Coupling between ionic defect structure and electronic conduction in passive films on iron, chromium and iron–chromium alloys

Electrochimica Acta 45 (2000) 2029 – 2048 www.elsevier.nl/locate/electacta

Coupling between ionic defect structure and electronic conduction in passive films on iron, chromium and iron–chromium alloys M. Bojinov a,b,*, G. Fabricius b, T. Laitinen a, K. Ma¨kela¨ a, T. Saario a, G. Sundholm b a

b

VTT Manufacturing Technology, Kemistintie 3, P.O. Box 1704, FIN-02044 VTT, Espoo, Finland Laboratory of Physical Chemistry and Electrochemistry, Helsinki Uni6ersity of Technology, P.O. Box 6100, FIN-02015 HUT, Espoo, Finland Received 2 July 1999; received in revised form 12 November 1999

Abstract A quantitative kinetic model is presented for the steady-state passive films on Fe, Cr and Fe – Cr alloys. It emphasises the coupling between the ionic defect structure and electronic conduction. According to the model, the passive film can be represented as a heavily doped n-type semiconductor– insulator-p-type semiconductor junction. At low potentials in the passive state, the positive defects injected at the metal/film interface play the role of electron donors. At high positive potentials, the negative defects injected at the film/solution interface play the role of electron acceptors. At sufficiently high positive potentials the concentration of these ionic defects and corresponding electron holes reaches high enough values for the film to transform into a conductor. This enables transpassive dissolution of Cr and oxygen evolution on the film surface. Equations for the electronic conductivity of the passive film, as depending on the concentration of point defects, are derived. The proposed model is compared with experimental data obtained for pure Fe, pure Cr, Fe–12%Cr alloy and Fe – 25%Cr alloy passivated in 0.1 M borate solution (pH 9.2) using rotating ring-disk voltammetry, photocurrent and impedance spectroscopy and in situ dc resistance measurements by the contact electric resistance (CER) technique. © 2000 Elsevier Science Ltd. All rights reserved. Keywords: Ionic defects; Electronic conduction; Passive film; Impedance spectroscopy; Contact electric resistance technique

Nomenclature a bi B1, B2 ci(L) ci(0) ci(x)

atomic jump distance (cm) Tafel coefficients of the interfacial reactions (i = 1 – 4, 1i, 3i) (V − 1) constants in Eq. (23) (mol cm − 3) concentration of interstitial cations at the metal/film interface (mol cm − 3) concentration of interstitial cations at the film/solution interface (mol cm − 3) concentration profile of interstitial cations in the film (mol cm − 3)

* Corresponding author. Fax: + 358-9-456-5875. E-mail address: [email protected] (M. Bojinov) 0013-4686/00/$ - see front matter © 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 4 6 8 6 ( 9 9 ) 0 0 4 2 3 - 5

2030

cM(L) cM(0) cM(x) cmin cO(L) cO(0) cO(x) Dc(x, v) C CF/S CM/F De Di D %i DM D %M DO D %O d E E F Ji JM JO j ki k 0i K31, K1,i 3,i L m M2i + MIII M OO R1, R2 x V2O+ V3M− Z Ze Zf ZF/S Zion ZM/F aF/S o o0 fM/F fF/S f 0F/S fF me rd r0 se se(x)

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concentration of cation vacancies at the metal/film interface (mol cm − 3) concentration of cation vacancies at the film/solution interface (mol cm − 3) concentration profile of cation vacancies in the film (mol cm − 3) minimum defect concentration in the film (mol cm − 3) concentration of oxygen vacancies at the metal/film interface (mol cm − 3) concentration of oxygen vacancies at the film/solution interface (mol cm − 3) concentration profile of oxygen vacancies in the film (mol cm − 3) complex amplitude of the concentration (mol cm − 3) capacitance of the film or of the space charge layer (F cm − 2) capacitance of the film/solution interface (F cm − 2) capacitance of the metal/film interface (F cm − 2) diffusion coefficient of electrons (cm2 s − 1) diffusion coefficient of interstitial cations (cm2 s − 1) apparent diffusion coefficient of interstitial cations (cm2 s − 1) diffusion coefficient of cation vacancies (cm2 s − 1) apparent diffusion coefficient of cation vacancies (cm2 s − 1) diffusion coefficient of oxygen vacancies (cm2 s − 1) apparent diffusion coefficient of oxygen vacancies (cm2 s − 1) relative penetration depth of the conductivity profile in the Young’s model applied potential (V) electric field strength (V cm − 1) Faraday constant (96487 C mol − 1) flux of interstitial cations in the film (mol cm − 2 s − 1) flux of cation vacancies in the film (mol cm − 2 s − 1) flux of oxygen vacancies in the film (mol cm − 2 s − 1) imaginary unit rate constants of the interfacial reactions (i = 1 – 4, 1i, 3i) (mol cm − 2 s − 1) standard rate constants of the interfacial reactions (i = 1 – 4, 1i, 3i) (mol cm − 2 s − 1) constants in Eqs. (13a) and (13b) thickness of the film or of the space charge layer (cm) metal atom in the metal phase divalent metal interstitial in the anodic film trivalent metal in a metal position in the anodic film oxygen in an oxygen position in the anodic film constants in Eq. (23) (cm − 1) distance within the film (cm) oxygen vacancy in the anodic film metal vacancy in the anodic film total impedance (V cm2) impedance due to electron transport through the film (V cm2) impedance of the passive film (V cm2) impedance of the film/solution interface (V cm2) impedance due to ion transport through the film (V cm2) impedance of the metal/film interface (V cm2) polarizability of the film/solution interface dielectric constant of the film dielectric permittivity of vacuum local potential drop at the metal/film interface (V) local potential drop at the film/solution interface (V) local potential drop at the film/solution interface at E= 0 (V) potential drop in the film (V) mobility of electronic charge carriers (cm2 V − 1s − 1) parameter in Eq. (19) (V cm) parameter in Eq. (19) (V cm) electronic conductivity (V − 1 cm − 1) electronic conductivity profile in the film (V − 1 cm − 1)

M. Bojino6 et al. / Electrochimica Acta 45 (2000) 2029–2048

t v

2031

time constant in the Young’s model (s) angular frequency (rad s − 1)

1. Introduction The resistance of metallic construction materials to both generalised and localised corrosion in industrial environments is most often in correlation with the transport of matter and charge through the thin passive films formed on metal surfaces. Because transfer of matter through anodic films can be described in terms of defect transport, understanding the mechanism of generation and transport of ionic and electronic defects is of paramount importance to corrosion prevention. Quantitative correlations between the electronic and ionic transport properties of passive films, their composition and structure are yet to be established. Recently, the chemical and crystallographic nature of anodic layers on iron, chromium and iron–chromium alloys has been revealed using spectroscopic techniques based on synchrotron radiation and scanning probe microscopy [1–7]. This progress and the close parallelism between the anodic films on these materials and thin oxide films prepared by sputter deposition [8–11] has opened new possibilities to the search for such correlations. Particularly, it has been demonstrated that anodic layers on iron have a spinel-like nanocrystalline structure [3]. Thus, the well-developed theories of point defects in crystalline materials can be used, albeit with some caution, to describe the transport properties of thin passive films on metals and alloys. Concerning the conduction mechanisms in passive films on metals, much modelling work has been done to disclose the electronic (or semiconductor) properties of the films [12–16]. Also, the growth and dissolution kinetics of the films, which is related mainly to the ionic transport through them, has been a subject for several studies [17–24]. The prevailing contemporary view depicts the steady-state passive film as a thin highly doped semiconductor layer with multiple donor/acceptor levels or a continuous distribution of donor/acceptor states in the apparent band gap. The donor/acceptor density varies both with the applied potential and the distance within the film. It seems to be a function of the chemical composition of the oxide and its crystallographic defect structure. The paramount role of crystallographic point defects in the growth mechanisms of compact anodic oxide films on metals has been discussed in detail in the point defect model (PDM) developed by Macdonald and co-workers during the last two decades [20–27]. It has also been recognised in the models proposed by Kirchheim [17], Castro [18], Guo et al. [19]. Within the frames of the PDM, an attempt has been made to correlate the transport of point defects (vacancies/inter-

stitials) and the electronic conductivity of anodic films on tungsten and zirconium [26,27]. Recently, the so-called surface charge approach to anodic film growth and dissolution was proposed by one of us [28 – 32]. In this model, the anodic film was represented as a highly doped n-type semiconductor– insulator-p-type semiconductor (n –i –p) junction. This junction is formed due to the injection of electron donors (positive ionic charges, e.g. oxygen vacancies) at the metal/film interface, and electron acceptors (negative ionic charges, e.g. metal vacancies) at the film/solution interface, correspondingly. In analogy to the PDM, concentration profiles of ionic defects are assumed to exist in the steady-state anodic film and govern the distribution of the electronic charge carriers within it [29,30]. On the other hand, as early as in the 1950s, Young [33] explained the impedance versus frequency dependence of niobium oxide films assuming an exponential conductivity profile within the film. The model of Young was suggested to be valid for passive iron in borate buffer by Cahan and Chen [13]. The validity of the Young model for passive films on stainless steels was inferred recently by Hakiki et al. [15,16]. Very recently we demonstrated its applicability to the anodic films on Cr and Fe – Cr alloys in acidic sulphate solutions [34,35]. In the present paper, a kinetic model for the anodic passive film on metals is presented emphasising the relationship between the concentration of ionic defects (vacancies and interstitials) and the transport of electronic charge carriers. It is shown that profiles of electronic conductivity and their dependence on potential can be derived by means of estimating the concentration profiles of ionic defects in the film. The validity of the model for the passive state of Fe, Cr, Fe – 12%Cr and Fe – 25%Cr alloys in neutral borate buffer solution is supported by an extensive set of experimental data. The data were obtained using ring-disk voltammetry, impedance spectroscopy, photocurrent spectroscopy and dc resistance measurements by means of the contact electric resistance (CER) technique.

2. Theory

2.1. A physical model of the metal/film/electrolyte system To present equations for the profiles of ionic defects in the film, the possible reactions and transport phe-

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nomena in the system are defined first. A simplified scheme of the processes taking place in the metal/oxide/ electrolyte system according to the proposed approach is shown in Fig. 1. In order to be able to check the applicability of the model to the films on iron and iron-based alloys, it is assumed below, that the passive film has a spinel-like structure with a certain excess concentration of octahedral metal interstitials and tetrahedral metal vacancies in the metal ion sublattice of the spinel. Further, a small concentration of oxygen vacancies is assumed to be maintained in the oxygen sublattice as well. On the basis of previous results, it seems reasonable to assume that primary passive films (e.g. films that are not formed via deposition from the solution) grow via the motion of oxygen ions inwards [20,21,36]. Consequently, the processes taking place in the film and at the interfaces can be described as follows. At the metal/film interface, the formation of normal metal positions, reactions (I) and (II), the annihilation of tetrahedral metal vacancies (I), injection of octahedral metal interstitials (I,i) and of oxygen vacancies (II) takes place: k1

− m+V3M− “ MIII M +3e k1,i

m “ M2i + +2e− k2

2+ − m “ MIII M +1.5VO +3e

k3

3− 3+ MIII M “ VM + Maq

(III)

k3,i

M2i + “ Mxaq+ + (x− 2)e− 1.5H2O+ 1.5V2O+ “ 1.5OO + 3H+

ks

M2O3 + 6H+ “ 2M3aq+ + 3H2O

In the bulk film, high-field assisted migration of defects is assumed to proceed. At the film/solution interface, metal vacancies are generated via abstraction of metal positions in the oxide lattice, reaction (III), the interstitials are dissolved (either oxidatively or isovalently, III,i) and the oxygen vacancies react with adsorbed water (IV). Reaction (IV), together with reaction (II) and transport of oxygen vacancies outwards (i.e. motion of oxygen ions inwards as indicated above), results in film growth:

= − DO −



2FaE(x, t) DO cO(x, t) sinh RT a

= −Di −



(cO(x, t) 2FaE(x, t) cosh (x RT

Ji(x, t)



(ci(x,t) 2FaE(x, t) cosh (x RT



2FaE(x, t) Di ci(x, t) sinh RT a

JM(x, t) = −DM +

Fig. 1. A scheme of the processes taking place in the metal/anodic layer/solution system according to the proposed model.

(V)

If the reaction of chemical dissolution of the film is very slow, the true steady-state will be reached only after long-term polarisation. However, a very slow chemical dissolution implies a very slow growth of the film to compensate for it. Thus, the system can be regarded as being in a quasi-steady-state in the time scale of the electrochemical measurements described in Section 4. The transport of the point defects (interstitials and vacancies) generated at the interfaces via the electrochemical charge transfer reactions (I) to (IV) is assumed to be governed by the generalised transport equations proposed by Fromhold and Cook [37]: JO(x, t)

(II)

(IV)

In order to preserve the steady-state thickness of the film for a given potential, the growth reaction has to be balanced by chemical dissolution of the film

(I) (I,i )

(III,i )

k4

n

n

n

n (2)



(cM(x, t) 3FaE(x, t) cosh (x RT



(1)

3FaE(x, t) DM cM(x, t) sinh RT a

n

n (3)

A detailed description of the symbols is given in the nomenclature. The subscript M is used for metal vacancies (negative defects) and the subscript i for metal interstitials, respectively. In analogy to the PDM and the surface charge approach, it is assumed that the field strength in the film is homogeneous and independent of potential [20 – 32]. This means that the film thickness is proportional to the applied potential. Thus, the formation of space charges at the interfaces and their influence on the distribution of the potential drop in the system is not taken into account in the present treatment of the ion transport. This is clearly a simplification with regard to the exact mechanism of the electronic processes. However, as the main goal of

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the present paper is to discuss the coupling between the ionic defect structure and the electronic conduction, this assumption seems reasonable as a first approximation. The applied potential is assumed to be distributed as interfacial and bulk potential drops as follows [20–32]: (4)

E=fM/F +EL+fF/S where fF/S =aF/SE+f 0F/S fM/F =(1−aF/S)E−EL−f 0F/S

aF/S is the polarisability of the film/solution interface [20–25] and L is the thickness of the film. As the fluxes of the defects are also related to the kinetics of the interfacial charge transfer reactions, the following boundary conditions are defined: Ji(x =L)= − k1,i

(5a)

Ji(x =0)= −k3,ici(x=0)

(5b)

JM(x=L) =k1cM(x=L)

(5c)

JM(x=0) =k3

(5d)

JO(x =L)= − (3/2)k2

(5e)

JO(x =0)= −(3/2)k4cO(x=0)

(5f )

The steady-state solutions of Eqs. (1)–(3) subject to the boundary conditions (Eqs. (5a), (5b), (5c), (5d), (5e) and (5f)) when using the high-field approximation (i.e. zFaE/RT1, tanh(zFaE/RT)=1 and sinh(zFaE/ RT)=exp(zFaE/RT)/2 are 1 cO(x) =k2[(k − 4 −1.5a/D %O) exp(−x/a)+1.5a/D %O]

(6a) ci(x) =k1,i[(k

−1 3,i

cM(x)=k3[(k

−a/D %) i exp(− x/a)+a/D %] i

−1 1

(6b)

−a/D %M) exp((x−L)/a)+a/D %M] (6c)

where D% =D exp(zFaE/RT)/2 and analogous subscripts as in Eqs. (1)–(3) are used. The interfacial rate constants are thought to obey an exponential dependence on the potential drops at the appropriate interfaces kj =k 0j exp(bjfN )

(7) 0 j

where j =1, 1i, 2, 3, 3i, 4, N=M/F, F/S, k is a standard rate constant and bj =aj F/RT is a Tafel coefficient. Eqs. (6a), (6b), (6c) and (7) give the possibility to compute the concentration profiles of the point defects as a function of the applied potential with the rate constants and diffusion coefficients as parameters. The concentration of positive ionic defects has its maximum at the inner interface, and its profile decays towards the outer interface in an exponential fashion. On the other

2033

hand, the concentration of negative ionic defects has its maximum at the outer interface, and its profile decays towards the inner interface in an exponential fashion.

2.2. Dc electronic conducti6ity of the metal/film/electrolyte system A correlation between the concentration of ionic defects and electronic conductivity can be made assuming that the point defects in the film play the role of electron donors or acceptors. Possible electron donors are the metal interstitials and/or oxygen vacancies generated at the metal/film interface, while possible acceptors are the metal vacancies generated at the film/solution interface. It can be assumed as a first approximation that the local electronic conductivity is proportional to the local concentration of ionic defects. Thus in the measurement of the dc resistance of the passive film, the main contribution to the resistance will come from the regions of the film with minimal concentration of ionic defects where the conductivity is also minimal. Following this line of reasoning, the measured electronic conductivity of the film is approximately proportional to the minimum concentration of the ionic defects: se 8 meFcmin

(8)

where me = FDe/RT is the mobility of electronic charge carriers, which is assumed to be constant. An expression for the minimum concentration of ionic defects can be obtained by means of searching for the minimum of the sum of contributions of different defects. Let us assume that the concentration of metal interstitials is much higher than that of oxygen vacancies, i.e. the interstitials are the major positive ionic defects. This assumption appears reasonable in the context of recent in-situ investigations of the structure of the passive film on iron in neutral solution by XANES and in-situ X-ray diffraction [1 – 4]. These studies have shown that the passive film has a spinel structure intermediate between that of magnetite Fe3O4 and maghemite g-Fe2O3. As the metal vacancies are the negative ionic defects, taking the sum of Eqs. (6b) and (6c) gives for the profile of the total defect concentration 1 c(x) =k1,i{k − exp(−x/a)]} 3,i exp(−x/a) + (a/D %)[1− i 1 +k3{(k − exp((x− L)/a) 1

+ (a/D %M)[1− exp((x− L)/a)]}

(9)

As the typical passive film thickness lies in the range 2 – 4 nm, and the values of the half-jump distance a are of the order of 0.3 – 0.4 nm, the following assumptions appear reasonable: exp( −x/a) B1 and exp((x− L)/ a) B1. Thus, Eq. (9) simplifies to

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1 c(x)=k1,i{k − 3,i exp( −x/a)+a/D %} i 1 +k3[(k − exp((x−L)/a)+ a/D %M} 1

(9%)

Differentiating in respect to the distance results in the expression dc(x)/dx=(−1/a)(k1,i/k3,i) exp(−x/a) +(1/a)(k3/k1) exp((x−L)/a)

(10)

tration are given by Eqs. (6a), (6b) and (6c), can be derived as follows. Assuming that the transport of both ionic and electronic defects contributes to the impedance, the film impedance is given by the sum of the impedances due to electronic and ionic conduction in parallel 1 −1 −1 Zf = (Z − e + Z ion )

(16)

The film thickness versus potential dependence can be adopted from the PDM [24]:

To derive an equation for the electronic contribution to the film impedance at low potentials, it is assumed as above that the concentration of metal interstitials is much higher than that of oxygen vacancies. In addition, it is assumed that metal interstitials dominate over metal vacancies at low potentials. This assumption is again based on the results of XANES and in-situ X-ray diffraction investigations [1 – 4]. The profile of the electronic conductivity in the film is then expressed by (see Eqs. (6b) and (8)):

L= LE = 0 +(1 −aF/S)E/E

se(x) = (F 2De/RT)ci(x)

Searching for a minimum (dc(x)/dx=0) gives xmin = 0.5{L−a ln[(k3/k1)/(k1,i/k3,i)]}. Inserting this in Eq. (9%), we obtain the expression for the minimum defect concentration as follows cmin =2[k3k1,i/k1k3i]1/2 exp(−L/2a)+k1,ia/D %i +k3a/D %M

(11)

(12)

where LE = 0 is the thickness of the film at E=0. From Eqs. (4), (7) and (12) it follows that

= (F 2De/RT)

k3/k1 =K31 exp(b3aE)

(13a)

k1,i/k3,i =K1,i 3,i exp( −b3,iaE)

(13b)

Combining Eqs. (13a) and (13b) with Eq. (11) predicts that the minimum concentration of the defects decreases with increasing potential, passes through a minimum and starts to increase at high enough potentials. Then the dependence of the minimal electronic conductivity of the film on potential can be obtained on the basis of Eqs. (8) and (11) under the assumption that the mobility of electronic charge carriers is constant.

2.3. Ac impedance response of the metal/film/electrolyte system The overall impedance of the metal/film/electrolyte system is defined as the sum of the interfacial impedances at the metal/film and film/solution interfaces and the transport impedance in the bulk film Z=ZM/F +Zf +ZF/S

(14)

The faradaic impedances of the metal/film and film/ solution interfaces have in general the form of charge transfer resistances which are functions of the rate constants and Tafel coefficients of the interfacial reactions (for derivation, see for example Ref. [32]): 1 Z− M/F =(1− aF/S)F(3b1k1 +4/3b2k2 +2b1,ik1,i)+jvCM/F (15a) 1 Z− F/S =aF/SF(3b3k3 +4/3b4k4 +b3ik3i)+jvCF/S

(15b)

These transfer functions are thus obtained in terms of the interfacial rate constants and Tafel coefficients. The transfer function describing the impedance response of a film, in which the profiles of defect concen-

1 k1,i{k − exp(−x/a)]} 3,i exp(−x/a) + (a/D %)[1− i (17)

and if exp(−x/a) 1, 1 se(x) = (F 2De/RT)k1,i[k − 3,i exp(−x/a) + a/D %] i

(18)

Using the notation rd = RTk3,i/F 2Dek1,i and r0 = 2 RTD %/aF Dek1,i to group the parameters depending i only on potential but not on the distance within the film, we arrive at 1 1 se(x) = r − exp(−x/a) + r − d 0

(19)

Accordingly, rd is proportional to the ratio of the rate constants of consumption and generation of divalent metal interstitials, while r0 is proportional to the ratio of the rates of transport and generation of major ionic charge carriers. Following the derivation of the impedance of a layer for which the model of Young is valid [38], we obtain Ze = L

&

1

1 1 d(x/L)/{r − exp[−(1/d)x/L]+jvoo0 + r − d 0 }

0

(20) where L is either the film thickness or the thickness of the space charge layer formed in the system, d= a/L is a constant, o is the dielectric constant of the layer and o0 is the dielectric permittivity of free space. The integration leads to the following transfer function Ze = [d/( jvC+1/r0L)] ln[(1 +jvt exp(1/d))/(1+jvt)] (21) where C is the capacitance of the layer or the space charge capacitance (C =ooo/L) and t is a time constant (t= rdoo0).

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An analogous transfer function can be derived for the contribution of the electronic transport in the passive film at high potentials, assuming that above a certain potential the role of metal vacancies as ionic defects dominates over that of metal interstitials. This assumption is based on the fact that with increasing potential the concentration of divalent metal interstitials in the film decreases strongly because of their consumption at the film/solution interface (reaction (IIIi)). On the other hand, metal vacancies are generated by the dissolution of trivalent metal ions from the oxide at the same interface (reaction (III)). The rates of both reaction (IIIi) and reaction (III) are expected to increase exponentially with increasing potential. Now the profile of the electronic conductivity in the film is described by the following equation (see Eqs. (6c) and (8)): 1 se(x) =(F 2De/RT)cM(x)=(F 2De/RT)k3[(k − 1 −a/D %M)

exp(− (L− x)/a)+a/D %M]

(22)

In this case, rd from Eq. (19) is proportional to the ratio of the rate constants of consumption and generation of metal vacancies, while r0 is again proportional to the ratio of the rates of transport and generation of major ionic charge carriers. The next step is to evaluate the impedance due to the ionic transport through the film Zion. If the transient concentration gradient of an ionic defect transported through the film is significant, a Warburg-like impedance is to be expected in accordance with the PDM and other related treatments [18–21]. In order to obtain the form of the Warburg function, we write the small amplitude ac solution of Eqs. (1)–(3) in the form [18]: Dc(x, v)=B1 exp(R1x)+B2 exp(R2x) R1 = [−1/a+(a

−2

+4jv/D%) ]/2

R2 = [−1/a−(a

−2

+4jv/D%)0.5]/2

(23)

0.5

These equations are valid for the transport of any kind of ionic defect. If Dc(x, v) refers to the concentration of oxygen vacancies, the use of Eqs. (5a), (5b) and (5c) to evaluate the constants B1 and B2 leads to the usual Warburg impedance 2 −2 1 Z− + ion =[(2F) D %O/RT]{cO(L)(1−aF/S)[−1/a+(a

4jv/D %O)0.5]/2}

(24)

On the other hand, if the transient concentration gradients of the transported defects can be neglected in comparison with the migration (see for example Ref. [19]), the impedance due to the transport of ionic defects reduces to a resistance 2 1 Z− ion =[(2F) (D %O/a)/RT]cO(L)(1−aF/S)

(25)

It is worth mentioning that if the transport rate of all kinds of ionic defects in the film in the steady-state is

2035

much smaller than that of the transport of electronic defects, then the impedance due to the ionic transport can be neglected. On the other hand, the impedance due to the electronic transport has a minor contribution, if the electronic conductivity of the layer is minimal. The rest of the present paper is devoted to experimental results, which give support to the ideas presented and can thus be used to verify the model approach proposed.

3. Experimental Working electrodes were made of 99.5% Fe (Goodfellow), 99.7% Cr (Goodfellow) and laboratory made high purity Fe – 12%Cr and Fe – 25%Cr alloys. For the CER measurements, electrode tips with a 2.5-mm diameter were used. In the other experiments disk electrodes fixed in PTFE holders with epoxy resin (area 0.2 cm2) were employed. The rotating ring-disk studies were accomplished with electrodes featuring a disk made of the material under investigation (area 0.16 cm2) and a ring made of Au (99.9%, ring area 0.1 cm2). The electrodes were abraded mechanically and rinsed with MILLI-RO® water before the measurements. In a conventional threeelectrode cell, a Pt sheet was used as counter electrode and a Hg/Hg2SO4/K2SO4 (sat.) electrode as a reference electrode. All the potentials are given vs. the standard hydrogen electrode (SHE). 0.1 M Na2B4O7 (pH 9.2) was prepared from reagent grade chemical and MILLI-RO® water. The measurements were carried out at room temperature (20 91°C) in naturally aerated solutions. A Wenking LB 81 potentiostat and a HITEK PPR1 function generator were used for potential control in the CER measurements, the CER equipment being supplied by Cormet Ltd. Impedance spectra were obtained with a Solartron 1287/1260 system controlled by ZPlot software (Scribner Associates) in a frequency range of 0.01 – 30 000 Hz at an ac amplitude of 20 mV (rms). The reproducibility of the impedance spectra was 9 2% by amplitude and 93° by phase. The program for the fitting of impedance spectra to selected transfer functions was written by one of the authors (M.B.). An Oriel 300 W Xe-lamp combined with an Oriel Multispec 257 monochromator and a chopper (Bentham model 218) were used as a modulated light source (wavelength 250 – 700 nm) and the light was focused through a lens onto the quartz window of the photoelectrochemical cell. The photocurrent was registered with a PAR 5208 lock-in amplifier coupled to a PAR 273 potentiostat. Quantum efficiencies were calculated by correcting the measured photocurrent for the spectral dependence of the lamp/monochromator light power measured by an optical power meter (Grazeby Optronics). The reflections at the metal/film interface were not taken into account.

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M. Bojino6 et al. / Electrochimica Acta 45 (2000) 2029–2048

Fig. 2. Linear sweep voltammogram of Fe in 0.1 M Na2B4O7 (sweep rate 10 mV s − 1, rotation rate 1200 rpm) together with (a) the ring current for the detection of Fe(II) by their oxidation at the Au ring to Fe(III) (ring potential 0.44 V), (b) the ring current for the detection of Fe(III) by their reduction at the Au ring to Fe(II) (ring potential −0.46 V).

4. Results and discussion

4.1. Release of soluble products during formation and reduction of the anodic film Figs. 2–4 show voltammograms for pure Fe (Fig. 2a,b), Fe–12%Cr (Fig. 3a,b) and Fe–25%Cr (Fig. 4a,b). They also show the detection of soluble Fe(II) by its oxidation to Fe(III) at the Au ring (ring potential

0.44 V) and the detection of soluble Fe(III) and/or soluble Cr(VI) by their reduction at the Au ring (ring potential − 0.46 V). The disk voltammograms demonstrate the similarity of the behaviour of iron and the Fe – 12%Cr alloy, except for the smaller currents in the case of the alloy. On the other hand, the voltammogram of the Fe – 25%Cr alloy shows significant transpassive oxidation, which is typical of pure chromium. During the positive-

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going potential scans, divalent iron is released from the studied materials in the potential region where a peak is observed in the disc voltammogram of pure Fe ( − 0.6 to − 0.1 V). This release decreases with addition of Cr in the alloy. The release of small amounts of trivalent iron from all the materials is observed during the second anodic peak in the disc voltammograms (at ca. − 0.15 to + 0.1 V). Also, the release of hexavalent chromium is observed for the Fe–25%Cr alloy at ca.

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0.9 V which coincides with the corresponding anodic peak on the disc voltammogram of the alloy. Such a feature was not reproducibly observed for the Fe – 12%Cr alloy. During the negative-going potential scan, two maxima are observed in the ring current versus disc potential curve (Ering = 0.44 V). The first maximum at − 0.3 to − 0.7 V coincides roughly with the broad reduction peak observed in the disc voltammogram, while the

Fig. 3. Linear sweep voltammogram of Fe–12%Cr in 0.1 M Na2B4O7 (sweep rate 10 mV s − 1, rotation rate 1200 rpm) together with (a) the ring current for the detection of Fe(II) by their oxidation at the Au ring to Fe(III) (ring potential, 0.44 V), (b) the ring current for the detection of Fe(III) and/or Cr(VI) by their reduction at the Au ring to Fe(II) and/or Cr(VI) (ring potential, −0.46 V).

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Fig. 4. Linear sweep voltammogram of Fe–25%Cr in 0.1 M Na2B4O7 (sweep rate 10 mV s − 1, rotation rate 1200 rpm) together with (a) the ring current for the detection of Fe(II) by their oxidation at the Au ring to Fe(III) (ring potential, 0.44 V), (b) the ring current for the detection of Fe(III) and/or Cr(VI) by their reduction at the Au ring to Fe(II) and/or Cr(III) (ring potential, − 0.46 V).

second one is situated in the region where a fast increase in the cathodic current is observed at the disc at low potentials. These two ring current maxima are in agreement with the two-step reduction of the anodic films on iron resulting in the release of Fe(II) in the solution [39–42]. An important point to mention is that for pure Fe the second ring current peak is more prominent than the first one, whereas the opposite trend is observed for the alloys. This might be related to the incomplete reduc-

tion of the inner part of the oxide layer which contains significant amounts of Cr when formed on Fe – Cr alloys. These RRDE voltammetric results demonstrate the fact that divalent iron is released at low potentials, while trivalent iron and hexavalent chromium are released at high potentials. These reactions are included in the general kinetic scheme of the reactions at the film/solution interface as proposed by the model (see Fig. 1).

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4.2. Electronic conducti6ity of the anodic film Fig. 5a,b show the dc resistance of the film on pure Fe, Cr, Fe–12%Cr and Fe–25%Cr as a function of potential in 0.1 M Na2B4O7. It was measured with the CER technique during film formation when sweeping the potential in the positive direction at 1 mV s − 1.

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The curve for pure Cr is analogous to those published earlier [43,44]. For pure Fe, a steep increase of the resistance starts at ca. − 0.55 V, i.e. shortly after a positive current starts to flow in the cyclic voltammograms (cf. Fig. 2a,b). A short arrest is observed in the resistance vs. potential curve of Fe at − 0.3 V which coincides with the maximum release of soluble

Fig. 5. Dependence of resistance on potential for (a) pure Fe and Cr, (b) Fe – 12%Cr and Fe – 25%Cr alloys during a positive sweep (sweep rate 1 mV s − 1) in 0.1 M Na2B4O7, (c) pure Fe, Cr, Fe – 12%Cr and Fe – 25%Cr during a negative-going sweep (sweep rate 1 mV s − 1) following 1 h polarisation in the passive region (0.24 V for pure Cr, 0.64 V for pure Fe, Fe – 12%Cr and Fe – 25%Cr) in 0.1 M Na2B4O7.

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Fig. 5. (Continued)

Fe(II) during a positive potential sweep (Fig. 2a). Relatively high and stable resistance values are measured in the range 0.9–1.2 V. Subsequently, the resistance starts to decrease slowly together with the increase of the positive current due to the oxygen evolution reaction (Figs. 2a and 5a). For Fe–12%Cr (Fig. 5b), the arrest in the R versus E curve at −0.3 V is situated at a higher value of the resistance than for pure Fe. This is in accordance with the smaller amount of soluble products detected by the ring-disc electrode during oxidation in comparison to pure Fe (Figs. 2a and 3a), i.e. greater part of the positive current is consumed for film growth when compared to pure Fe. The highest values of the resistance are reached earlier for Fe–12%Cr than for pure Fe. In the oxygen evolution region, no difference between the curves for pure Fe and Fe–12%Cr can be detected (Fig. 5a,b). In the curve for Fe–25%Cr, no arrest is observed at −0.3 V, and higher resistance values are measured for this alloy when compared to Fe–12%Cr in the potential range − 0.3 to 0.2 V. Conversely, the values of the resistance in the oxygen evolution region (above 1.1 V) are lower for Fe–25%Cr than for both Fe and Fe–12%Cr, i.e. the film on the Cr-rich alloy is more conductive. This is probably due to the formation of more Cr(VI), and accordingly, more metal vacancies in the film in the case of the higher Cr content of the alloy substrate (cf. the curve for pure Cr, Fig. 5a). The general shape of the R versus E curves is in good agreement with the predictions of the model approach described in Section 2: the electronic conductivity of the

film is good at low potentials, it becomes poor in the middle of the passive region, and at high potentials it again increases. As a demonstration of the ability of the present model approach to reproduce experimental results, a simulated R versus E curve (see Fig. 5a) was computed from Eqs. (8) and (11) for Fe in 0.1 M Na2B4O7. The kinetic parameters used to construct the simulated curve are given in Table 1. The thickness of the film was estimated using the equation L =1.10 − 7 + 0.18E, which is based on the reflectance data obtained by Table 1 Parameters used to simulate the experimental dc resistance vs. potential curve for Fe shown in Fig. 5a Parameter

Value

DMv (cm−2 s−1) DMi (cm−2 s−1) k 01 (cm s−1) b1 (V−1) k 01,i (mol cm−2 s−1) b1,i (V−1) k 03 (mol cm−2 s−1) b3 (V−1) k 03,i (cm s−1) b3,i (V−1) aF/S E (mV cm−1) a (cm) LE = 0 (cm) De (cm−2 s−1)

1×10−16 3×10−16 1×10−8 17 1×10−10 9 1×10−11 26 5×10−9 55 0.5 3.5 3.5×10−8 1×10−7 5×10−12

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Buechler et al. [45] for Fe in borate buffer (pH 8.4). The assumption aF/S =0.5 leads to the estimate for the field strength in the film. This is of the same order of magnitude than that estimated by other authors [46]. The order of magnitude of the diffusion coefficients for metal vacancies and interstitials was taken from Refs. [18,19,54], while the value of the half-jump distance a is typical for high-field assisted transport [30,31]. Unfortunately, no data are available in the literature for the diffusion coefficient of electronic charge carriers De. However, assuming a polaron glass model to be valid for the electronic conduction [48] in a spinel-type oxide with significantly less Fe2 + than in magnetite [8,49] gives a reasonable estimate for De. The shape of the curve and the absolute values of the resistance are influenced mainly by the magnitude of the Tafel coefficients of the interfacial reactions of defect generation/ annihilation. The magnitude of the rate constants has a much less pronounced effect. Fig. 5c presents the dc resistance vs. potential curves during negative-going potential sweeps following a 1-h polarisation of the studied materials in the passive region (0.24 V for pure Cr, 1.14 V for pure Fe and both alloys). The resistance starts to decrease at ca. 0 V for pure Fe and the alloys, whereas the first decrease for Cr is detected at −0.5 V. The shape of the resistance versus potential curve indicates that the reduction of the passive film is a two-step process in accordance with voltammetric and spectroscopic data from other authors [10,11,39–42,45] and RRDE results presented above. The arrest in the resistance vs. potential curve at −0.7 to −0.3 V is observed at the same potentials as the first peak of the ring current, corresponding to the release of Fe(II) from the studied materials (Figs. 2a, 3a, 4a). The resistance value at the arrest is the greater, the higher the Cr content in the alloy (Fig. 5c). Further insight into the mechanism of electronic conductivity was sought by measuring photocurrent spectra of the passive film formed on Fe, Cr and both the studied alloys. Typical spectra for pure Fe in the range 0.56–1.16 V after polarisation for 0.5 h at 0.56 and 1.16 V are shown in Fig. 6a,b. For pure Cr corresponding spectra in the range −1.14 to 0.66 V after polarisation for 0.5 h at 0.66 V are presented in Fig. 6c. The spectra for the alloys after polarisation for 0.5 h at 0.56 and 1.36 V are shown in Fig. 6d,e (Fe–12%Cr, 0.56–1.36 V) and Fig. 6f (Fe–25%Cr, 0.36–1.26 V). The spectra are in qualitative agreement with data obtained by other authors (see for example Refs. [14–16,45,50,51]). One main conclusion is that several times higher quantum efficiencies are obtained if the photocurrent spectra are measured after polarisation in the oxygen evolution region (cf. Fig. 6a,b and d–f). The increase of the quantum efficiency cannot be attributed to the increase in film thickness alone, which is of the order of

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50% in the studied potential interval [8,45,49,52]. Instead, the significant increase in the quantum efficiency can be qualitatively explained on the basis of the model approach proposed in Section 2. It has been demonstrated that in the oxygen evolution region on pure Fe, the rate of Fe(III) dissolution increases with potential [12]. Also the formation of a near-surface layer containing Fe(IV) in the anodic film on Fe has been postulated [13]. The formation of Cr(VI) in the outermost layers of the anodic film on Fe – Cr alloys at high enough potentials has also been documented [55]. As a result of the formation of high-valent species, the concentration of metal vacancies near the film/electrolyte interface is expected to increase. According to the present model, the electronic conductivity at that interface is proportional to the concentration of metal vacancies. Thus, if a film with a higher concentration of metal vacancies is formed at high potentials, it is expected to have better electronic conductivity and hence higher quantum efficiency. This is exactly what was observed experimentally.

4.3. Ac impedance response of the metal/film/electrolyte system Fig. 7 shows impedance spectra of (a) pure Fe, (b) pure Cr, (c) Fe – 12%Cr and (d) Fe – 25%Cr measured after 1 h of polarisation in the passive region (0.84 V for pure Fe, 0.66 V for pure Cr, 0.84 V for Fe – 12%Cr and 0.64 V for Fe – 25%Cr). The results could be best fitted using the transfer function given for the electronic contribution to the conduction in Eq. (21). The best-fit results are presented with solid lines. The fair correspondence between the fit and the experiment suggests that for the particular materials studied and experimental conditions the impedance response is governed mostly by the electronic conduction through the passive film. The ionic resistivity of the films is thus many orders of magnitude greater than the electronic resistivity. This is well in line with the very low metal dissolution currents (‘corrosion currents’) observed in the studied systems. The parameter r0 obtained from the fit of the impedance results to Eq. (21) was close to infinity for the frequency range covered by our impedance measurements for almost all the systems. As commented above, this parameter contains the ratio of the rates of generation and transport of the predominant ionic charge carriers. Accordingly, when it approaches infinity, ionic transport is much slower than the rate of generation of ionic defects. The very high value of r0 supports the conclusion drawn above, i.e. that electronic conduction through the film dominates the impedance response. The other parameters determined from the fit, i.e. C as C − 2 (the capacitance of the anodic film or the space charge layer) and rd are

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Fig. 6. Photocurrent vs. photon energy spectra after polarisation of (a) pure Fe at 0.56 V, (b) pure Fe at 1.16 V, (c) pure Cr at 0.66 V, (d) Fe – 12%Cr at 0.56 V, (e) Fe–12%Cr at 1.16 V and (f) Fe – 25%Cr at 0.36 V in 0.1 M Na2B4O7 for 1 h. (A spectrum measured at 0.86 V after 1 h polarisation at 1.36 V is also shown in Fig. 6f.)

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Fig. 6. (Continued)

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Fig. 7. Impedance spectra after polarisation of (a) pure Fe at 0.84 V, (b) pure Cr at 0.66 V,(c) Fe – 12%Cr at 0.84 V and (d) Fe–25%Cr at 0.64 V for 1 h in 0.1 M Na2B4O7. Points — experimental data; solid lines — best-fit results according to Eq. (21).

plotted vs. potential in Fig. 8a,b for all the materials studied. It is worth noting that the parameter d (not shown in the figures) which describes the relative permeation depth of the corresponding concentration profile (Eq. (20)) is of the order of 0.045–0.05 for pure Fe and Fe–12%Cr, and 0.035 for Fe–25%Cr and pure Cr. The capacitance C presented versus potential in Mott–Schottky coordinates in Fig. 8a follows the trend observed by other authors [12,14–16,51,53], indicating that it represents most probably the capacitance of the space charge layer and not that of the whole film. Predominantly acceptor-type conductivity is found for

the film on pure Cr, while the films on the two alloys contain mainly donors as ionic defects in the potential range − 0.2 to 0.5 V (Fe – 12%Cr) and − 0.2 to 0.3 V (Fe – 25%Cr), in analogy to pure Fe. However, the Mott – Schottky plots are non-linear suggesting a change in the donor type and/or the donor density with potential. For pure Fe and the Fe – 12%Cr alloy, the parameter rd (Fig. 8b) increases with potential in the region −0.36 to 0.24 V and is approximately constant at higher potentials. As discussed above, when metal interstitials are the major ionic defects, the parameter rd is proportional to the ratio of the rate constants of the

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reaction of consumption (III,i) and production (I,i) of the interstitials. Accordingly, the present results indicate that with increasing potential the rate of consumption of interstitials becomes higher than the rate of their production. This observation is supported by both capacitance and XANES measurements by other authors [2,8,49]. They have shown that the concentration of divalent metal interstitials decreases with increasing potential due to their oxidation to Fe(III). For pure Cr, the parameter rd decreases with increasing potential in the potential range −0.1 to 0.66 V. As the presence of Cr(VI) in the passive film on Cr prior to transpassive dissolution has been well documented [9], it can be assumed that it is segregated in the outermost

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layer of the passive film [54]. The formation of Cr(VI) may be written as a reaction analogous to reaction (III): − 3− CrIII Cr “ Cr(VI)F/S + VM + 3e

(III%)

Accordingly, this reaction leads to an increased production of metal vacancies at the film/solution interface. The proposed model predicts that when cation vacancies are the predominant ionic defects in the film, the parameter rd is proportional to the ratio of the rate of consumption of metal vacancies (reaction (I)) and production of these defects (reaction (III%)). The increase of the potential in the positive direction favours the reaction of production of defects. Thus, the

Fig. 8. (a) Mott – Schottky plot of the film capacitance C as calculated from the best-fit results according to Eq. (21) for pure Fe, Cr, Fe – 12%Cr and Fe – 25%Cr; (b) the parameter rd, as calculated from the best-fit results according to Eq. (21) for pure Fe, Cr, Fe–12%Cr and Fe – 25%Cr.

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Fig. 9. Impedance spectra after polarisation of (a) pure Fe at 1.14 V, (b) Fe – 12%Cr at 1.14 V and (c) Fe – 25%Cr at 1.14 V for 1 h in 0.1 M Na2B4O7.

parameter rd is predicted to decrease with increasing potential, as it was found out experimentally (Fig. 8b). For the Fe–25%Cr alloy, the parameter rd increases with increasing potential at low potentials, and an opposite trend is found at higher potentials (Fig. 8b). Thus it can be argued that the behaviour of the passive film in terms of electronic conduction is intermediate between that of the films on pure Fe and Cr. This means that at low potentials in the passive region divalent metal interstitials are the predominant ionic defects acting as electron donors. On the other hand, metal vacancies are the dominating ionic defects at high potentials. The most probable source of the increased production rate of metal vacancies at high potentials is once again reaction (III%), because the formation of

Cr(VI) in the passive film on a Fe – 26%Cr alloy has been detected by XANES [55]. It was suggested above that the increase of the photocurrent efficiency of the passive film after polarisation at high potentials is due to the formation of high-valency defects. Further proof for this was sought by registering impedance spectra for pure Fe and the studied alloys at high potentials after polarisation in the oxygen evolution region (1.14 V). Such spectra are presented in Fig. 9a – c. The transpassive dissolution of pure Cr proceeds in this region, as already reported by us earlier [43,44]. In addition to the low frequency response seen also in Fig. 7a – d, a high-frequency time constant appears in the spectra. This time constant is the most pronounced

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for pure Fe (Fig. 9a) and it has been found also by other authors for Fe [47,56]. For Fe–25%Cr it is detected only after careful deconvolution of the spectrum (Fig. 9c). The relationship between this time constant and the formation of Fe(IV) species or hydroxyl radicals and incorporated peroxide [39] in the near-surface layer has been suggested on the basis of spectroscopic ellipsometric measurements [47,52]. It has been further substantiated by transient photocurrent and intensity modulated photocurrent spectroscopic (IMPS) measurements [56,57]. On the basis of the ideas outlined in Section 2, it can be suggested that the high-frequency time constant corresponds to the formation of these high-valency species at the film/electrolyte interface (cf. Eq. (15b)). This suggestion is supported by the exponential dependence of the resistance associated with this time constant, as found by other authors [47]. However, for the model to account for the presence of intermediates of the oxygen evolution reaction on Fe and Fe–Cr alloys, further analytical data on these intermediates are needed.

5. Conclusions In the present paper, a model approach to the steadystate anodic passive films on Fe, Cr and Fe–Cr alloys is proposed. It emphasises the coupling between the ionic defect structure and electronic conduction in the metal/film/electrolyte system. Equations to demonstrate the model approach have been derived for the case of a spinel oxide, in which divalent metal interstitials and cation vacancies are the major ionic defects. The assumption concerning the establishment of a steady-state requires that the rates of film growth and dissolution involving transport of ionic defects equal each other. Thus, a definite ionic defect structure is reached in the film as depending on potential and the compositions of the substrate metal and the electrolyte. Expressions for the concentration profiles of defects in the passive film can be derived starting from the steadystate solution of the generalised differential equations for field assisted defect transport in anodic oxides. It is assumed in the model that positive ionic defects generated at the metal/film interface and consumed at the film/electrolyte interface play the role of electron donors, while negative defects generated at the film/ electrolyte interface and consumed at the metal/film interface play the role of acceptors. The electronic conductivity of the steady-state passive film is considered to be proportional to the concentration of ionic defects and thus depends on potential and distance within the film. From this starting point, equations for the potential dependence of the minimal dc conductivity of the passive film and the impedance response due to the existing electronic conductivity profiles were

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derived. For the sake of completion, the impedance response due to ionic motion was also derived, and limiting cases leading to a Warburg-like transport impedance or high-field migration resistance were discussed. The predictions of the proposed model are compared to an extensive set of experimental data on the passive state of Fe, Cr, Fe – 12%Cr and Fe – 25%Cr alloys in neutral borate solution. The model was found to be able to account quantitatively for both the dc resistance versus potential and the impedance response of the studied systems. Also, it was found to be in qualitative agreement with the photocurrent spectroscopic results and the observed release of soluble products during formation and reduction of the passive film. It can be concluded that electrical and electrochemical properties of the passive films on Fe, Cr and Fe – Cr alloys in neutral solutions are governed by the rates of generation and transport of ionic point defects. These defects act as donor and/or acceptor species ensuring the electronic conduction through the films. Thus quantitative correlations between the crystallographic structure of the passive film, the chemical reactions at the interfaces with the substrate metal and the electrolyte, and the rates of ionic and electronic transport through it may be established. The exact mechanism of electronic conduction requires, however, a treatment of the problem from the viewpoint of semiconductor electrochemistry and solid state physics. This treatment is beyond the scope of the present paper in which mainly the chemical nature of the processes giving rise to electronic conduction is discussed. On the basis of the model, quantitative assessment of the effect of alloying elements such as Cr on the conduction mechanism in the passive film on Fe becomes possible. Thus, a more direct relationship between compositional and structural data for passive films and their susceptibility to general and localised corrosion may be established.

Acknowledgements The authors are grateful to the Finnish Ministry of Trade and Industry and the Radiation and Nuclear Safety Authority, Finland, (Research Programme on Operational Safety and Structural Integrity 1999-2002, FINNUS). The financial support of the Academy of Finland is also gratefully acknowledged.

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