Effect of Heat Treatment of Martensitic Stainless Steel on Passive Layer Growth Kinetics Studied by Electrochemical Impedance Spectroscopy in Conjunction with the Point Defect Model

Abstract

Martensitic stainless steels are widely used materials. Their mechanical and corrosion properties are strongly influenced by their microstructure and thereby can be affected by heat treatment. In the present study, the effect of different austenitizing temperatures on the passive film growth kinetics of martensitic stainless steel is studied by electrochemical impedance spectroscopy. The data was further fitted by the point defect model to determine kinetic parameters. We show that an increasing austenitizing temperature leads to a more protective passive film and slows down passive film dissolution in sulfuric acid.

1. Introduction

The passivity of stainless steel is one of the most important phenomena of our modern metal-based civilization [1]. Various studies over recent decades on stability [2,3,4], structure [5,6,7] and semi conductive properties [8,9,10,11] of passive films underline their significance in research and industry.

Heat treatment of materials can affect the microstructure and thus has not only a crucial effect on the mechanical properties but also on the corrosion properties [12]. The influence on the different corrosion mechanisms; however, is not easy to distinguish in many cases. For example, it is well known that austenitizing a martensitic steel can increase its hardness but the influence on corrosion behavior is still a matter of debate. Candelaria et al. showed that with increasing austenitizing temperature up to a critical temperature, the weight loss due to corrosion increases [13]. At higher austenitizing temperatures, the mass loss decreases again. In contrast, Rosemann et al. observed increasing critical pitting potentials with heat treatment temperature [14] indicating a better resistance against corrosion.

In a recent study, Bösing et al. showed that different corrosion mechanism are affected differently by the austenitizing procedure [12]. Several microstructure changes occur due to the heat treatment: Increasing temperature leads to the formation of martensite, the dissolution of iron carbides, to the dissolution of chromium carbides, the diffusion of chromium, and the growth of the former austenite grain size.

These microstructure properties can affect the passive layer and thus the corrosion behavior. An increasing martensite content leads to a more defective passive layer and an accelerated dissolution [15]. This is attributed to a higher lattice distortion due to the martensite crystal structure [13]. The dissolution of carbides increases the carbon content in the crystal structure which also leads to higher lattice distortions and can increase corrosion. The dissolution of chromium carbides on the other hand increases the amount of chromium which is available for the formation of the passive layer.

It was shown that the lattice distortions, which increase with austenitizing temperature, lead to a decreasing pitting potential (the potential at which pitting starts), which indicates a higher susceptibility to pitting corrosion. Contrary to these findings, a decreasing corrosion rate and decreasing passive current could be observed. Both phenomena (decreasing corrosion rate and decreasing passive current) suggest slower corrosion during steady state and a more passivated surface. This behavior was addressed to the dissolution of chromium carbides and thus a higher amount of chromium in the passive film [12].

A description of passive film formation requires detailed information about the physio-electrochemical properties and reactions of the investigated system [16]. It has been frequently studied using the point defect model (PDM) [1,17,18,19,20]. This model describes oxide film growth by a set of interfacial reactions and transport of point defects through the film, which leads to the growth and decomposition of the passive layer. Macdonald et al. developed the PDM from its early beginnings in the 1980s [21] until today and added reactions for the dissolution of the passive layer, the formation of metal and oxygen interstitials as well as vacancies [19]. It was concluded that the rate-limiting step for film growth is the formation of anion vacancies at the metal/oxide interface, which is driven by the potential drop at this interface [19]. Since film growth is limited by interfacial potentials and electric field strength is assumed to be independent from the applied potential (because of the buffering action of band-to-band or band-to-interbandgap state tunneling [1]), the PDM is considered an interface model (IFM) that can reliably describe oxide growth. In 2002, Olsson et al. showed, by using a electrochemical quartz crystal microbalance, that oxide growth on stainless steel follows kinetics described by IFM rather than so-called high field models [22]. Such models assume the film growth to be limited by ion conductivity through the oxide and the electrical field across the passive film as a function of the applied potential [23].

In the actual PDM, the typical set of 7 interfacial reactions is used to describe various oxide layers on different metals, such as n-type oxides (through mainly oxide vacancies and metal interstitials) and p-type oxides (through mainly metal vacancies and oxygen interstitials) as well as the formation of defects by interfacial reactions [19]. The influence of heat treatment on passivation kinetics, however, has not been described yet.

In the present study we investigate the passive film growth kinetics of martensitic stainless steel samples which were heat-treated at different austenitizing temperatures. By analysis of electrochemical impedance spectroscopy (EIS) measurement and parameter optimization using the PDM, we explain the effect of austenitizing on the reaction kinetics at the metal/passive film interface in order to explore the effects of material treatment on corrosion behavior. To do this, we investigated a set of 6 stainless steel samples with different heat treatment. EIS measurements were performed using sulfuric acid for which the passive behavior of stainless steel is known to show a broad passive area that clearly distinguishes from the active/passive and transpassive region [24]. By a combination of comprehensive electrochemical measurements and modeling using a full as well as a reduced PDM film growth model, we gained new insight into the passivation processes that go beyond the findings of measurements alone.

2. Materials and Methods

2.1. Material and Sample Preparation

The stainless steel samples were produced at the Max Planck Institut für Eisenforschung (Düsseldorf, Germany) with a composition as listed in

Table 1. Element distribution of investigated stainless steel samples in ma.-%.

Alloying Element	C	Cr	Fe	Mn	O	P	S	Si
Laboratory alloy	0.497	13.700	bal.	0.372	0.0033	0.0022	0.0031	0.0075

. The samples were heat-treated at 5 different austenitizing temperatures (T_A: 1000, 1050, 1100, 1150 and 1200 °C) for 2 h and compared to an untreated sample (T0). The austenitizing time was chosen to ensure dissolution of the carbides and to provide time for the diffusion of chromium and for grain coarsening. The austenitizing temperatures were chosen to provide two temperatures below the dissolution temperature of Cr₇C₃ (1000 and 1050 °C), two temperatures around the dissolution temperature (1100 and 1150 °C), and one temperature at which dissolution is ensured (1200 °C) [25]. Furthermore, due to the different austenitizing temperatures, the grain growth accelerates with increasing temperature [26,27]. By quenching with high temperature gradients (50 K/s), new carbide formation was excluded, and martensite formation was achieved. The samples were subsequently glued into a Polyvinyl chloride (PVC) embedding using two-component adhesive.

The X-Ray diffraction measurements were performed using a MZ IV diffractometer (GE Inspection Technology, Ahrensburg, Germany). For signal detection a position-sensitive detector “Miostar 2” (Photron-X) was used. Kβ-Filtration was carried out by using Vanadium filter. The measurements were performed with a scan step of 0.05°.

To determine the former austenite grain size, the samples were etched with a mixture of 1.5% picric acid, 10 drops of 100% hydrochloric acid and 0.9 g sodium dodecylbenzenesulfonate. The line intersect method was used to determine the grain size.

2.2. Electrochemistry

Each EIS measurement was repeated 3 times. Before each measurement the working electrodes (martensitic stainless steel samples, sample area: 0.785 cm²) were wet grinded with SiC grinding paper with grain sizes from 600 to 200 µm, degreased in ethanol, and cleaned in an ultrasonic bath containing deionized water.

All electrochemical measurements were carried out at room temperature using a standard three electrode cell using a Metrohm Autolab potentiostat (Utrecht, Netherlands) with the PGSTAT204 and the FRA32 modules. A platinum electrode served as counter electrode and a $\mathrm{Ag}/\mathrm{AgCl}$ 3M $\mathrm{KCl}$ electrode as reference electrode, while the investigated material samples act as working electrodes. All potentials in this paper are referred to the $\mathrm{Ag}/\mathrm{AgCl}$ electrode.

Linear sweep voltammetry (LSV) measurements were performed in 0.5 M sulfuric acid, the scan velocity was set to 100 mV/s, the starting potential was set to −500 mV, and the stop potential lies at 1600 mV. The high scan rate was chosen to show the last oxidation peak which does not occur at typical low scan rates.

The EIS measurements were also performed in 0.5 M sulfuric acid after polarization for 1800 s at 600 mV during which a passive layer was formed. The measurements were performed in the frequency range between 10⁵ and 10⁻² Hz with an AC amplitude of ±10 mV_rms.

2.3. Fitting the Point Defect Model

The PDM [21] describes oxide film growth on metals by interfacial reactions and point defect transport through the film. The interfacial reactions as well as the defect transport are controlled by potential drops at the metal–film and film-solution interfaces and the potential drop over the film (Figure 1a).

Some additional assumptions were made in this manuscript to simplify the fitting of the impedance data to the model:

• The passive layer solely consists of ${\mathrm{Me}}_2{\mathrm{O}}_3$ species, namely ${\mathrm{Fe}}_2{\mathrm{O}}_3$ and ${\mathrm{Cr}}_2{\mathrm{O}}_3$(which leads to $\chi$=3 in the reaction scheme (Figure 1)).
• The dissolution of the oxide layer does not lead to a change of oxide state of the metals and ${\mathrm{Me}}_2{\mathrm{O}}_3$ dissolves to ${\mathrm{Me}}^{3+}$ (Reactions 4, 5, and 7 do not play a role regarding the faradaic impedance).
• The defects leading to the n-type structure of the passive film on stainless steel are oxygen vacancies. Hence, metal interstitials are neglected for the purpose of this paper and for simplification and so are Reaction 3 and 5. Both currents, produced by the formation of metal interstitials and by the formation of oxygen vacancies, cannot be distinguished and the calculated rate constants could be composite of both reactions. To ensure n-type behavior at the given potential Mott-Schottky-Analysis are done and can be seen in the supplementary materials (Figures S1 and S2).
• The inner barrier layer is the protecting passive layer and the reactions at the barrier layer-metal interface are described by the PDM.

The model comprises two competing parallel electron-generating reactions and three non-electrochemical reaction. The electrochemical reactions are Reaction 1 and 2, which describe the consumption of metal vacancies and formation of metal ions in the oxide (Reaction 1) and formation of the oxide by the production of metal ions in the oxide layer and oxygen vacancies (reaction 2). The driving force of the electrochemical reaction is the potential drop at the metal–film interface ${\phi }_{\mathrm{mf}}$. The potential drop is equal to the external potential $U_{\mathrm{ext}}$ minus the potential drop over the passive film ${\phi }_{\mathrm{f}}$ (which is the product of the electric field strength inside the film and the film thickness, ${\phi }_{\mathrm{f}}=E_{\mathrm{F}}L$) and the potential drop at the film-solution interface ${\phi }_{\mathrm{fs}}$ (Figure 1b):

$${\phi }_{\mathrm{mf}}=U_{\mathrm{ext}}-{\phi }_{\mathrm{f}}-{\phi }_{\mathrm{fs}}$$

(1)

Please note that for various pH and/or various external potentials the nature of ${\phi }_{\mathrm{fs}}$ must be considered in the model (${\phi }_{\mathrm{fs}}=\alpha U_{\mathrm{ext}}-\beta \mathrm{pH}-{\phi }_{\mathrm{fs}}^0$). In the present case (constant pH and external potential) ${\phi }_{\mathrm{fs}}$ can be considered to be a constant and can be combined with the base rate constant to form the standard rate constant: $k_i^0=k_i^{\prime 0}\exp \left(-{\phi }_{\mathrm{fs}}\right)$. The potential drop across the passive film can be calculated by the electric field strength inside the film $E_{\mathrm{F}}$ and the film thickness $L$. By this, the rate constants $k_1$ and $k_2$ can be calculated as

$$k_i=k_i^{\prime 0}\exp \left(\frac{{\alpha }_i{n}_{i}F}{RT}\left(U_{\mathrm{ext}}-E_{\mathrm{F}}L-{\phi }_{\mathrm{fs}}\right)\right)=k_i^0\exp \left(\frac{{\alpha }_i{n}_{i}F}{RT}\left(U_{\mathrm{ext}}-E_{\mathrm{F}}L\right)\right);i=1,2$$

(2)

Here, $k_i^{\prime 0}$ represents the base rate constant of reaction $i$, $k_i^0$ stands for the standard rate constant of reaction $i$, ${\alpha }_i$ for the charge transfer constant of reaction $i$, $F$ for Faradays constant, $n_i$ for the number of electrons involved in the reaction, $R$ is the universal gas constant and $T$ the absolute temperature. The admittance $Y_{\mathrm{F}}$ of the electrochemical system can be calculated by the total differential of the current [28]:

$$Y_{\mathrm{F}}=\frac{1}{Z_{\mathrm{F}}}=\frac{\partial I}{\partial U}={\left(\frac{\partial I}{\partial U}\right)}_{L,C_{\mathrm{c}}}\frac{\partial U}{\partial U}+{\left(\frac{\partial I}{\partial L}\right)}_{C_{\mathrm{c}},U}\frac{\partial L}{\partial U}+{\left(\frac{\partial I}{\partial C_{\mathrm{c}}}\right)}_{L,U}\frac{\partial C_{\mathrm{c}}}{\partial U}$$

(3)

Here $Z_{\mathrm{F}}$ is the faradayic impedance, $I$ the current, $U$ the potential and $C_{\mathrm{c}}$ the concentration of cation vacancies. The complete derivation of the admittance can be found in the original literature of Macdonald et al. – for example ref. [29,30].

The partial differential $\partial L/\partial U$ can be neglected if the change of $L$ is less than the atomic radius of the chromium and iron atoms in the oxide [30]. Adding an admittance for the oxide layer by a Constant Phase Element (CPE) yields

$$Y_{\mathrm{tot}}=Y_{\mathrm{F}}+Y_{\mathrm{cap}}$$

(4)

With

$$Y_{\mathrm{cap}}=Y_{\mathrm{CPE}}=Q{\left(j\omega \right)}^N$$

(5)

where $j$ is the imaginary number and $\omega$ the angular frequency. The CPE parameters $Q$ and $N$ are related to the film thickness $L$ by

$$\frac{\epsilon {\epsilon }_0}{L}=Q{\left({\omega }_{\mathrm{m}}^{″}\right)}^{N-1}$$

(6)

with ${\omega }_{\mathrm{m}}^{″},$ the frequency at which the imaginary part of the impedance has its maximum [31] and ${\epsilon }_0,$ the permittivity of the vacuum (8.85 × 10⁻¹⁴ AsV⁻¹cm⁻¹) as well as $\epsilon$, the dielectric constant of the oxide ($\epsilon =12$ [32]). Combining Equations (3)–(6), the total admittance can be written as

$$Y_{\mathrm{tot}}={\left(\frac{\partial I}{\partial U}\right)}_{L,C_{\mathrm{c}}}+{\left(\frac{\partial I}{\partial C_{\mathrm{c}}}\right)}_{L,U}\frac{\partial C_{\mathrm{c}}}{\partial U}+\frac{\epsilon {\epsilon }_0}{L{\left({\omega }_{\mathrm{m}}^{″}\right)}^{N-1}}{\left(j\omega \right)}^N$$

(7)

The partial derivative $\partial I/\partial U$ can be found as

$$\frac{\partial I}{\partial U}=3F{a}_1{k}_1{C}_{\mathrm{c}}+3F{a}_2{k}_2$$

(8)

with $a_i={\alpha }_{i}nF/RT.$ Differentiating the current $I$ with respect to the cation vacancies concentration $C_{\mathrm{c}}$ yields

$$\frac{\partial I}{\partial C_{\mathrm{c}}}=3F{a}_1{k}_1$$

(9)

The partial derivate $\partial C_{\mathrm{c}}/\partial U$ at the point $x=L$ ($\partial {\mathrm{C}}_{\mathrm{c}}^L/\partial U$) can be derived by the flux of cations [30] and can be written as

$$\frac{\partial C_{\mathrm{c}}}{\partial U}=C_{\mathrm{c}U}^L+{\mathrm{C}}_{\mathrm{c}L}^{\mathrm{L}}\frac{\partial L}{\partial U}$$

(10)

with

$$C_{\mathrm{c}U}^L=\frac{(b_{1V}{a}_{22}-b_{2V}{a}_{12})\exp \left(r_{1}L\right)+(b_{2V}{a}_{11}-b_{1V}{a}_{12})\exp \left(r_{2}L\right)}{a_{11}{a}_{22}-a_{12}{a}_{21}}$$

(11)

$$C_{\mathrm{c}L}^L=\frac{-a_{12}{b}_{2L}\exp \left(r_{1}L\right)+a_{11}{b}_{2L}\exp \left(r_{2}L\right)}{a_{11}{a}_{22}-a_{12}{a}_{21}}$$

(12)

and

$$r_{1,2}=\frac{3K\pm \sqrt{3^2{K}^2+4i\omega /D_{\nu }}}{2}$$

(13)

$$\begin{gathered} a_{11}=\left(r_1-3K\right)D_{\nu };a_{12}\left(r_2-3K\right)D_{\nu } \\ {a}_{21}\left[\left(r_1-3K\right)D_{\nu }+k_1\right]\exp \left(r_{1}L\right);a_{22}\left[\left(r_2-3K\right)D_{\nu }+k_1\right]\exp \left(r_{2}L\right) \end{gathered}$$

(14)

$$b_{1\mathrm{U}}=-k_4{a}_4;b_{2U}=-k_1{a}_1{C}_{\nu }^L;b_{2L}=k_1{C}_{\nu }^L{b}_1$$

(15)

in which $K=F{E}_{\mathrm{F}}/RT$ and $D_{\nu }$ the diffusion coefficient of the vacancies and $C_{\nu }^L$ the vacancies concentration at $L$. By adding the electrolyte resistance we finally find the impedance

$$Z_{\mathrm{tot}}=R_{\mathrm{el}}+\frac{1}{Y_{\mathrm{tot}}}$$

(16)

By solving the differential equation for film growth (considering a constant H⁺ concentration ($k_7^{\prime }{C}_{{\mathrm{H}}^{+}}=k_7$)),

$$\frac{\mathrm{d}L}{\mathrm{d}t}=\mathsf{\Omega }{k}_2-\mathsf{\Omega }{k}_7^{\prime }{C}_{{\mathrm{H}}^{+}}=\mathsf{\Omega }{k}_2^0\exp \left(\frac{{\alpha }_{2}nF}{RT}\left(U_{\mathrm{ext}}-E_{F}L\right)\right)-\mathsf{\Omega }{k}_7$$

(17)

with $\mathsf{\Omega }=14.6$ cm³mol⁻¹, the molar volume of the oxide, it is possible to calculate the current film thickness $L$

$$L={\displaystyle \int }\left(\mathsf{\Omega }{k}_2^0\exp \left(\frac{{\alpha }_{2}nF}{RT}\left(U_{\mathrm{ext}}-E_{F}L\right)\right)-\mathsf{\Omega }{k}_7\right)\mathrm{d}t$$

(18)

The steady state film thickness $L_{\mathrm{ss}}$ is found by setting $\mathrm{d}L/\mathrm{d}t=0,$

$$L_{\mathrm{ss}}=\frac{U}{E}-\frac{1}{\frac{{\alpha }_{2}nF}{RT}{E}_{\mathrm{F}}}\ln \left(\frac{k_7}{k_2^0}\right)$$

(19)

Taking these dependencies into account, the model parameters can be optimized to match model calculation and measurements. The fitted variables are $k_1^0$, ${\alpha }_1$, $k_2^0$, ${\alpha }_2$ and $N$. $L$ is calculated by Equation (6) and $k_7$ is calculated by Equation (19). The parameter $E_{\mathrm{F}}$ is guessed as 10⁶ V/cm.

To guarantee finding the global optimum in model parameter identification, a two-step minimization was performed based on the impedance data. For the optimization step the faradaic impedance resulting from the PDM, $Z_{\mathrm{PDM}},$ was summarized and the overall impedance was calculated. Using the dependencies (Equations (6), (8) and (19)) it was possible to calculate the single kinetic parameters from the overall impedance $Z_{\mathrm{PDM}}$. At first a genetic algorithm (in MATLAB R2015a by The MathWorks Inc) was employed to obtain initial values for the final nonlinear programming step using Matlab’s sqp solver. As constraint for the optimization it was assumed that the measured current after polarization results only from Reaction 1 and 2 ($i_{\mathrm{pol}}=3F{k}_1{\mathrm{C}}_{\mathrm{c}}+3F{k}_2$) additional to this constraint physically reasonable boundaries were used for all fitted parameters. The goodness of the fit was assessed by the $R^2$ value.

3. Results and Discussion

3.1. Microstructure

The XRD results of the samples are shown in Figure 2. Based on the location, the $\alpha$(011), $\alpha$(002) and the $\alpha$(211) peaks are the only visible diffraction peaks. All samples show clear martensitic diffraction pattern and no significant austenite phase. Intensity variations are among other reasons attributable to the variation in the former austenite grain size.

The former austenite grainsize of the material increases with austenitizing temperature. It was determined using the line intersect methods of optical micrographs presented in a former study [12]. The average grain size in µm was calculated from the grain size number G [33] (Figure 3). The amount of chromium carbide and the austenite content (during heat treatment) is calculated by ThermoCalc calculations (version 2018b,

Table 2. Austenite phase and mass amount of dissolved chromium.

TA /°C	Amount of Austenite Phase (During Austenitizing) /ma.-%	Mass Amount of Dissolved Cr /ma.-%
1000	97.46	11.94
1050	98.44	12.65
1100	99.60	13.45
1150	99.97	13.69
1200	99.97	13.69

). It should be noted that these calculations are based on equilibrium conditions. It is assumed that 2 h of austenitizing is sufficient to dissolve all carbides and reach equilibrium (regarding the dissolution).

Cr₂₃C₆ carbides are dissolved at temperatures above ca. 930 °C while the Cr₇C₃ carbides completely dissolve above 1110° [25]. Therefore, the amount of dissolved chromium as well as the amount of dissolved carbon increases with austenitizing temperature. While a higher chromium amount leads to a more protective passive film, an increasing amount of carbon leads to lattice stress which can have a negative effect on the passive film formation [12,13,15].

3.2. Linear Sweep Voltammetry

LSV of a selected sample was done to identify the required potential for passive film formation before each EIS measurement. The LSV measurements (Figure 4) are in agreement with literature [24]. The cathodic current peaking at −450 mV (vs. $\mathrm{Ag}/\mathrm{AgCl}$) is a result of ${\mathrm{H}}_2$ evolution, which also overlays the anodic dissolution of iron and chromium. At higher potentials, ${\mathrm{H}}_2$ evolution decreases, and the dissolution of iron and chromium to ${\mathrm{Fe}}^{2+}$ and ${\mathrm{Cr}}^{2+}$ is accelerated. The current peak at −300 mV is due to oxidation of chromium to ${\mathrm{Cr}}^{3+}$ that leads to the formation of chromium oxide and a formation of a passive region. The passive area is characterized by a low anodic current as a result of the inhibited dissolution of chromium to $\mathrm{Cr}\left(\mathrm{III}\right)$ and iron to $\mathrm{Fe}\left(\mathrm{II}\right)$ and $\mathrm{Fe}\left(\mathrm{III}\right)$ for potentials over 350 mV. The transpassive area starts at a potential of 1000 mV. Here, the anodic current increases due to the oxidation of the chromium species to $\mathrm{Cr}\left(\mathrm{VI}\right)$ in soluble ${\mathrm{Cr}}_2{\mathrm{O}}_7^{2-}$, followed by ${\mathrm{O}}_2$ evolution due to water electrolysis.

While stationary polarization curves only show one plateau for the $\mathrm{Cr}\left(\mathrm{VI}\right)$ oxidation, the LSV measurements show two peaks, one at 1200 mV and one at 1500 mV.

3.3. Extraction of Kinetic Parameters from EIS Data Via the PDM

To understand the film growth kinetics as a function of austenitizing temperature the PDM parameters were fitted to the electrochemical measurements to simulate the impedance data. In a first approach, the impedance data of a test sample were fitted by the model including five interfacial reactions and the transport of anion and cation vacancies through the layer. In a second (simplified) approach, the fitting results were compared to an impedance model without consumption of cation vacancies (Reaction 1) and the cation transport through the film. This simplification, which was already done by others [30,34], seems justifiable because of two reasons. First, Reaction 1 and 2 are two competing parallel reactions, hence either can occur without the other whereas Reaction 2 must occur for film growth. Secondly, passive films on stainless steels show n-type semiconducting behavior [35] and thus the dominant defect species are oxygen vacancies and metal interstitials (compare also MS plot in the SI). Using the first full model approach, fit results (Figure 5) and fit quality ($R$² value in

Table 3. Fitted parameter.

Parameter	Full Model	Simplified Model
$k_1^0$	2.34 × 10−14	–
${\alpha }_1$	0.13	–
$k_2^0$	7.25 × 10−11	7.66 × 10−11
${\alpha }_2$	0.028	0.028
$k_4^0$	2.56 × 10−14	–
${\alpha }_4$	0.01	–
$k_5^{\prime }$	2.93 × 10−10	3.05 × 10−10
$E_{\mathrm{F}}$	1.12 × 106	1.0 × 106
${\phi }_{\mathrm{fs}}$	0.1	0.1
$D_{\mathsf{\nu }}$	7 × 10−16	–
$n$	0.875	0.875
$R^2$	0.9974	0.9970

) are slightly better compared to the simplified model. Nevertheless, also without taking the transport of defects into account, the fit quality of the reduced model is still very good. Comparison of the fitting parameter (Table 3), the standard rate constant of the film formation $k_2^0$, the rate constant of film dissolution $k_7,$ and the charge transfer coefficient of the film formation reaction show very good agreement for both models.

Referring to these findings the simplified model was used to keep the set of fitting parameters as low as possible and to avoid overfitting. Hence the following admittance is used:

$$Y_{\mathrm{F}}={\left(\frac{\delta I}{\delta U}\right)}_{L,C_{\mathrm{c}}}\frac{\delta U}{\delta U}+\frac{\epsilon {\epsilon }_0}{L{\left({\omega }_{\mathrm{m}}^{″}\right)}^{N-1}}{\left(j\omega \right)}^N$$

(20)

The impedance data of the 6 different samples (Figure 6) reveals clear differences between the samples. The lowest overall impedance can be seen in the untreated sample (T0) (Figure 6a).

Up to a value of T_A = 1100 °C, the overall impedance monotonically increases with austenitizing temperature. For T_A = 1150 °C, the impedance decreases compared to the lower temperatures but a further increase in temperature increases the impedance again, so that the maximum is reached at T_A = 1200 °C (Figure 6b).

The fitting of the impedance data can be done by the equivalent electrical (EEC) circuit presented in Figure 7. The EEC consists of an electrolyte resistance $R_{\mathrm{el}}$, a charge transfer resistance $R_{\mathrm{ct}}$, a CPE for the double layer (and the porous electrolyte filled outer layer) $CP{E}_{\mathrm{dl}}$, a CPE representing the capacitive behavior of the passive layer $CP{E}_{\mathrm{pl}}$, and an impedance element describing the interfacial reactions predicted by the PDM $Z_{\mathrm{PDM}}$.

The PDM calculations show good agreement with the measurements (Figure 8). The reaction coefficients for passive film growth (Reaction 2) and passive film dissolution (Reaction 7) as well as the charge transfer coefficient ${\alpha }_2$ for the film growth reaction and the passive film thickness $L_{\mathrm{ss}}$ are shown in

Table 4. Fitted parameters for all martensitic stainless steel samples.

${\mathit{T}}_{\mathit{A}}/°\mathbf{C}$	${\mathit{k}}_2^0$$/\mathbf{m}\mathbf{o}\mathbf{l}/\mathbf{c}\mathbf{m}²\mathbf{s}$	${\mathit{\alpha }}_2$	${\mathit{k}}_5^{\prime }$$/\mathbf{m}\mathbf{o}\mathbf{l}/\mathbf{c}\mathbf{m}²\mathbf{s}$	${\mathit{L}}_{\mathbf{s}\mathbf{s}}/\mathbf{nm}$
T0	9.61 × 10−11	0.0247	3.60 × 10−10	1.50
1000	5.91 × 10−11	0.0298	3.09 × 10−10	1.32
1050	3.68 × 10−11	0.0282	1.92 × 10−10	1.07
1100	1.80 × 10−11	0.0353	1.38 × 10−10	1.16
1150	2.69 × 10−11	0.0368	2.66 × 10−10	0.76
1200	9.46 × 10-12	0.0352	8.44 × 10-11	0.77

.

The crucial factor for the steady state film thickness is the ratio $k_2^0/k_7$ The higher the ratio the thicker the passive film; whereas the same ratio, which is achieved at higher rate constants, leads to the same film thickness but accelerated dissolution (for similar $\alpha$ values) thereby approaching the steady state faster. An increasing $\alpha$ value leads to thinning of the passive film at slightly less dissolution (assumed $k_2^0$ and $k_7^{\prime }$ remain constant).

Generally, with increasing austenizing temperature, $\alpha$ increases, the standard rate constant $k_2^0$ for film formation (Figure 9a) decreases, and of rate constant $k_7$ for film dissolution (Figure 9b) decreases. A decrease in $k_2^0$, which means decreasing metal dissolution at the metal-passive film interface, indicates a slower corrosion rate. Thus, samples with lower $k_2^0$ show a better passivation and lower reaction rates at the metal surface due to a more protective passive layer. A further indicator for a more stable and protective passive film is the decreasing rate constant for film dissolution $k_7$. This behavior can be attributed to the higher amount of chromium at higher austenitizing temperature and a more homogeneous chromium distribution due to the higher diffusion at higher temperatures.

However, the sample T_A = 1150 °C shows deviations from this trend by a slightly increased rate constant for film formation and a drastically increased rate constant for film dissolution leading to the thinnest passive film. This might be connected to the dissolution of Cr₇C₃ carbides [25], which on the one hand brings more chromium to the matrix but on the other hand leads to further lattice distortion due to the dissolved carbon. Taking the different stoichiometry of the carbides into account (Cr₂₃C₆ dissolving at 930 °C and Cr₇C₃ dissolving at 1150 °C) the ratio of dissolved carbon to dissolved chromium is higher at 1150 °C (Cr₇C₃) compared to the Cr₂₃C₆ carbides. This leads to higher lattice stress for the same amount of chromium and thus explains this behavior.

One can further observe a thinning of the passive film thins with increasing austenitizing temperature (Figure 10). The untreated sample shows the thickest passive film but also the lowest CPE parameter, N = 0.76, which indicates a less homogeneous layer [36]. Thus, the passive film thickness is not crucial for the protective effect of the passive layer in this case. Rather, a protective passive layer leads to a deceleration of metal dissolution at the metal–film interface and thus to a thinner but more protective film.

The simulation via the PDM shows a deviation from the impedance data for low frequencies (10⁻²  Hz). These differences could be because the model does not incorporate the transport of oxygen vacancies through the film. This vacancy transport from the metal-film interface towards the film-solution interface could be overcome by adding a Warburg impedance to the EEC [30] at the cost of at least one additional fitting parameter.

Fitting the passive film kinetics to the obtained impedance data provide a very good explanation for the observed dependency of corrosion resistance and austenitizing temperature [14]. Due to the higher chromium content and the more homogeneous chromium distribution the passive film is more protective and thus slows down the dissolution of the metal (decreasing rate constant for film formation) and of the film itself (decreasing rate constant for film dissolution). The results also match with our previous findings with respect to an increasing critical pitting potential [12] which indicates a slower dissolution by local corrosion attacks with increasing austenitizing temperature. Next to the increasing critical pitting potential we previously found decreasing passive current densities and decreasing corrosion rates. The fitting delivers a clear explanation for this behavior due to the decreased film formation in the passive state. Nevertheless the findings are contradictory to the findings of Candelaria et al. [13] and the decreasing start potential for pitting corrosion with increasing austenitizing temperature [12]. The presented fitting of the passive layer growth kinetics by the PDM provides detailed insight to the electrochemical behavior of the passivated samples. The phenomena investigated by Candelaria result from breakdown of passivity in the presence of halides. It is important to distinguish both mechanisms, the steady state metal loss due to growth of the passive film itself and the local corrosion attacks.

4. Conclusions

The electrochemical behavior of martensitic stainless steel was investigated with respect to the austenitizing temperature. By modeling impedance data of high reproducibility by the PDM it was possible to describe all data sets with good agreement and small standard deviations which were clearly below 10% in all cases. This agreement between calculation and data confirms that the PDM is sufficient to describe the kinetics of these passive film growths and to compare the parameters of different materials regarding their passive behavior. Furthermore, it was found that austenitizing temperature significantly influences all four kinetic parameters of passive film growth. More specifically the standard rate constant for film formation and the rate constant for film dissolution decreases with increasing austenitizing temperature. This indicates a better protection of the metal by slowing down the film growth and a more stable passive film by reducing the dissolution rate of the film itself. Both phenomena can be addressed to an increasing amount of dissolved chromium and a more homogeneous chrome distribution. This study underpins that modeling of passive film growth in combination with comprehensive electrochemical measurements has great potential for both the understanding of passivation processes and developments in materials science.