Diffusional growth of multiphase scales with variable composition

Materials Science and Engineering, A I20 (1989) 69-75

69

Diffusional Growth of Multiphase Scales with Variable Composition* M. DANIELEWSKI Institute of Materials Science, Academy of Mining and Metallurgy, Al. Mickiewicza 30, 30-059Cracow (Poland) (Received March 14, 1989)

Abstract

The paper presents a quantitative theory for the growth of multilayered scales formed of phases with variable compositions, molar volumes and diffusivities. The theory enables the determination of moving-boundary state growth kinetics for the scales formed of phases showing high non-stoichiometry. It completely describes the multilayered (multiphase) planar scale growing on pure metal and allows for the calculation of the total as well as partial reaction rates, the ratio of thickness of the phases formed and the metal concentration distribution in the scale. I. Introduction

The Wagner theory of metal oxidation and other classical developments of the parabolic rate constant for the quasi-steady state growth of a single- as well as multilayer scale on metals neglect the effect of scale thickness growth on the diffusional flux of metal. The F e - F e j _ y O - F e 3 0 4 Fe203, C u - C u 2 0 - C u O and C o - C o O - C 0 3 0 4 systems are classic examples of the formation of a sequence of oxide layers in the scale. Quantitative and qualitative theories for double- and multilayered scale formation have been presented by a number of authors [ 1-7]. In all of these theories, the authors have assumed that the growth of each layer is diffusion-controlled. This present treatment, however, differs from their assumptions. The Fromhold, Hsu, Kofstad and all other classical developments of the parabolic rate constant for the growth of a multilayered scale on metals [2-7] neglect the effect of scale thickness growth on the diffusional flux of metal (i.e. they

*Invited paper. 0921-5093/89/$3.50

neglect the moving-boundary state growth and assume the quasi-steady state growth) and they do not allow for the calculation of the metal (defect) distribution in the growing scale. Hirth and Rapp [1] presented modified equations for the parabolic rate constant for highly defected monolayered scales with large variation in molar volume. However, they used these equations assuming the defect distribution in the growing scale, which limits the possible use of such expressions. Such a simplified treatment may lead to high systematic errors. A good example is the F e Fel_yO-Fe304 system where all the classical treatments neglect the effect of wustite composition change (at 1473 K its composition through the scale varies in the range Feo.9450 to Feo.8440) while they include the detailed analysis of the effect of the reaction at the FeO-Fe304 interface (adequate compositions of FeO and Fe304 at this interface are respectively Fe0.g~O and Feo.751O). This present treatment includes the movingboundary state growth conditions for the scales formed of phases showing high non-stoichiometry [8, 9]. It is not necessary to make any assumptions regarding properties and the diffusional mechanism in the layers formed. For simplicity the treatment is illustrated in Fig. 1 for triple-layered planar scales formed by the diffusion of metal through the scale to react with oxidant X at interface 13. The theory consists of two "independent" parts: (a) the model of the reaction, which relates the rate of the total as well as all partial reactions (and their constants) with movement of interfaces in the product layer; (b) the transport equation which allows for calculation of fluxes, distribution of diffusant in © ElsevierSequoia/Printed in The Netherlands

70

d~t

~)l(x) =

= x'

/)12 = X2

CMe x)

Me[: rIO

X"

=

=o

i

dXl

= x~

vi, = d--Y~;,

t

!

(1)Me,iX

"

jhe(x))

,

(2)Me,2X

J~t¢

)

i

¢

) ) * - - - - x~

X

e.

! x3

i 1•

) )

)i

( .( i

)

)i

X2

i

!

t

l

Interfacex

Interface°

i

Interface

I(x) a!~~)

a~x

pOx~=/,o

PL =P

P~: =P'

aM.(X)

|

X3

ao

aM~= 1 a ° .

I

l I

)i

XI

X2(g)

J~io )

)

: X

4

~- X t

(3)Me~3x

It

I) .

x,

!

-!

VI3 = X

InterfacC 12

Interface3 13

P~, = f

P~ = f

J

J

!

,

i

,

i

a~e [a~,

a~e i a~e

a~. i

Fig. 1. Moving-boundary state growth model of the simultaneous formation of triple-layered scales on pure metals by diffusion of Me through the scale to react with oxidant X at interface 13.

the product layer and rate of movement of isocomposition surfaces (including interfaces I'). The final equations integrate these two parts.

2. Model Figure 1 shows the schematic diagram of this system. The parabolic rate law is usually expressed as dX k' X' = - - = dt X

t

(2)

= kp = X'X

For the multilayered scales, the parabolic rate constant for the growth of the ith layer is given by t

t

,

t

k i = XiJ(**. = ( x ; - x i _

Vl(x)

Vl(x)

Of(x)

K~ = X~ (excl)Xi = {x~ - x ; _ , (excl) }(xi - xi_ I) (4)

(1)

where k' is expressed in cm 2 s-i. It follows that the total parabolic rate constant is a function of the rate of vi~ of Mea3X-Xz(g ) interface movement and the total scale thickness is given by k"

The present treatment also allows for the calculation of the exclusive growth rate of scale-forming layers, K~. In such a case the metal flux through the ith layer is a result of the reaction at the Mea0.-oX-Mea~X interface only (jo-1)°= 0), growth of oxide Me~X is a result of oxidation of the lower oxide and the parabolic rate constant for the exclusive growth of the ith layer is

• l )(Xi

--

Xi--

1)

(3)

3. Transport equation The total metal flux for the compound with a high concentration of defects in the cationic sublattice, making no assumption concerning defect structure, is given by the equation JMe = --/~Me VCMe

(5)

The chemical diffusion coefficient, /~M,, may be related to the self-diffusion coefficient, DM,, by the

71 and defect distribution in the scale to the rate of the reaction:

formula [ 1O] 0 In aMe /~Me = DMe 0 In CM~

(6) jM,(X) = CMe(X)-dX ~

where aMe is the thermodynamic activity and CM~ the molar ratio, (CMe = N C M e , N being the concentration of all atoms per unit volume). The incremental advance of interface 13 represents moles of X incorporated into M%3X at interface 13 as a result of metal flux j ~ , . The incremental advance of isocomposition surface I(x) represents moles of Me incorporated into the scale as a result of its thickness growth (adjustment of the metal distribution in the scale). For the case under discussion, Fick's second law has the form OjM, (X)t~X , =

where x' is the rate of isocomposition surface movement. It follows that

From eqn. (8) the metal flux for any distance, x, from the Me-MeaiX interface is given by

X' dCMe

Thus

combining

eqns.

(9)

and

(13)

and

cM,(X)

(14)

f CMedx 0 where ?Me is the average metal concentration. From eqns. (5), (7), (13) and (16), the transport equation now has the form

ODM¢ OCM~ /~M~O2CM~ 1 dX OCM~ (~X OX (~X2 X X dt Ox

0CMe(Y) - # fit)

(10)

Oy

where y =x/X. Thus from eqns. (7) and (10), rates of the isocomposition surface movement (V~(x)= x') and scale growth are related by

CMe(X)

(15)

(9)

As the concentration profile CMc(X) is timeinvariant when plotted against x/t 1/2, it follows that

x X 2 OUl~c~4¢=CMe(X)

02CMe

[ x dX dBMo\ | dt +X---d-x-x) OcM~ /

-

~2CMe 0CMe Ox 2 + F ( x ) ~

(11)

For the diffusion-controlled growth of oxides (sulphides) it is necessary to relate the metal flux

=0

(16)

The boundary conditions for the system are given by (Fig. 1)

cMo(O) = c~o cMo(X) = c~o ~CMe

=0

(13)

=XXcMo(X)

XCMc= ~ X dCMc+ ~g CM~dx gives

cre

O(x/X) _ 1 Ox Ot cie(x/X) X Ot

JMe(X)

X

It is a second-order differential equation with variable coefficients

CMe(X)

f

(12)

(8)

djMe(X) = X' dCMe(X)

JMe(x) = J r ~ +

X' = X x '

(7)

0x

= CM~(X)X"

By introducing eqn. (12) into eqn. (1 l) the rate of movement of any isocomposition surface, I(x), is now expressed by

jM+(X) =j~+ XCM+(X) _j~+

OCMe(X)otx = "~'Ox cMe(*)OCMe(X)ox t

= x ' dCMo(X)

cm~(x)

OX x=O

:

J~ BMo(c~o) j~e

OCMo OX x = X

--

c~e

/~Me (C ~e ) CM: (X')

72 The solution satisfying eqn. (16) is x

Ferrous oxide. Deviations from stoichiometry in FeO (its dependence on oxygen pressure and temperature) are described by [11]

x

CMe(X)=fexp(-fF(x)dx)dx 0

(17)

( ~ - 9)6(2--2-~,)-- e x p ( - 1 . 5 ]npo 2

0

which, by introducing boundary conditions for x = 0, has the form x

CM¢=

96544.5 ) T +481227

JMe .1, r [ f/ (J~e) 2 x -- ~ J exp -- j k,k~ ) 2 / ~ M e 0

The metal concentration in Fel_yO is N~(1 --y)

0 CMe - -

+ ~x (In/~Me)) d x ] dx all- C ~vle.

(18)

From boundary conditions for x = X, k' and ?M~(X) values are given by the following equations which have to be satisfied for all values of J~e"

{

15 e /-~- ~vle C- M -e ( S )

1,2 YMe

x

k,~Me(X)2 jDMe

= exp

dx

0

- _I d(ln/~iMe)},

(19)

DMe x -

(21)

x

cE,o) ff,

x

VFeO

=

f(po~T)

(22)

where 17F~o is the molar volume. Its dependence on the non-stoichiometry can be approximated by the equation 1TEe° = (1 --Y)MFe + Mo bl(1 - y ) + b 2

(23)

where MFe and Mo are the relative atomic masses of iron and oxygen and b l and b2 are the coefficients of a linear equation describing the wustite density as a function of its composition. The data of Chen and Peterson [ 12] have been utilized in order to describe the self-diffusion coefficient of iron in Fel_y O as a function of nonstoichiometry. The empirical equation

DEe ----B l y 2 -I.- B2y + B3

=

0

0

/d, lnMo,]

approximates DM¢ for the temperatures 1073, 1273 and 1473 K,

/~Me

dx

(24)

(20)

z3~o Equations (19) and (20) can be solved numerically without any restrictions concerning the dependence of the diffusion coefficient, the molar volume and the metal concentration on oxidant activity.

/

1~73 K 1.03 "~" 1.02

K~/K'~Cm/

K~ -quasi-steady-stafe-growfh~ / K'p(A)-moving-boundory-stot~rowfh/ with const scale /

/

X 'v

4. Applicationand discussion Complete testing of the proposed equations was not the purpose of this paper. However, for the F e - F e l _ y O - F e 3 0 4 system, the diffusion as well as physio-chemical properties of the phases formed necessary for a partial test of this theory are available. The following data were used for the present work.

1.01

,

10-6

,

10-s

I0-~

Po2 , Po Fig. 2. The calculated ratios of the kp of the oxidation of iron for the moving-boundary and quasi-steady state growth models.

73 1

/

Fe

I

I

I

Iron

Fe~J,/,'

Fe30~,

" ~ . O ~ " ~

10.6

o a Q---

1(]6 -

T

~

~1o-7

M.M, Davies et o [ , 1951 3 Poidossi, 1958 R.D Shaw, 1972 ~. _ - R D Show,1974 (colcut~ed) O~m-this work

ld 7 _ 'T

104

~ / / I 100 101

I

o

I

102 103 Po~/ PFe/FeO

\.\A

Fig. 3. The calculated kpH values of the oxidation of iron as a function of oxygen fugacity at 1073, 1273 and 1473 K (moving-boundary state growth model).

10-9

'\ •

MM

10-10

Magnetite. The average density of magnetite (5.18 g c m - 3) was used. All values have been corrected for thermal expansion and a value g T = l . 5 X 10-Sdeg -1 has been used to correct the molar volume and the density. Finally, the most recent point defect formation and diffusion data [13] have been used in this work.

5. Results

From the comparison of k' values calculated for the moving-boundary and quasi-stationary state growth approximation of wustite scale on iron, it follows that neglecting the movingboundary condition causes a systematic error which is a function of the oxygen partial pressure and has a maximum value (about 4%) at the FeO-Fe304 oxygen equilibrium partial pressure (Fig. 2). Figure 3 presents the dependence of kp of iron oxidation on oxygen fugacity defined by Poz Po2 (Fe-FeO)

A

&'\ Oxidation of wustite:

ao2

I

oxidation:

(25)

/kDavies, 1951 × - C.E. Birchenot, 1953 ,A--3.Poidossi. 1958 ----- - this work

"~

°\

•

I

I

I

I

I

07

o,B

09

10

1.1

1/T" 103, K-1

Fig. 4. Comparison of experimental and calculated values for the parabolic rate constants in the iron-oxygen system as a function of temperature.

It demonstrates that k~ is a complex function of oxygen pressure and cannot be described by simplified formulae. In Fig. 4, the results of k' and K' calculations are compared with experimental data [ 14-17]. As is evident, the present theory not only agrees with experimental data but allows for a more precise description of scale growth on metal. Table 1 shows the calculated thickness ratio and k' values of iron oxidation at 1073 and 1273 K with Po2 = 10-6 Pa. For comparison, the values calculated by Yurek et al. [2] have been included. The comparison of calculated k' and Xi values with those of Yurek et al. [2] and the measured

TABLE 1 Calculated p a r a b o l i c rate c o n s t a n t s and v a l u e s o f Xr,~oJXv, ,_,o for the oxidation of iron with Poz ----10-6 Pa Temperature (K)

1073 1273

k' = k~otal (10 -8 cm 2 s - l )

K~ = K~(Fe 304) (10-1° em 2 s - t )

XI/X2 = Xveo/XF=~04: '

This work

Ref 2

This work

Ref 2

This work

Ref 2

1.18 19.7

I. 76 21.13

24.1 685.7

0.15 19.85

97.5/2.5 96.7/3.3

99.91/0.09 99.04/0.96

74

disorder in Fe304) results in a relatively strong dependence of Ea on nonstoichiometry. As can be seen, the present theory allows for an accurate prediction of the rate of the reactions in the iron-oxygen system. Thus, when utilizing available experimental data for reactions in this system this theory proves to be consistent with the most recent defect diffusion data. This would seem to demonstrate the validity of defect models in wustite and magnetite.

35 E~

~3c

I

(a)

0.06

I

I

008 Y

I

I

~0

60

\ (b)

I

0

i

O0O5

Fig. 5. The comparison of the average (T= 1073-1473K) activation energies of self-diffusion and k' for (a) wustite growth on iron and (b) magnetite formation on wustite (K').

k' and X at 1073K [18, 19] where the ratio XFe3oJXFeoequals 0.026 (as opposed to the value 0.001 calculated in ref. 2) demonstrates how the present theory provides values which are approximately the same as those obtained experimentally, Differences in the calculated k' values for the exclusive growth of magnetite may also be a result of differing defect diffusion data taken into the calculations. Equations (18)-(20) allow for the calculation of the average as well as local activation energies of reactions in the multiphase systems. In Fig. 5 the average (T = 1073-1473 K) activation energies of self-diffusion (in FeO and Fe304) and oxidation of iron and wustite are compared. As can be predicted that the complex defect structure in these oxides (defect clusters in wustite and Frenkel-type

6. Conclusions (1) A quantitative theory for the growth of multilayered oxide and sulphide scales on pure metals has been developed by considering the moving-boundary state growth of the product layer. (2) The theory completely describes the multilayered scale growing on pure metal and allows for the calculation of reaction rates, the thickness ratio of the phases formed and metal concentration distribution in the scale (e.g. k', k~, K~, X~/X and (3) The present theory may be adopted for other reaction systems. (4) The general form of the proposed equations allows for the calculation of diffusion properties of phases formed when the rate of the reactions, the layers thicknesses and the component concentrations in the product layer are known. (5) The theory is consistent with all available experimental data in the iron-oxygen system. It confirms the validity of wustite and magnetite diffusional data.

Acknowledgments This work has been supported by the Ministry for the Promotion of Progress in Science and Technology under Contract 6.6.64. The authors wish to thank Professor Stanistaw Mrowec for his helpful comments and indefatigable sense of humour during moments of authorial cation crisis.

References 1 J. P. Hirth and R. A. Rapp, Oxid. Met., 11 (1977) 57. 2 G. J. Yurek, J. P. Hirth and R. A. Rapp, Oxid. Met., 8 (1974) 265. 3 P. Kofstad, in High Temperature Corrosion, Elsevier, London, 1988, p. 45. 4 C. Wagner, .4eta Metall., 17(1969) 99.

75 5 H. Schmalzricd, in G.-M. Schwab (ed.), Reactivity of Solids, Elsevier, Amsterdam, 1965, p. 204. 6 A. T. Fromhold, Jr., and N. Sato, in Proc. 1st Int. Conf. on Transport in Non-Stoichiometric Compositions, Mogiluny, August 27-30, 1980, p. 81. 7 H. S. Hus, Oxid. Met., 26(1986) 315. 8 M. Danielewski, The theory of multiphase scale formation on metal, in Proc. DIMETA-88, 5 - 9 Sept. 1988, Balatonfiired, Hungary, in press. 9 M. Danielewski, Diffusional growth of scales with variable composition, in Proc. DIMETA-88, 5 - 9 Sept. 1988, Balatonfiired, Hungary, in press, 10 G. E. Murch, Atomic Diffusion Theory in Highly Defective Solids, Trans Tech, Aedermannsdorf, Switzerland, 1980, p. 109.

11 M. R~kas and S. Mrowec, Solid State Ionics, 22 (1987) 185.

12 W. K. Chen and N. L. Peterson, J. Phys. Chem. Solid& 36 (1975) 1097. 13 R. Dicckmann, M. R. Hilton and T. O. Mason, Ber. Bunsenges. Phys. Chem., 91 (1987) 59. 14 R. D. Shaw and R. Rolls, Corrosion Sci., 14 (1974) 443. 15 R. D. Shaw, Ph.D. Thesis, University of Manchester, 1972. 16 J. Paidassi, Acta Metall., 6(1958) 184. 17 M. M. Davies, M. T. Simnad and C. E. Birchenall, J. Met., 3 (1951) 889. 18 J. Paidassi, Rev. Metall. (Paris), LIV (1957) 569. 19 A. G. Goursat and W. W. Smeltzer, Oxid. Met., 6 (1973) 101.