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Page 25 of 57
Accepted Manuscript
25
Figure 4
(Nyquist plot of parallel impedance)
Figure 5
(Bode magnitude)
Figure 6
(Bode phase angle)
Figures 4 to 6 were calculated by using the values obtained from the preliminary
optimization parameters. The main question that arises here concerns the relative values of the
electronic and Randles impedances as a function of frequency; i.e. whether it is possible to
neglect one of these contributions when calculating the total parallel impedance. In order to
answer this question, we calculated the electronic and Randles impedances separately and we
display these terms along with their sum (parallel impedance) in Figures 5 and 6. Figure 5
clearly shows that with the exception of low frequencies (
f
< 10
-2
Hz) the electronic and Randles
impedances are of the same order of magnitude, i.e. in the general case, both of these terms must
be taken into account when calculating the parallel impedance. At very low frequencies, the
parallel impedance reduces to the value of the charge transfer resistance of the electrochemical
cathodic reaction,
R
ct
, (in our case
R
ct
= 0.308×10
10
Ω.cm
2
, see above).
Examination of the data plotted in Figures 5 and 6 show that at low frequencies (
f
< 10
-2
Hz), the magnitude of the parallel electronic impedance is of the order of 10
6
Ω cm
2
. Numerous
optimization trials performed in this study show that this is sufficiently high with respect to the
faradaic plus defect Warburg impedance that sufficient current flows through the latter that the
optimization yield values for the parameters contained therein that are the same as those obtained
by setting the parallel electronic impedance arbitrarily to 10
17
Ω cm
2
, as was done in our
previous work. In other words, the parallel electronic impedance is sufficiently large that it has
negligible impact on the impedance of the interphase.
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Accepted Manuscript
26
4. Results and discussion
4.1. Extraction of model parameter values from EIS data
The Genetic-inspired Differential Evolution (GDE) curve fitting approach was selected
for optimizing the PDM on the EIS experimental data, in order to extract values for the model
parameters. Briefly, curve fitting (“optimization”) is the process of obtaining a representation of
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
reconcile models to observations and for developing optimal solutions to different kinds of
problems, such as 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, (
k
i
0
), the transfer coefficients for the point
defect generation and annihilation at the barrier layer interfaces (
α
i
) , and the constant
Φ
0
BOI
,
(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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Accepted Manuscript
27
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, ©1988-2010 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 (
k
i
0
), transfer
coefficients (
α
i
) (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).
Figure 7 shows typical experimental electrochemical impedance spectra for the passive
state on iron in borate buffer solution [0.3 M
H
3
BO
3
+ 0.075 M
Na
2
B
4
O
7
] + 0.001 M
EDTA
[Ethylenediaminetetraacetic acid, disodium salt] (pH = 8.15 and 10, T = 21
o
C) 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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Accepted Manuscript
28
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 reactions rate 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
(
k
0
2
) compared with the Reaction 3 (
k
0
3
), 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
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
Page 29 of 57
Accepted Manuscript
29
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.
3
2
k
k
F
I
(71)
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 enhances the 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 (
L
ss
) increases with the applied potential, as is predicted by the
PDM, and shows good agreement with the experimental results. Likewise, the calculated steady-
state current density (
I
ss
) is essentially independent of voltage and is close to the experimental
value, as is shown in Figure 9.