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Diffusion-controlled growth of solid solution scales on nickel-cobalt alloys
Corrosion Science, 1976, Vol. 16, pp. 57 to 69. Pergamon Press. Printed in Great Britain
D I F F U S I O N - C O N T R O L L E D G R O W T H OF SOLID SOLUTION SCALES ON NICKEL-COBALT ALLOYS* B. D. BASTOW,D. P. WHITTLEt and G. C. WOOD Corrosion and Protection Centre, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester, England Abstract--Cation concentration profiles in solid-solution scales on Ni-Co alloys have been calculated by making use of Wagner's theoretical analysis. As a result of the computation method used, it becomes unnecessary to incorporate experimental values in the calculation for either the parabolic rate constant, kp, or the boundary conditions of the cation profile. The agreement between the independently determined theoretical and experimental cation profiles provides strong sui~port for Wagner's model. The calculated variation of kp with alloy composition is also in agreement with the experimental data. I. INTRODUCTION A THEORETICALdescription of binary alloy oxidation for those systems where a solid solution scale is formed has recently been developed by Wagner 1 and modified by Coates and Dalvi. * This general treatment shows that the segregation o f one of the scale components towards the scale-gas interface, observed previously by Wood and Ferguson 3 in scales on Ni-Co alloys, is related to the inequality of the cation diffusion rates in the scale. Various workers have recently applied Wagner's theoretical treatment to a number of alloy systems ~-7 for which the conditions for the application of the model are fulfilled. However, in all these cases it has been necessary to use data from the experimentally measured profiles, together with the experimental value for the parabolic rate constant, to establish the boundary conditions for the calculation of the theoretical cation profile. Consequently the agreement found between the theoretical and experimental profiles partly arises because the boundary conditions are the same and these results do not constitute a complete test of the validity of Wagner's model. In order to carry out a more rigorous test of the model, it is necessary to determine as many as possible of the required boundary conditions independently of the experimental profile. The profile calculated by combining these conditions with the equations describing the profile shape can then be compared with the experimental data. Agreement between the two profiles and the experimental and calculated values of the parabolic rate constants would provide a more satisfactory demonstration of the applicability of Wagner's model. It is often true in a diffusion controlled oxidation reaction that interdiffusion in the alloy is relatively unimportant in comparison with the cation diffusion in the scale. Recent work by the present authors a has demonstrated that when this assumption is valid then the boundary condition for the cation profile at the alloy-scale • Manuscript received 4 March 1975. "l'Departmentof Metallurgy and Materials Science, University of Liverpool, P.O. Box 147, Liverpool, England. 57
58
B.D. BASTOW,D. P. Wmrrt.E and G. C. WOoD
interface can be estimated froth the alloy composition if the ratio of the cation diffusivities in the scale is known. The necessary preliminary condition is satisfied for a number of systems to which the Wagner model is applicable a and in particular for the Ni-Co alloys considered in the present work. Combining this boundary value with the thermodynamic description of the exchange reaction between alloy and scale, a value can be obtained for the oxygen activity at the alloy-scale interface. Since the composition of the oxidizing gas is known, the oxygen activity at the scalegas interface is also defined. Hence three boundary conditions are known independently of the measured profile and these are sufficient for the solution of the equations, given below, which describe the interdependence of the cation and oxygen activity profiles in the scale. 2. THEORETICAL DESCRIPTION AND SOLUTION OF EQUATIONS The original equations given by Wagner 1 apply to a general case which is described in terms of the activities of the components in the scale. However, the requisite thermodynamic information is often unavailable and in order to calculate the cation profile across the scale a relationship between the cation concentration and the activity in the scale must be assumed. In the case of the (Ni, Co)O mixed oxide there is some indirect evidence that the solid solution exhibits ideal behaviour. The position of the phase boundary between the (Ni, Co)O solid solution and the two phase field (solid solution ÷ CoaO4) is consistent with approximately ideal behaviour 9 and a comparison of tracer and interdiffusion data in mixed oxides shows that the thermodynamic factor (1 ÷ t~ In 7i/t9 Ni) in the Darken equation is close to unity. ~°,u In addition, Dalvi and Coates 4 have already noted that related pseudo-binary oxide systems form ideal solutions. The equations are then considerably simplified and, in the notation of Dalvi and Coates, 4 they become:
d2~
--
k lool,v (1
dlnao dy where
y
=
E
fd~12
-- l/v)
--y
[p -- (p --
p
1)~]~d~
=
O,
Jdy
k' ' dE] 1 ~ a o l / v - ( p - 1)~y P - - ( P - -
~'
(1) (2)
1)
x/x~, the ratio of the distance from the alloy-scale interface to the total
scale thickness, is the equivalent cation fraction of component B in the scale, a 0 is the oxygen activity, k' = k/D °, the ratio of the corrosion constant to the diffusion coefficient of B in BO in equilibrium with oxygen at unit activity. The meaning and interrelationship oftbe symbols p, {3and v is outlined below in equations (5-10). Boundary conditions of the oxygen activity and cation profiles are indicated by a superscript prime' and double prime ~ for the alloy-scale and scale-gas interfaces respectively.
Diffusion-controlled growth of solid solution scales on nickel--cobaltalloys
59
A third relationship is necessary in order to obtain acomplete solution of equations (1) and (2) in terms of ~, ao and k' and this is provided by a mass balance at one of the scale boundaries. Since the balance at the aHoy-scale interface has already been used in the estimation of the inner boundary condition, ~',s the third equation is obtained by considering the formation of scale at the scale-gas interface, in order to avoid any inconsistency in the treatment. This results in :4 d~] k' ~" (1 -- ~" )(p -- 1) dyy y=l = p ~"-X (a")x/"
(3)
The most convenient way to solve this equation is to estimate k' by assuming that the oxygen activity profile in the scale is not very dependent on the gradient of the cation profile. Then equation (2) can be integrated between y = 0, ao = a' a n d y = l, a0 ----a~ to give: k' = v~[-x [p -- (p -- 1)~] [(ag)~'v -- (a~)X/,l,
(4)
where ~ is the average value of ~ in the scale. If diffusion in the alloy is negligible it can be assumed that ~ = N ° where N ° is the bulk alloy composition. Using a trial value for ~" the magnitude of [d~/dy] y = 1 can be estimated and writing this gradient in finite difference form, a value for ~ after the first interval can be calculated. The remainder of the oxygen activity and cation profiles can then be calculated using equations (1) and (2) in finite difference form, the result being independent of the step length used. It is emphasized that the first values of k' and ~" used are only approximations. The values must be subsequently modified, k' such that the oxygen activity profile starts at a0 and finishes at a'o and ~" such that the cation profile terminates at ~'. Hence the final result is independent of the initial estimates for k' and ~" and the procedure described is only used to facilitate the rapid arrival at the correct values without calculating an unduly large number of intermediate profiles. In practice exact agreement of the calculated profiles with the inner boundary conditions is unnecessary and the iteration of k' and ~" can be terminated when It' -- ~ l c [ ~ 0.005 and [log a 0 -- log a~. ~c [ --< 0.3 since there is no noticeable effect on either the shape or position of the oxygen activity and cation profiles for variations within these limits. The changes in k' and ~, which would result in the calculated values, ~ l c and do. ~ , approaching the fixed boundary values, ~' and a~, with any greater accuracy would be much smaller than any errors in the experimental measurements, particularly in the case of k' where the change required would be negligible. The parameters ~, v andp in equations (1-3) describe, in the nomenclature of Dalvi and Coates, 4 the variation of the cation diffusivities with the composition of the scale. If it is assumed that the compositional variation of the diffusivity is only related to the change in vacancy concentration and that the effects of variations in the lattice parameter and the rate of vacancy-cation exchange are negligible in comparison then the appropriate relationship can be developed. The variation of the diffusivity of cation B in the mixed oxide is given by: =
N
/N
o,
(5)
B . D . BASTOW,D. P. WIffrrL~ and G. C. WOOD
60
where N~o is the mole fraction of cation vacancies, with a charge ~ relative to the lattice, in pure BO at unit oxygen activity. At constant temperature the mole fraction of vacancies with charge ~ in the mixed oxide, N% is a function of both the oxide composition and the oxygen partial pressure. If the activation energy for vacancy formation, AHs, is proportional to the oxide composition then:
(6)
AHf = (1 -- ~) AH~° -q- ~hHfl °,
where AH~.° and AHfl° are the activation energies for vacancy formation in the pure terminal oxides at unit oxygen activity. Since the vacancy concentration is an exponential function of AHI, then the variation of the ratio N~/N~o with composition is given by:
N~/N~o = ~-1,
(7)
ct Gt where ~ = NBo/N,4 o, the ratio of the vacancy mole fractions in the pure oxides and N~ is the vacancy mole fraction at the composition ~, all the vacancy mole fractions referring to unit oxygen activity. The variation of N~o with oxygen partial pressure is given by :10
nl/2 (a+1) ( N,~Bo)p/(N =~o)p=l =*'o,
(8)
Since the oxygen activity a0 = po~, then combining equations (5-8) results in: DB = Do ~--1
aollV,
(9)
where v = (~ q- 1). This expression is the same as that used previously by Dalvi and Coates4 and for other oxide solid solutions 5,7 in simplified forms, based on experimental results. If it is assumed that the rates of vacancy-cation exchange for the A and B ions vary in the same proportion with composition and oxygen activity then: DA = p D , . 3. E X P E R I M E N T A L
CATION
PROFILES
IN SCALES ON Ni-Co
(lO) ALLOYS
Wood and Ferguson 3 have measured cation concentration profiles in scales formed during the oxidation of a Ni-10.9%Co alloy. The oxidation of other Ni-Co alloys has also been studied by these workers1~-14 using the same experimental methods and the results of other workers have been critically reviewed./3 The scales on some of these alloys have been re-examined in the present work and profiles have been measured in greater detail using a Cambridge "Microscan V" electron probe microanalyser. Analyses for both Co and Ni were carried out at fixed intervals across each scale using an accelerating voltage of 25 kV. The background intensity and counter dead time were taken into account when calculating the intensity ratios which were converted to wt% concentrations by applying the relevant corrections for atomic number, absorption and fluorescence effects.
Diffusion-controlled growth of solid solution scales on nickd--cobalt alloys
61
An interpretation of the profile in the scale on the Ni-10.9%Co alloy has already been given by Dalvi and Coates 4 using the Wagner model. However, in order to carry out their calculation, it was necessary to use boundary conditions taken from the experimental profiles and only the profile shape in the central region of the scale was computed. Using the calculation procedure outlined above, in conjunction with recently published data for cation diffusion in the (Ni, Co)O solid solution, theoretical profiles can now be computed throughout the scale for comparison with the experimental data. 4. NUMERICAL DATA FOR THE CALCULATION OF PROFILES The bulk compositions of the oxidized alloys, N °, are summarized in Table 1 together with the values of ~' calculated from the relationship 8 ~ ' = pN°/ [l q - N ° ( p - 1)] using a value of p = 0.2 (see below). Making the reasonable assumption that both the alloy and oxide solid solutions are ideal 4,9 the thermodynamics of the exchange reaction at the alloy-scale interface (NiO (in oxide soln) hCo (in alloy soln) = Ni (in alloy soln) q- CoO (in oxide soln), at 1000°C AG° = 4,050 cal 1") results in the expression a~ -----(4.95 -- 3.95 ~') 1-2 × l0 -6. The calculated values of a ' for each value of ~' are included in Table l, together with the value a o = l, since all the oxidation experiments were carried out in pure oxygen at 1 atmosphere pressure. Tracer diffusion coefficients of Ni and Co ions have recently been measured in both NiO and CoO by Crow 16 and Chen and Peterson. 1° The latter workers have also measured values in the mixed oxide solid solution. As a result of these measurements, it should be possible to obtain more reliable estimates for the parameters ~ and p in equations (9) and (10) since, in each case, the required data have now been measuredin two separate, internally consistent, investigations. In contrast, the values used by Dalvi and Coates 4 were derived from a number of isolated investigations and as a result of differences in experimental procedures the data may not be strictly comparable. From equation (9) it is clear that {3 = ao~'~ ---con°/~-co/nNi°and, using data from the work noted above, an average value of [3 ---- 125 4- 25 is obtained at 1000°C. It is noted that while equation (9) describes a linear relationship betweenlog Dco and ~ the data of Chen and Peterson suggest that there is a positive deviation from the line in the mixed oxide region. Consequently equation (9) is only an approximate representation of the variation of Dco with ~ b u t the small deviations will have very little effect on the results of the calculations. TABLE 1. DIFFUSION DATA AND BOUNDARY CONDITIONS USED IN THE CALCULATION OF CATION AND OXYGEN ACTIVITY PROFILES IN SCALES ON N [ - C o ALLOYS WITH THE FINAL ITERATED VALUES OF k '
Diffusion data Alloy composition Initial Boundary Conditions Iterated rate constant
p =0-2 4-0.I, 13 = 125 4- 25, v =2-5 4- 0.5, Do° =2.4 x 10-' N° ~' a~ x 10° a~ k' x 103
0"109 0"024 5"8 1 8.25
0"206 0.049 5"7 I 13.6
0-384 0-111 5-4 1 34.7
0.800 0.444 3.8 1 600
62
B.D. BASTOW,D. P. WHrrrLE and G. C. Wood
A value o f p is obtained using equation (10) and for pure CoO the alternative sets of data give a consistent value of p = 0-2 ± 0.01. However, the two sets of data for pure NiO suggest contradictory effects of composition on p, giving values of p = 0"10le and p = 0.27. z° The data for an oxide of composition Ni0.~ Co0.47 O gives p ----0.31 z° which is consistent with the latter trend for p increasing with NiO content. However, because this trend is inconsistent with the data of Crow, an average value o f p = 0.2 has been used for all scale compositions. It is noted however that the values o f p from this recent data are both more consistent with each other than the values quoted by Dalvi and Coates and are also considerably lower than the average value (p = 0.5) used in their calculation. The consequences of using a lower value are discussed below. The dependence of the cation diffusivity on oxygen partial pressure in the oxides is given by equation (9) in terms of an exponent v = (~ + I). There is considerable experimental evidence z° that the relative charge on the cation vacancy, ~, varies between 1 and 2 as the oxide composition varies from CoO to NiO. Hence v varies from 2 to 3 and an average value of v = 2.5 is used in the present calculation. However it is clear f r o m Fig. 1 that the shape of the cation profile is insensitive to the valued used. The average value for D ° in equation (9) is 2.4 × l0 -9 cm 2 s -1 at 1000°C. z°ae The values o f all the parameters used in the calculation of the profiles are summarized in Table 1. The calculated cation profiles across the scales on the four N i - C o alloys listed in Table 1 are shown in Figs. 1 and 2 where they are compared with profiles determined by microprobe analysis. The effects of changing each of the variables p, ~ and v I
I
I
I
I
I
I
I
I
• Microprobe analysis results, N i - I O ' 9 w t . %Co alloy Theorellcol" I ~ p , v Mean values, /9= 125_4-25 I - - - /9,u Mean values, p = 0-I i Cation "i - - - - / 9 , v Meon values, p = 0 . 3 -ii
0.3
j -;-_ ,,p Meon val o,,
Profiles
--
p,/9
eon va es, v = 3
•
0.2
/
•
"/
i
-"
.@"
.~'"
-
/ /
.I"
O, I
J
//,/
..i~.
• _e,,7 I 0
I 0'2
I
I 0"4
I
I 0 6
I
I 08
I
Y
FIG. 1.
A comparison between experimental and theoretical cation profiles in scales
formed on a Ni-10.9 %Co alloy in pure oxygen at 1000°C. The variation in the profile shape is also shown when different values are used for the parameters p and v in " the calculation. Profiles calculated using different values for 13are indistinguishable from each other and are coincident with the solid line.
Diffusion-controlled growth of solid solution scales on nickel--cobaltalloys i
,
Microprobe] •
i
63
i
N1-20,6wt * / . C o alloy Analysis ~ • Ni-38.4wt %Co alloy profiles / T Ni-8Owt%Co alloy 1.0 ~ Theoretical cation profiles T V
0
vT/ 0.6
r /
•
D • • 0o~
oco
(3.4
./
/
.. F o
o'.2
/ ~ / ~ Ni-20Co
.J o!4
o'.6
o!8
y ]~o. 2. A comparison between e×perimental and theoretical cation profiles in oxide scales formed at 1000°C on Ni-Co alloys containing 20.6~0, 38"4~o and 80~Co
respectively.
separately, within the limits of uncertainty of the experimental diffusion data given in Table 1, while the other two parameters retain their average values, are shown in Fig. 1 for the Ni-10.9%Co alloy. The oxygen activity profiles associated with each cation profile have the same shape as that given by Dalvi and Coates. 4 Each profile commences at a~ = 1 by definition and finishes at a value of a~ within a factor of two of the relevant value given in Table 1. This agreement is considered to be sufficiently accurate since both the profile shape and the iterated value of k' are insensitive to minor changes in the value of a~. It is worth remarking at this point that the diffusion data summarized in Table 1 was chosen solely from considerations of reliability and no attempt has been made to modify the data in order to improve the agreement between the experimental and calculated profiles. 5. D I S C U S S I O N (a) Cation concentration profiles The experimental and theoretical profiles show good agreement in view of the various approximations in the calculations. The uncertainties in the diffusion data given in Table 1 and the scatter in the cation concentrations obtained by microprobe analysis shown in Figs. 1 and 2 are both important factors which prevent complete agreement.
64
B.D. BaSrOW,D. P. WmrrLEand G. C. WOOD
The probe analysis results at the scale boundaries are of lower accuracy than those in the central region because of the finite size of the volume analysed. A further problem in the reliable analysis of cation concentrations arises from the porosity in these scales and the total concentration was found to deviate by up to 3 % from the theoretical value of 78.5 wt%. Hence the disagreement with the theoretical curves at the scale boundaries is considered to be acceptable, especially when a further potential inaccuracy is considered which appears to have an increasingly important effect as the cobalt content increases. At the alloy-scale interface the boundary condition ~' is estimated on the assumption that (d~/dy)y=o ~ O. s The experimental curves in Fig. 2 show that the condition is not satisfied for the Co-rich alloys and consequently the estimated values for ~' disagree with the experimental values in these cases. If the value of ~' used in the calculation were increased in each ease to coincide with the experimental value each profile would be raised but such an increase cannot be justified by the experimental diffusion data. Agreement with the complete experimental profile would be improved, although it is noted that the profiles would not be raised by a constant amount across the scale. By using equations (9) and (10) to describe the cation diffusion rates in the present calculation, it is assumed that the parameters p, [3 and v do not vary with scale composition. Comparing the shapes of the theoretical and experimental profiles, and the variation in their agreement as the average cobalt content increases, suggests that the differences between them would be reduced ifp increased with cobalt content. While this would result in an increase in ~' and a flattening of the profile such an increase cannot be justified by the available diffusion data. The effects on the cation profile of varying eacll of the three parameters within the limits of experimental uncertainty given in Table 1 are shown in Fig. 1. The variation in [3 between the limits 125 -k- 25 has a negligibl e effect on the profile shape and the results of Using different values cannot be distinguished from the profile calculated using the average values of the parameters given in Table 1. This is because a change in [3 changes the concentration dependence of the diffusion rates of both cations by the same amount. However, from equations (9) and (10), as [~increases the cation diffusion rates decrease (since ~ < 1) and this results in a decrease in the calculated value of k'. The effect of varying v is also very small because the parameter, like [3, effects both diffusion rates equally. However in this ease as v increases the diffusion rates, and consequently the value of k', increase. With reference to the possible variation of v, Dalvi and Coates4 stated that a meaningful result could not be obtained when v ---- 2. It is clear that this is not confirmed by the present results and the reason for their difficulties is not understood. The effects of a variation in the value of p are more complicated. While the difference between the cation diffusion rates (i.e. increase in p) and the degree of segregation decrease together, the inner boundary conditions, ~', determined from a subsidiary relationship8 increases. Hence while the profile is raised at the inner boundary it is also flattened, as shown in Fig. 1. In addition a decrease in p reduces the total diffusion flux through the scale and results in a decrease in k'. When the differences between the profiles calculated using different values ofp within the l~mits of experimental uncertainty are superimposed on the scattered experimental data, it
Diffusion-controlledgrowth of solid solution scales on nickel--cobaltalloys
65
can be concluded that there is good agreement between the theoretical and experimental results in this case. For the profiles in the other scales, Fig. 2, it is clear that the estimated values of ~' are consistently low and this contributes to the disagreement with the experimental results at this boundary. Additional errors may also arise in the calculation because of the assumption that the oxide scale is an ideal solid solution4,~-11 but the qualitative effect on the profiles of any deviation from ideality is uncertain. However, it is noted that the maximum deviation between the experimental and calculated profiles occurs in the composition range 40-60%COO where the deviation from ideality is greatest, ix Furthermore the variation in the cation vacancy concentration across the scale will tend to produce an additional enrichment of Co at the scale-gas interface but the effect will probably be negligible in this system because the vacancy concentration is small, x7 When all the possible sources of error are taken into account it is considered that the experimental and theoretical profiles show substantial agreement and that the results provide support for Wagner's model. ~ (b) The Dalvi and Coates solution 4 Dalvi and Coates were unable to obtain "a meaningful result" using a value v = 2 but the present work shows that the variation in v has very little effect on the cation profile and it is useful to make a critical comparison between the profile obtained using the alternative boundary conditions and diffusion data of Dalvi and Coates for the Ni-10.9%Co alloy. Dalvi and Coates obtained good agreement between the theoretical and experimental profiles by fitting the cation profile between the boundary conditions ~" = 0.315 and ~' = 0.041 taken from the experimental profile. The parameters in equations (9) and (10) had the values p = 0.5, 13 = 62 and v = 2.5, together with k' -~ 0.036 (the values of 13and k' are uncertain since they were adjusted slightly during the course of their calculation in order to satisfy the inner boundary condition, %). Hence, although the value ofp differs from that used in the present work by a factor of 2.5, the total concentration change across the scale in each case is approximately the same. Using the Dalvi and Coates values for p and 13 in the procedure described in the present work, an oxygen profile is obtained which agrees closely with their published result. In contrast, the total cation concentration change across the scale is only 0.12 compared with the experimental value of 0.274. The associated value of k' = 0.035, while an estimation of the final value used by Dalvi and Coates in their calculation is k' -~ 0.068. Hence the higher value of p, which results in a smaller concentration change across the scale, appears to have been compensated for by using a value for k' which was higher than the one originally quoted. It is thought that the contradiction between the results of the present work, Fig. 1, and those of Dalvi and Coates 4 arises because the latter's profile is artificially constrained to lie between the boundary conditions ~' and ~". In the present work only ~' is fixed and the value of ~", consistent with both g' and eqns (1) and (2), is free to vary in the range ~' < ~" < 1. Boundary conditions have been obtained from experimental data in other examinations of Wagner's model s-7 and as a result the applicability of the model is not tested completely. However, as shown in Section (d) below, in the case of Fe-Co alloys7 this has very little effect on the cation profile.
66
B. D. BASTOW,D. P. WHn'rLEand G. C. Wood
(c) Parabolic rate constants An important aspect of the present calculations is that the value of k' is obtained as a direct result of the iteration procedure and is not introduced as part of the initial conditions as in earlier work. 4,~ Values of the parabolic rate constant, kp, were calculated from the iterated values of k' given in Table 1 using the expression4 kp ---k' D°o x 512/V~to, where VMo is the molar volume of the oxide MO. The calculated variation of kp with the cobalt content of the alloy is compared with the more reliable experimental data x~-l~in Fig. 3. The theoretical values of k n for pure Ni and Co were calculated using the expression given by Wagnerxs for the formation of single phase, homogeneous scales. In the case of the Ni-10-9%Co alloy each profile in Fig. 1 is associated with a different value of k'. Consequently the value of kp varies over the range enclosed by the error bars, the maximum and minimum values corresponding to values o f p = 0.3 and p = 0.1 respectively. The good agreement between the theoretical and experimental values of k~ in both magnitude and compositional variation constitutes further confirmation of the applicability of Wagner's model. The model assumes that diffusion occurs through the cation sub-lattice and that there is no interference from short-circuiting mechanisms. Since the diffusion data used in the calculations refers to single crystal oxides the agreement obtained suggests that the growth of the oxide scale on these alloys at 1000°C occurs by bulk diffusion of cations and that the contribution of oxide grain boundary diffusion is negligible. In the present work the estimate of k' from equation (4) was within a factor of two of the calculated value. This suggests that the shape of the cation profile has very little influence on the value o f k ' . The approximation used in equation (4) that ~ -~ N °, was i
i
I
i
i
i
l
i
i
i
• Oxyqen I aim, ref.t2 1Ex er'm i • Air ref. 14 ~ p I ento • Air ref.I 3 (doto of Frederick I~ Cornet J doto -Theoretlcol curve
io-iC
/1
'
io-II Pure Ni '
"o
'
4"
' G/ I Wt.% Co in Ull0y
8/
I Pure Co
F1o. 3. The variation of the parabolic rate constant, /~,, with composition for Hi-Co ' alloys oxidized at 1000°C. The points arc experimental values from Refs. ]2-14 and the solid line is the theoretical variation.
Diffusion-controlled growth of solid solution scales on nickel-cobalt alloys
67
demonstrated by integrating between y = 0 and 1 under the cation profile and earnparing the area, ~, with the bulk alloy composition. Hence if these conditions are satisfied a reliable estimate of k ' can probably be obtained for other systems without resorting to a lengthy profile calculation. (d) The cation profile in the scale on a Co-10wt%Fe alloy ~ As pointed out in Section (b), earlier calculations 4 are inconsistent with those in the present work and it is of interest to make similar comparisons for another representative alloy. Some of the disagreement between experimental and theoretical results in Fig. 2 is attributable to the errors in the estimated values of ~'. In order to eliminate this effect the Fe-Co system was chosen for the comparison since the approximation used is reliable s in this case. A cation profile was calculated for the Co--10wt%Fe alloy which exhibits the maximum Fe segregation across the scale. The values of the relevant parameters at 1200°C are given by Mayer and Smeltzer7 and the value of a 0 is obtained from the thermodynamic data for the exchange reaction, Fe -t- CoO = Co + FeO: 15 N ° = 0.096; a0 = 1 × 10-5; a 0 = 3.2 × 10-2 p = 0.6; [3 ~ 15; v = 6.8. It is assumed by Mayer and Smeltzer that for dilute solutions Dre ~ ~ and not DF= ~ [3~-I as in equation (3). The value of (3 given above is chosen so that the compositional variation of DFc is approximately the same using either of these relations. The good agreement between the calculated profile and the experimental results 7 is shown in Fig. 4. The associated oxygen activity profile also agreed closely with that calculated previously. This provides additional confirmation of the validity of the computation procedure used in the present work and reinforces the conclusions relating to the result of Dalvi and Coates 4 given in Section (b). A value of k ' = 0.157 was obtained by iteration. Using an extrapolated value I O.~
~
I
I
I
Theoretical profile T
Range of experimental data 7'
o
O"
"I-I--I--I-" I"Ijl/l, , , , ] 0'2
0"4
0-6
0"8
1
y
FIG. 4.
Experimental and theoretical cation profiles in a scale formed during the oxidation of a Co-10wtyoFe alloy at 1200°C in 10-a atm oxygen.
68
B.D. BASTOW,D. P WHITTLEand G. C. WOOD
of DFe Feo = 4"6 × 10 - e c m ~ s -1 19 the resulting calculated value of kp = 2.4 × 10 -6 gZ cm ~ s-X. This is in reasonable agreement with the experimental value ofkp = 8 × 10 -~ g2 cm 4 s-X although this may be partly fortuitous since the value of Fe DFc o is dependent on the stoichiometry of wustite and the equilibrium relationships in the CoO-Fex_xO system relevant to the formation of solid solution scales are uncertain. 6. CONCLUSIONS Wagner's model for the formation of solid solution oxide scales gives rise to equations which can be solved in a finite difference form without introducing experimental values for either the terminal cation concentrations or the parabolic rate constant as boundary conditions. The comparison of the theoretical and experimental cation profiles provides a satisfactory demonstration of the applicability of the model and the results m a y be summarized as follows: (1) The theoretical and experimental cation profiles show good agreement in view of their independent origins. The observed discrepancies are attributable to: (a) Inaccuracies in the cation concentrations measured by microprobe analysis, particularly at the scale boundaries • (b) The use of an approximate relationship to estimate the boundary condition, ~' which results in the differences between the estimated and measured values increasing with the cobalt content (c) Uncertainties in the values of the parameters describing the cation diffusion coefficients. The possible error in p is reflected in the decreasing accuracy of the estimate of ~' noted in (b) (d) The assumption that the oxide solid solution exhibits ideal behaviour. (2) The predicted compositional variation of the parabolic rate constant, kp, is in good agreement with the experimental results. The overall agreement between experimental and theoretical cation profiles and kp values supports Wagner's theoretical model and demonstrates that the underlying assumptions are satisfied for the oxidation of N i - C o alloys at 1000°C. The authors would like to thank the Science Research Council for the financial support of this work and Professor C. Wagner for a useful discussion. Acknowledgements
REFERENCES 1. C. WAGNER,Corros. Sci. 9, 91 (1969). 2. D. E. COATESand A. D. DALW, Oxidation of Metals 2, 331 (1970). 3. G. C. WOODand J. M. Fm~GUSON,Nature 208, 369 (1965). 4. A. D. DALWand D. E. COATES,Oxidation of Metals 3, 203 (1971). 5. A. D. DALVIand W. W. SM~LTZVaLJ. electrochem. Soc. 121, 386 (1974). 6. P. MAYERand W. W. SMELTZER,J. electrochem. Soc. 119, 626 (1972). 7. P. MAYERand W. W. SMELTZER,J. electrochem. Soc. 121, 538 (1974). 8. D. P. Wm'rn:E, B. D. BASTOWand G. C. WOOD,Oxidation of Metals 9, 215 (1975). 9. R. J. MOOm~.and J. WHITE, J. Mater. Sci. 9, 1393 (1974). 10. W. K. ~ and N. L. I~rEV.SON, J. Phys. Chem. Solids 34, 1093 (1973). 11. J. J. STmUCH,J. B. COHLmand D. H. WmTMO~, J. Am. Ceram. Soc. 56, 119 (1973). 12. J. M. I~RGUSON,Ph.D. Thesis, University of Manchester (1967). 13. G. C. WOOD,I. G. WmonT and J. M. FEROUSON,Corros. Sci. 5, 645 (1965). 14. G. C. WOODand I. G. W~GHT, Corros. ScL 5, 841 (1965).
Diffusion-controlled growth of solid solution scales on nickel-cobalt alloys
69
15. J. F. ELLIOT'r and M. GLEISER, Thermochemistry for Steelmaking, p. 173, Addison-Wesley, Reading, MA (1960). 16. W. B. CRow, Report ARL-70-0090, Project No. 7021, Office of Aerospace Research, U.S.A.F., Wright-Patterson Air Force Base, Ohio (1970). 17. T. EI~CSSON,Corros. Sci. 11, 249 (1971). 18. C. WAGNER,Atom Movements, A.S.M. Monograph, p. 153 (1951). 19. L. I-IIMMEL,R. F. MEHI.,and C. E. BIRCHENALL,Trans. metall. Soe. AIMES, 827 (1953).
D I F F U S I O N - C O N T R O L L E D G R O W T H OF SOLID SOLUTION SCALES ON NICKEL-COBALT ALLOYS* B. D. BASTOW,D. P. WHITTLEt and G. C. WOOD Corrosion and Protection Centre, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester, England Abstract--Cation concentration profiles in solid-solution scales on Ni-Co alloys have been calculated by making use of Wagner's theoretical analysis. As a result of the computation method used, it becomes unnecessary to incorporate experimental values in the calculation for either the parabolic rate constant, kp, or the boundary conditions of the cation profile. The agreement between the independently determined theoretical and experimental cation profiles provides strong sui~port for Wagner's model. The calculated variation of kp with alloy composition is also in agreement with the experimental data. I. INTRODUCTION A THEORETICALdescription of binary alloy oxidation for those systems where a solid solution scale is formed has recently been developed by Wagner 1 and modified by Coates and Dalvi. * This general treatment shows that the segregation o f one of the scale components towards the scale-gas interface, observed previously by Wood and Ferguson 3 in scales on Ni-Co alloys, is related to the inequality of the cation diffusion rates in the scale. Various workers have recently applied Wagner's theoretical treatment to a number of alloy systems ~-7 for which the conditions for the application of the model are fulfilled. However, in all these cases it has been necessary to use data from the experimentally measured profiles, together with the experimental value for the parabolic rate constant, to establish the boundary conditions for the calculation of the theoretical cation profile. Consequently the agreement found between the theoretical and experimental profiles partly arises because the boundary conditions are the same and these results do not constitute a complete test of the validity of Wagner's model. In order to carry out a more rigorous test of the model, it is necessary to determine as many as possible of the required boundary conditions independently of the experimental profile. The profile calculated by combining these conditions with the equations describing the profile shape can then be compared with the experimental data. Agreement between the two profiles and the experimental and calculated values of the parabolic rate constants would provide a more satisfactory demonstration of the applicability of Wagner's model. It is often true in a diffusion controlled oxidation reaction that interdiffusion in the alloy is relatively unimportant in comparison with the cation diffusion in the scale. Recent work by the present authors a has demonstrated that when this assumption is valid then the boundary condition for the cation profile at the alloy-scale • Manuscript received 4 March 1975. "l'Departmentof Metallurgy and Materials Science, University of Liverpool, P.O. Box 147, Liverpool, England. 57
58
B.D. BASTOW,D. P. Wmrrt.E and G. C. WOoD
interface can be estimated froth the alloy composition if the ratio of the cation diffusivities in the scale is known. The necessary preliminary condition is satisfied for a number of systems to which the Wagner model is applicable a and in particular for the Ni-Co alloys considered in the present work. Combining this boundary value with the thermodynamic description of the exchange reaction between alloy and scale, a value can be obtained for the oxygen activity at the alloy-scale interface. Since the composition of the oxidizing gas is known, the oxygen activity at the scalegas interface is also defined. Hence three boundary conditions are known independently of the measured profile and these are sufficient for the solution of the equations, given below, which describe the interdependence of the cation and oxygen activity profiles in the scale. 2. THEORETICAL DESCRIPTION AND SOLUTION OF EQUATIONS The original equations given by Wagner 1 apply to a general case which is described in terms of the activities of the components in the scale. However, the requisite thermodynamic information is often unavailable and in order to calculate the cation profile across the scale a relationship between the cation concentration and the activity in the scale must be assumed. In the case of the (Ni, Co)O mixed oxide there is some indirect evidence that the solid solution exhibits ideal behaviour. The position of the phase boundary between the (Ni, Co)O solid solution and the two phase field (solid solution ÷ CoaO4) is consistent with approximately ideal behaviour 9 and a comparison of tracer and interdiffusion data in mixed oxides shows that the thermodynamic factor (1 ÷ t~ In 7i/t9 Ni) in the Darken equation is close to unity. ~°,u In addition, Dalvi and Coates 4 have already noted that related pseudo-binary oxide systems form ideal solutions. The equations are then considerably simplified and, in the notation of Dalvi and Coates, 4 they become:
d2~
--
k lool,v (1
dlnao dy where
y
=
E
fd~12
-- l/v)
--y
[p -- (p --
p
1)~]~d~
=
O,
Jdy
k' ' dE] 1 ~ a o l / v - ( p - 1)~y P - - ( P - -
~'
(1) (2)
1)
x/x~, the ratio of the distance from the alloy-scale interface to the total
scale thickness, is the equivalent cation fraction of component B in the scale, a 0 is the oxygen activity, k' = k/D °, the ratio of the corrosion constant to the diffusion coefficient of B in BO in equilibrium with oxygen at unit activity. The meaning and interrelationship oftbe symbols p, {3and v is outlined below in equations (5-10). Boundary conditions of the oxygen activity and cation profiles are indicated by a superscript prime' and double prime ~ for the alloy-scale and scale-gas interfaces respectively.
Diffusion-controlled growth of solid solution scales on nickel--cobaltalloys
59
A third relationship is necessary in order to obtain acomplete solution of equations (1) and (2) in terms of ~, ao and k' and this is provided by a mass balance at one of the scale boundaries. Since the balance at the aHoy-scale interface has already been used in the estimation of the inner boundary condition, ~',s the third equation is obtained by considering the formation of scale at the scale-gas interface, in order to avoid any inconsistency in the treatment. This results in :4 d~] k' ~" (1 -- ~" )(p -- 1) dyy y=l = p ~"-X (a")x/"
(3)
The most convenient way to solve this equation is to estimate k' by assuming that the oxygen activity profile in the scale is not very dependent on the gradient of the cation profile. Then equation (2) can be integrated between y = 0, ao = a' a n d y = l, a0 ----a~ to give: k' = v~[-x [p -- (p -- 1)~] [(ag)~'v -- (a~)X/,l,
(4)
where ~ is the average value of ~ in the scale. If diffusion in the alloy is negligible it can be assumed that ~ = N ° where N ° is the bulk alloy composition. Using a trial value for ~" the magnitude of [d~/dy] y = 1 can be estimated and writing this gradient in finite difference form, a value for ~ after the first interval can be calculated. The remainder of the oxygen activity and cation profiles can then be calculated using equations (1) and (2) in finite difference form, the result being independent of the step length used. It is emphasized that the first values of k' and ~" used are only approximations. The values must be subsequently modified, k' such that the oxygen activity profile starts at a0 and finishes at a'o and ~" such that the cation profile terminates at ~'. Hence the final result is independent of the initial estimates for k' and ~" and the procedure described is only used to facilitate the rapid arrival at the correct values without calculating an unduly large number of intermediate profiles. In practice exact agreement of the calculated profiles with the inner boundary conditions is unnecessary and the iteration of k' and ~" can be terminated when It' -- ~ l c [ ~ 0.005 and [log a 0 -- log a~. ~c [ --< 0.3 since there is no noticeable effect on either the shape or position of the oxygen activity and cation profiles for variations within these limits. The changes in k' and ~, which would result in the calculated values, ~ l c and do. ~ , approaching the fixed boundary values, ~' and a~, with any greater accuracy would be much smaller than any errors in the experimental measurements, particularly in the case of k' where the change required would be negligible. The parameters ~, v andp in equations (1-3) describe, in the nomenclature of Dalvi and Coates, 4 the variation of the cation diffusivities with the composition of the scale. If it is assumed that the compositional variation of the diffusivity is only related to the change in vacancy concentration and that the effects of variations in the lattice parameter and the rate of vacancy-cation exchange are negligible in comparison then the appropriate relationship can be developed. The variation of the diffusivity of cation B in the mixed oxide is given by: =
N
/N
o,
(5)
B . D . BASTOW,D. P. WIffrrL~ and G. C. WOOD
60
where N~o is the mole fraction of cation vacancies, with a charge ~ relative to the lattice, in pure BO at unit oxygen activity. At constant temperature the mole fraction of vacancies with charge ~ in the mixed oxide, N% is a function of both the oxide composition and the oxygen partial pressure. If the activation energy for vacancy formation, AHs, is proportional to the oxide composition then:
(6)
AHf = (1 -- ~) AH~° -q- ~hHfl °,
where AH~.° and AHfl° are the activation energies for vacancy formation in the pure terminal oxides at unit oxygen activity. Since the vacancy concentration is an exponential function of AHI, then the variation of the ratio N~/N~o with composition is given by:
N~/N~o = ~-1,
(7)
ct Gt where ~ = NBo/N,4 o, the ratio of the vacancy mole fractions in the pure oxides and N~ is the vacancy mole fraction at the composition ~, all the vacancy mole fractions referring to unit oxygen activity. The variation of N~o with oxygen partial pressure is given by :10
nl/2 (a+1) ( N,~Bo)p/(N =~o)p=l =*'o,
(8)
Since the oxygen activity a0 = po~, then combining equations (5-8) results in: DB = Do ~--1
aollV,
(9)
where v = (~ q- 1). This expression is the same as that used previously by Dalvi and Coates4 and for other oxide solid solutions 5,7 in simplified forms, based on experimental results. If it is assumed that the rates of vacancy-cation exchange for the A and B ions vary in the same proportion with composition and oxygen activity then: DA = p D , . 3. E X P E R I M E N T A L
CATION
PROFILES
IN SCALES ON Ni-Co
(lO) ALLOYS
Wood and Ferguson 3 have measured cation concentration profiles in scales formed during the oxidation of a Ni-10.9%Co alloy. The oxidation of other Ni-Co alloys has also been studied by these workers1~-14 using the same experimental methods and the results of other workers have been critically reviewed./3 The scales on some of these alloys have been re-examined in the present work and profiles have been measured in greater detail using a Cambridge "Microscan V" electron probe microanalyser. Analyses for both Co and Ni were carried out at fixed intervals across each scale using an accelerating voltage of 25 kV. The background intensity and counter dead time were taken into account when calculating the intensity ratios which were converted to wt% concentrations by applying the relevant corrections for atomic number, absorption and fluorescence effects.
Diffusion-controlled growth of solid solution scales on nickd--cobalt alloys
61
An interpretation of the profile in the scale on the Ni-10.9%Co alloy has already been given by Dalvi and Coates 4 using the Wagner model. However, in order to carry out their calculation, it was necessary to use boundary conditions taken from the experimental profiles and only the profile shape in the central region of the scale was computed. Using the calculation procedure outlined above, in conjunction with recently published data for cation diffusion in the (Ni, Co)O solid solution, theoretical profiles can now be computed throughout the scale for comparison with the experimental data. 4. NUMERICAL DATA FOR THE CALCULATION OF PROFILES The bulk compositions of the oxidized alloys, N °, are summarized in Table 1 together with the values of ~' calculated from the relationship 8 ~ ' = pN°/ [l q - N ° ( p - 1)] using a value of p = 0.2 (see below). Making the reasonable assumption that both the alloy and oxide solid solutions are ideal 4,9 the thermodynamics of the exchange reaction at the alloy-scale interface (NiO (in oxide soln) hCo (in alloy soln) = Ni (in alloy soln) q- CoO (in oxide soln), at 1000°C AG° = 4,050 cal 1") results in the expression a~ -----(4.95 -- 3.95 ~') 1-2 × l0 -6. The calculated values of a ' for each value of ~' are included in Table l, together with the value a o = l, since all the oxidation experiments were carried out in pure oxygen at 1 atmosphere pressure. Tracer diffusion coefficients of Ni and Co ions have recently been measured in both NiO and CoO by Crow 16 and Chen and Peterson. 1° The latter workers have also measured values in the mixed oxide solid solution. As a result of these measurements, it should be possible to obtain more reliable estimates for the parameters ~ and p in equations (9) and (10) since, in each case, the required data have now been measuredin two separate, internally consistent, investigations. In contrast, the values used by Dalvi and Coates 4 were derived from a number of isolated investigations and as a result of differences in experimental procedures the data may not be strictly comparable. From equation (9) it is clear that {3 = ao~'~ ---con°/~-co/nNi°and, using data from the work noted above, an average value of [3 ---- 125 4- 25 is obtained at 1000°C. It is noted that while equation (9) describes a linear relationship betweenlog Dco and ~ the data of Chen and Peterson suggest that there is a positive deviation from the line in the mixed oxide region. Consequently equation (9) is only an approximate representation of the variation of Dco with ~ b u t the small deviations will have very little effect on the results of the calculations. TABLE 1. DIFFUSION DATA AND BOUNDARY CONDITIONS USED IN THE CALCULATION OF CATION AND OXYGEN ACTIVITY PROFILES IN SCALES ON N [ - C o ALLOYS WITH THE FINAL ITERATED VALUES OF k '
Diffusion data Alloy composition Initial Boundary Conditions Iterated rate constant
p =0-2 4-0.I, 13 = 125 4- 25, v =2-5 4- 0.5, Do° =2.4 x 10-' N° ~' a~ x 10° a~ k' x 103
0"109 0"024 5"8 1 8.25
0"206 0.049 5"7 I 13.6
0-384 0-111 5-4 1 34.7
0.800 0.444 3.8 1 600
62
B.D. BASTOW,D. P. WHrrrLE and G. C. Wood
A value o f p is obtained using equation (10) and for pure CoO the alternative sets of data give a consistent value of p = 0-2 ± 0.01. However, the two sets of data for pure NiO suggest contradictory effects of composition on p, giving values of p = 0"10le and p = 0.27. z° The data for an oxide of composition Ni0.~ Co0.47 O gives p ----0.31 z° which is consistent with the latter trend for p increasing with NiO content. However, because this trend is inconsistent with the data of Crow, an average value o f p = 0.2 has been used for all scale compositions. It is noted however that the values o f p from this recent data are both more consistent with each other than the values quoted by Dalvi and Coates and are also considerably lower than the average value (p = 0.5) used in their calculation. The consequences of using a lower value are discussed below. The dependence of the cation diffusivity on oxygen partial pressure in the oxides is given by equation (9) in terms of an exponent v = (~ + I). There is considerable experimental evidence z° that the relative charge on the cation vacancy, ~, varies between 1 and 2 as the oxide composition varies from CoO to NiO. Hence v varies from 2 to 3 and an average value of v = 2.5 is used in the present calculation. However it is clear f r o m Fig. 1 that the shape of the cation profile is insensitive to the valued used. The average value for D ° in equation (9) is 2.4 × l0 -9 cm 2 s -1 at 1000°C. z°ae The values o f all the parameters used in the calculation of the profiles are summarized in Table 1. The calculated cation profiles across the scales on the four N i - C o alloys listed in Table 1 are shown in Figs. 1 and 2 where they are compared with profiles determined by microprobe analysis. The effects of changing each of the variables p, ~ and v I
I
I
I
I
I
I
I
I
• Microprobe analysis results, N i - I O ' 9 w t . %Co alloy Theorellcol" I ~ p , v Mean values, /9= 125_4-25 I - - - /9,u Mean values, p = 0-I i Cation "i - - - - / 9 , v Meon values, p = 0 . 3 -ii
0.3
j -;-_ ,,p Meon val o,,
Profiles
--
p,/9
eon va es, v = 3
•
0.2
/
•
"/
i
-"
.@"
.~'"
-
/ /
.I"
O, I
J
//,/
..i~.
• _e,,7 I 0
I 0'2
I
I 0"4
I
I 0 6
I
I 08
I
Y
FIG. 1.
A comparison between experimental and theoretical cation profiles in scales
formed on a Ni-10.9 %Co alloy in pure oxygen at 1000°C. The variation in the profile shape is also shown when different values are used for the parameters p and v in " the calculation. Profiles calculated using different values for 13are indistinguishable from each other and are coincident with the solid line.
Diffusion-controlled growth of solid solution scales on nickel--cobaltalloys i
,
Microprobe] •
i
63
i
N1-20,6wt * / . C o alloy Analysis ~ • Ni-38.4wt %Co alloy profiles / T Ni-8Owt%Co alloy 1.0 ~ Theoretical cation profiles T V
0
vT/ 0.6
r /
•
D • • 0o~
oco
(3.4
./
/
.. F o
o'.2
/ ~ / ~ Ni-20Co
.J o!4
o'.6
o!8
y ]~o. 2. A comparison between e×perimental and theoretical cation profiles in oxide scales formed at 1000°C on Ni-Co alloys containing 20.6~0, 38"4~o and 80~Co
respectively.
separately, within the limits of uncertainty of the experimental diffusion data given in Table 1, while the other two parameters retain their average values, are shown in Fig. 1 for the Ni-10.9%Co alloy. The oxygen activity profiles associated with each cation profile have the same shape as that given by Dalvi and Coates. 4 Each profile commences at a~ = 1 by definition and finishes at a value of a~ within a factor of two of the relevant value given in Table 1. This agreement is considered to be sufficiently accurate since both the profile shape and the iterated value of k' are insensitive to minor changes in the value of a~. It is worth remarking at this point that the diffusion data summarized in Table 1 was chosen solely from considerations of reliability and no attempt has been made to modify the data in order to improve the agreement between the experimental and calculated profiles. 5. D I S C U S S I O N (a) Cation concentration profiles The experimental and theoretical profiles show good agreement in view of the various approximations in the calculations. The uncertainties in the diffusion data given in Table 1 and the scatter in the cation concentrations obtained by microprobe analysis shown in Figs. 1 and 2 are both important factors which prevent complete agreement.
64
B.D. BaSrOW,D. P. WmrrLEand G. C. WOOD
The probe analysis results at the scale boundaries are of lower accuracy than those in the central region because of the finite size of the volume analysed. A further problem in the reliable analysis of cation concentrations arises from the porosity in these scales and the total concentration was found to deviate by up to 3 % from the theoretical value of 78.5 wt%. Hence the disagreement with the theoretical curves at the scale boundaries is considered to be acceptable, especially when a further potential inaccuracy is considered which appears to have an increasingly important effect as the cobalt content increases. At the alloy-scale interface the boundary condition ~' is estimated on the assumption that (d~/dy)y=o ~ O. s The experimental curves in Fig. 2 show that the condition is not satisfied for the Co-rich alloys and consequently the estimated values for ~' disagree with the experimental values in these cases. If the value of ~' used in the calculation were increased in each ease to coincide with the experimental value each profile would be raised but such an increase cannot be justified by the experimental diffusion data. Agreement with the complete experimental profile would be improved, although it is noted that the profiles would not be raised by a constant amount across the scale. By using equations (9) and (10) to describe the cation diffusion rates in the present calculation, it is assumed that the parameters p, [3 and v do not vary with scale composition. Comparing the shapes of the theoretical and experimental profiles, and the variation in their agreement as the average cobalt content increases, suggests that the differences between them would be reduced ifp increased with cobalt content. While this would result in an increase in ~' and a flattening of the profile such an increase cannot be justified by the available diffusion data. The effects on the cation profile of varying eacll of the three parameters within the limits of experimental uncertainty given in Table 1 are shown in Fig. 1. The variation in [3 between the limits 125 -k- 25 has a negligibl e effect on the profile shape and the results of Using different values cannot be distinguished from the profile calculated using the average values of the parameters given in Table 1. This is because a change in [3 changes the concentration dependence of the diffusion rates of both cations by the same amount. However, from equations (9) and (10), as [~increases the cation diffusion rates decrease (since ~ < 1) and this results in a decrease in the calculated value of k'. The effect of varying v is also very small because the parameter, like [3, effects both diffusion rates equally. However in this ease as v increases the diffusion rates, and consequently the value of k', increase. With reference to the possible variation of v, Dalvi and Coates4 stated that a meaningful result could not be obtained when v ---- 2. It is clear that this is not confirmed by the present results and the reason for their difficulties is not understood. The effects of a variation in the value of p are more complicated. While the difference between the cation diffusion rates (i.e. increase in p) and the degree of segregation decrease together, the inner boundary conditions, ~', determined from a subsidiary relationship8 increases. Hence while the profile is raised at the inner boundary it is also flattened, as shown in Fig. 1. In addition a decrease in p reduces the total diffusion flux through the scale and results in a decrease in k'. When the differences between the profiles calculated using different values ofp within the l~mits of experimental uncertainty are superimposed on the scattered experimental data, it
Diffusion-controlledgrowth of solid solution scales on nickel--cobaltalloys
65
can be concluded that there is good agreement between the theoretical and experimental results in this case. For the profiles in the other scales, Fig. 2, it is clear that the estimated values of ~' are consistently low and this contributes to the disagreement with the experimental results at this boundary. Additional errors may also arise in the calculation because of the assumption that the oxide scale is an ideal solid solution4,~-11 but the qualitative effect on the profiles of any deviation from ideality is uncertain. However, it is noted that the maximum deviation between the experimental and calculated profiles occurs in the composition range 40-60%COO where the deviation from ideality is greatest, ix Furthermore the variation in the cation vacancy concentration across the scale will tend to produce an additional enrichment of Co at the scale-gas interface but the effect will probably be negligible in this system because the vacancy concentration is small, x7 When all the possible sources of error are taken into account it is considered that the experimental and theoretical profiles show substantial agreement and that the results provide support for Wagner's model. ~ (b) The Dalvi and Coates solution 4 Dalvi and Coates were unable to obtain "a meaningful result" using a value v = 2 but the present work shows that the variation in v has very little effect on the cation profile and it is useful to make a critical comparison between the profile obtained using the alternative boundary conditions and diffusion data of Dalvi and Coates for the Ni-10.9%Co alloy. Dalvi and Coates obtained good agreement between the theoretical and experimental profiles by fitting the cation profile between the boundary conditions ~" = 0.315 and ~' = 0.041 taken from the experimental profile. The parameters in equations (9) and (10) had the values p = 0.5, 13 = 62 and v = 2.5, together with k' -~ 0.036 (the values of 13and k' are uncertain since they were adjusted slightly during the course of their calculation in order to satisfy the inner boundary condition, %). Hence, although the value ofp differs from that used in the present work by a factor of 2.5, the total concentration change across the scale in each case is approximately the same. Using the Dalvi and Coates values for p and 13 in the procedure described in the present work, an oxygen profile is obtained which agrees closely with their published result. In contrast, the total cation concentration change across the scale is only 0.12 compared with the experimental value of 0.274. The associated value of k' = 0.035, while an estimation of the final value used by Dalvi and Coates in their calculation is k' -~ 0.068. Hence the higher value of p, which results in a smaller concentration change across the scale, appears to have been compensated for by using a value for k' which was higher than the one originally quoted. It is thought that the contradiction between the results of the present work, Fig. 1, and those of Dalvi and Coates 4 arises because the latter's profile is artificially constrained to lie between the boundary conditions ~' and ~". In the present work only ~' is fixed and the value of ~", consistent with both g' and eqns (1) and (2), is free to vary in the range ~' < ~" < 1. Boundary conditions have been obtained from experimental data in other examinations of Wagner's model s-7 and as a result the applicability of the model is not tested completely. However, as shown in Section (d) below, in the case of Fe-Co alloys7 this has very little effect on the cation profile.
66
B. D. BASTOW,D. P. WHn'rLEand G. C. Wood
(c) Parabolic rate constants An important aspect of the present calculations is that the value of k' is obtained as a direct result of the iteration procedure and is not introduced as part of the initial conditions as in earlier work. 4,~ Values of the parabolic rate constant, kp, were calculated from the iterated values of k' given in Table 1 using the expression4 kp ---k' D°o x 512/V~to, where VMo is the molar volume of the oxide MO. The calculated variation of kp with the cobalt content of the alloy is compared with the more reliable experimental data x~-l~in Fig. 3. The theoretical values of k n for pure Ni and Co were calculated using the expression given by Wagnerxs for the formation of single phase, homogeneous scales. In the case of the Ni-10-9%Co alloy each profile in Fig. 1 is associated with a different value of k'. Consequently the value of kp varies over the range enclosed by the error bars, the maximum and minimum values corresponding to values o f p = 0.3 and p = 0.1 respectively. The good agreement between the theoretical and experimental values of k~ in both magnitude and compositional variation constitutes further confirmation of the applicability of Wagner's model. The model assumes that diffusion occurs through the cation sub-lattice and that there is no interference from short-circuiting mechanisms. Since the diffusion data used in the calculations refers to single crystal oxides the agreement obtained suggests that the growth of the oxide scale on these alloys at 1000°C occurs by bulk diffusion of cations and that the contribution of oxide grain boundary diffusion is negligible. In the present work the estimate of k' from equation (4) was within a factor of two of the calculated value. This suggests that the shape of the cation profile has very little influence on the value o f k ' . The approximation used in equation (4) that ~ -~ N °, was i
i
I
i
i
i
l
i
i
i
• Oxyqen I aim, ref.t2 1Ex er'm i • Air ref. 14 ~ p I ento • Air ref.I 3 (doto of Frederick I~ Cornet J doto -Theoretlcol curve
io-iC
/1
'
io-II Pure Ni '
"o
'
4"
' G/ I Wt.% Co in Ull0y
8/
I Pure Co
F1o. 3. The variation of the parabolic rate constant, /~,, with composition for Hi-Co ' alloys oxidized at 1000°C. The points arc experimental values from Refs. ]2-14 and the solid line is the theoretical variation.
Diffusion-controlled growth of solid solution scales on nickel-cobalt alloys
67
demonstrated by integrating between y = 0 and 1 under the cation profile and earnparing the area, ~, with the bulk alloy composition. Hence if these conditions are satisfied a reliable estimate of k ' can probably be obtained for other systems without resorting to a lengthy profile calculation. (d) The cation profile in the scale on a Co-10wt%Fe alloy ~ As pointed out in Section (b), earlier calculations 4 are inconsistent with those in the present work and it is of interest to make similar comparisons for another representative alloy. Some of the disagreement between experimental and theoretical results in Fig. 2 is attributable to the errors in the estimated values of ~'. In order to eliminate this effect the Fe-Co system was chosen for the comparison since the approximation used is reliable s in this case. A cation profile was calculated for the Co--10wt%Fe alloy which exhibits the maximum Fe segregation across the scale. The values of the relevant parameters at 1200°C are given by Mayer and Smeltzer7 and the value of a 0 is obtained from the thermodynamic data for the exchange reaction, Fe -t- CoO = Co + FeO: 15 N ° = 0.096; a0 = 1 × 10-5; a 0 = 3.2 × 10-2 p = 0.6; [3 ~ 15; v = 6.8. It is assumed by Mayer and Smeltzer that for dilute solutions Dre ~ ~ and not DF= ~ [3~-I as in equation (3). The value of (3 given above is chosen so that the compositional variation of DFc is approximately the same using either of these relations. The good agreement between the calculated profile and the experimental results 7 is shown in Fig. 4. The associated oxygen activity profile also agreed closely with that calculated previously. This provides additional confirmation of the validity of the computation procedure used in the present work and reinforces the conclusions relating to the result of Dalvi and Coates 4 given in Section (b). A value of k ' = 0.157 was obtained by iteration. Using an extrapolated value I O.~
~
I
I
I
Theoretical profile T
Range of experimental data 7'
o
O"
"I-I--I--I-" I"Ijl/l, , , , ] 0'2
0"4
0-6
0"8
1
y
FIG. 4.
Experimental and theoretical cation profiles in a scale formed during the oxidation of a Co-10wtyoFe alloy at 1200°C in 10-a atm oxygen.
68
B.D. BASTOW,D. P WHITTLEand G. C. WOOD
of DFe Feo = 4"6 × 10 - e c m ~ s -1 19 the resulting calculated value of kp = 2.4 × 10 -6 gZ cm ~ s-X. This is in reasonable agreement with the experimental value ofkp = 8 × 10 -~ g2 cm 4 s-X although this may be partly fortuitous since the value of Fe DFc o is dependent on the stoichiometry of wustite and the equilibrium relationships in the CoO-Fex_xO system relevant to the formation of solid solution scales are uncertain. 6. CONCLUSIONS Wagner's model for the formation of solid solution oxide scales gives rise to equations which can be solved in a finite difference form without introducing experimental values for either the terminal cation concentrations or the parabolic rate constant as boundary conditions. The comparison of the theoretical and experimental cation profiles provides a satisfactory demonstration of the applicability of the model and the results m a y be summarized as follows: (1) The theoretical and experimental cation profiles show good agreement in view of their independent origins. The observed discrepancies are attributable to: (a) Inaccuracies in the cation concentrations measured by microprobe analysis, particularly at the scale boundaries • (b) The use of an approximate relationship to estimate the boundary condition, ~' which results in the differences between the estimated and measured values increasing with the cobalt content (c) Uncertainties in the values of the parameters describing the cation diffusion coefficients. The possible error in p is reflected in the decreasing accuracy of the estimate of ~' noted in (b) (d) The assumption that the oxide solid solution exhibits ideal behaviour. (2) The predicted compositional variation of the parabolic rate constant, kp, is in good agreement with the experimental results. The overall agreement between experimental and theoretical cation profiles and kp values supports Wagner's theoretical model and demonstrates that the underlying assumptions are satisfied for the oxidation of N i - C o alloys at 1000°C. The authors would like to thank the Science Research Council for the financial support of this work and Professor C. Wagner for a useful discussion. Acknowledgements
REFERENCES 1. C. WAGNER,Corros. Sci. 9, 91 (1969). 2. D. E. COATESand A. D. DALW, Oxidation of Metals 2, 331 (1970). 3. G. C. WOODand J. M. Fm~GUSON,Nature 208, 369 (1965). 4. A. D. DALWand D. E. COATES,Oxidation of Metals 3, 203 (1971). 5. A. D. DALVIand W. W. SM~LTZVaLJ. electrochem. Soc. 121, 386 (1974). 6. P. MAYERand W. W. SMELTZER,J. electrochem. Soc. 119, 626 (1972). 7. P. MAYERand W. W. SMELTZER,J. electrochem. Soc. 121, 538 (1974). 8. D. P. Wm'rn:E, B. D. BASTOWand G. C. WOOD,Oxidation of Metals 9, 215 (1975). 9. R. J. MOOm~.and J. WHITE, J. Mater. Sci. 9, 1393 (1974). 10. W. K. ~ and N. L. I~rEV.SON, J. Phys. Chem. Solids 34, 1093 (1973). 11. J. J. STmUCH,J. B. COHLmand D. H. WmTMO~, J. Am. Ceram. Soc. 56, 119 (1973). 12. J. M. I~RGUSON,Ph.D. Thesis, University of Manchester (1967). 13. G. C. WOOD,I. G. WmonT and J. M. FEROUSON,Corros. Sci. 5, 645 (1965). 14. G. C. WOODand I. G. W~GHT, Corros. ScL 5, 841 (1965).
Diffusion-controlled growth of solid solution scales on nickel-cobalt alloys
69
15. J. F. ELLIOT'r and M. GLEISER, Thermochemistry for Steelmaking, p. 173, Addison-Wesley, Reading, MA (1960). 16. W. B. CRow, Report ARL-70-0090, Project No. 7021, Office of Aerospace Research, U.S.A.F., Wright-Patterson Air Force Base, Ohio (1970). 17. T. EI~CSSON,Corros. Sci. 11, 249 (1971). 18. C. WAGNER,Atom Movements, A.S.M. Monograph, p. 153 (1951). 19. L. I-IIMMEL,R. F. MEHI.,and C. E. BIRCHENALL,Trans. metall. Soe. AIMES, 827 (1953).