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Electrochimica Acta 52 (2007) 7475–7483

Estimation of the parameters of oxide film growth on nickel-basedalloys in high-temperature water electrolytes

Martin Bojinov a,∗,1, Anouk Galtayries b, Petri Kinnunen c,Alexandre Machet b, Philippe Marcus b,1

a Department of Physical Chemistry, University of Chemical Technology and Metallurgy, 1756 Sofia, Bulgariab Laboratoire de Physico-Chimie des Surfaces, CNRS-ENSCP(UMR 7045), Universite Pierre et Marie Curie,

Ecole Nationale Superieure de Chimie, F-75231 Paris Cedex 05, Francec VTT Technical Research Centre of Finland, P.O. Box 1000, FI-02044 VTT Espoo, Finland

Received 12 February 2007; received in revised form 29 May 2007; accepted 6 June 2007Available online 12 June 2007

bstract

The Mixed-Conduction Model (MCM) has been adapted to obtain the main kinetic and transport parameters for passive film growth on nickel-ased alloys in high-temperature electrolytes. For the purpose, a procedure for the calculation of these parameters from electrochemical impedancepectroscopic data measured on Ni–15% and Ni–20%Cr alloys in 0.1 M Na2B4O7 at 300 ◦C in a wide potential range has been devised. Thebtained sets of parameters have been used to successfully predict the steady-state current density versus potential dependence for the two Ni–Cr

lloys, as well as the thickness versus time dependence for films grown on Alloy 600 at times ranging between a few minutes and a few thousandours. The calculations are supported by results of previous XPS measurements of film growth on Alloy 600 showing that the barrier layer growingn the early stage consists of Cr2O3.

2007 Elsevier Ltd. All rights reserved.

netic

ameiaiaeo(rd

eywords: Nickel-based alloy; Oxide film growth; High-temperature water; Ki

. Introduction

The alloy materials used in a commercial nuclear power plant,uch as stainless steels and nickel-based alloys, rely on the prop-rties of their passivating oxide films to ensure the structuralntegrity and lowest activity build-up of the plant. The prop-rties of the film, being a result of the manufacturing and thenvironment during the life of the component must be good andemain good for the safest and most economical operation ofhe plant. The need to optimise the oxide film properties callsor deterministic modelling of activity build-up and localisedorrosion phenomena. Such modelling needs more fundamen-

al work describing the elementary processes during oxide filmormation, and developments have been pointed out in severalecent studies. The theoretical understanding has today reached

∗ Corresponding author.E-mail address: martin@uctm.edu (M. Bojinov).

1 ISE Member.

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c

013-4686/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.oi:10.1016/j.electacta.2007.06.002

model

level where quantitative modelling of the oxide film transportechanism can be performed [1–7]. In addition, considerable

fforts have been done in order to measure the surface reactionsn high-temperature water loops and to understand their mech-nisms [8–11]. The common objective of all these approachess to enable quantitative modelling of the oxide film behaviourllowing prediction of activity build-up and corrosion phenom-na controlled by the oxide film. In particular, knowledge of thexidation processes on Ni alloys in Pressurised Water ReactorPWR) coolant is of major importance for at least two practicaleasons: (i) the radioactivity of the primary circuit is primarilyue to cations released by corrosion of the steam generator tubesnd (ii) the oxidation process is important in the mechanisms ofhe initiation of intergranular stress corrosion cracking in Alloys00, 82 and 182. Understanding the root causes of this crackinghould also allow the safety margins offered by the Alloys 690,

2 and 152, which have replaced the former ones, to be evaluatedore accurately.Very recently, using a combination between in situ electro-

hemical impedance spectroscopic and ex situ Auger electron

7 imica

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476 M. Bojinov et al. / Electroch

pectroscopic data and the Mixed-Conduction Model (MCM)or oxide films, some of us have presented estimates of theinetic parameters of growth and restructuring of the passive filmn AISI 316L(NG), a typical construction material in nuclearower plants, in a high-temperature electrolyte [6]. Also veryecently, unique experiments and new data on the initial stagesf passive oxide film growth on nickel-based Alloys 600 and 690n a simulated PWR coolant electrolyte using a micro-autoclaveechnique combined with ex situ XPS and STM characterisa-ions have been reported [12,13]. In that connection, the aimf the present paper is to combine these results with previous initu electrochemical impedance spectroscopic measurements oni–Cr alloys already presented by some of us [14]. The goal is

o find reliable estimates for the kinetic and transport parametersf film growth and restructuring on nickel-based materials. First,he MCM is adapted to the present system using the qualitativeicture presented in recent studies based on the interpretation ofhe results obtained using the micro-autoclave technique [13].econd, the relevant electrochemical impedance data are brieflyescribed and the procedure for obtaining estimates of the kineticnd transport parameters by fitting them to the transfer functionf the MCM is outlined. Third, the obtained parameters are usedo reproduce both the initial stage and the long-term kinetics oflm growth on Alloy 600 in simulated PWR water and theirelevance is discussed.

. Theoretical background

Fig. 1 presents a simplified scheme of the processes dur-ng exposure of a Ni–Cr alloy in high-temperature water. Thischeme is constructed on the basis of the qualitative picture ofhe film growth process proposed in Refs. [13,15], as well as theramework of generation, transport and consumption of pointefects that is believed to control the oxidation process accordingo the MCM [4,6]. According to those treatments, the continuousarrier-like layer that forms on the alloy in the initial stages of

xposure is assumed to be Cr2O3. Such an assumption is backedp by the XPS and STM characterisations of the films formed inimulated PWR coolant electrolyte in a micro-autoclave [13,15]hat have proved that the initial surface oxide film consists of

ig. 1. A model for the mechanism of formation of the oxide film on Ni–Crlloys in high-temperature water.

rh

as

bti

Acta 52 (2007) 7475–7483

r2O3. Accordingly, the predominant point defects in the hexag-nal structure of Cr2O3 are the cation and oxygen vacancies. Therowth of Cr2O3 which was expressed in Refs. [13,15] by theeneralised equation

r + Cr(OH)3 → Cr2O3 + 3H+ + 3e− (1)

nvolved Cr transport through the layer. One way to describe thisransport in terms of point defects is according to the equations16]

Crmk2−→CrCr + 1.5 VO(M/F)

•• + 3e−m

1.5 VO••DO, �E−→1.5 VO(F/E)••

1.5 VO•• + Cr(OH)3

k4−→CrCr + 3OO + 3H+(2)

n which a Kroger–Vink notation is used for the chromiumnd oxygen positions in the lattice, as well as for the predomi-ant point defects. For a complete list of symbols, the reader iseferred to the Nomenclature section. The presence of Cr(OH)3s not shown in Fig. 1 for simplicity reasons and the reactions2) can be written in a simpler form as

CrMk2−→CrCr + 1.5 VO

•• + 3e-M

1.5 VO(M/F)••DO, �E−→1.5VO

••

1.5 VO(F/E)•• + 1.5H2O

k4−→1.5OO + 3H+(3)

In addition, as also proposed in Refs. [13,15], Ni is trans-orted through the film, forming Ni(OH)2 in contact with thelectrolyte and/or releasing Ni2+ ions into the electrolyte. Withinhe frames of the formalism used above, this process can be rep-esented as a sequence of generation, transport and consumptionf cation vacancies

Ni′IICr + 2H2Ok3−→Ni(OH)2 + V′′′

Cr(F/E) + 2H+

V′′′Cr(F/E)

DM, �E−→V ′′′Cr(M/F)

V′′′Cr(M/F) + Nim

k1−→Ni′IICr + 2e−m

(4)

It is worth mentioning that the first reaction in (4) is in fact aesult of lumping nickel dissolution as aquo-ions and subsequentydrolysis:

Ni′IICr → Ni2+aq + V′′′

Cr(F/E)

Ni2+aq + 2H2O → Ni(OH)2 + 2H+

The transport equations of the MCM in their low-fieldpproximation in analogy to previous work on AISI 316 stainlessteel [6] have been employed

JO = −DO∂cO(x)

∂x− 2F �EDO

RT

JM = −DM∂cM(x)

∂x+ 3F �EDM

RT

(5)

They were solved under a small amplitude ac current pertur-ation around a quasi-steady-state making use of the equality ofransport and electrochemical reaction fluxes at the respectiventerfaces (x = 0 at the film/electrolyte interface and x = L at the

imica

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L

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fpu

M. Bojinov et al. / Electroch

lloy/film interface). The use of a quasi-steady-state approxima-ion is deemed appropriate due to the fact that the increase oflm thickness within the time span of an impedance measure-ent can be neglected. This assumption will be verified for the

ystems under study in the Results section.As a result, the following equation for the impedance of the

lm is obtained [6]

Zf = (Z−1e + Z−1

i )−1

Ze ≈ p

jωCln

(1 + jωρdεε0 e1/p)

1 + jωρdεε0, p = 1

2KL, K = F �E

RT,

ρ−1d = F2De

RT

(k3

k1+ k2

k4

)

Zi = Rct + (Z−1i,M + Z−1

i,O)−1

Zi,M = σM tan h(jωτM)1/2

(jω)1/2 , σM = RT

F2(1 − α)cM(L)√

81DM

Zi,O = σO tan h(jωτO)1/2

(jω)1/2 , σO = RT

1.5F2(1 − α)cO(L)√

32

In these equations, the impedance of the film is representeds a parallel combination of the impedance due to the elec-ronic properties of the oxide Ze which can be regarded as aon-ideal capacitance of a semiconductor layer with defect den-ities depending on both potential and distance within the film4–6], and the impedance of the transport of ionic defects in thexide Zi. This impedance is described as a parallel connection ofwo finite length transport impedances Zi,M and Zi,O correspond-ng to the transport of cation and oxygen vacancies through thexide, respectively.

In the expression (6), the interfacial reactions of electronransfer are exponentially dependent on the potential drop athe respective interface: ki = k0

i exp (biφN), i = 1. . .4, N = M/Fr F/E. As it has been previously assumed [6] that the poten-ial drop at the metal/film interface does not depend on thexternal applied potential in the steady-state, only the reactionate constants 3 and 4 (cf. Fig. 1) are exponentially dependentn potential: k3 = k0

3 exp (b3αE), k4 = k04 exp (b3αE), where b3

nd b4 are the respective exponential factors.It is worth mentioning that the form of the transport

mpedance Eq. in (6) is analogous to those derived in the Pointefect Model for Ni in phosphate and borate buffer solutions

t room temperature [17], the difference being that in the MCMhe rate constants at the film/electrolyte interface play a moremportant role [4–6,16].

The electrochemical reactions occurring at thelm/electrolyte interface due to the presence of redox couplesuch as H2/H2O and H2O2/H2O, as well as the transpassivexidation of Cr(III) in the film to soluble Cr(VI) have to beaken into account because of their increasing importance in

igh-temperature electrolytes, with respect to the film growthnd dissolution reactions [18,19]. In the present treatment, aslready mentioned in Ref. [4], the interfacial impedance dueo the presence of such redox reactions is modelled as the

deQa

Acta 52 (2007) 7475–7483 7477

= LE=0 + 1 − α

�E E,

cM(L) = k3

k1

cO(L) = k2

k4e−2KL

(6)

mpedance of a one-step reaction. This gives for the impedancet the film/electrolyte interface

−1F/E = αF (3b3k3 + 4

3b4k4 + bredoxkredox) + jωCF/E (7)

The first term on the right hand side can be representeds a generalised charge transfer resistance R−1

F/E = αF (3b3k3 +/3b4k4 + bredoxkredox) and the second term involves the capac-tance of the film/electrolyte interface CF/E.

The total impedance is then written as a series combinationf the impedance of the film and that of the film/electrolytenterface (as within the frames of the MCM it is assumed thathe potential drop at the inner interface is not a function of thexternal potential, then the impedance at the inner interface iseglected):

= Rel + Zf + ZF/E (8)

As a next step, assuming that the growth rate of the innerayer is proportional to the transport flux of oxygen vacancies,he growth law can be adapted from recent treatments using theDM [20,21] in the form

L(t) = L(t = 0) + 1

bln [1 + Ωk2b e−bL(t=0)t],

b = 3α2F �ERT

(9)

The next chapters are devoted to the comparison of thequations of the model to experimental data in order to obtainstimates for the kinetic parameters.

. Experimental

The experimental details on the impedance measurementsor Ni–Cr alloys have been published earlier [14]. Briefly, high-urity Ni–Cr alloys (15 and 20 wt.% Cr, Goodfellow) weresed as working electrodes. Disc-shaped electrodes of 5 mm

iameter were sealed in polytetrafluoroethylene (PTFE). Thelectrodes were mechanically polished and rinsed with Milli-

purified water. A Pt wire was used as a counter electrodend a Pd electrode electrochemically charged with hydrogen

7478 M. Bojinov et al. / Electrochimica Acta 52 (2007) 7475–7483

F g trani

aatwaAtfapsiNwl(1wbtar≤bsiaKatt

6e61audd

PeawcaotHtoato1ww2

4

afttvcitad[fad

ig. 2. Extrapolated experimental (points) and calculated by a Kramers–Kronimpedance for Ni–15%Cr in 0.1 M Na2B4O7 at 300 ◦C and −0.88 V.

pproximating the reversible hydrogen electrode was employeds a low-impedance reference electrode in order to improvehe high-frequency response. The potential of the Pd electrodeas checked versus a 0.1 M KCl/AgCl/Ag electrode balanced

gainst external pressure (EAgCl/Ag = −0.04 VSHE at 300 ◦C).ll the potentials are given on the standard hydrogen elec-

rode scale (SHE) and corrected for the IR drop determinedrom the high-frequency intercept of the impedance spectrand the steady-state current density. The measurements wereerformed at 300 ± 1 ◦C and ca.10 MPa in a 0.1 M Na2B4O7olution. The pH of the solution was 9.1 at 300 ◦C. Exper-ments were performed in a static Ti-clad autoclave under a

2 (99.999%) atmosphere. Continuous bubbling with nitrogenas used during the measurements, the maximum hydrogen

evel in such conditions at high-temperatures being ca. 600 ppb10 cm3 kg−1 STP) and the maximum oxygen level roughly0–20 ppb ((3–7) × 10−7 mol dm−3). Impedance measurementsere carried out with a Solartron 1287/1260 system controlledy ZPlot Software (Scribner Associates). The impedance spec-ra were measured in a frequency range of 0.01–30,000 Hzt an ac amplitude of 20 mV (rms) after a steady-state cur-ent density was reached (variation during the measurements2%). The validation of the impedance spectra was performed

y checking the linearity condition, i.e. measuring spectra atignal amplitudes between 2 and 20 mV (rms), and by check-ng the causality using a Kramers–Kronig compatibility testccording to the method of Boukamp [22]. An example of theramers–Kronig transformation of the data for Ni–15%Cr atpotential of −0.88 V is shown in Fig. 2 and demonstrates

he reliability of the obtained experimental data with respecto causality.

The measurements of the initial stages of oxidation of Alloy00 using a micro-autoclave technique have also been publishedlsewhere [13,15]. Polycrystalline samples of commercial Alloy00 (Ni–16Cr–9Fe wt.%) were disks of 15 mm diameter andmm thickness. The coupons were mechanically polished to

1 �m diamond finish, and further cleaning was performed inltra high vacuum conditions with argon ion sputtering. The oxi-ation was performed at 325 ◦C and ca. 14 MPa in a speciallyesigned titanium micro-autoclave in simulated unsaturated

tm

f

sform using the procedure of Boukamp (lines) real and imaginary parts of the

WR primary water conditions. This set-up allows controlledxposure of samples for very short times (a few tens of secondsnd up). The oxide layers formed at 325 ◦C in high-temperatureater are analysed ex situ at room temperature. However, the

ooling was rapid (ca. 2 min) and was performed under argon,s well as the transfer to the XPS, so that the possible changesf the surface composition were minimised. The aqueous solu-ion contained 2 mg dm−3 Li as LiOH and 1200 mg dm−3 B as

3BO3. A hydrogen overpressure of 0.03 MPa was maintainedo ensure a dissolved H2 concentration of 35 cm3 kg−1 and a lowxygen content of <30 �g kg−1. The pH of the solution was 7.1t 325 ◦C and the potential was −0.808 VSHE as calculated fromhe H2/H+ equilibrium potential, close to the corrosion potentialf Alloy 600. For XPS characterisation, Ni 2p, Cr 2p, Fe 2p, Os, C 1s, B 1s and Li 1s core level spectra have been recordedith a VG ESCALAB Mk II X-ray photoelectron spectrometer,ith a Mg K� radiation (hν = 1253.6 eV), at a pass energy of0 eV.

. Results and discussion

The impedance spectra of the Ni–15%Cr and Ni–20%Crlloys at 300 ◦C in 0.1 M Na2B4O7 at different potentials rangingrom −0.9 to 0.2 V are presented in Figs. 3 and 4, respec-ively. In the passive region (below −0.3 V [14]), a total ofhree time constants could be detected by preliminary decon-olution of the impedance spectra. The highest frequency timeonstant being associated with the electronic properties of thenner, continuous layer of the oxide film, the medium frequencyime constant reflecting the charge transfer and redox processest the film/electrolyte interface and the low-frequency shoul-er assigned to the transport of point defects in the inner layer14]. In the transpassive region (above −0.3 V [14]), the inter-acial process starts to dominate over the transport process,nd finally at the highest potentials a single time constant isetected, probably because the oxide layer becomes a conduc-

or, as demonstrated also by earlier contact electric resistance

easurements [14].The solid lines in Figs. 3 and 4 represent fits to the transfer

unction described by the Eqs. (6)–(8) and demonstrate the abil-

M. Bojinov et al. / Electrochimica Acta 52 (2007) 7475–7483 7479

00 ◦C

iAoph

))−

we

Rascfi(old

at

p

Ttbiatoniu

Fig. 3. Impedance spectra of Ni-15%Cr in 0.1 M Na2B4O7 at 3

ty of the MCM to account for the present experimental data.three-step calculation procedure has been devised in order to

btain statistically reliable values of the kinetic and transportarameters [6]. First, the spectrum at each individual potentialas been fitted to a simplified version of Eqs. (6)–(8), namely

Z = Rel + (R−1F/E + jωCF/E)

−1 +⎡⎣( p

jωCln

(1 + jωρdεε0 e1/p

1 + jωρdεε0

σ = σOσM

σO + σM

ith Rel, p, C, CF/E, RF/E, Rct, ρdε, σ and τ as adjustable param-ters.

The values of the uncompensated electrolyte resistance,el, preserved constant values of ca. 3.5 � cm2 for bothlloys at all potentials. Due to the overlap of the time con-tants involved, the values of Rct (15 ± 12 � cm2) have beenomputed with a large error and will not be commentedurther. On the other hand, it was necessary to replace thenterfacial capacitance CF/E with a constant phase element

CPE) reflecting the geometric and energetic heterogeneityf the film/electrolyte interface. The values of the equiva-ent capacitance calculated from the fit were typical for aouble-layer capacitance, lying between 40 and 60 �F cm−2,

lgdt

Fig. 4. Impedance spectra of Ni-20%Cr in 0.1 M Na2B4O7 at 300 ◦C

. Points – experimental values, solid lines—best fit calculation.

1

+(

Rct + σ tan h(jωτ)1/2

(jω)1/2

)−1⎤⎦

−1

(10)

nd did not exhibit any meaningful dependence on poten-ial.

The dependences of p, C, RF/E, ρdε, σ and τ onotential are shown in Fig. 5 for the two Ni–Cr alloys.

he dependence of the oxide capacitance on potential suggestshat the films behave as n-type semiconductors for potentialselow ca.−0.6 V and as p-type semiconductors above this value,n analogy to room-temperature results [23]. The switch betweenpparent n- and p-type semiconductivity of the inner and con-inuous barrier sublayer of the oxide (identified as chromiumxide by XPS) can be explained as follows. In fact, at veryegative potentials, the oxygen vacancies are the predominantonic defects that act as donors and form an impurity band sit-ated close to the conduction band. As the potential is made

ess negative, the model predicts that the concentration of oxy-en vacancies will decrease and finally the film will be almostepleted of charge carriers (at a potential of ca.−0.6 V). Abovehis potential, there is an apparent p-type behaviour of the bar-

. Points—experimental values, solid lines—best fit calculation.

7480 M. Bojinov et al. / Electrochimica Acta 52 (2007) 7475–7483

F he expi ointsl

rcrwgrarcet

segald

tdwmafpmsttcbo

ig. 5. Dependence of the main parameters of the transfer function used to fit tn 0.1 M Na2B4O7 at 300 ◦C: C (a), RF/E (b), p−1 (c), ρdε (d), σ and τ (f). Pines—predicted curves according to the Mixed-Conduction Model.

ier sublayer of the oxide which is due to the predominance ofation vacancies acting as acceptor states and forming an impu-ity band close to the valence band (their concentration increasesith increasing potential according to the model). These sug-estions are supported by the dependence of the charge transferesistance at the film/electrolyte interface RF/E on potential if it isssumed that the redox reaction dominates over the ionic transfereactions at this interface. Indeed almost identical exponentialoefficients of 11 ± 0.5 V−1 are obtained indicating a single-lectron transfer and thus a simple redox reaction dominatinghis interfacial impedance.

Second, the p−1, ρdε and σ versus E dependences have beenimultaneously fitted to the equations of the model to obtainstimates of k1, k2, k3, k4, b3, b4, De, DO, DM, �E, and α. The

ood correspondence between the values of the parameters �d�nd � obtained in the first stage fitting (points) and those calcu-ated on the basis of the model equations (solid lines) in Fig. 5emonstrates the ability of the model to account correctly for

T

tc

erimental impedance spectra on potential for Ni-15%Cr and Ni-20%Cr alloys—values, calculated from the fitting of the experimental impedance spectra,

heir dependences on potential. Additionally, the p−1 versus Eependence (Fig. 5) is found to be almost linear in agreementith the predictions of the model, giving the possibility to esti-ate the film thickness at E = 0 (LE=0). Finally, a global fit of

ll the impedance data at a given temperature to the transferunctions has been performed using the values of the kineticarameters obtained in the second calculation step as initial esti-ates. Statistical weighting was used for the experimental data

et and the errors of parameter estimation were multiplied byhe square root of the reduced chi-square value resulting fromhe fit. The standard error of estimate of the kinetic parametersalculated by the procedure did not exceed ± 10% which cane considered a reasonable value. The values of the parametersbtained as a result of the calculation procedure are collected in

able 1.

As a further test of the validity of the obtained parameters,he estimated values were used to predict the quasi-steady-stateurrent densities for the Ni–15%Cr and Ni–20%Cr in 0.1 M

M. Bojinov et al. / Electrochimica

Table 1Kinetic and transport parameters for the Ni-15%Cr and Ni–20%Cr alloys in0.1 M Na2B4O7 at 300 ◦C determined by the calculation procedure outlined inthe text

Parameter Ni–15%Cr Ni–20%Cr

1011 De (cm2 s−1) 1.2 1.21015 De (cm2 s−1) 0.7 1.01017 DM (cm2 s−1) 4.0 3.01011 k2 (mol cm−2 s−1) 12.0 7.2107k0

4 (cm s−1) 1.7 3.0109 k1 (cm s−1) 17 2.6109k0

3 (mol cm−2s−1) 9.0 1.7b4 (V−1) 3.0 3.4b3 (V−1) 7.5 6.2α 0.91 0.87E

L

Nipt

i

c−IegiAco

p

FfP

•

•

•

•

� (MV cm−1) 0.50 0.60¯

E=0 (nm) 11.1 11.6

a2B4O7 at 300 ◦C measured after completion of the respectivempedance spectra during the same experiments. For the pur-ose, an equation for the steady-state current density analogouso that derived earlier [6] was used:

= 3FKDOk0

2

k04

e−b4αEe−2KL + 2Fk02

+3FKDMk0

3

k01

eb3αE (11)

A rather good correspondence between experimental and cal-ulated data is observed in Fig. 6 for potentials higher than0.5 V, which demonstrates once again the validity of the model.

n order to reproduce the shallow minimum observed in thexperimental data, it was concluded that the reactions involvingeneration, transport and consumption of interstitials should bentroduced to the model, as shown previously for the oxide onISI 316 [6]. However, this was judged to lead to further compli-

ation of the calculation procedure, thus the simplified approachutlined above was preferred at this stage of the investigation.

The following main conclusions can be drawn from thearameter values listed in Table 1:

ig. 6. Dependence of the steady-state current density on potentialor Ni–15%Cr an Ni–20%Cr alloys in 0.1 M Na2B4O7 at 300 ◦C.oints—experimental values, solid lines—best-fit calculation.

•

rtia(f

Acta 52 (2007) 7475–7483 7481

The values of the diffusion coefficients of ionic currentcarriers are in fair agreement with those estimated earlierin high-temperature electrolytes, being significantly smallerthan those determined by us for the oxide on stainless steel[6]. The values of the diffusion coefficient of cation vacan-cies are rather close to a preliminary estimate of the diffusioncoefficient of Cr in Cr2O3 using XPS data obtained in theinitial stages of film growth, as reported earlier [14].The calculations show that the exponential coefficient b3 isconsiderably higher than b4. This result points to a muchbigger influence of the applied potential on the reactionsinvolving cation vacancies. On the other hand, k2 k1, k0

4 �k0

3 and also DO � DM. These inequalities may be interpretedby the hypothesis that film growth is probably controlled bythe ability of injection of oxygen vacancies at the inner inter-face. Moreover, cation vacancies injected at the outer interfaceare retained in the film because of their slower transport andplay the role of acceptor states, ensuring the apparent p-typeconductivity of the barrier sublayer at potentials higher thanca. −0.6 V. Such a result is at variance to the results obtainedpreviously on austenitic stainless steel in the same tempera-ture range [6].The mean electric field strength is ca. an order of magnitudesmaller at high-temperatures than at room temperature. Thehigh-temperature values are well in accordance with the valid-ity of the low-field limit of the transport equations confirmingthe basic hypothesis of the model in that temperature range.However, the field strength in the oxide on the nickel basedalloys is significantly higher than that on stainless steel, whichcan be taken as a proof of the different nature of the barriersublayer of the film on the two types of materials.The values given in Table 1 for De most probably representthe minimum value of the mobility in the depletion layer ofthe semiconductor phase in the oxide, which is the primarybarrier for the transport of electronic current carriers. A tenta-tive explanation of the low values of the diffusion coefficientof electronic current carriers could be that the mobile ionicdefects in the high-temperature films act as electron traps,immobilising and later releasing electronic carriers that, as aresult, spend a long time in such traps.The main differences between the parameter sets calculatedfor the oxide films on the two alloys are found between the val-ues of the rate constants at the alloy/film interface. This maypoint out to an important role of the structure of this inter-face for the electrochemical behaviour and oxidation rate ofnickel-based alloys and is in line with the previously advancedhypothesis of the blocking effect of Ni on the diffusion of Crfrom the alloy to the film during the initial stages of oxidationof Alloy 600 in PWR water [13,15].

In an attempt to reproduce the film thickness versus timeelationship for Alloy 600 on the basis of the presented model,he obtained values for k2 and �E were used as initial estimates

n Eq. (9). An L(t = 0) value of 1.5 nm has been employed inccordance to the experimental data reported in Ref. [13]. Eq.9) was fitted to the experimental thickness vs. time relationshipor Alloy 600 at 325 ◦C in simulated PWR water (data taken

7482 M. Bojinov et al. / Electrochimica

Fig. 7. Thickness of the oxide film vs. oxidation time for Alloy 600 at 325 ◦Cin simulated PWR water (data taken from Refs. [13,15] for oxidation times upto 400 h). The data for the long-term (1000–8000 h) oxidation of Alloy 600 at2T

ftiweiFslbaiNTcci6ts

Tavooafatiaoett

owhtbatgvet

5

tetpraapt6eocoaottm

A

aFLp

A

b

b

c

c (x) concentration profile of oxygen vacancies in the film

60 ◦C in simulated PWR secondary side water has been taken from Ref. [24].he solid line is the best-fit calculation according to Eq. (9).

rom Refs. [13,15] for oxidation times up to 400 h). Data forhe long-term (1000–8000 h) oxidation of Alloy 600 at 260 ◦Cn simulated PWR secondary side water taken from Ref. [24]ere also included. The thickness of the oxide in short term

xperiments (up to 10 min) has been estimated from the XPSntensities supporting a layer model similar to that shown inig. 1 (an outermost layer of Ni(OH)2 crystals covering a barrierublayer of Cr2O3) [13]. On the other hand, the thickness atonger exposure times (up to several thousands of hours) haseen estimated from XPS depth profiles by computing the deptht which the oxygen signal of the oxide form has reached half ofts maximum value [15,24]. This depth estimation is validated byRA measurements of the oxygen in the whole passivation layer.he mean thickness of the films on Ni–15%Cr at a potentialorresponding to the hydrogen line at 300 ◦C and a pH of 9.1omputed from the model calculations discussed above is alsoncluded. It is rather similar to those obtained by XPS on Alloy00 which can be taken as an indication that the films on thewo materials in the two high-temperature electrolytes are alsoimilar.

The calculational result is shown in Fig. 7 with a solid line.he quality of the fit is reasonable enough to claim that the modelgrees with the experimental data. The only parameter that wasaried was the rate constant of the reaction of generation ofxygen vacancies at the alloy/film interface, the estimated valuef which was a factor of 4 larger than that for Ni–15%Cr. Suchchange is considered to be reasonable taking into account the

act that the data stem from the oxidation of a different alloy indifferent medium at a different temperature. It is worth noting

hat in a previous paper [15] the experimental data up to 400 hndicated a plateau in the thickness versus time relationship (seelso Fig. 7) which was well reproduced by an empirical equationf the same form as Eq. (9). However, a somewhat different

quation was employed to reproduce the data for short oxidationimes (up to 10 min) [25]. On the other hand, the evolution of thehickness of the film at short oxidation times, as well as the form

c

Acta 52 (2007) 7475–7483

f the curve at very long oxidation times (data from Ref. [24]) areell reproduced by the present model, which lends support to theypothesis that the oxidation process of the alloy is controlled byhe transport properties of a very thin barrier-like layer, showny XPS to be Cr2O3 up to at least several thousands of hours (cf.lso [13,15]). Another parameter that has been estimated fromhe fit shown in Fig. 7 is the transfer coefficient of the reaction ofeneration of oxygen vacancies at the alloy/film interface α2. Aalue of 0.39 has been obtained, which is once again a reasonablestimate and supports the assumptions used in the adaptation ofhe model to the oxidation of nickel-based alloys in PWR water.

. Conclusions

In this paper, the MCM for passive films has been adapted tohe oxide film growth on nickel-based alloys in high-temperaturelectrolytes and a procedure to determine the main kinetic andransport parameters has been devised. With the obtained set ofarameters, the steady-state current density and the impedanceesponse during oxide film growth on Ni–15%Cr and Ni–20%Crlloys in 0.1 M Na2B4O7 at 300 ◦C have been predicted inbroad potential range. In addition, a self-consistent set of

arameter values has been found to reproduce successfully thehickness versus time relationship determined by XPS for Alloy00 in simulated PWR water for oxidation times up to sev-ral thousands of hours. This result supports the view that thexidation of nickel-based alloys in high-temperature water isontrolled by generation, solid-state transport and consumptionf point defects in a barrier sublayer (which would be Cr2O3ccording to XPS data) which thickness is much smaller than thatf the whole film, especially after long exposure times. In turn,his film most probably controls the susceptibility of the alloyo localised corrosion and the initiation rate of stress corrosion

odes.

cknowledgements

M.B. and P.K. are grateful to the Finnish Ministry of Tradend Industry, the Radiation and the Nuclear Safety Authority,inland for the funding of this work within the frames of theWROXI project as a part of SAFIR, Safety of nuclear powerlants—Finnish national research programme 2003–2006.

ppendix A. Nomenclature

j exponential coefficients of the interfacial reactions(j = 1–4) (V−1)

redox exponential coefficient of the redox reaction at thefilm/electrolyte interface (V−1)

M (x) concentration profile of cation vacancies in the film(mol cm−3)

O(mol cm−3)

¯O (L) steady-state concentration of oxygen vacancies at thealloy/film interface (mol cm−3)

imica

c

C

CCCDDDEE

JJjk

k

k

LN

NORRR

R

TVVZZZ

Gα

α

ε

ε

φ

φ

σ

σ

τ

τ

Ω

ω

R

[[[

[

[[

[

[[[

[[

[[

J. Electroanal. Chem. 504 (2001) 29.[24] S.E. Ziemniak, M. Hanson, Corros. Sci. 48 (2006) 498.

M. Bojinov et al. / Electroch

¯M (L) steady-state concentration of cation vacancies at thealloy/film interface (mol cm−3)capacitance of the film or of the space charge layer(F cm−2)

F/E capacitance of the film/electrolyte interface (F cm−2)rCr chromium ion position in the oxide film latticerM chromium atom in the alloy phasee diffusion coefficient of electrons (cm2 s−1)M diffusion coefficient of cation vacancies (cm2 s−1)O diffusion coefficient of oxygen vacancies (cm2 s−1)

applied potential (V)� electric field strength at the metal/film interface

(V cm−1)M flow of cation vacancies in the film (mol cm−2 s−1)O flow of oxygen vacancies in the film (mol cm−2 s−1)

imaginary unitj rate constants of the interfacial reactions (i = 1–4)

(mol cm−2 s−1 or cm s−1)0j rate constants of the interfacial reactions (i = 1–4) at

E = 0 (mol cm−2 s−1or cm s−1)redox rate constant of the redox reaction at the film/electrolyte

interface (mol cm−2 s−1)thickness of the film (cm)

i′IICr nickel ion in a chromium position in the oxide filmlattice

im nickel atom in the metal phaseO oxygen position in the oxide film

universal gas constant (8.314 J mol−1 K−1)el uncompensated electrolyte resistance (� cm2)ct charge transfer resistance at the alloy/film interface

(� cm2)F/E charge transfer resistance at the film/electrolyte inter-

face (� cm2)absolute temperature (K)

O•• oxygen vacancy in the oxide film lattice

′′′Cr cation vacancy in the oxide film lattice

total impedance (� cm2)f impedance of the bulk film (� cm2)F/E impedance of the film/electrolyte interface (� cm2)

reek letterspolarizability of the film|electrolyte interface

i transfer coefficients of the interfacial reactions (i = 1–4)dielectric constant of the film

0 dielectric permittivity of vacuum (8.85 × 10−14

F cm−1)

M/F local potential drop at the metal/film interface (V)F/E local potential drop at the film/electrolyte interface (V)O Warburg constant for the transport of oxygen vacancies

(� cm2 s−1/2)

[

Acta 52 (2007) 7475–7483 7483

M Warburg constant for the transport of cation vacancies(� cm2 s−1/2)

O time constant of the transport impedance of cation oxy-gen vacancies (s)

M time constant of the transport impedance of cationvacancies (s)molar volume of the oxide film, 29.2 cm3 mol−1

(Cr2O3)angular frequency (rad s−1)

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