Electrochemical impedance spectroscopy on oxide films formed on zircaloy 4 in high temperature water

Elecrrochm~ca Aca. Vol 39, No 3. pi

455-465, 1994

Copyright 0 1994 Ehlct Sctctta Ltd PnnkdUIGreatBntatnAll nghtsrraervcd 0013-4686/945600+000

Pergamon

ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY ON OXIDE FILMS FORMED ON ZIRCALOY 4 IN HIGH TEMPERATURE WATER C

BATAILLGN

and S

BRUNET

Commtssartat a

L’Energe Atomtque, DTA/CEREM/DTM/Servlce de Corrosion, d’Electrochlmle et de Chlmle des FluIdes, B P no 6,92265 Fontenay-aux-Roses-Cedex, France (Received 21 January 1993, m revlsedform 31 August 1993)

Abstract--Complex capacitance spectra obtamed on oxide films formed by high temperature water 0x1datlon of Zlrcaloy 4 exhIbIted dlelectnc losses which have been ldentdkd urlth those derived from the cluster model The complex capacitance showed a Low Frequency I)lsperslon (LFD) A descnptlon of oxide film defects and defect transport are presented The couphng between dynanuc behavlour of defects . and solution pH 1s shown

Key words oxide film, capaatance, cluster, pH, Impedance spectroscopy

particularly detrlmental It 1s well established that the oxldatlon rate of wrcomum alloys m high temperature water 1s controlled by mass transport through a more or less protective oxide film[l] The purpose of this paper 1s to determme, using EIS, the characteristics of mass transport through oxide films which have been formed m the vlcmlty of the kinetic transition The cluster model has been used to describe hoppmg conduction and dlelectnc loss m the films pH dependencies of both hopping conduction and dlelectrlc characteristics have been studied

NOMENCLATURE pure capacitance complex capacitance

real part of the complex capacitance lmagmary part of the complex capaatante constant oxide film thickness applied frequency dlstmctlve frequencies pure imaginary (fi) exponents E [0, l] pure resistance electrolytic resistance dlmenslonless pulsation complex impedance exponent dlelectrlc constant of oxide film dlelectnc constant of vacuum phase angle ionic resistivity of solutions complex dielectric susceptlblhty real part of x(w) imaginary part of x(w) amplitudes of dlelectrlc losses pulsation ( = 2$) dlstmctlve pulsation

EXPERIMENTAL Maternal and oxtdatron condztzons

modulus absolute value Gaussian hypergeometrlc function INTRODUCTION Oxrdatlon of Zlrcaloy 4 1s the hnutmg factor for burning up enhancement of fuel cladding m Pressurized Water Reactors (PWR) The sharp increase m oxldatlon rate beyond the kmetrc transition 1s

Cyhndrlcal specimens were cut from a rod of Zlrcaloy 4 (Sn 144%, Fe 021%, Cr 0 12%, 0 0 113%) The side and the extremities of the cylinders were polished with 8C up to grade 1200 and etched for 30s m a mixture containing 100cm3 HF (d = 1 13, 40%), 600cm3 HNO, (d = 133,52 5%) and 3OOcm’ water After this, the samples were heat-treated at 625°C for 1 h m glass, sealed under vacuum Finally, the sealed glass was air-cooled The oxide films were formed by high temperature oxldatlon at 360°C under 18 6 MPa pressure in water containing 2 ppm Ll as lithium hydroxide and 1OOOppm B as boric acid (PWR primary water condltlons at the middle of a burn cycle) The first sample was oxidized for 2 months and was covered with a 2 18pm thick oxide film The second sample was oxldlzed for 3 months and was covered with a 2 46 pm thick oxide tilm The second sample was oxldlzed until the kinetic transition The thickness of each oxide film was calculated from the weight gam assummg that no zlrcomum species have passed mto the oxldmng solution and ZrO, was the oxide formed (oxide density = 5 70)

455

456

C BATAILLON and S BRUNET

DATA ANALYSIS

Impedance measurements

EIS measurements were carried out at least twice for each sample m three test solutions Chemical composltlon, lomc reslstlvlty (p) and pH values for each solution are hsted below Solution A KOH 125 x lo-*M+H,BO, 18 x lo-‘M p = 967QcmpH = 76 Solution B KOH 5 x 10e3 M p=980RcmpH=114 Solution C KOH 5 x 10-l M p = 10RcmpH = 134 It has been shown m a previous study on similar oxide films[2], that impedance spectra were mdependent of applied potential as long as no dc current flowed through the oxide film Therefore in this study, spectra were collected under potentlostatic control at OmV vs the Hg/HgO/KOH 0 5 M reference electrode (+ 131 mV/nhe) The temperature of the electrochemical cell was momtored at 25 + 0 2°C and the solution was deaerated by nitrogen bubbling before and durmg measurement runs The state of the electrochermcal system (Zlrcaloy 4/oxide lilm/ solution) was monitored in comparmg successive spectra collected from day to day Steady-state was considered to be achieved when two successive spectra were identical over the whole frequency range (65 kHz-1OmHz) Generally, a polarization of at least 100 h was necessary to a&eve a steady-state For all expenments, no dc current flowed through the cell To ensure that the long term exposures to test solutions did not change the structure of the oxide film, EIS measurements were repeated at least tHnce m each test solution m the following order solution C + solution B + solution A + solution B + solution C + solution B and finally solution A For each test solution, the results presented below are the mean value of at least 2 runs and error bar corresponds to the deviation from this mean value

Re =

675.4115 Ohm cm2 I4 I in Ohm cm21

General processzng

The strategy for the ldentlficatlon of an equivalent clrcult was to use only actual and well defined physlcal elements with the simplest arrangement The Impedance spectra collected m this study showed capacitance characterlstlcs In the Bode representa-

tlon, the impedance modulus llZ[l mcreases contmuously as the frequency decreased (Ftg la) whilst the phase angle 4 decreases from about 90” (Fig lb) This capacitance characteristic 1s due to the oxide layer which 1s a dielectric material As starting relation, the umversal law for the dielectric response was used[3,4] C(w)= C, + AC(JW)“-’ = IIClle@,

lIzlIe-‘+,

C-w)-z(+~)l=---&=

’

I 90

10 (a)

88 86 84 82 80

5

-

4

-

3

-

78 76 74

”

2-I’

-3 -2

-1

0

logtfrequency

” 1

’

2

3 lo

4 Hz3

72 5

(2)

where Z( + 00) 1s the hmlt of the complex Impedance as the frequency tends towards mfimty In practice Z(+ co) 1s equal to the electrolytic resistance R, which 1s a cell constant The value of R, was systematically subtracted from Z(o) before any analysis The phase angle-4 was used because electrochemlsts have the use to reverse the sign of the lmagmary part of Z(w) In usmg the findings presented m the Appendix and from the shape of vanatlons with the frequency of both log IlZll and the phase angle #J (Fig I), two simple equivalent clrcults may be tried to fit expenmental data

Surface = 12 99 cm2 Phase angle 0 In

lo9Cl

(1)

where w = 2rrf IS the pulsation and f the applied frequency,j = &i, C(o) a complex capacitance, C, a pure capaatance, AC a constant and n an exponent (n E [0,11) In the above relation C, corresponds to the defectless oxide whereas the next term corresponds to the defect response leading to dlelectrlc loss The complex capacitance C(w) IS related to the complex Impedance Z(W) by the followmg relation

-3-2-l

0 logtfrequency

12

3 in

4

5

Hz3

Fig 1 Bode representation of unpedance data (-) collected m solution B for Z~rcaloy 4 oxldlzed for 3 months m PWR pnmary water eondlttons (---- ) calculated Hnthn = 0 981, AC = 10 53 nFcm-* and ) calculated mth n =0851, AC= 523nFcme2, C, = 74nFem-’ and R=55Gflcm’ ( R = 1 SGncm’

Electrochemical impedance spectroscopy on oxide films

Firstly, let us consider a Constant Phase Element (cpe) m parallel with a pure resistance R The value for the exponent n was determined from the plateau level on the phase angle vs logf graph m the high frequency range (Fig lb) The value of the constant AC was calculated from the linear fitting of log llZl[ vs logf(relatlon Al6 m the Appendix) The value of the pure resistance R was adjusted m such a way that the phase angle was equal to the expenmental one for the lowest frequency of the expenmental spectrum The resultmg fit IS added m the Fig 1 (dashed line) A careful exammatlon of Fig lb reveals that the phase angle 4 evolves quasi-linearly with log f m the high frequency part of the graph (lOHz-1OkHz) Thts corresponds to a variation of Type II for C(w) (see Appendix) Then, let us consider a complex capacitance C(o) given by the relation (1) m parallel with a pure resistance R The most suitable way to determine the values of n and C, 1s to transform the experimental impedance data corrected from R, into experimental capacitance data m usmg the reverse of the relation (2) It follows from the relation (l), that c’ =

c, +

(n-lb

c” cot -

[

2

1

where C’ and C” are, respectively, the real and the Imagmary parts of C(o) Therefore, the values of n and C, were calculated usmg the linear fitting of the high frequency capacitance data on a complex plane representatlon[C” vs C’] (Fig 2) The value of AC was determined usmg the hnear fitting of log llZl[ vs logf (relation (A12) m the Appendix) Finally the value of the resistance R was adJusted to obtain the best fit m the low frequency range The resulting fit 1s

-7

Cm

457

added m Fig 1 (dotted lme) It can be seen that this equivalent circuit leads to a markedly better fit than the previous one except m the very high frequency range (f> 10 kHz) From a physical vlewpomt, the first equivalent clrcmt (cpe//R) assumes that the oxide films formed were not very protective even at room temperature because the impedance spectrum corresponds solely to the defect response This implies a high defect concentration m the films The second equivalent clrcult (C,//cpe//R) assumes that the oxide films formed were much more protective at room temperature because the Impedance spectrum corresponds to a mixed response of defects (cpe and R) and defectless oxide (C,) This lmphes a moderate defect concentration m the films Remembering that the oxldatlon rate at high temperature was of the order of a pm per month, the second equivalent clrcult seems to be the most appropriate one As a consequence, the dlscrepanctes observed m the very high frequency range (Fig lb) which corresponds also to the discrepancies from linearity in Fig 2 would come from devices From a practical ylewpoint, It was observed that the non linear range (Fig 2) extended to lower frequencies as the pure capacitance C, and/or R, Increased Concerning the poor fittmg of experlmental data m the low frequency range, It can be noticed that the varlatlon of 4 vs logs 1s agam quasi-linear m this range This seems to correspond to a second vanatlon of Type II for C(w) with a higher value of the exponent n (section Type II m the Appendix) In other words, the fitting of expenmental data would need the use of the relation (1) with two different values of the exponent n, one m the high frequency range and the other m the low frequency range

8

9

10

C’ (nf/cm2) Fig 2 High frequency range of complex capacitance spectrum collected m solution B for Z~rcaloy 4 0x1for 3 months m PWR pnmary water condltrona (Nyqmst reprcsentatlon CC”vs C’l) EA39:3-K

C BATAILLONand S BRUNET

458

To clarify this, the value of C, correspondmg to the defectless oxide, was subtracted from C(w) As the oxide films were thin, the planar relation between the capacitance and the dielectric units was used C-d

and x(f)

=

d(C(f)- Cd =

x’

+

,f

(3)

Eo

where so IS the dielectric constant of vacuum, E, the dlelectnc constant of oxide film, d the film thickness and x(f) the dielectric susceptlblhty A typical example of a dlelectnc susceptlblhty spectrum IS presented m Frg 3 This spectrum corresponds solely to the defect response (dlelectnc loss) and shows a typical Low Frequency Dispersion (LFD) which means that both real and lmagmary parts of the complex susceptlblhty increase as the frequency decreases151 From this representation of expenmental data (log x’, log 1” vs logf), it can be clearly seen that the relation (1) wth two values of exponent n has to be used to fit this data From relations (1) and (3), it follows that d x’ = - AC cos (n - 1) E Eo

2

[

1 &l-l

and d x” = - AC sm (n - 1) n

so

2

[

1 w”-l

(4)

Cluster model

The cluster model has been denved by L A Dlssado and R M Hill m the 80s Two distinct classes of dielectric response have been ldentlfied, namely that of bound charges[6-91 and that of potentially mobile charges[lO] A typical example of the former class IS that of CaF, doped with Err1 l] Electroneutrahty implies F- ions m mterstltlal posltlons around the dopants A typical example of the latter class IS that of Hollandltes or K,Al,T1,_,0,,) m which (K,Mgx,,Tl,-,,,% K+ ions are located along one dlmenslonal channels blocked by structural lrregularltles[ 121 Clusters are areas of material m which the local structure IS distorted from the pure host one Imperfect macroscopic materials are described as a cluster array Two cooperative processes have been considered one IS the mtra-cluster relaxation mechanism and the second IS the inter-cluster exchange mechanism A correlation index has been associated to each mechanism In consequence of mtra- and inter-cluster processes, each cluster evolves contmuously Hrlth time and at any gven time all possible cluster configurations may exist As a result of macroscopic constraints (chemical compoatlon, electroneutrahty, etc ) a macroscopic matenal must be regarded as a steady-state cluster dlstrlbutlon From this descnptlon of dielectric materials, L A Dlssado and R M Hill have derived complete expressions for the dlelectrlc susceptlblhty x(f) The reader will find m Cl33 a complete desmptlon of the cluster model For bound charges, x(f) IS gven by n-1 x(f)=X(O) l+,f ( fi> 1 x *F1 1 - n, 1 - m, 2 - n, -

I1(3

So, the lmes corresponding to vanatlons of log x’ and log 1” vs logfwould be parallel and spaced out from log{tan[(n - 1)x/2]} This kmd of variation IS clearly noticeable m Fig 3 both m the high (0 l10 kHz) and m the low (0 01-l Hz) frequency ranges l+,f [ To our knowledge, only the cluster model IS able to f, descrrbe such a LFD m volume[S] (re a cpe wth two where *FI[, , ,] IS the Gaussian hypergeometrlc function, f,the relaxation frequency, n the index of correvalues of exponent) ExperImental

18

curve

16 14 12 10 08 06 0 4 02 00 -0 2 -3

-2

-1

0

1

log (frequency

2

3

4

5

in Hz)

Fig 3 Complex susceptlhllty spectrum collected m solution B for Zrcaloy 4 oxldued for 3 months in PWR pnmary water condltlons (log[d(f)] emptysquare,log[x”(f)] asterzsk,cluster model he) LFD p=O3O,q=O67,f,=O13Hz,X,= 165 NDR m=O5O,n=O88,/,= lSOHsx(O)=450

459

Electrochenucal impedance spectroscopy on oxide films latlon for the mtra-cluster process, m the index of correlation for the inter-cluster process and x(O) the amplitude of the dielectric loss The asymptotic limits of the index n are such that n = 0 defines a material m which dipoles remam independent whereas n = 1 refers to fully correlated dipoles m the clusters The asymptotic limits of the index m are such that m = 0 corresponds to isolated identical clusters whereas m = 1 corresponds to an efficient inter-cluster process which leads to a broad cluster dlstrlbutlon This function (equation 5) shows a similar shape to that of the pure Debye relaxation which may be regarded as a hmltmg case where n = 0 and m = 1 Further, It wdl be called Non Debye Relaxation (NDR) For potentially mobile charges, x(j) 1s gven by

/

F14-1

X(f)=L(l+Jf-)

1-

q, 1 + p. 2 -

4, l+J’

1

,

(6)

f, I

where all symbols are identical to those listed above, except that q replaces n and p replaces m to prevent any confusion This function shows a Low Frequency Dispersion (LFD) for f < f, where f, 1s a chstmctlve frequency related to the time constant of inter-cluster process xe characterizes the amplitude of the dielectric loss and 1s related to the effective charge transported from cluster to cluster (cooperative hopping conduction) Further, this function will be called LFD Fzttzng

procedure

The numerical estimation of the Gaussian hypergeometric functions was carried out according to a computer program published by L A Dssado et al [9] Senal and parallel arrangements were tested for several parameter values Only the parallel arrangement led to the shape of spectrum presented m Fig 3 where lines correspond to the cluster model This indicated that the dielectric loss processes ran m parallel As a rule, the fitting of experimental data were carned out using the simplest arrangement As the general shape of spectra corresponded to the LFD one, the fitting procedure began in using equation (6) The index p was determined m the low frequency range of the lmagmary part of x(f) The frequency f, was chosen m a frequency range one order of magnitude higher than the one used to determine p The index q was determined m the high frequency range of the real part of x(f) When only the LFD equation might fit experimental data, the values f,and xc were chosen m order that the best fitting was obtained for medium frequencies This was only the case for the specimen oxidized for 2 months and tested m solutions A and B When the LFD equation alone could not fit the experimental data, the index q was chosen as the highest one m order that the simulated spectrum should be nowhere higher than the expenmental one At this point, expenmental data were fitted m the low frequency range and underestimated m the high frequency one Then the residue between expenmen-

tal and simulated data was calculated for each frequency The residue spectrum m the high frequency range presented a more or less well-defined pattern corresponding to NDR Accorchngly, the fitting of the residue spectrum was carned out m using equation (5) Rough estlmatlons of m and fp were generally available At the end of thrs stage, the NDR simulated data were subtracted from the expenmental data to generate a second residue spectrum corresponding to the LFD part of the dielectric loss Precise estimations of j,, xE and q were then obtained Next, the LFD simulated data were subtracted from the experimental data to generate a third residue spectrum correspondmg to the NDR part of dlelectnc loss Thus, precise estimations of m, f, and x(O)were obtained A recursive fitting of the susceptlblhty spectrum with respectively LFD and NDR equations led to a more and more accurate fitting of experlmental data especially for medium frequencies The fitting procedure was stopped when no change m parameter values was achieved after a recurrent loop EXPERIMENTAL

RESULTS

The complex capacitance spectra obtained m this study exhibited a LFD It has been shown[14] that from a theoretical viewpomt the LFD might correspond to a fractal porous film In this case, the datmctive frequency would be inversely proportional to the solution reslstivlty and as the p and q indexes are purely morphologcal, then values would be mdependent of the solution composltlon The expenmental results presented below rule out the porosity of these oxide films Except for the specimen oxldlzed for 2 months and tested m solutions A and B, both LFD and NDR dielectric losses were identified For the specimen oxidized for 3 months, variations off, and x(O) with the solution pH are presented, respectively m Figs 4 and 5 Figure 5 shows that the level of NDR dielectnc loss has increased with the solution pH Since the level of NDR dielectric loss for the specimen oxidized for 2 months and tested m the solution C (pH = 13 4) was about 06, that IS 25

7

8

9

10

11

12

13

14

PH

Fig 4 Vanation of the dMmctwe frequency .&, of NDR dlelectnc loss wth solution pH for oxide film formed after a 3 month oxldatlon

460

C BATAILLONand S BRIJNET

times lower than the level for the specimen oxldlzed for 3 months, It has been concluded that the NDR dielectric loss for the spe.clmen oxldlzed for 2 months and tested m solutions A and B would be too low

loo1

g 0 5 I-----! “‘& _ -... 04 '.. --__ 9-03 --__ ---_______;_____----02 01

OOL. 7

”

‘.

9

B

s

”

10

11

m 12

“.

’

13

14

PH

Fig 8 Vanations with solution pH of p (0) and q (0) indexes for oxide film formed after a 2 month oxidation. D (B) and q (0) indexes for oxide film formed after month oxidation

I

11

7

8

9

10

12

11

13

14

PH

Rg 5 Varlatlon of amplitude x(O)of NDR dlelectrlc loss with solution pH for oxide film formed after a 3 month

10

oxidation

09 08

07 06

7

6

9

11

10

12

13

J 14

PH

Fig 6 Eqmvalent clrcmt for oxide film formed on ikcaloy 4 m lngh temperature water

Fig 9 Variation with solution pH of n (0) and m (D) indexes for oxide film formed after a 3 month oxidation

25

0 2 nonths+

242,

-

2 22: '\

‘\ ‘.

,’ ‘.

_I’

I’

: I' : I' : ,' ,' ,'

2 I

I’

l

ID-

7

10

ii

14

PH

Fig 7 Vanation of the dielectric constant (E,) wth solution pH Oxide film formed after a 2 month oxldatlon (0) Oxide film formed after a 3 month oxidation (B)

n

3 months 8

9

10

11

12

13

PH

Fig 10 Vanation of the distmctive frequency f, of LFD chelectnc loss with solution pH (0) oxide film formed after a 2 month oxidation and (m) oxide film formed after a 3 month oxldatlon

Electrochemrcal Impedance spectroscopy on oxide films II 1

F t

_.

.._._+A(

*”

11 7

.

8

9

10

11

12

13

14

PH

Fig 11 Vanatlon of the level xs of LFD dlelectrlc loss with solution pH (0) oxide film formed after a 2 month oxldatlon and (m) oxide film formed after a 3 month oxldatlon and therefore hidden by the LFD dlelectnc loss Thus, the eqmvalent clrcmt presented m Fig 6 was assumed to be vahd for all the cases examined m this study The variations of structural parameters (E, , p, q, n and m) with the solution pH are presented m Figs 7, 8 and 9 For the oxide film obtained after a 2 month oxldatlon, the duzctlons of vanatlons did not reverse m the studied pH range For the oxide film obtamed after a 3 month oxldatlon, curves showed “extrema” around pH 9-10 The variations of parameters f, and xE (LFD dlelectnc loss) with the solution pH are presented, respectively, in Figs 10 and 11 For the oxide film formed after a 2 month oxldatlon, the curves showed “extrema” around pH 9-10 For the oxide film formed after a 3 month oxldatlon, decreasing varlatlons m the studied pH range were achieved The shape of variations of all parameters (E, , p, q, n, m,JC and x,) for the films formed after the 2 and 3 month oxldatlons are m opposltlon No physical reason has been found to explain this opposltron and further experimental mvestlgatlons are needed to specify this point, considered for the present to be a casual result

461

Assuming that electrons and oxygen vacancies crossed through the oxide film along gram boundanes, the latter became electronic and ionic conductive paths Smce the oxide films were dc insulator at room temperature, the conductive paths were necessarily fragmented by more or less msulatmg areas This pattern IS equivalent to those of Hollandltes and may account for the LFD dlelectrlc loss of oxide films f, was the mean frequency of charge transltlon between two successive conductive channels through an insulating area In the framework of cluster model, xc 1s related to the effective charge transported As xe was of the same order of magnitude for both oxide films (8-3, see Fig 1 l), f, might be considered as a reverse measure of the protective efflclency of oxide film For the oxide film formed after a 2 month oxldatlon, f, ranged from 2 to 8 Hz For the oxide film formed after a 3 month oxldatlon,f, ranged from 0 1 to 0 3 Hz Assuming that the oxldatlon rate was controlled by the charge transport through the oxide film, these results are m quahtatlve agreement with the weight gam results because the oxldatlon rate was indeed a decreasing function of oxrdatlon time (ozto2g-1 = t-O”) le the oxide film formed after a 3 month oxldatlon was more protective than the oxide film formed after a 2 month oxldatlon However this agreement assumes that the grading of protective eft?aencles of oxide films was not modified between 360°C (temperature for the weight gam results) and 25°C (temperature for the EIS results) Dependence of the dtelectrrc constant on the solution

PH The results presented m Fig 7 show clearly that the dielectric constant E, of oxide films has varied with the solution pH These expenments have been repeated at least twice m such a way that the expenmental results for one pH value were never successive Error bars have been represented m the figures It can therefore be concluded that these experimental results were not an artefact coming from the experimental procedure Consldermg that the lsoelectrlc point (IEP)[17] and the value of the dielectric constant s,[18] for monochmc ZrO, are, respectively, about 65-7 and around 22, It can be concluded from the results obtained for the oxide DISCUSSION film formed after a 2 month oxidation, that the bulk structure of the oxide seems to be related to its The LFD dzelectrrc loss surface structure For the film formed after a 2 It IS well established that the we@t gam law for month oxldatlon, the lower value of E, was 22 7 for oxldatlon of Zlrcaloy 4 m high temperature water 1s pH = 7 The E, value increased as the solution pH a power law t” Hrlth a = 0 25-O SO[lS] In this study, moved away the IEP For the film formed after a 3 month oxldatlon the a was equal to 0 29 Such a power law with an exponent lower than 3 1s generally explamed by a mass lower value of E, was around 20 for a pH around transport along short-arcmts of dlffuslon located at 9-10 A decrease of E, and a shift of the lsolectrlc point to basic pH were observed for this oxide film gram boundanes m oxide films[l6] In the case of non porous oxide films, oxldatlon of It 1s well known that oxldatlon of zlrcomum alloys induces compressive stresses m the oxide films Zr at the metal/oxide interface yields oxygen (Pdhng-Bedworth ratio = 1 52-1 56)[19] Moreover, vacancies V0 and electrons It has been shown using X-ray scattering that comZr+(Zr+1V)t,+2V,+4e(7) pressive stresses became very high at the kmetlc transition (200-6OOMPa)[20] There 1s a relation These charges must cross through the oxide film to between the value of the dlelectrlc constant E, and react with water at the oxide/water interface the level of compressive stresses This relation has 2V,+4e+2H,O-+(O’-);+4H (8) been studied experimentally and theoretically for

462

C BATAILLON and S BRUNET

several oxide structureg21) For all structures studied, E, decreases as the compressive stress increases Therefore, the low value of E, for the oxide film formed after a 3 month oxidation may be explained by a high increase of compressive stresses It is slightly more dtffrcult to explam the basic shift of IEP since knowledge m this field is not clearly understood The results pubhshed by Ardlzzone et al [22] may be pointed out These authors have heated crystalline monochmc zlrcoma powder up to 800°C to dehydrate the oxide surface and have measured the value of IEP as a function of lmmerslon time m the solution They have observed a decrease of the IEP value from 9 1 after a 16 h immersion to 6 5 after a 250 h immersion They have attributed this decrease to a slow hydration of zlrcoma leading to the formation of Zr(OH), on the gram surface On the other hand, the theoretical value of the IEP for monochmc zlrcoma may be calculated by usmg the electrostatic model of Yoon et al [23] and the structural parameters of heptacoordmated Zr+” m the monochmc zmoma[24] The value calculated was IEP = 8 65, which was a more basic value than those generally accepted Therefore, it can be proposed that the compressive stresses m the oxide film have caused a dehydration of the outer layer of the film leading to a basic shift of the IEP of the oxide film Dependence of the ac conductzvrty of the oxrdefilms on the s01ut10npH The results presented m Figs 10 and 11 show that L and xc have vaned Hrlth the solution pH, le the nc conductlvlty of oxide film have varied with the solution pH The following pattern may be put forward to explain this experimental result This pattern brings m hydrogen and proton It IS well known that hydrogen penetrates mto the oxide films and the underlying metal during the high temperature oxldatlon[25] In oxldes, hydrogen may exist as proton, atom or hydride The latter two are actual chemical species but the proton cannot exist m the oxide as a bare partnle[26] It penetrates mto the electronic cloud of 02- to form the defects (OH-), or (OH,), The monoclinic zlrcoma consists of an arrangement of ZrO, polyhedral m such a way that the oxygen sub-lattice IS an alternate layered structure (O,O,O,O,, )[24] In the 0, plane, the oxygen ions are close packed Along the c axa, the oxygen-oxygen distance (a,,_,) IS equal to 2 62A (2 x r,,*- = 2 64-2 72&2fl) Along the b-axa do-o = 2 66A Therefore, the protons can move easily from electronic cloud to electronic cloud m the 0, plane In the 0, plane, the oxygen ion arrangement IS much more irregular Only 2/3 of the oxygen ions are m close contact Thus the protons can move from one plane to another In short, the protons can tunnel from cloud to cloud mslde grams at room temperature The gram boundaries contam oxygen vacancies and electrons coming from the oxldatlon of underlyrng metal whereas the grams contain protons coming from the solution At some areas along the gram boundanes, electrons and protons may react to form hydrogen atoms The gram boundanes are then fragmented mto two electronic conductive channels where electrons move freely, separated by an msulat-

mg area where electrons are trapped m hydrogen atoms (LFD dlelectrlc loss) As the solution pH changed, the proton concentration m the oxide grams change as well, therefore there were more or less hydrogen atoms m the insulating areas and electron transfer between conductive channels was more or less easy This is a rough pattern since the msulatmg areas are not identified and/or located along the gram boundaries and since the proton transport m oxide film depends on crystallographic data which are influenced by a great number of parameters such as impurities or stresses It is, for the present, lmpossable to explain precisely the results presented m Figs 10 and 11 But this pattern may explam qualitatively the relation observed expenmentally between the UC electronic conduction of oxide films and the solution PH

CONCLUSIONS Zlrcaloy 4 oxide films formed by high temperature water oxldatlon have been analysed by EIS It has been shown that the oxide films were compact and exhibited a Low Frequency Dlsperslon (LFD) ddectrlc loss The cluster model has been used to describe the vanatlons of both the real and imaginary parts of the complex dlelectrlc susceptlbdlty over about SIX decades of frequencies A coupling between the bulk oxide characteristics and the solution pH was shown The followmg pattern for the interpretation of the experimental results has been proposed Oxygen vacancies and electrons commg from the anodlc process at the metal/oxide interface are located at gram boundarles forming conductive channels insulated from one another by less conductlve areas containing hydrogen atoms The coupling between the oxide film characteristics and the solution pH IS ensured by (1) a proton exchange at the oxide/solution interface, (11)a direct proton tunnelhng transfer m grams, and (m) a trappmg m msulatprocess) separating the mg areas (cathodic conductive channels Further mvestlgatlons are needed to add more details to thrs crude pattern and are m progress m our laboratory The direct ldentlficatlon of conductive channels and insulating areas 1s very dlflicult since then estimated sizes are very small (m the order of nanometers) and therefore only made possible by STEM Unfortunately, the STEM mvestlgatlons required refined preparations of samples and a high vacuum which may destroy or greatly modify these structural Items Because of this, the use of EIS ISJustified as a non-destructive method for the mvestlgatlon of charge transport through the oxide films in the field of corrosion science

REFERENCES I B Cox, In

Advances

m Corrosion

Science

and

Tech-

(Edlted by M G Fontana and R W Staehle) Vol 5, p 295 Plenum Press, New York (1976) 2 C Batadlon and S Brunet, EurocorrPf, Budapest 21-25 October, Vol II p 670 (1991) nology,

463

Electrochenucal Impedance spectroscopy on oxide films 3 A K Jonscher, m Advances m Research and Develop ment, (Edited by G Hass and M H Francombe) Vol 11, p 217 Acadermc Press, New York (1980) A K Jonscher, Physics of Dlelectrw, p 22 Inst Phys Conf Ser No 58 (1980) A K Jonscher, Efectrochrm Acta 35, 1595 (1990) L A Dlssado and R M Hdl, Nature 279,685 (1979) L A Dissado and R M Hill, Phdos Mag B 41, 625 (1980) L A Dissado and R M Hdl, Proc R Sot Lond A 390,131(1983)

10 11 12 13 14 15 16 I7 18 19 20

L A Dissado, R R Nigmatulhn and R M Hill, DynamIcal Processes rn Condensed Matter, p 253 Wiley Interscience, New York (1985) L A Dissado and R M I-Id], J Chem Sot, Faraday Trans 2 80,291(1984) E L Kitts and J H Crawford, Phys Rev Lett 30, 443 (1973) H U Beyeler, Phys Rev Lett 37, 1557 (1976) L A Dissado, Physlca ScrIpta Tl, 110 (1982, L A I)lssado and R M I-MI,Sohd State lonzcs 22, 331(1987) R M I-Id1and L A Dissado, Sohd State Ionlcs 26, 295 (1988), J Appl Phys 66, 2511 (1989), B Sapoval, Sohd State Ionlcs 23,253 (1987) G A Eloff, C J Greyhng and P E VdJoen, J Nucl Mat 199,285 (1993) G P Sabol and S B Dalgaard, J Eiectrochem Sot 122,316 (1975) J Randon, A Larbot, C Gulzard, L Cot, M Lmdheimer and S Partyka, Collards and Surfaces 52, 241 (1991) A K Vyh, J Mat SCI Lett 7, 245 (1988) B Cox, m Advances an Corrosron Science and Technology, (E&ted by M G Fontana and R W Staehle) Vol 5, p 314 Plenum Press, New York (1976) F Garzarolh, H Sadel, R Tricot and J P Gras, Zw

log o or logfand the other concerning the vanation of the phase angle $ vs log o or logf Three Types of vanatlons of both log ljZll and Q,vs log w or logfhave been denved Let us consider relations (I) and (2) C(w) = C, + ACow)“-’ = llClld

cm4 - Z( + co)] =

,& =IIZlle-N

IIZII = -&

and

Q = i+

0

Inter SCI 70,483 (1979)

24 D K Smith and H W NewkIrk, Acta Cryst

18, 983

(Al)

The meaning of symbols 1s listed m the nomenclature From the relation (l), it follows that AC~i{(n-l)~]&-L tan 0 = Cm+ACco[(n-1)ik-l

(A2)

Next, the followmg denvative 1s obtained from the second relation (Al)

d4 -=-=-dlogw

d0 dlogw

0 d tg 0 1 + tg* 0 do

(A3)

From (A2) the followmg expressions are denved

1+tan2 0 =

(A4)

and A M Grade) p 395, Amencan Society for Testing and Mater&, Phdadelphla (1991), and J Godlewslci and R Cadalbert, Proceedmgs of the lnternatlonal Sym-

G Cludictumo, A Golemme and M 22 S Ardzone, Radaelh, Croatrca Chenuca Acta 63, 545 (1990) 23 R H Yoon, T Salman and G Donnay, J Co/lords and

(2)

Then

comum m the Nuclear Industry Nmth lnternatlonal Symposrum, ASTM STP 1132 (Edited by C M Eucken

posrum m Nuclear Emwonment MATERIALS CHEMISTRY ‘92, p 3, Tsukuba, Japan (1992) 21 G A Samara, J Appl Phys 68,4214 (1990), and Phys Rev B 13,4529 (1976)

(1)

and

C,AC s,{(n - 1) i]c#-’ ~=(n-l)(c_+Acco{(n_l)~~-l~

(A5)

Finally, substltutmg (A4) and (A5) into the relation (A3), it follows that db d log o

(1965)

25 B Cox, m Advances In Corrosion Science and Technology, (Echted by M G Fontana and R W Staehle) Vol 5, p 343 Plenum Press, New York (1976) 26 T Norby, m Selected TOPICS m lflgh Temperature Chewstry (Edited by 0 Johannesen and A G Andersen) pp 101 Elsevler, Amsterdam (1989) 21 The Oxide Handbook, second e&on, p 1, G V Samsonov, IFI/Plenum, New York (1982, Handbook of Chenustry and Physics, 54th edition, F-194, R C Weast, CRC Press, Cleveland (1974), and Nouveau Tralte de Chrnue Mmerale, P Pascal, Tome XIII, p 314, Masson et c”, Pans (1960)

APPENDIX The purpose of this se&on 1s to ldentlfy on the Bode representabon, the universal law for the response of ddectnc matenals The Bode reprmtatlon consists of a set of two graphs, one conazrnmg the vanation of log llZ/ vs

C,AC sm[(n-1)

.

:lo”-’

I

C2,+2C,ACco

(A6) The denonunator of the above expression IS equal to ljC\j’ The vanatlon of llClj* with respect to w”-’ hn log/log scale 1s presented m the Rg Al (hne) Let us consider the followmg loganthmlc derivative of llCjj2 d hllCll* =_-co-’ dllCl12 d log on-’ llC1j2do”-’ on-’ + 2AC*w*‘“-” =

(A7) w”-’ + AC*W~‘“-~’

As w”-l vanishes, d log/lCI12/d log w”-’ tends towards _ zero, re I(C(I’ may be approximated to a constant

C BATAILL~N and S BRUNET

464

For each Type, the vanatlons of both 6 and IogllZIJ with respect to log 0 may be approximated to a simple analytlcal expression

Using the approxlmatlon A6, it follows that

dQ

-z(n-

1) g

d log w

Type I for IIC11’m the relation

sin

UJ”-’ > 0 (AlO)

m

for this Type C, % AC&‘-’

Vanatton of logl/C/1* (hne) and of the three approxtmate equations (dotted line) with respect to log UP-’ (I) logllC/l~ = log c:, (II) logllCl1~= log 2C, AC{ 1 + cos[(n - 1)~ - 2]}0”- ’ , (III) loglJC11’= log AC202’“-” (As the exponent n - 1 1s negative, the high frequencies are located on the left of the graph ) logl(Cl12 x log Ci As ID”-’ tends towards mfimty, d log((CII’/d log w”‘-’ tends towards 2, I llCl12 may be approximated to a parabolic equation expressed wrth on-’ log AC202’“-1~ These two approxunatlons correspond m log/log scale, to the strmght lmes labelled, respectively, I and III m the Rg Al It can be seen on this figure that for mediyrn;pes of UP-‘, IlCll’ may be approxyty equation expressed with i:gllc;2 m log Constant UP-’ (straight line labelled II) To determme this linear equation, the value of on-l for whch d loglJC112/dlog w”-’ IS equal to 1 has been calculated ze o$’ = C-/AC This hnear equation IS the tangent to the curve logllCllz vs log w”-’ at o$- ’ So the Constant IS equal to 2C, AC{ 1 + cos[(n - 1)42]} Then, for medmm values of co”-‘, llCl12may be approximated to the followmg hnear equation logllC112z log 2C, AC{ 1 + cos[(n - 1)742]}w”- ’ Summamed, the vanatlon of logllCl12with respect to log u”-’ may be approxunated to the followmg three sunple equations

oJ&’
C,~2AC~os[ln-l~*,Z]w”-~

This ISequivalent to

log UP-1

UY-’
and

logllCJ12z IogC’,

(g-1

_“-

O”O_L

I

1

4 1 and o”o-‘* 2 co

so

and then

d4 --e*nd log o

II

where 1x I represents the absolute value of the real x In other words, for n close to 1 the vanatlon of 4 with respect to log o is of the order of a few degrees per decade from 90” Moreover from the relation Al, it follows that 1 IlZll % -cc WC,

Type I

o-1

This Type of vanations corresponds to those of a quaslpure capacitance (C,)

< w;l/ll: on-l

logllClJ2 % log ZC,AC

Type II Type

uJ&:
logl/CJ12 m log ACzw2”-”

11

Type III

where w;,; ’ and o~;,j,: are, respectively, the mtersectlons of straight lines (I and II) and (II and III) r-1

%I

=

Using the approximation Type II for llC/12m the relation A6, it follows that

d--F d4

n-l

2AC{l+c:&1);]~=2{l+co~-l);]}

L48)

The varlatlon of 4 with respect to log w IS hnear From the first relation Al, It follows that

and 2C,{l

+co{(“-

I,;]}

4mf =

(A121 = 20”;~‘(1 +:.(n

- 1) ;,)

(A9)

This approximation is only valid m a range around o0 To estimate the width of this range, the relative error E(w”-I)

465

Electrochemical Impedance spectroscopy on oxide films

T y p e 111 Usmg the approximation Type Ill for llCll' in the relation A6, A6, it tion it follows follows that that

has been calculated in using the followmg relation

Etco"-') =

7{

~17

d._.._~0 ~ (n - 1) ~-~ sin (n - 1) to d log to 2

= [2C® aCco " - i - C~ - AC2co'('-i~l

(AI3)

IICll~ Remembenng that AC = Coo/co[-t and using x = oY'-i/col-i as dimensionless pulsation, the above relation becomes 1 2 x - 1 -x21

E(x) l+2co

(n-l)

(AI4)

x+x 2

The variations of E(x) for several values of n are presented in the Fig. A2 If values of E(x) lower than 20% are cons~dered to be quite satisfactory to vahd the above approximaUon, the width of the vahdlty range for co zs 0 6 decade for n = 0 but rises to 3 84 decades of frequenoes for n=08

l-m

> 0,

(A15)

for this Type A C o J ' - i ~ • Coo which Is equivalent to co,,- l/oto- 1 ~. 1 Then A~sin[(n_l)

2]coi_" < C ~ c o , _ , co[-' AC = co,,-1 '~ 1

So d0/d log co ~ In - II, le the variation of 0 is of the order of a few degrees per decade for n close to 1 If n ]s not too low then cos[(n - 1)x/2] ]s of the order of unity and C® can be neglected in the denominator of the relation A2 So in this case tan 0 ~ tg~(n - 1) ; ] which imphes that ~ = n(lr/2) From the first relation A1 IlZll ~

1

ACCO"

~ co-"

(AI6)

n-08

~n=06

/

n=04 n=02

= x uJ

2

1

lo9

I0 x

Fig A2 Vanatlons of the relative error E(x) with the dimensionless pulsation x for the approximation Type II Iog[llzll3 I

This Type of vanauon with respect to log co corresponds to those of a Constant Phase Element (cpe) The complete variations of both IogllZl[ and ~ vs I o g f are presented on the Fig A3 The three types of variations can be seen For high frequencies, ~b tend towards 7r/2 whilst [IZll evolves like 1If (Type I) For medium fiequencles, ~ evolves hnearly with respect to l o g f w h l l s t IIZll evolves hke f - ( ' + u/, (Type II) For low frequenoes, 0 is nearly frequency independent and equal to n0r/2) whilst IIZU evolves hke f - " For a fixed frequencies range, the Type of variations observed depends on the relatwe value of coo with respect to the frequenoes range If coo ]s higher than the maximum frequency of the range, Type I is observed If coo hes in the frequencies range, Type II is observed and if ¢oo is lower than the minimum frequency of the range, Type III is observed From a physical viewpoint coo characterizes the ratm between the response of the defectless material ( C J and the dielectric loss coming from the defects m the matenal (AC) Phase

angle

(a)

I

I III

I II

IIl

II

I

log[frequency]

IogEfrequency]

Fig A3 Variations with frequency of Iogl]Zll (a) and the phase angle ~b (b) calculated from the followmg relation IIZU e -r* = wlthn=08, C®= 100andAC=05

1

Jo)[C~ -J- AC(Jos) n- 1]