j.electacta.2013.03.143

Pk is the analog of the Peclet number (from the theory of convective mass transfer) for k-th defect. We see that Pk is the ratio of the intensity of migration to that of diffusion.

As follows from Equation (57), for Pk >> 1, migration dominates in the positive direction and practically in all of the volume of the barrier layer, and hence we have with great accuracy

Ck Ck0 with the exception of the very thin transient area near x = 0 (bl/ol interface). In the

opposite case, for Pk<< -1, migration dominates in negative direction and practically in all

volume of theAcceptedbarrier layer we have with great accuracy Ck Ck with the exception of the very thin transient region near x = L (m/bl interface).

Let us consider the case of metal interstitials (k = 1). If, for example, χ = 3, 2.3 106

V/cm, T = 295 K, L 10-5cm Equation (57) yields P1 ≈ - 1810, i.e. P1 has a large, negative value

and with great accuracyC1 C1L , with the exception of the very thin transient area near the bl/ol

interface (with the thickness ~ 10-3L), where the concentration changes sharply with distance. Accordingly, for the partial cation interstitials flux density, we have:

and, for the concentration of cation interstitials inside the barrier layer we have:

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Please, note that, in the coordinate system used here (increasing x from right to left), with the

Equations (59) and (60) allow us to estimate the electronic impedance.

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It must be noted that electronic impedance can be measured, but only in conjunction with the reaction impedance represented by Randles circuit (see Figure 3).

solution, and Rct is the charge transfer resistance of the redox reaction. The Warburg coefficient,

where CO and CR are the bulk concentrations of oxidized and reduced components, respectively, of the redox couple and DO and DR are the corresponding diffusion coefficients.

Our calculations show that, due to the extremely low concentration of the oxygen in the

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It is evident that this reaction, in the written direction, has no diffusion limitation, and accordingly we can neglect the Warburg Impedance (Zw ≈ 0).

On the other hand, the current density corresponding to Reaction (64) is:

For example, calculations performed by using OLI commercial software [46] yields B = 3.92×10-

oC and calculations yield Rct = 0.308×1010Ω.cm2 at E = 0.044 (SHE) and Rct = 0.5938×1014Ω.cm2 at E = 0.544 (SHE).

For the case of Zw ≈ 0, the Randles impedance has the form:

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By using Equations (68) to (70) and Equation (66) for Rct, we can easily calculate the modulus:

components, as well as a modulus [Equation (71)] and phase angle [Equation (72)], all as a function of frequency. Because the real and imaginary components are frequency-dependent, they must be incorporated into the optimization procedure.

As notedAcceptedabove, the electronic impedance in parallel with the barrier layer is a complex number [Equation (68)], which yields real [Equation (69)] and imaginary [Equation (70)]

Figures 4 to 6 show typical Nyquist and Bode plots for the parallel impedance for the case of iron in borate buffer solution [0.3 M H3BO3 / 0.075 M Na2B4O7, as appropriate] + 0.001 M EDTA [Ethylenediaminetetraacetic acid, EDTA, disodium salt], pH = 8.15, T = 21 oC and E = 0.044 V (SHE). The parameter values used in the calculations are given in Table 2.

Table 2 (Parameter values).

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4. Results and discussion

4.1. Extraction of model parameter values from EIS data

a multivariate data set by an “objective function” that describes a physico-electrochemical system, at least as employed here. The main objective of optimization is to find the set of parameter values that minimize the total error determined from the difference between the observed dependent variable values, Z( ) and those calculated from the derived parameter values over the considered data set. After selecting a functional form and setting up the error metrics, curve fitting becomes an optimization problem. It is a common method used to

The Genetic-inspired Differential Evolution (GDE) curve fitting approach was selected for optimizing the PDM on the EIS experimentalManuscriptdata, in order to extract values for the model parameters. Briefly, curve fitting (“optimization”) is the process of obtaining a representation of

reconcile models to observations and for developing optimal solutions to different kinds of problems, suchAcceptedas simulation and statistical inference [51, 52].

The optimization procedure ends if the result satisfies the selected convergence criteria and the following requirements: (1) All the parameter values are physically reasonable and should exist within known bounds; (2) The calculated Z’(ω) and Z”(ω) should agree with their respective experimental results in both the Nyquist and Bode planes; (3) The parameters, such as the polarizability of the barrier layer/outer layer interface (BOI) (α), the electric field strength across barrier layer (ε), the standard rate constants, (ki0), the transfer coefficients for the point defect generation and annihilation at the barrier layer interfaces (αi) , and the constant Φ0BOI, (standard potential drop across the barrier layer/outer layer interface) should be approximately potential-independent; and (4) The calculated current density and passive film thickness, as

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estimated from the parameter values obtained from the optimization, should be in reasonable agreement with the steady-state experimental values (note that these values are not used in the optimization and hence provide for an analytical test of the model and the optimization

procedure). The Igor Pro (Version 6.2.1.0, ©1988Manuscript-2010 WaveMetrics, Inc.) software with a

custom software interface powered by Andrew Nelson’s “gencurvefit”[54] package was used in this work for optimization, so as to obtain values for the standard rate constants (ki0), transfer coefficients (αi) (for the i elementary interfacial reactions), the polarizability of the barrier layer/outer layer interface (α), the electric field strength across barrier layer (ε), and other parameters as described below. A freely distributed interface is now available to effectively leverage gencurvefit for the optimization of complex impedance functions [37].

Figure 7 (Nyquist and Bode plots showing comparison of experimental and calculated

impedance).

FigureAccepted7 shows typical experimental electrochemical impedance spectra for the passive state on iron in borate buffer solution [0.3 M H3BO3 + 0.075 M Na2B4O7] + 0.001 M EDTA

[Ethylenediaminetetraacetic acid, disodium salt] (pH = 8.15 and 10, T = 21oC) in the form of Nyquist and Bode planes in the passive potential range. It should be mentioned that the quality of the EIS data was checked both experimentally and theoretically. The data were checked experimentally by stepping the frequency from high-to-low and then immediately from low-to- high. The quality of the impedance data were also checked using the Kramers-Kronig transforms. These integral transforms test for compliance of the system with the linearity, stability, and causality constraints of linear systems theory (LST). [30, 47-50]. The solid lines show the best-fit result calculated based upon the PDM equations and the parameter values determined by optimization. It can be seen that the agreement between the experimental results

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and those calculated from the PDM is very good, except at high frequencies, indicating that the PDM provides a reasonable account of the experimental data. The discrepancy at high frequencies arises from unaccounted-for capacitance, which is not of primary interest in this

study. The extracted parameters including reactionsManuscriptrate constants, transfer coefficients,

diffusivity of iron interstitials, steady-state thickness and current density are listed in Table 3 for pH = 8.15 and 10.

Table 3. Parameter values.

Comparison of the obtained kinetic parameters from PDM optimization as a function of applied potential is shown in Figure 8. As can be seen, the kinetic constants and transfer coefficients are almost independent of applied potential, in conformity with electrochemical theory. Another important finding is the higher magnitude of the rate constant for Reaction 2

(k0 ) compared with the Reaction 3 (k0 ), Figure 1, which confirms that iron interstitials are the

predominant defects in the defective barrier oxide layer over the entire potential range and that passive film has an n-type semiconductor character. For the sake of comparison, the results reported by Marx [55] for reaction rate constants at pH= 8.4, room temperature, are incorporated into Figure 8(a). A very good level of agreement between results proves the reliability of the model in predicting the oxide layer behavior.

Figure 8 (comparison of kinetic parameters)

4.2. Determination of Steady-state Current Density and Barrier Layer Thickness

2 Accepted3

Figure 9 shows the comparison between the calculated steady-state current density and thickness of the barrier layer with the measured values. For the sake of comparison, data reported by Bojinov et al. [44] , Büchler et al. [56] and Marx [55] for the thickness of the

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passive film on iron in pH = 8.4 borate buffer solution without adding EDTA are shown in this figure. In the current work, steady-state thickness and current density values were calculated from the following Equations (10 and 71) using the parameters obtained from the PDM

optimization, as presented in Table 3.

The simulated thickness of barrier layer is close to the values measured by spectroscopic ellipsometry (SE) [18]. Although good agreement is obtained between the results of this work and those reported by Marx [55], there is a small difference between the calculated and measured thickness obtained in the present work when compared with those reported by Bojinov et al. [44] and Büchler et al. [56]. This could arise from the impact of the outer layer in the later works, presumably because EDTA removed the outer layer in the present study, and also because EDTA

probably enhancesAcceptedthe rate of the barrier layer dissolution. Thus, since, in this study, EDTA was used in order to prevent the formation of the outer layer, we expected to find a difference

between our results and those reported by the other researchers identified above. However, the work of Liu et al. [6] has shown that the thickness of the passive film in the presence of EDTA is thinner than without it, which is in agreement with the results obtained in this work, because the standard rate constant for Reaction (7), Figure 1, is expected to be higher. The calculated thickness of the barrier layer (Lss) increases with the applied potential, as is predicted by the PDM, and shows good agreement with the experimental results. Likewise, the calculated steadystate current density (Iss) is essentially independent of voltage and is close to the experimental value, as is shown in Figure 9.

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