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Electrochimica Acta 53 (2007) 1737–1742

Interpretation of EIS data on passive steel surfaces in aqueous sulfuricacid solution in terms of carrier migration and recombination

Gabor Nagy 1, Zsolt Kerner 1, Robert Schiller ∗Central Research Institute for Physics, Atomic Energy Research Institute, P.O.B. 49, Budapest H-1525, Hungary

Received 22 March 2007; received in revised form 9 August 2007; accepted 9 August 2007Available online 15 August 2007

bstract

A model is proposed for the description of the interplay of charge carrier migration, effected by external voltage and combination within very thinassive oxide layers. This theory is used to describe the electrochemical impedance spectra of passive stainless steel in sulfuric acid, a prototypef such surfaces. A simple Rs − Zmr||Rp circuit is considered with Zmr accounting for dispersion and Rp for the Faraday process. Zmr is based oncontinuous time random walk (CTRW) expression and it is seen to be independent of voltage. Rp shows a strong voltage dependence which cane understood in the negative voltage range as a Butler-Volmer-Erdey-Gruz process, whereas in the positive range it can be described in terms of

unneling across the oxide. Curve fitting results in parameters characteristic to oxide structure and process kinetics. All fitted quantities are seen toe realistic with the exception of charge carrier density which turned out to be much too high, indicating that mobility might not obey the classical,requency-independent law.

2007 Elsevier Ltd. All rights reserved.

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eywords: Impedance; Passive oxide; Migration; Random walk; Recombinatio

. Introduction

In a recent paper [1], we presented the similarities and differ-nces in the passivation and surface oxidation of 08H18N10T,ISI304 and AISI316L stainless steels, focusing mainly on the8H18N10T steel, as this is the main structural material of therimary circuits of the Russian type pressurized water reactorsVVER), among them the Paks NPP in Hungary. With the help ofccelerated corrosion tests of electrochemical experiments, weanked these materials according to their resistance against uni-orm corrosion. In the present paper, we aim to discuss the chargearrier transport in the passive surface oxide layer by analyzinglectrochemical impedance spectra of the 08H18N10T steel inulfuric acid solution.

The protective layers on the surfaces of valve metals are semi-

onducting oxides. The oxides of iron and its alloys are n-typeaterials. Nevertheless, electric conduction across the layer has

wo parallel modes. One is effected by electrons (n) or holes

∗ Corresponding author. Tel.: +36 1 3922270; fax: +36 1 3922684.E-mail address: schiller@sunserv.kfki.hu (R. Schiller).

1 ISE member.

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013-4686/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.oi:10.1016/j.electacta.2007.08.022

p), the other by ions or ion vacancies. This broad picture, noweing part of common wisdom [2], is the basis of a numberf detailed models and theories of charge motion and chemicalransformations in the layer and at the solid/liquid interface.

Macdonald [3] specified the type of ion transport devising theoint defect model (PDM) by which he stated that the passivexide layer on a valve metal surface is “a cation conductor that,nder anodic polarization conditions, involves the irreversiblejection of cations from the film”. He succeeded to describe thempedance spectra obtained for Ni-oxide surfaces in terms ofDM if also a potential dependent capacitance was added inarallel with the interfacial impedance. Photoelectrochemicaltudies performed by Goosens et al. [4] indicate that holes andxygen ion vacancies migrate simultaneously within the layer.

Simultaneous electron and ion transport within the oxideayer was observed also in Zr(Nb)-oxides by Vermoyal et al.5]. Schultze and Macagno [6] discussed the kinetics of cathodiceduction of Fe3+ at a passive Ta-oxide electrode solely in termsf semiconductor processes finding that both tunneling and clas-

ical transition take place through the solid/liquid interface.erreira et al. [7] investigated the semiconducting properties ofassive layers on stainless steels both by Mott-Schottky analy-is and photoelectrochemical response, methods which enabled

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738 G. Nagy et al. / Electrochim

he authors to elucidate the heterogeneous structure of the oxide.assel and Lohrengel [8] considered semiconducting, tunneling

nd proton conduction for the understanding of cathodic break-own of thin Al-oxide films stating, among other aspects, that theeparation of electronic and protonic conduction usually poses aroblem. They attribute the breakdown to an autocatalytic pro-ess which is started by a rearrangement of the Debye layer athe solid/liquid interface and involves hydrogen evolution. Dro-owska et al. [9] performed impedance studies on stainless steeln near-neutral aqueous solutions stating that oxide formation isue to both charge and mass transfer, i.e. electronic and ionicrocesses. Schultze and Lohrengel [2] gave a detailed accountoth on the properties of and processes in passive films pay-ng special attention to the distinction between ion and electronransfer processes.

There are several competing theories for the mathematicalescription of the impedances of surface oxides. Young [10]ealt with Nb-oxide films of thicknesses greater than 2 nm andxplained his findings in terms of an exponential distribution ofhe local specific resistivities of the oxide. That is, he considered

series of elementary, parallel RC circuits the time constantsf which varied as the local resistivities did. He did not specify,owever, that the change in resistivity is due to a variation of den-ity or mobility of the charge carriers. Gohr et al. [11] made usef Young’s impedance as a constituent of an equivalent circuitor the impedances of porous oxide layers on Zircaloy surfaces.

Fromhold and Cook [12] proposed and discussed a differen-ial equation for charge motion in a periodic potential field of arystal under the aggregate effect of a stationary concentrationradient and of an external voltage. The total amount of mobileharge carriers were taken to be constant, i.e. neither carrierormation nor recombination were considered. The steady con-entration gradient was thought to be effected by constant ratesf injection at one side and of absorption (chemical reaction) athe other side of the crystal layer.

A simplified but time dependent version of this differentialquation was used by Guo et al. [13] for modeling one of sixlements in the equivalent circuit for the impedances of passivelms on carbon steel in nitric acid solutions. This element, whichas seen to play a role in the very low frequency domain, turnedut to be similar to the Warburg impedance, i.e. the phase angleid not exceed 45◦.

Also the work of Bojinov et al. [14] on passive films on iron,hromium and their alloys was written in the spirit of Fromholdnd Cook’s ideas considering also the point defect model. Theuthors found a coupling between the ionic defect structure andhe electronic conduction of the metal/oxide/electrolyte system.his is brought out by their finding that the electrode impedance

s to be described as a parallel circuit made of the electron andon impedances.

Most of the above models and calculations rest upon the clas-ical ideas of charge migration and particle diffusion. It was,owever, demonstrated that these tenets do not hold for all the

ystems studied. Nyikos and Pajkossy [15–17] considered theffect of fractal geometry of the solid/liquid interfaces both oniffusion and migration suggesting a physical understanding ofhe almost ubiquitous constant phase element (CPE), the admit-

taop

cta 53 (2007) 1737–1742

ance of which is given as YCPE = σ(iω)α with ω being the angularrequency and i = √−1; the authors expressed α by the frac-al dimension. For a Warburg impedance, α = 1/2 holds and theoefficient is given as σ = √

2D0c(ze)2/kBT , where D0 meansiffusion coefficient, c the concentration, z the number of ele-entary charges of the diffusant, e the elementary charge, kB

he Boltzmann constant and T is the absolute temperature.Bisquert and Compte [18] discussed diffusion impedance

n thin layers considering three types of anomalous diffusion:a) Fick’s Law is replaced by a fractional differential which iselated to continuous time random walk theory; (b) Fick’s Laws replaced by a fractional integral which is related to a non-

arkovian behaviour of random walk; (c) Fick’s Law is used ints classical form but the number of diffusing particles is not con-erved. It seems to be interesting that the phase angles in most ofhe diffusion types and boundary conditions do not exceed 45◦ ifhe frequency is not too low. The non-conservation assumptioneems to be quite rare among impedance theories; Gerischer’s19] and Bisquert’s [20,21] impedances might be considered asistant relatives of this idea.

More recently, we suggested a random walk model [22] whichccounted for both diffusion and charge carrier recombination,.e. in this case the number of diffusants was not conserved.or the diffusion coefficient, D0, the Einstein-Smoluchowskiquation was taken, as

0 = s2

6τD(1)

here s2 is the square average jump length of the charge car-ier and τD is the waiting time between two successive jumps.his diffusion coefficient is obviously independent of time, i.e.oes not show any dispersion (frequency dependence). Theharge carrier recombination was thought to obey quasi-first-rder kinetics with a rate coefficient

s = 1

τC. (2)

These elementary expressions were combined in terms of theontinuous time random walk theory [23,24] which rests on thedea that the waiting time between two successive jumps is notonstant but is given by a distribution function. We could showhat although τD is taken as constant the effect of recombinations such that the interplay between D0 and ks can be given in termsf a dispersive diffusivity of the form

= D0τ

τD

1 + iωτD

1 + iωτ, (3)

here the symbol 1/τ = 1/τD + 1/τC is used.We combined this equation with the usual Warburg expres-

ion of diffusion impedance and applied to the impedancepectra of hydrothermally grown, thick Zr-oxides (thicknesses ofhe order of 1 �m). Our assumptions were that: (a) the impedancepectra can be attributed to charge carrier motion and reac-

ion within the oxide layer and (b) the oxide layer is thicknd the charge carrier density high enough to screen the bulkf the oxide from the external voltage, hence the carrier trans-ort is purely diffusive. If these assumptions hold one finds the

ica Acta 53 (2007) 1737–1742 1739

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G. Nagy et al. / Electrochim

iffusion-recombination admittance for unit area, Ydr as

dr = B

(iωτ

1 + iωτD

1 + iωτ

)1/2

(4)

ith B = [c(ze)2/kBT ](D0/τD)1/2 = [c(ze)2/√

6kBT ](lrms/

D), where lrms is the root-mean-square displacement of thearticle performing random walk.

The situation is much different with extremely thin oxideslayer thicknesses in the order of several nm), layers which formharacteristically on stainless steel surfaces at room temperature25,26]. Here, the main mode of charge transfer is migrationnder the influence of the external field and the admittance forlayer of unit area and of thickness l is given as

/Z = Y = ecμ/l, (5)

here μ is the mobility and c is the charge carrier concentration.he Nernst-Einstein equation, i.e. the proportionality betweenand μ,

= eD

kBT(6)

olds even if the transport is dispersive. Hence, if migration andecombination proceed simultaneously in a thin oxide layer, Eq.3) can be combined with Eqs. (5) and (6) in order to obtain theayer admittance finding

1

Zmr= Ymr = ec

lμ0

τ

τD

(1 + iωτD

1 + iωτ

)(7)

We will try to apply this expression to our experimental data.s far as the nature of the charge carrier is concerned we haveacdonald’s [3,4] point diffusion model (PDM) in mind. Thus,e think, also in line with our previous publication on zirconiumxide [23] that positive ionic charge carriers, presumably oxygenon vacancies, are drifted by the external potential gradient andhese carriers can recombine with the majority charge carriersf the n-type material, electrons, in a pseudo-first-order process.

. Results and discussion

A series of comparative studies regarding the electrochem-stry of different stainless steels, used in PWRs, was completedy us [1] lately. Since we could show that all those materialsehave similarly from corrosion point of view, in this commu-ication we will deal only with the steel type 08H18N10T in itsassive state. The composition of this alloy is 0.07% C, 0.31%i, 0.35% P, 1.43% Mn, 17.85% Cr, 10.42% Ni and 0.66% Ti.he practical importance of this alloy lies with its use in theussian made nuclear power stations type VVER, four units ofhich are in operation also in Hungary.The experiments were performed according to the standards

STM G5-95, G1-03. Deaerated, 0.5 M sulfuric acid solution

as used as electrolyte with platinum mesh counter electrode

nd standard calomel reference electrode; disk-shaped steel elec-rodes of 2.7 cm2 contact area were used in the so-called hanging

eniscus configuration. The measurements were performed

w

b

ig. 1. Cyclic voltammogram measured in 0.5 M sulfuric acid with a scan ratef 100 mV/s.

ith an Autolab PGSTAT30 potentiostat. Further experimentaletails are given in ref. [1].

A cyclic voltammogram, taken with a scan rate of 100 mV/s,s given in Fig. 1, showing the steel to be in the passive staten the +400 to −400 mV (SCE) range. Nevertheless, the insethows that the current starts to increase abruptly with decreasingotential from around −200 mV.

Impedance spectra were measured between E = +400 and500 mV (SCE) in the frequency range 1–105 Hz. Two exam-

les of the observed spectra, given in terms of the real part,′, the imaginary part, Y′′, and phase angle, ϕ, of the admit-

ance are given in Figs. 2 and 3. No significant changes in thempedance spectra were observed beyond the applied frequencyimits. Nevertheless, at much lower frequencies some Faradayrocesses may be revealed. This problem, however, was beyondur present task.

That is also the reason why a comparatively high scan rateas applied in the CV measurement.For the interpretation of impedance spectra, we consider

simple three-element equivalent circuit of –Rs − (Zmr||Rp)–,here Rs represents the ohmic resistance of the electrolyte,p corresponds to a Faraday process, whereas Zmr accounts

or charge recombination and migration under the effectf the external field as described in connection with Eqs.5)–(7).

Hence, the admittance can be calculated by the expression

1

Z − Rs= Ymr + 1

Rp

=(

bω2ττD + 1

ω2τ2 + 1+ 1

Rp

)+ i

(bω(τ − τD)

ω2τ2 + 1

)

= Y ′ + iY ′′ (8)

ith

= ecμ0τ

lτD= c

l

e2s2

6kT

τ

τ2D

and ϕ = arctgY ′′

Y ′ . (9)

1740 G. Nagy et al. / Electrochimica Acta 53 (2007) 1737–1742

Y′′, ϕ

bsbabaRpia1v

ettrl

of magnitude. The semilogarithmic plot, log(1/Rp) versus E isgiven in Fig. 4. Its slope corresponds to a cathodic process.A sharp break can be seen at around −200 mV, i.e. near the

Table 1Fitted parameters of the equivalent circuit –Rs − (Zmr||Rp)– by Eq. (7) at anumber of potentials, E

E (mV, SCE) b (� cm−2) τD (s) τ (s) Rp (� cm−2)

−500 1.1 × 10−4 6.18 2.4 × 10−3 160−400 1.2 × 10−4 6.16 2.4 × 10−3 659−300 1.3 × 10−4 6.20 2.4 × 10−3 1,992−200 1.3 × 10−4 6.34 2.6 × 10−3 5,454−100 1.2 × 10−4 6.98 2.6 × 10−3 6,804

0.0 1.4 × 10−4 6.22 2.8 × 10−3 7,425

Fig. 2. Real and imaginary admittances together with the phase angle (Y′,

The series resistance, Rs is always in the order of 1 �. It cane taken from the high frequency limit and it can be directlyubtracted. Thus, there are four parameters to be fitted: τD, τ,and Rp. We performed independent fitting procedures for Y′

nd Y′′ finding the parameters τD, τ and b virtually the same foroth the real and imaginary part. These parameters were useds fixed in the curve fitting for ϕ, finding the last free parameter,p. This treatment was inevitable because the fitting of the realart turned out to be very insensitive to Rp whereas the imag-nary part does not contain this element. The errors of τD, τ

nd b were about 1–3% whereas the error of Rp was as high as0–50% and the relative uncertainty increased with increasingoltage.

A list of the fitted parameters is given in Table 1. The param-ters which characterize the dissipative element, Ymr, are found

o be independent of the potential. This is in accordance withhe expectations based on the model: neither the mobility nor theeactivity of the ionic charge carriers within the passive oxideayer are thought to depend on the potential.

) taken at E = 400 mV (SCE). Curves fitted according to Eqs. (8) and (9).

The parallel ohmic element, however, varies some two orders

100 1.5 × 10−4 6.00 2.8 × 10−3 8,532200 1.4 × 10−4 6.26 2.6 × 10−3 9,774300 1.4 × 10−4 6.06 2.4 × 10−3 13,660400 1.1 × 10−4 6.78 2.2 × 10−3 14,800

G. Nagy et al. / Electrochimica Acta 53 (2007) 1737–1742 1741

Fig. 3. Real and imaginary admittances together with the phase angle (Y′, Y′′, ϕ)

Fig. 4. Logarithm of the parallel resistance as a function of the electrode poten-tial.

pdccsts

cnlbai

I

lt

taken at E = −500 mV (SCE). Curves fitted according to Eqs. (8) and (9).

oint where the cathodic current shows an abrupt increase withecreasing voltage, as given in Fig. 1. Below this voltage, onean attribute the current to a Butler-Volmer-Erdey-Gruz pro-ess [27]. Indeed, along the steeper branch of the Tafel plot thelope is β = −αe/(2.3 kBT) ≈ −0.0051 mV−1, where α denoteshe transfer coefficient. By this one finds α ∼= 0.3, a not unrea-onable value for a conventional electrode process.

The slope of the more flat anodic branch, above −200 mV,annot be understood in the same line. However, it can be ratio-alized in terms of the tunnel effect, i.e. electron migration fromiquid to solid across the oxide layer. In case of a rectangulararrier, the tunneling current, It, as a function of the height, U,nd width, λ, of the barrier, with m denoting the electron mass,s given [28] as

t = KU exp[−(2/h)λ√

2mU]. (10)

Let us take U = e(E + E0), where E0 is defined by the Fermievel in the metal and by the standard oxidation/reduction poten-ial of the electron acceptor and let us identify Rp with the

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742 G. Nagy et al. / Electrochim

ifferential resistance of the tunneling current. Further let usssume that the barrier width is equal with the thickness of thexide layer, i.e. λ = l. Thus, one finds

−1p = dIt

dE= Ke(1 − 0.5lγ

√E + E0) exp(−lγ

√E + E0),

(11)

here the notation γ = (2/h)√

2me is used.By fitting Eq. (11) to the points above −200 mV in Fig. 4,

ne obtains l ∼= 0.6 nm and E0 ∼= 2.5 V. Since no passivation peakan be observed on the CV curve (Fig. 1), one can reasonablyssume that the layer thickness, l, is constant, independently ofhe applied voltage throughout the experiments. One would per-aps expect a somewhat higher value for the layer thicknessnd a lower one for E0, nevertheless, given the approxima-ions involved by the use of Eq. (10), these values seem to becceptable.

The time constants, being in reasonable agreement withimilar data obtained for Zr-oxide [23], show a slow charge car-ier migration and comparatively fast recombination, the lattereing characterized by the pseudo-first-order rate coefficient,s = 1/τC ∼= 1/� ∼= 4 × 102 s−1.

By making use of Eq. (9) and the fitted parameters τ, τD andone can evaluate the combination A = l/cs2, a quantity of area

imension. One finds A ∼= 5 × 10−19 cm2, a value which giveshe possibility to calculate s and c, with l already obtained above,n an intuitively direct way. The resulting charge carrier concen-ration, c is in the order of 1025 cm−3, a figure much too high. Theeason of this difficulty lies most probably with the oxide layero be inhomogeneously structured, a fact which was pointedut several times [8,29]. Thus, the assumption of the classicalaw of migration may be an oversimplification. A similar effectas already considered in a previous publication [30].

. Conclusions

A model is proposed for the description of the interplay ofharge carrier migration and recombination within very thinassive oxide layers. Electrochemical impedance spectra of pas-ive stainless steel samples, immersed into 0.5 M sulfuric acid,ere discussed in terms of this modification of the continuous

ime random walk theory. An equivalent circuit of two paral-el elements was suggested: one of them, Zmr, accounted forhe interplay between charge carrier recombination and motionithin the oxide layer under the sole influence of the exter-al voltage, whereas the other one was a non-dispersive ohmicesistor, Rp.

The log(1/Rp) versus voltage plot resulted in two straight linesf different slopes intersecting at about −200 mV (SCE), a volt-ge which coincided with a sharp dc current increase observedy cyclic voltammetry. The negative voltage branch was treated

n terms of the Butler-Volmer-Erdey-Gruz equation and resultedn a transfer coefficient of about 0.3. The positive voltage branchas attributed to tunneling current; here parameter fitting ren-ered an oxide layer thickness of about 0.6 nm.

[

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cta 53 (2007) 1737–1742

The analysis revealed that Zmr, the element which accountedor the dispersion, was independent of voltage. The time con-tants for random walk jumps and chemical reactivity, obtainedy parameter fitting, were in the expected range. However, theombination of layer thickness, jump length and time constantsesulted in much too high charge carrier concentration values,fact which indicates that use of the classical law of migra-

ion may be an oversimplification, due to the non-homogeneoustructure of the oxide.

cknowledgements

The work was partially supported by the Hungarian Nationalcience Research Fund (OTKA) under the projects nos.049202 and F049839 and by the 6th Framework Program of

he European Union within the PERFECT project under contracto. FI6O-CT-2003-508840.

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47.[5] J.J. Vermoyal, A. Hammou, L. Dessemond, A. Frichet, Electrochim. Acta

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48 (2006) 1307.26] C.-O.A. Olsson, D. Landolt, Electrochim. Acta 48 (2003) 1093.27] A.J. Bard, L.R. Faulkner, Electrochemical Methods, second ed., Wiley,

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28] W. Sacks, Scanning probe microscopy: beyond the Images, in: C.

Joachim, S. Gauthier (Eds.), Les Editions de Physique, Les Ulis, France,1991.

29] J. Hafele, B. Heine, R. Kirchheim, Z. Metallkd. 83 (1992) 395.30] G. Nagy, R. Schiller, Phys. Chem. Chem. Phys. 4 (2002) 791.