of using equivalent circuits for a Volmer-Heyrovsky mechanism with chemical desorption. Other interesting works in this respect are those of Harrington33,34 who proposed a general theory for the analysis of impedance results of multistep mechanisms that takes into account the number of independent reaction steps and the number of independent reactions of the soluble species. From these considerations, a mathematical expression for the faradaic admittance can be constructed from the division of two determinants, which include different derivatives of electrochemical reactions rates on the potential and on the different species concentration. Therefore, most identification procedures are in principle numerical computer techniques, and therefore, sophisticated software is necessary for this calculation. This sophistication is reduced by means of the techniques of graphical analysis.35 The aim of this work is two-fold: modeling of the processes which take place by means of two consecutive irreversible electron transfers followed by a step of solubilization dealt with as a first-order chemical step; graphical analysis of the theoretical impedance function in order to design a parametrical identification procedure with a very high accuracy in the calculation of the kinetic parameters, allowing the EIS simulation in a subsequent step.
10.1021/jp0467954 CCC: $30.25 © 2005 American Chemical Society Published on Web 02/18/2005
Graphical Analysis of EIS
J. Phys. Chem. B, Vol. 109, No. 10, 2005 4585
Theoretical Impedance Function Considering the mechanism of eq i, the theoretical faradaic impedance function for two consecutive irreversible singleelectron transfers is deduced. k1
k2
-1e
-1e
k3
M(0)(Γ0) 9 8 M(I)(Γ1) 9 8 M(II)(Γ2) 98 M2+ -
(i)
This function has been derived by assuming the following hypotheses: 1. All three steps of this mechanism are elementary steps, and the kinetics of these reactions follows a first-order law. 2. The applied potential during the EIS experiment can be emulated by means of the following expression36
h + ∆E )E h + ∆E e jωt ) E dE ) jω∆E dt
function in formal kinetics37, eq 9, with respect to the applied potential.
I ) k1Γ0 + k2Γ1 FA 2 1 dI -I0ω + jω[(σ + R)I0 - Kγ] + ) FA dE (σ + R)jω - ω2 + (σR + B) ((σR + B)I0 + γK(X - R) - MXγ)
(σ + R)jω - ω2 + (σR + B)
(11)
K ) k10
(12)
M ) k20
(13)
γ ) k20b2Γ1
(14)
X ) k10
(15)
σ ) k3
(16)
R ) k20 + k10
(17)
B ) k10k20
(18)
(1) (2)
(3)
where ki is the kinetic constant of the transfer i. 4. The potential excursion ∆E taken is small enough so that eq 4 and the kinetic constants can be expanded in power series bounded to the first order of the increments ∆’s, eq 5.
dΓi ∆Γi ) dE ∆E
(4)
ki ) ki0(1 + bi∆E + ...)
(5)
dΓ1 ) k1Γ0 - k2Γ1 dt dΓ2 ) k2Γ1 - k3Γ2 dt
(7)
0
(8a)
dΓ2 k2 b2Γ1(jω + k1 + k2 ) ) dE (jω + k )(jω + k 0 + k 0) + k 0k 0 3 1 2 1 2
(8b)
k20b2Γ1(jω + k20) dΓ0 )dE (jω + k3)(jω + k10 + k20) + k10k20
(8c)
0
2 2 2 dE (-ω + R)(S + Tω ) + δφω ) + dI ω2φ2 + (S + Tω2)2 (φ + Tδ)ω3 + ω(Sδ - φR) (19) j ω2φ2 + (S + Tω2)2
where
R ) (Rσ + B) S ) (σR + B)I0 + γK(X - R) - MXγ
(20) (21)
φ ) (R + σ)I0 - Kγ
(22)
δ ) (R + σ) T ) -I0
(23) (24)
(6)
dΓ1 k1 k2 b2Γ1 )dE (jω + k3)(jω + k10 + k20) + k10k20 0
where A is the electrode surface area and F is the Faraday constant. The faradaic impedance function (Z) is obtained by taking the inverse of the admittance function
FA
where Γi is the surface concentration of species i. 5. Finally, the steady state is also considered, and at the same time, the bi are considered to be similar. In these conditions, the differential equations governing the time dependence of the surface concentrations are established from the kinetic laws (eqs 6 and 7), and accounting for eq 3, the complex ratios (dΓi/dt) (eq 8) are derived
0
(10)
I0 ) k10b1Γ0 + k20b2Γ1
where ω is the angular frequency signal, t is the time, E h is the study potential, ∆E is the amplitude of the potential perturbation, j is the square root of -1, and ∆E is the variation of the study potential. 3. The kinetic constants of the electron transfers have a Butler-Volmer dependence on the applied potential.37
ki ) ki0 ebi (Eh +∆E) ) ki0 ebi ∆E
(9)
The steady-state surface concentrations considered in eq 10 obey the following expressions
Γ1 )
0
Γ2 )
where Γi is the mean surface concentration of the species i, equal to its steady-state value at potential E. Considering these derivatives, the faradaic admittance (Y), eq 10, is obtained from the differential of the current density
k3k1 k1k2 + k3(k1 + k2) k2k1 k1k2 + k3(k1 + k2)
( ( )
Γ 0 ) Γ0 1 - 1 +
Γ0
(25)
Γ0
(26)
k2 k3k1 k3 k1k2 + k3(k1 + k2)
)
(27)
where Γ0 is the initial total surface concentration of unoccupied electroactive sites that keeps constant during the impedance experiment.
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The electrical or measured impedance (Ze) should also include the contributions of double layer capacitance (Cdl) and the electrolyte resistance (Ru), eq 28.
Ze )
Zf + Ru 1 + jωCdl Zf
(28)
Graphical Analysis of the Impedance Function Once the faradaic impedance function is derived, the problem still remains to analyze the different plots corresponding to this function. From the analysis of the characteristic points and zones in the different plots, it is possible to obtain most of the kinetic parameters. However, to make the understanding of the obtained equations easier, several approaches will be considered. The first one is that the kinetic constant for the first electron transfer is much larger than the other kinetic constants. This is a common case in metal dissolution processes.4 The second one is the steady-state hypothesis expressed by eqs 25-27. Hence, it can be written as
I ) Vdissolution ) k1Γ0 ) k2Γ1 ) k3Γ2 2FA
(29)
since the partial currents through the first and second steps are summed in the total dissolution current. In this equation, Vdissolution represents the overall reaction rate. Nyquist Plot. Figure 1 shows the theoretical impedance spectrum corresponding to a process with two consecutive single-electron transfers, without the contribution of the electrolyte resistance and the double layer capacitance. Five characteristic points can be identified in this plot: 1. The point where the frequency tends to infinity. When the frequency tends to infinity (initial point of the capacitive loop, Figure 1), the faradaic impedance shows the following value
FAZ ωf∞ imag ) 0 FAZ ωf∞ real )
(30)
b1)b2 1 1 ) FARct (31) ) 0 k1 Γ0(b1 + b2) 2k1 b1Γ0 0
This characteristic point mainly depends on the rate of the first electron transfer and corresponds to the charge-transfer resistance (Rct). 2. The zero frequency limit of the impedance. When the frequency tends to zero (intersection point of the inductive loop and second characteristic point, Figure 1), the imaginary part of the impedance becomes zero, and the real part of the faradaic impedance depends on all of the kinetic parameters of the faradaic processes that take place on the working electrode, eq 33. Furthermore, this point corresponds to the polarization resistance (Rp).
FAZ ωf0 imag ) 0 FAZ ωf0 real
k10.k20,k3
≈
(
1
(32)
)
k3 - k20 b2 k1 b1Γ0 1 + k + k 0 b1 0
3
2
b1)b2
)
k3 + k20 2k3k10 b1Γ0
)
1 1 + ) FARp (33) 2k10 b1Γ0 2k3 b1Γ1
Figure 1. Nyquist plot of the theoretical impedance function deduced in eq 19. The kinetic parameters of this simulation are k10 ) 4000 s-1, k20 ) 5 s-1, k3 ) 21 s-1, b1 ) 16 V-1, b2 ) 14 s-1, Γ0 ) 4 × 10-4 mol m-2 y, A ) 0.5 cm2.
3. The point where the imaginary part of the impedance is zero. The third characteristic point is characterized by the fact that the imaginary part of the faradaic impedance is equal to zero
ωZimag)0 ) xk10(k3 - k20) imag)0 FAZ Zreal
k10.k20,k3
≈
1 k1 b1Γ0 0
(34) (35)
The frequency value of this characteristic point depends on the ratio of all of the kinetic constants, eq 34, whereas the value of the real part of the faradaic impedance function only depends on the kinetic parameters of the first faradaic transfer, eq 35. It is interesting to point out that eq 34 indicates that this characteristic point could not exist if the kinetic constant k3 is smaller than the kinetic constant k2. In this case, the inductive loop at low frequencies disappears, and a capacitive loop appears instead. In fact, since the pioneering work in the field,38 many authors have attributed the presence of inductive loops to an electrochemical mechanism implying the relaxation of the electrode coverage by an adsorbed reaction intermediate. According to this, the inductive or capacitive character depends on the values of the kinetic parameters. In other works, the capacitive or inductive nature of this loop appears as a consequence of the sign of a parameter like in the analysis suggested by Cao.30 It also proves interesting to introduce the hypothesis of the steady state, eqs 25-27, in this analysis. According to that, the inductive loop at low frequencies appears if the surface concentration of ZnI (Γ1) is larger than the surface concentration of ZnII (Γ2). The following characteristic points are calculated by differentiating the expression for the imaginary part of 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. 4. The point where the imaginary part of the impedance is a maximum.
Graphical Analysis of EIS
J. Phys. Chem. B, Vol. 109, No. 10, 2005 4587
The impedance and frequency values at the maximum (inductive loop) of the imaginary part are
ωZimagmaximum
k10.k20,k3
≈
imagmaximum FAZ Zimag
(k3 + k20) + (k3 - k20)
k10.k20,k3
≈
1 b1 k3 + k20 2k10b1Γ0 +1 b2 k - k 0
(( ) ) 3
k3 - k20 0
4k3k1 b1Γ0 imagmaximum FAZ Zreal
k10.k20,k3
≈
b2 b2)b1 ) 2k3 (36) b1
)
(
4k3k10b1Γ0
2
1 1 (37) 4k1 b1Γ0 4k3b1Γ1
)
1 b2 k3 - k20 1+ b1 k + k 0 3
)
)
0
1 1+ 0 2k1 b1Γ0
k3 + k20
b1)b2
2
b1)b2
)
1 1 + (38) 4k10b1Γ0 4k3b1Γ1
The frequency of this characteristic point is a direct estimation for the kinetic constant of the transport step if b1 ) b2, eq 36. On the other hand, the imaginary part of this characteristic point indicates that this one is a minimum when k3 < k2 and, therefore, the inductive loop is transformed into a capacitive loop, eq 37. 5. The point where the imaginary part of the impedance is a minimum. At this minimum (capacitive loop), the faradaic impedance shows the following relationships with the kinetic parameters
ωZimagminimum
imagminimum FAZ Zimag
imagminimum FAZ Zreal
k10.k20,k3
≈
k10.k20,k3
≈
k10.k20,k3
-
≈
k10 b1)b2 k10 ) b2 2 1+ b1 1
( )
b1)b2
) -
b1 2k10b1Γ0 1 + b2
(
1 1+ 2k10b1Γ0
1 1+
)
b2 b1
(39)
1 4k1 b1Γ0
)
depends on the overall reaction rate (Vdissolution), b1, and the kinetic constants of the second electron transfer and the transport step. Real Admittance-Imaginary Admittance Plot (Admittance Plot). Although other kinds of plots will represent the same experimental results as the Nyquist plot, the information that can be extracted from the graphical analysis may be slightly different, and in some cases an expression is found that allows the evaluation of one parameter with smaller uncertainty. That way, Figure 2 shows the theoretical faradaic admittance function for a process (i), eq 10. In this plot, there are five characteristic points which will be analyzed 1. The zero frequency limit of the impedance. In this characteristic point, the faradaic admittance function shows the following theoretical expression
0
(40) b1)b2
Figure 2. Admittance plot of the theoretical impedance function deduced in eq 19. The kinetic parameters of this simulation are k10 ) 4000 s-1, k20 ) 5 s-1, k3 ) 21 s-1, b1 ) 16 V-1, b2 ) 14 s-1, Γ0 ) 4 × 10-4 mol m-2 y, A ) 0.5 cm2.
3 4k10b1Γ0 (41)
The frequency at this minimum is a direct estimation for the kinetic constant of the first electron transfer when b1 ) b2, eq 39. The analysis of these expressions allows us to conclude that the diameter of the capacitive loop at high frequencies, eq 31, depends on the overall reaction rate and on the thermodynamic parameters, while the frequency for the minimum mainly depends on the first kinetic constant, k1. The diameter of the inductive loop can be evaluated algebraically as the difference between eq 35 and eq 33 and corresponds to (k3 - k20/2k3k10b1Γ0), which is exactly, as expected, twice the height of this loop, eq 37. It is interesting to note that the diameter of this loop could be positive or negative, depending on the relative values of k3 and k20. In fact, a negative diameter (k20 > k3) means that this loop is capacitive instead of inductive. To summarize, the diameter of this loop
(
Y ωf0 imag ) 0
(42)
)
k 0.k20,k3 k3 - k20 b2 Y ωf0 real 1 ≈ k10b1Γ0 1 + FA k + k 0 b1 3
2k3
(
2
b1)b2
)
)
1 1 k1 b1Γ0 ) + 0 0 k 3 + k2 2k1 b1Γ0 2k3b1Γ1 0
-1
(43)
2. The point where the frequency tends to infinity. When the frequency tends to infinite, the faradaic admittance function depends only on the first single-electron transfer, eq 45.
Y ωf∞ imag ) 0
( )
b2 Y ωf∞ real ) k10b1Γ0 1 + FA b1
(44) b1)b2
) 2k10b1Γ0
(45)
3. The point where the imaginary part of the admittance is zero. The third characteristic point, eqs 46-47, as commented above, does not exist when the kinetic constant k3 is smaller than the kinetic constant k2.
4588 J. Phys. Chem. B, Vol. 109, No. 10, 2005
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ωYimag)0 ) xk10 (k3 - k20)
(46)
imag)0 Y Yreal ) k10 b1Γ0 FA
(47)
The previous three characteristic points of this plot are the exact counterparts of the first three characteristics points of the Nyquist plot, and no new information can be extracted. This is because the imaginary part of impedance in these points is always zero. However, for the following points, the imaginary part of impedance is not zero, and then the analysis of the inverse may yield a different result. 4. The point where the imaginary part of the admittance is maximum. The admittance function presents the following expressions at the maximum of this function k10.k20,k3
≈
ωYimagmaximum
k10
(48)
0 imagmaximum k 0.k 0,k k10b2Γ0 Y Yimag 1 2 3 k b Γ 2 2 1 ≈ ) FA 2 2
(
)
imagmaximum k 0.k 0,k b2 Y Yreal 1 2 3 ≈ k10b1Γ0 1 + FA 2b1
b1)b2
)
(49)
3 0 k b Γ (50) 2 1 1 0
The frequency of the fourth characteristic point, eq 48, gives a direct estimation of the kinetic constant of the first singleelectron transfer, and the diameter and height of this loop depend on the overall reaction rate. 5. Finally, the point where the imaginary part of the admittance is a minimum. At this characteristic point, the faradaic impedance value is
ω
Yimagminimum
k10.k20,k3
≈
(k3 + k2 ) 0
(51)
2
-
(
(k3 - k20) 2(k3 + k20)
k10b2Γ0 (52)
)
imagminimum k 0.k 0,k b2 k3 - k20 Y Yreal 1 2 3 ≈ k10b1Γ0 1 + FA 2b1 k + k 0
3
2
(53)
The value of the frequency at this minimum is a direct measure of k3 + k20, while the diameter and height of this loop depend on the overall reaction rate (Vdissolution) but also on the kinetic constants k3 and k20. From these points, it is also possible to explain mathematically why the inductive loop changes to a capacitive one when k3 < k20, eq 52. By comparison with the Nyquist plane, the analysis in the admittance plane has the advantage that it allows the faster identification of k10 and k3 + k20 from the values of frequency at maximum and minimum of the impedance function, respectively. Cole-Cole Plot. In a Cole-Cole plot, the impedance function is transformed into an equivalent complex capacitance mode by means of the following expression
C)
1 Zωj
The Cole-Cole plot represents the imaginary part of the capacitance with respect to the real part of this function (Figure 3). Two characteristic points of this plot will be analyzed. 1. The zero frequency limit of the impedance. The first characteristic point of this plot shows the following ratio with the kinetic parameters
C ωf0 imag ) -∞ FA
(54)
(55)
(
)
0 k 0.k20,k3 k b Γ b2 C ωf0 real 1 1 1 0 0 2k (k - k2) ≈ 2 FA b1 3 (k + k )2 3
imagminimum k 0.k 0,k (k3 - k20) 0 Y Yimag 1 2 3 k2 b2Γ1 ) ≈ FA 2(k + k 0)
3
Figure 3. Cole-Cole plot of the theoretical impedance function deduced in eq 19. The kinetic parameters of this simulation are k10 ) 4000 s-1, k20 ) 5 s-1, k3 ) 21 s-1, b1 ) 16 V-1, b2 ) 14 s-1, Γ0 ) 4 × 10-4 mol m-2 y, A ) 0.5 cm2.
2
(56)
2. The point where the real part of the capacitance is a maximum. The second characteristic point is a maximum and it is calculated by differentiating the real part of the capacitance with respect to the frequency and equating to zero. The capacitance at this point is equal to
ωCrealmaximum
k10.k20,k3
≈
xk10(k3 - k20)
realmaximum k 0.k 0,k C Creal 1 2 3 ≈ b2Γ0 FA realmaximum k 0.k 0,k C Cimag 1 2 3 ≈ FA
k10b1Γ0
xk10(k3 - k20)
(57) (58)
(59)
According to previous analysis, this point exists only if an inductive loop appears in the Nyquist plot (k3 > k20), and looking at eq 57, this frequency corresponds to the third characteristic point in the Nyquist plot (when Zimag ) 0). It is also interesting to note that eq 58 gives a direct estimation for b2Γ0. Logarithm Real and Imaginary Plot. Figure 4 shows the shape of the expected log |Zreal| and log |Zimag| vs log frequency (f) plot for the faradaic impedance function hereby analyzed. In the plot of the imaginary part of the impedance, two characteristic points and two characteristic straight lines or linear
Graphical Analysis of EIS
J. Phys. Chem. B, Vol. 109, No. 10, 2005 4589 impedance at lower frequencies is determined by the sign of k3 - k20, positive for inductive loops and negative for capacitive loops in the Nyquist plot. At higher frequencies (ωv), the faradaic impedance dependence with respect to the angular frequency is established by
FAZ ωv imag
k10.k20,k3
≈
-
b1)b2
1
( )
b1 b2Γ0 +1 ω b2 2
) -
1 (61) 4b2Γ0ω
In this case, the calculation of the slope of Zimag vs 1/ω may be used to obtain a first approach for the parameter Γ0 The dependence of the Zreal on the frequency (Figure 5) shows five characteristic points and two characteristic linear intervals. The linear intervals correspond to the dependence of the faradaic impedance at high and low frequencies. At high frequencies, the real part of the impedance should vary with the frequency according to Figure 4. Logarithm real and imaginary plot of the theoretical impedance function deduced in eq 19. The kinetic parameters of this simulation are k10 ) 4000 s-1, k20 ) 5 s-1, k3 ) 21 s-1, b1 ) 16 V-1, b2 ) 14 s-1, Γ0 ) 4 × 10-4 mol m-2 y, A ) 0.5 cm2.
FAZ ωv real
(
2
k10 1 1+ b2 b1 k10b1Γ0 +1 ω2 +1 b1 b2
k10.k20,k3
( )
≈
)
( )
(62)
And, at low frequencies
FAZ ωV real
k10.k20,k3
≈
(
)
b2 ω2 + (k3 + k20) (k3 + k20) + (k3 - k20) b1 1 (63) 2 0 b k1 b1Γ0 2 0 0 2 ω + (k3 + k2 ) + (k3 - k2 ) b1
(
Figure 5. Bode mode capacities plot of the theoretical impedance function deduced in eq 19. The kinetic parameters of this simulation are k10 ) 4000 s-1, k20 ) 5 s-1, k3 ) 21 s-1, b1 ) 16 V-1, b2 ) 14 s-1, Γ0 ) 4 × 10-4 mol m-2 y, A ) 0.5 cm2.
intervals will be analyzed. Characteristic points, which correspond to the maximum and minimum in the Nyquist plot, have been analyzed previously. At lower frequencies (ωV), the imaginary part of impedance can be written as
FAZ ωV imag
k10.k20,k3
≈
ω (k3 + k20) b1 0 0 (k3 - k2 )k2 b2Γ1 +1 (k - k 0) b2
(
ω 2
0
k2 b2Γ1
4k3
k3 - k20
3
2
ω
) 0
k1 b2Γ0
)
It is difficult to obtain a direct measure of any kinetic parameter from these equations. It is only possible to obtain information from the limits of eqs 62 and 63, which correspond to characteristic points in the Nyquist plot, eqs 31 and 33, respectively. The five characteristics points of this plot are as follows: 1. The zero frequency limit of the impedance, which corresponds to eqs 32 and 33. 2. The point where the frequency tends to infinity, which corresponds to eqs 30 and 31. 3. The point where the real part of the impedance is a maximum, which corresponds to eqs 34 and 35. Nevertheless, the other two characteristic points correspond to inflection points in this plot and are calculated by setting the second differential of the theoretical faradaic impedance function with respect to the angular frequency equal to zero. 4. Inflection point at low frequencies. The fourth characteristic point presents the following ratio with the kinetic parameters.
b1)b2
2
4k32
)
ωinflection ωV
k10.k20,k3
(60)
k3 - k20
Therefore, the slope of Zimag vs ω at lower frequencies gives information on the overall reaction rate and on kinetic constants k20 and k3. Besides, the sign of the imaginary part of the
)
ωV FAZ inflection real
≈
(
(k3 + k20) + (k3 - k20)
)
b2 b1
x3
b1)b2
)
2k3
x3
(64)
k10.k20,k3
≈
(
3 1 + 0 3 4k1 b1Γ0
(k3 + k20)
)
b2 (k3 + k20) + (k3 - k20) b1
(65)
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The frequency of this inflection point is a direct estimation for the kinetic constant of the transport state, k3, if b1 ) b2. 5. The inflection point at high frequencies. This fifth characteristic point shows the following ratio with the kinetic parameters
ωinflection ωv
k10
k10.k20,k3
x3 1 + ωv FAZ inflection real
k10
b1)b2
( ) ( ) ( )
≈
k10.k20,k3
b2 b1
3
≈
4k10b1Γ0 1 +
)
b2 b1
2x3
4 b2 + 3 b1
(66)
b1)b2
)
This frequency is also a quasi-direct measure of parameter k10, the phase angle of this characteristic point depends only on the thermodynamic parameters of both electron transfers, and it is constant and equal to 70.2° when both parameters are equal. This point allows us to evaluate separately the thermodynamic and kinetic parameters of the electrodic mechanisms that take place through two consecutive single-electron transfers. 2. The point where the phase angle is a maximum. The theoretical expression of the second characteristic point of the Bode C plot is
ωmaximum ωV
x
7 (67) 8k1 b1Γ0
ωv k10.k20,k3
|C| FA
( )
k10b1Γ0 1 +
≈
b2 b1
ω
b1)b2
)
2k10b1Γ0 ω
(
ω
ψminimum ωv
k10.k20,k3
≈
arctg
≈
(
)
This characteristic point depends on all of the processes that take place on the working electrode. The two characteristic linear intervals of this plot correspond to the values of the phase angle at high and low frequencies. Hence, the characteristic linear interval at high frequencies shows the following ratio with the kinetic parameters
(68) ψωv
2k3 ω(k3 + k20)
(69)
In both cases, the modules of capacitance depend linearly on the inverse of the angular frequency, and from the slopes of this plot it is possible to obtain the overall reaction rate, whereas, from the ratio between both slopes, it is possible to obtain the parameter k3/k20. That way, if k3 ) k20, then the high frequency slopes or capacitances will be twice the low frequency slopes or capacitances, for the case of an inductive loop at low frequencies, k3 > k20, and the ratio between slopes will be smaller than 2 in the case of a capacitive loop at low frequencies, k3 < k20. It is another way to analyze the presence of inductive or capacitive loops at low frequencies in this system. The plot of the absolute value of the phase angle vs the frequency (Figure 5) shows two characteristic points and two characteristic linear intervals. The characteristic points are: 1. The point where the phase angle is a minimum. In this point it is found out that k10.k20,k3
k10.k20,k3
(72)
k10.k20,k3
≈
ω
1 b1 -k10 b1 + b2
b1)b2
)
2ω -k10
(74)
2
k10b1Γ0
minimum ωv
x2k3(k3 + k20)
ψmaximum ωV ≈ 2xb1 arctg (k + k20)((k3 + k20)b1 + (k3 - k20)b2) 0 x 3 b2(k3 - k2 ) (73)
0 0 0 0 |C|ωV k1 .k2 ,k3 0 (k3 + k2 )b1 + (k3 - k2 )b2 b1)b2 ) ≈ k1 Γ0 FA ω(k + k 0) 3
≈
(k3 + k20)((k3 + k20)b1 + (k3 - k20)b2) b1)b2 ) b1
0
It is possible to say that this latter point depends only on the first single-electron transfer. Furthermore, when b1 ) b2, the frequency of this inflection point is also an estimation of the kinetic constant of the first single-electron transfer. Capacitance in Bode Mode Plot. Capacitance is calculated from faradaic impedance function by means of eq 54 and represented the capacitance module and the phase angle (ψ) vs frequency (Figure 5). The high and low frequency dependencies of modules of capacitance on the frequency correspond to these expressions
k10.k20,k3
x
2
k10 b1 b1)b2 k10 ) b 1 + b2 x2
)
-2xb1(b1 + b2) b2
(70)
b1)b2
)
arctan(-2x2) (71)
which is another possibility for a fast estimation of k10. However, the characteristic linear interval at low frequencies corresponds to
ψ
ωV
k10.k20,k3
≈
(k3 + k20)((k3 + k20)b1 + (k3 - k20)b2) b1)b2 ) ω(k3 - k20)b2 2k3(k3 + k20) ω(k3 - k20)
(75)
It is interesting to note that none of these characteristic points or intervals related to the phase angle depend on the overall reaction rate and, therefore, on the electroactive surface of the working electrode. Bode Mode Impedances Plot. The Bode mode impedances plot (Bode Z) represents the module and phase angle of the faradaic impedance with respect to the angular frequency, (Figure 6). The plot of the logarithm of the faradaic impedance with respect to the angular frequency of a process with two consecutive single-electron transfers (Bode Z, Figure 6) has five characteristic points: 1. The zero frequency limit of the impedance. 2. The point where the frequency tends to infinity. These two characteristic points are the same points as the points calculated in the Nyquist plot. 3. The point where the module of the impedance is a maximum.
Graphical Analysis of EIS
J. Phys. Chem. B, Vol. 109, No. 10, 2005 4591
Figure 6. Bode mode impedances plot of the theoretical impedance function deduced in eq 19. The kinetic parameters of this simulation are k10 ) 4000 s-1, k20 ) 5 s-1, k3 ) 21 s-1, b1 ) 16 V-1, b2 ) 14 s-1, Γ0 )4 × 10-4 mol m-2 y, A ) 0.5 cm2.
This point coincides with the first characteristic point in the Nyquist plot (when Zimag ) 0). 4. Inflection point at low frequencies. The expressions for this characteristic point are inflexion ωV
k10.k20,k3
ω
FA|Z|
inflexion ωV
k10.k20,k3
≈
(
≈
1+
3 4
(k3 + k20)
x3
)
(k3 - k20) b2 (k + k 0) b1 3 2 0 k1 b1Γ0
(76)
b1)b2
)
k3 3 (77) 2 (k + k 0)k 0 b Γ 3 2 1 1 0 This characteristic frequency is related to k3 and k20 and the faradic impedance with these two kinetic constants and with the overall reaction rate (Vdissolution). 5. The inflection point at high frequencies. The expressions obtained for this characteristic point are
ωinflexion ωv
k10.k20,k3
≈
FA|Z|inflexion ωv
x(
2
b1k10 b2 3 1+ b1
k10.k20,k3
≈
)
b1)b2
)
1 3 4 k 0b Γ 1 1 0
k10
x6
(78)
(79)
This point gives only information on the k10 and on the overall reaction rate. The analysis of the phase angle vs frequency plot corresponds exactly to the analysis made for phase angle in the capacitance phase plot since there is a relationship between the phase angles in the two plots. Discussion and Conclusion Considering the results of the previous section, the graphical analysis using the theoretical faradaic impedance function allows
Figure 7. Flowchart of the graphical analysis of the theoretical faradaic impedance function of a process that takes place by two consecutive electron transfers followed by a transport step.
us to establish the dependence of the EIS spectra on the different elemental steps of a process that consists of two consecutive irreversible single-electron transfers followed by a transport step. So, the theoretical impedance function depends mainly on the kinetic parameters of the first electron transfer at high frequencies, whereas, at low frequencies, the impedance function depends on all the elementary steps of this electrodic mechanism. As a result, these systems show two consecutive capacitive loops when k3 < k2 and a capacitive loop at high frequencies followed by a inductive loop at low frequencies when k3 > k2. The graphical analysis of the theoretical impedance function establishes also that the minimum of the phase angle of the Bode C (or Bode Z), eq 71, depends only on the bi parameters of the processes that involve electron transfer. This procedure of parametrical identification is established in the flowchart shown in Figure 7. In this procedure, the graphical analysis is done with a very high accuracy, since the error on these kinetic parameters is only tight to the accuracy in the determination of the characteristic points. The angular frequency of the maximum of the imaginary admittance, eq 48 (admittance plot), is a direct measurement of the kinetic constant of the first single-electron transfer. Alternatively, the impedance when the imaginary part of the theoretical impedance function is equal to zero, eq 35 (Nyquist plot), is a direct measure of the reaction rate. The equations system to calculate the kinetic constants k2 and k3 is established from the third characteristic point of the Nyquist plot (eq 34, when the imaginary impedance is equal to zero) and the fifth characteristic point of the admittance plot (eq 51, when the imaginary admittance is a minimum). The measurement of the parameters bi of the first and second single-electron transfer can be easily carried out by means of
4592 J. Phys. Chem. B, Vol. 109, No. 10, 2005
Garcı´a-Jaren˜o et al.
the dependence of the kinetic constants with respect to the stabilization potential.
ki0 ) ki0 ebi Eh
(80)
Γ0, initial total surface concentration of unoccupied electroactive sites. Γi, surface concentration of species i. Γi surface concentration of species i at the steady state. Ψ, phase angle. References and Notes
0
where ki is a constant. Consequently, as the graphical analysis is mainly performed at low frequencies, it is correct to consider that the nonfaradaic impedance can be simulated by means of a double layer capacitance and an electrolyte resistance, eq 28. Therefore, the error on the calculation of the characteristic points is minimized, since the faradaic impedance can be separated easily from the nonfaradaic impedance. The latter can be directly calculated from the experimental impedance at high frequencies. Finally, this procedure allows the numerical simulation of the EIS spectra through eq 19, and therefore, it is a promising way for validating kinetic models of technologically important reactions. Acknowledgment. This work has been supported by CICyTMAT/2000-0100-P4. D. Gime´nez-Romero acknowledges a fellowship from the Generalitat Valenciana, Program FPI. J. J. Garcı´a-Jaren˜o acknowledges the financial support by the “Ramo´n y Cajal” program (Ministerio de Ciencia y Tecnologia, Spanish Ministry of Science and Technology). Appendix: List of Symbols A, electrode surface area. bi, exponential factor of the kinetic constant of electron transfer i. C, capacitance. Cdl, double layer capacitance. E, applied potential during the EIS experiment. E h , study potential ∆E, variation of the study potential ∆E, amplitude of the potential perturbation. f, frequency. F, Faraday constant, 96 484.56 C° mol-1. I, current intensity. j, square root of -1. ki, the kinetic constant of the elemental step i. ki0, kinetic constant of the transfer i. ki0, preexponential factor of the kinetic constant of the transfer i. Rct, charge transfer resistance. Rp, polarization resistance. Ru, electrolyte resistance. t, time. w, angular frequency. Y, faradaic admittance. Z, impedance. Ze, experimental impedance. Zf, faradaic impedance.
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