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Calculation of precipitate dissolution zone kinetics in oxidising binary two-phase alloys
Vol. 44, No. 10, pp. 4033-4038, 1996 Copyright ~ 1996 Acta Metallurgica Inc. Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 1359-6454/96 $15.00 + 0.00
Acta mater.
Pergamon PII S 1359-6454(96)00054-7
CALCULATION OF PRECIPITATE DISSOLUTION ZONE KINETICS IN OXIDISING BINARY TWO-PHASE ALLOYS P. CARTER, B. GLEESON and D. J. YOUNG School of Materials Science and Engineering, The University of New South Wales, Sydney. NSW 2052, Australia (Received 25 September 1995; in revised form 17January 1996)
Abstract~xidation of a binary alloy at appropriate oxygen potentials leads to the selective oxidation of one component. When that component is concentrated in a dispersed precipitate within a two-phase alloy, dissolution of the precipitate phase will occur when external oxidation takes place. In this paper a diffusional analysis is used to predict the width of the precipitate-dissolutionzone (Xd) formed during such an oxidation process. Solutions to the case of both a stationary and a mobile alloy/scale interface are obtained. The predicted values of X~ derived from both the present and previous theoretical treatments are then compared with recently obtained experimental results on the oxidation behaviour of a Ni-Ni~Si alloy. Predicted values of Xd calculated using the present model were in very close agreement with those obtained experimentally. This is in contrast to the values predicted from previously derived models, which significantly underestimated Xd. Copyright © 1996 Acta Metallurgica Inc.
I. INTRODUCTION Most of the existing theories on alloy oxidation are specific to single-phase alloys. However, many of the commercial alloys used for high-temperature applications are multiphase [1-16]. For example, nickelbase superalloys used for gas turbine components are typically based on a two-phase microstructure of ~,'-Ni3(AI, Ti) precipitates in a y-Ni matrix. In addition to multiphase alloys, many multiphase coatings are used for high temperature applications [14]. The most common of these are the plasma sprayed, N i - C o - C r - A 1 - Y overlay coatings, which are usually two-phase mixtures of the fl-NiAl and 7-Ni structures. Despite the considerable practical importance of multiphase alloys and coatings, there have been relatively few theoretical treatments of their oxidation behaviour [17-23]. Moreover, the treatments which do exist have not yet been compared to experimentally obtained results. Previous studies of multiphase alloy oxidation have been concerned primarily with determining the alloy scaling kinetics and characterising the resulting scale morphologies [1-12]. The results have shown that the oxidation of multiphase alloys is complex and highly variable. For instance, Gesmundo and Gieeson [17] have recently identified the following three broad categories of two-phase, binary-alloy oxidation behaviour: (1) alloys in which each phase oxidises independently forming a two-phase scale; (2) alloys in which the two phases oxidise cooperatively to form a homogeneous, single-phase scale; and (3) alloys in which the precipitate phase, rich in the more reactive solute element, acts as a reservoir for the continued, exclusive growth of the solute-metal oxide scale. This
last form of oxidation behaviour has been observed in a number of systems such as chromia-forming Co-Cr:3C6 alloys [16], and is characterised by the formation of an alloy subsurface zone in which the carbide precipitates are dissolved. The situation is shown schematically in Fig. 1. The aim of the present study is to provide a diffusional analysis for the kinetics of precipitate-dissolution zone formation. Previous theoretical treatments have been presented by Wahl [18], Wang et al. [19], Wang [20] and Gesmundo et al. [21-23] for the oxidation of a two-phase alloy in which the second phase acts as a solute-metal reservoir. All these treatments are semiquantitative, and are aimed at developing an expression for the critical solute-metal content, N~', necessary for the exclusive formation of its oxide scale. The treatments are analogous to but more complicated than Wagner's [24] treatment for predicting N* for a single-phase alloy. The treatment by Wahl [18] is the simplest of the multiphase treatments, as it assumes a linear concentration gradient of the reactive solute component in the precipitate-dissolution zone. As part of Wahl's treatment, the following expression for the depletionzone thickness, Xd, was obtained
x~ = ~ - - ° o ~/
Ng
'
(1)
where DE is the interdiffusion coefficient in the dissolution zone, t is time, and N~B and Ng are the atom fractions of B in the matrix and bulk alloy, respectively.
4033
4034
CARTER et al.:
PRECIPITATE DISSOLUTION ZONE KINETICS
A limitation of Wahl's treatment is that it fails to take into account the contribution of solute from the alloy matrix phase when establishing the mass balance between BO scale formation and consumption of solute metal B. Gesmundo and Gleeson [17] presented a modification of Wahl's treatment to account for the solute concentration in the matrix and found f 4D,N~Lt g0 = q 2 N ~ - N~B"
(2)
It is seen that if N~Bis negligible, equation (2) reduces to equation (1). Gesmundo et al. [21-23] considered the more realistic situation of a non-linear concentration gradient of the reactive solute component. However, this treatment did not lead to an expression for Xd. From a practical standpoint, an accurate expression for Xd is desirable since the service life of a component made from a multiphase material is likely to be dependent on the precipitate-dissolution zone thickness. This is because the dissolution of the precipitates will correspond to a reduction in the supply of solute for the formation of its protective oxide. Further, precipitates often provide strength to a multiphase alloy, and the formation of a deep precipitate-dissolution zone would weaken the alloy. In the following treatment, a diffusional analysis to predict Xd is presented. No assumption is made as to the shape of the solute-concentration gradient and both stationary and mobile alloy/scale interfaces are considered. The predicted values of X~ derived from both the present and previous treatments [equations (1) and 2)] are then compared with new experimental results for the oxidation behaviour of a Ni-Ni3Si alloy [25].
fraction of component B in the alloy, N~. It is assumed that the alloy contains B-rich precipitates of the fl-phase in equilibrium with an s-phase matrix of composition N~ (i,e. N~ > N~). As shown in Fig. 1, it is further assumed that the alloy oxidises to form an exclusive BO scale together with a near-surface, precipitate-free zone of width X~. As indicated in the figure, the composition of the alloy at the alloy/scale interface is N'B, which is assumed to remain constant with respect to time. This assumption is only precisely correct for parabolic kinetics and an immobile interface. Two cases are considered. In the first case, it is assumed that the alloy/scale interface is stationary with respect to time, there is a non-linear concentration profile in the precipitatefree zone, precipitate dissolution kinetics are rapid and, consequently, the rate at which the precipitatedissolution zone widens is diffusion controlled. This last assumption leads to the condition that the boundary compositions are time-invariant. The second case is identical to the first with the single exception that the alloy/scale interface is considered to be mobile with respect to time. It is recognised that the assumption of a stationary alloy/scale interface does not allow for a proper mass balance calculation at this interface, since the growth of a scale results in metal consumption. However, in instances where the rate o f scale growth is low, the amount of metal consumed is small and, hence, this error in the mass balance calculation may be considered negligible. This can be of value in simplifying the numerical analysis.
2.1. Stationary alloy~scale interface model The concentration profile of B in the single-phase region of the precipitate-dissolution zone is obtained by solving Fick's second law 3NB O2NB 0t - D ~ at x > 0
2. PROPOSED MODELS
We consider a two-phase binary alloy, in which components A and B form immiscible oxides AO and BO. It is assumed that no ternary oxides form, and that BO is much more stable than AO. The bulk composition of the alloy is defined by the atom
s~!e
alloy stanonary~
;if :
~
~ I I
/
/
I
I
',~J~
interface
(3)
for the boundary conditions Ns=N'B
at
x=0
NB=N',
at
x=Xd
for for
t>0 t~>0
where x = 0 corresponds to the original alloy/gas interface (Fig. 1). Gesmundo et al. [22, 26] have solved equation (3) for similar boundary conditions, except it was assumed that the concentration of solute B at the interface was zero. Modified for a finite interface composition, the solution for the concentration profile of B in the single phase dissolution-zone is of the form
I
erf(~x~
\ 2x/ Dst J N, = (N~ - N'a) 0 Fig.
X I. Schematic defining boundary conditions for diffusional analysis models.
eft(V)
+ Nh,
(4)
where Xd - 2~',,/OBt
(5)
CARTER et al.: PRECIPITATE DISSOLUTION ZONE KINETICS In these expressions, DB is the interdiffusion coefficient. The flux of B at the alloy/scale interface, JB, is given as (6)
J B - D B d N B ~=0 Vm dx '
where V,, is the molar volume of the alloy. Differentiating equation (41)with respect to x at x = 0 and substituting the solution into equation (6) gives D~ '-~ (N~ - N~) JB = - - 1,2 12 - Vm~
~/~ _ _
(7)
eft(y)
The instantaneous flux of B diffusing through the oxide away from the alloy/scale interface, Ja~o~), is given by (8)
where V~~ is the molar volume of the alloy at the alloy/scale interface and k~ is the corrosion constant of the alloy, a measure of its surface recession rate. For an oxide M.Ob it is related to the parabolic scaling rate constant kp by
4035
2.2. Moving alloy~scale interface model Maak [27] has solved Fick's second law [equation (3)] for the case of internal oxidation accompanied by the formation of an external scale, where both the alloy/scale interface and the internal-oxide/alloy interface are mobile. M a a k ' s solution was recently modified by Gesmundo et al. [22, 26] for the converse case of two-phase alloy oxidation with the formation of an exclusive external scale and a precipitate-dissolution zone. Both Gesmundo and M a a k assumed the concentration of solute B at one of the interfaces to be zero. In the present analysis the boundary conditions for Fick's second law [equation (3)] are NB=N~B
for
x=X
and
t>0
NB=N~
for
X=Xd
and
t>0,
where x = X corresponds to the alloy/scale interface and x = 0 is again the original alloy/gas interface (Fig. 1). The modified solution for the concentration profile of B in the single phase region is then
Ierf(
\ 2 \ / D , tJ
NB = (m~ - N~B)
[erf(~.) -- eft(u)]
+ APe(15)
kp = k J b M o y " \aV,.,]
(9)
where Mo is the atomic weight of oxygen. At steady-state JB = J~ox) so that, equating equations (7) and (8) yields, after rearrangement, N~ = ~'-'uerf(7) + APE,
(10)
where
It can be shown that this concentration profile reduces to the expression presented by Whittle and W o o d [28] for a single-phase alloy, by considering the limit as the volume fraction of B-rich phase tends toward zero, and the depletion zone becomes large. Substituting Ng for N~ in equation (15) and noting that for large Xo, erf(~,) --* 1, gives the expression for the concentration profile obtained by Whittle and Wood, i.e.
(11)
U ~
~ e r f ( - - - - . ~ x "] - e r f ( u ) 1 The mass balance at x = Xd, i.e. the a/~ +/~ interface, is
UB -- APB= L Ng - AP,
\ 2x/ D,t,/ erfc(u)
(16)
The flux at x = X = x/~ct, is given by
DB (dNB~ Fro ~,~ d x / v = Xd
_ (N~ - N~) dXd FI11
dt
(12)
DB(dNB~ J"=-~ dx }~.x
"
Hence, obtaining expressions for (dNB/dX) ..... , from equation (4) and dXd/dt from equation (5), one finds N~ - N~ N~ - N-~ - ~'~'~ exp(y2)erf(~')'
(13)
Substituting the expression for N~ from equation (10) into (13) and rearranging shows how the bulk alloy concentration of B and the scale/alloy interface value are related through the kinetic parameters ~, and u, i.e.
______L____ u N~° = y exp(~:) + n~"2uerf(~/) + APB.
(14)
(17)
= D~'2(N~ - APE) 1 Vmn'/2t ':2 exp(u2)[erf(7) -- err(u)] and the flux through the oxide, J~ox, is still given by equation (8). At steady state, JB = JB~ox),and therefore N~" = x','2u exp(u2)[erf(y) - eft(u) + APE. (18) AS before, the mass balance at x = Xd (the ~/u + fl interface) is given by equation (12) and therefore N~ -- APB= hE,27 exp(y2)[erf(?) _ eft(u)]. N.° - N~
(19)
4036
CARTER et al.:
PRECIPITATE DISSOLUTION ZONE KINETICS 2O i
-
-
-
Slatlonar/
•
~mnn~rli 1251
o4
Xd
•
•
/
•
•
4
/
•
•
•
pl
i
/ / i
s~"
0
~ o
,
~
,
I
t
L
,,
I, 100
5o x
Fig. 2. Cross sectional optical image of the Ni-16.2 at.%Si alloy oxidised for 8 h at l l 3 0 C in a 3% CO~ + Ar mixture with a total pressure of 1 atm [25].
Substituting for rearranging gives
N~
from
equation
(18)
and
+ n ~:u exp(u2)[erf(7) -- eft(u)] + N~.
(20)
N~ = u exp(u'-) "¢ exp(72)
3. APPLICATION OF THE MODELS The values of X~ predicted by both the present and previous [17, 18] treatments were compared with recent experimental results for the oxidation behaviour of a Ni-Ni3Si alloy of measured composition N i - 1 6 . 2 a t . % S i [25]. The alloy was oxidised in a thermogravimetric apparatus for up to 10h at 1130°C in a 3% C O . ~ + A r mixture with a total pressure of 1 atm. The specimens were found to form a scale which was identified by electron microprobe analysis ( E P M A ) as SiO~. Figure 2 shows a cross-section of the Ni-16.2 at.%Si alloy after 8 h oxidation. It is seen that a Ni3Si precipitate-dissolution zone of uniform width formed in the subsurface region of the alloy as a result of depletion of silicon to form the SiO: scale. Composition profiles within this dissolution zone were determined by E P M A , and interdiffusion coefficients calculated via the M a t a n o [29] technique. An example of a concentration profile is shown in Fig. 3, which was obtained after oxidation for 8h. Figure 3 shows that the silicon concentration varies from about 5.5 at.% at the alloy/scale interface to about 14 at.% at the y-Ni/),-Ni + fl-Ni3Si interface. These compositions were found to be time-invariant for the oxidation times studied, thus indicating that
, ,,
I, 150
,,, 200
(~)
Fig. 3. Experimental concentration profile of the Ni16.2 at.%Si oxidised for 8 h at 1130:'C in a 3% CO., + Ar mixture with a total pressure of 1 atm. O:. The two linear zones used for determining D~n are superimposed.
the steady-state condition existed in the precipitatedissolution zone. Inspection of the concentration profile in Fig. 3 shows that it consists essentially of two regions: one with a very steep gradient for about 30% of the total zone thickness and the other with a relatively shallow gradient for the remaining 70%. Using the M a t a n o [29] technique, the average interdiffusion coefficient was found [30] to be about 3.5 x 10 -~° and 1.2 x 10 -9 cm2/s in the regions of the steep and shallow gradient regions, respectively. Owing to the relatively constant value of D in these two regions, it was possible to estimate an effective interdiffusion coefficient, De~, using a weighted average. Values of Xd as a function of oxidation time were calculated for the Ni-16.2 at.%Si alloy, by solving equations (14) and (20), and using the experimentally determined parameters given in Table 1. The equations were simplified by substituting five-term Taylor series expansions for both the exponential and error function terms involving Xd. This resulted in a univariant polynomial in Xd and t, which was solved for Xd for values of t between 0 and l0 h, using a symbolic mathematics program ( M A P L E , Waterloo Software, Waterloo, Canada). The values of Xd calculated from the present and previous models are compared with the experimentally determined values in the parabolic plots in Fig. 4. The predicted and experimentally determined rate constants for dissolution-zone widening are summarised in Table 2. It is seen that the present treatment agrees well with the Table 1. Experimentally determined parameters used to solve equations (14) and (20) Vo, = 23.11 g/cm3 k~ = 1.55 x 10-~ cmZ/s Ng = 16.17 at.% V= = 6.63 g/cm3 D~e= 9.40 x 10-L°cmZ/s Nh = 5.5 at.%
CARTER
et al.:
4037
PRECIPITATE DISSOLUTION ZONE KINETICS
2rio I I
'
- --
Geamundoi~IGM111on[17J
~IO"
•
2OO
lOO
111 p
~o
Two-pb $ t~gJOn
Fig. 5. Schematic representation of the linear and error function concentration profiles. 0 40
j
I 8o
,
I ~2o
I
I ~eo
t
aoo
Time I/2 (s)
Fig. 4. Parabolic plots for the experimentally and theoretically determined Xn for Ni-16.2 at.%Si oxidised at ll3ffC in 7 x 10-6atm O.,. experimental results, whereas the previous models produce significant underestimates. 4. DISCUSSION In contrast with the models of Wahl [18] and Gesmundo and Gleeson [17], the present description provided very close agreement with Xd values obtained experimentally for the Ni-Nir Si system. The significant underestimations of Xd produced by the former models is explained by comparing their assumed linear concentration profile in the dissolution zone with the error-function profile of the present model, as shown in Fig. 5. The shaded area in this figure is equivalent to the mass of B which has diffused from the alloy to form the surface oxide BO. It is clear that the corresponding area for the linear profile is greater than that for the error function profile. It follows therefore that the depth of an actual dissolution zone, in which the concentration follows an error function profile, must be greater than would be predicted for a linear profile corresponding to the same amount of B consumption for the growth of a given scale thickness. The X~ values calculated from Gesmundo and Gleeson's model are greater than those from Wahl's model because the former model takes into account the rather high solubility of silicon in the 7-Ni matrix ( ~ 1 4 a t . % at 1130°C [31]). Notwithstanding this improvement, the assumption of a linear concentration profile is invalid and Table
2. Experimental and calculated parabolic dissolution-zone widening
Model Experimental [25] Stationary interface model Mobile interface model G e s m u n d o and Gleeson [17] Wahl [18]
constants
Parabolic constant k~, (cm"/s) 1.3 1.2 1.2 3.5 1.8
x x × x ×
10 -s 10-" 10 -s 10-' 10 -9
for
significantly underestimates the depth of solute consumption from the alloy. The present model's high degree of accuracy is somewhat surprising given it's simplifying assumptions of a constant diffusion coefficient and rapid precipitate dissolution. The validity and limitations of these assumptions are now discussed. The interdiffusion coefficient in fact varies by about an order of magnitude in the composition range of the precipitate-dissolution zone. In spite of this, the use of a constant (weighted average) interdiffusion coefficient gave excellent agreement between the experimental and calculated Xd values (Fig. 4). Figure 3 compares the calculated (based on D,g) and experimentally determined silicon concentration profiles in the depletion zone. The error function curve is seen to deviate from the experimental concentration profile; however, it does provide a reasonable approximation of the experimental profile at both the alloy/scale and the ";/? + // interfaces. It is for this reason that the calculations were reasonably accurate, since the model is constructed on the basis of mass balances at these interfaces. The compositions at these interfaces were time-invariant. Thus, changing the value of Den has the effect of changing the slopes of the error-function curve at the boundaries in order to provide the requisite flux of silicon as determined by the rate of scale growth. The simple weighting method adopted in this analysis to determine D~ was adequate. A more rigorous method does exist [32], but was not used in the present analysis due to the reasonable results obtained. In applying the model to other systems which show a more marked concentration dependence on D, the assumption of a constant Dc~ may not be appropriate. Due to the lack of appropriate experimental data for the oxidation or sulphidation of multiphase alloys, however, the applicability of the models to other systems cannot be assessed. The second assumption of rapid precipitate dissolution kinetics was appropriate for the Ni-Ni3Si system. A TEM study [33] of the Ni-Ni~Si system has shown that the Ni/Ni~Si interface is incoherent, a fact reflected in the globular shape of the Ni3Si
4038
CARTER et al.: PRECIPITATE DISSOLUTION ZONE KINETICS
precipitates shown in Fig. 2. The incoherency of the precipitate interfaces would allow for their rapid dissolution, and this is reflected in the fact that the silicon content of the matrix phase at the bulk alloy/dissolution-zone boundary was time-invariant [25]. Thus the rate of Ni3Si dissolution was sufficient to maintain local equilibrium, and did not contribute to the overall rate control. The planarity of the Ni/Ni + Ni3Si interface shows that this was uniformly the case, and the process was controlled by diffusion, not dissolution. The assumption of rapid precipitate dissolution kinetics will not always be valid in systems containing semicoherent or coherent precipitates [e.g. F e (Fe, Cr)23C6and Ni-Ni~AI]. If precipitate dissolution is too slow to sustain the exclusive growth of the solute-metal scale, formation of the nonprotective, solvent-metal oxide may occur. In the case where solvent-metal oxidation occurs because the precipitates fail to dissolve, the solute-rich precipitates are engulfed in the metal consumption zone so that no noticeable depletion zone exists. Alternatively, internal oxidation of the precipitates may result. In both cases the precipitates fail to dissolve sufficiently fast, thus rendering the model inapplicable. F r o m Fig. 4 it can be seen that both the stationary and the mobile-interface models are in close agreement, with the stationary interface model giving a slightly smaller value of Xd. The difference is small because of the relatively small kc value used. Only if the kr values are high, as in sulphidation, is the approximation of a stationary alloy/scale interface likely to be invalid.
5. CONCLUSIONS (1) A diffusional analysis can be used to accurately predict the growth kinetics of a precipitate-dissolution zone formed in a two-phase alloy during thermal oxidation. In this situation, the precipitate phase acts as a reservoir, providing a supply of the more reactive metal which leads to the exclusive formation of solute-metal oxide scale. (2) The rate o f Ni3Si precipitate-dissolution in ),-Ni matrix alloys is enough to maintain local equilibrium at the precipitate-dissolution front, and not contribute to overall rate control. (3) The assumption of a stationary alloy/scale interface is a reasonable approximation in the Ni-Ni3Si system where the rate of scale growth and, hence, metal consumption is sufficiently low. (4) Even though D varies over an order of magnitude in the precipitate-dissolution zone, an appropriate weighted average value is successful in predicting zone widening kinetics. Acknowledgements--This research was carried out with the assistance of a grant from the Australian Research Council. One of the authors, P. Carter, also gratefully acknowledges the receipt of an Australian Postgraduate Research Award
during the course of this research. Assistance in obtaining some of the experimental results was provided by Dr Mark Harper.
REFERENCES 1. J. G. Smeggil, Oxid. Metals 9, 31 (1975). 2. J. G. Smeggil, Oxid. Metals 9, 225 (1975). 3. J. Stringer, D. M. Johnson and D. P. Whittle, Oxid. Metals 12, 257 (1978). 4. M. E. El Dahshan and M. I. Hazzaa, Werkst. Korrs. 38, 422 (1987). 5. F, H. Stott, G. C. Wood and J. G. Fountain. Oxid. Metals 14, 31 (1980). 6. L. V. Mallia and D. J. Young, Oxid. Metals 22, 227 (1984). 7. N. Belen, P. Tomaszewicz and D. J. Young, Oxid. Metals 22, 227 (1984). 8. J. Doychak, J. A. Nesbitt, R. D. Noebe and R. R. Bowman, Oxid. Metals 38, 45 (1992). 9. D. E. Alman and N, S. Stoloff. in High Temperature Silicides and Refracto U, Alloys, Mat. Res. Soc. Symp. Proc. Vol. 322 (edited by C. U Briant, J. J. Petrovic, B. P. Bewlay, A. K. Vasudvan and H. A. Lipsitt), p. 255. Materials Research Society, Pittsburgh, PA (1994). 10. B. Gleeson, W. H. Cheung and D. J. Young, Corros. Sci. 35, 923 (1993). I1. S. Espevik, R. A. Rapp, P. L. Daniel and J. P. Hirth, Oxid. Metals 20, 37 (1983). 12. C. A. Barett and C. E. Lowell, Oxid. Metals 11, 199 (1977). 13. J. L. Gonzalez Carrasco, P. Adeva and M. Abell, Oxid. Metals 33, 1 (1990). 14. J. A. Nesbitt and R~ W. Heckel, Oxid. Metals 29, 75 (1988). 15. G. P. Wagner and G. Simkovich, Oxid. Metals 27, 157 (1987). 16. M. E. El Dahshan, J. Stringer and D. P. Whittle, Cobalt 4, 86 (1974). 17. F. Gesmundo and B. Gleeson, Oxid. Metals 44, 211 (1995). 18. G. Wahl, Thin Solid Films 107, 417 (1983). 19. Ge Wang, B. Gleeson and D. L. Douglass, Oxid. Metals 35, 333 (1991). 20. Ge Wang, J de Physique IV, Colloque C9, supplement to Journal de Physique I11 3, 873 (1993). 21. F. Gesmundo, F. Viani and D. L. Douglass, Oxid. Metals 39, 197 (1993). 22. F. Gesmundo, F. Viani and D. L. Douglass, Oxid. Metals 40, 373 (1993). 23. F. Gesmundo, F. Viani and D. L. Douglass, Oxid. Metals 42, 285 (1994). 24. C. Wagner, Z. Elektrochem. 63, 772 (1959). 25. P. Carter, B. Gleeson, D. J. Young and M. Harper, University of New South Wales, unpublished research. 26. F. Gesmundo and F. Viani, Oxid. Metals 25, 269 (1986). 27. F. Maak, Z. Metallk. 52, 545 (1961). 28. D. P. Whittle and G. C. Wood, J. electrochem. Sac. 115, 133 (1968). 29. C. Matano, Jap. J. Phys. 8, 109 (1933). 30. P. Carter, B. Gleeson, D. J. Young and M. Harper, University of New South Wales, unpublished research. 31. T. B. Massalski (ed.) Binary Alloy Phase Diagrams, p. 1754. ASM, Metals Park, OH (1986). 32. M. A. Dayananda, in Ordered lntermetallics from Physical Metallurgy and Mechanical Behaviour (edited by T. Liu, R. W. Cahn and G. Sauthoff), Vol, 213, p. 465. Kuiver Academic Publishers, London (1992). 33. M. Bartur and M.-A. Nicolets, J. electrochem. Soc. 131, 371 (1984).
Acta mater.
Pergamon PII S 1359-6454(96)00054-7
CALCULATION OF PRECIPITATE DISSOLUTION ZONE KINETICS IN OXIDISING BINARY TWO-PHASE ALLOYS P. CARTER, B. GLEESON and D. J. YOUNG School of Materials Science and Engineering, The University of New South Wales, Sydney. NSW 2052, Australia (Received 25 September 1995; in revised form 17January 1996)
Abstract~xidation of a binary alloy at appropriate oxygen potentials leads to the selective oxidation of one component. When that component is concentrated in a dispersed precipitate within a two-phase alloy, dissolution of the precipitate phase will occur when external oxidation takes place. In this paper a diffusional analysis is used to predict the width of the precipitate-dissolutionzone (Xd) formed during such an oxidation process. Solutions to the case of both a stationary and a mobile alloy/scale interface are obtained. The predicted values of X~ derived from both the present and previous theoretical treatments are then compared with recently obtained experimental results on the oxidation behaviour of a Ni-Ni~Si alloy. Predicted values of Xd calculated using the present model were in very close agreement with those obtained experimentally. This is in contrast to the values predicted from previously derived models, which significantly underestimated Xd. Copyright © 1996 Acta Metallurgica Inc.
I. INTRODUCTION Most of the existing theories on alloy oxidation are specific to single-phase alloys. However, many of the commercial alloys used for high-temperature applications are multiphase [1-16]. For example, nickelbase superalloys used for gas turbine components are typically based on a two-phase microstructure of ~,'-Ni3(AI, Ti) precipitates in a y-Ni matrix. In addition to multiphase alloys, many multiphase coatings are used for high temperature applications [14]. The most common of these are the plasma sprayed, N i - C o - C r - A 1 - Y overlay coatings, which are usually two-phase mixtures of the fl-NiAl and 7-Ni structures. Despite the considerable practical importance of multiphase alloys and coatings, there have been relatively few theoretical treatments of their oxidation behaviour [17-23]. Moreover, the treatments which do exist have not yet been compared to experimentally obtained results. Previous studies of multiphase alloy oxidation have been concerned primarily with determining the alloy scaling kinetics and characterising the resulting scale morphologies [1-12]. The results have shown that the oxidation of multiphase alloys is complex and highly variable. For instance, Gesmundo and Gieeson [17] have recently identified the following three broad categories of two-phase, binary-alloy oxidation behaviour: (1) alloys in which each phase oxidises independently forming a two-phase scale; (2) alloys in which the two phases oxidise cooperatively to form a homogeneous, single-phase scale; and (3) alloys in which the precipitate phase, rich in the more reactive solute element, acts as a reservoir for the continued, exclusive growth of the solute-metal oxide scale. This
last form of oxidation behaviour has been observed in a number of systems such as chromia-forming Co-Cr:3C6 alloys [16], and is characterised by the formation of an alloy subsurface zone in which the carbide precipitates are dissolved. The situation is shown schematically in Fig. 1. The aim of the present study is to provide a diffusional analysis for the kinetics of precipitate-dissolution zone formation. Previous theoretical treatments have been presented by Wahl [18], Wang et al. [19], Wang [20] and Gesmundo et al. [21-23] for the oxidation of a two-phase alloy in which the second phase acts as a solute-metal reservoir. All these treatments are semiquantitative, and are aimed at developing an expression for the critical solute-metal content, N~', necessary for the exclusive formation of its oxide scale. The treatments are analogous to but more complicated than Wagner's [24] treatment for predicting N* for a single-phase alloy. The treatment by Wahl [18] is the simplest of the multiphase treatments, as it assumes a linear concentration gradient of the reactive solute component in the precipitate-dissolution zone. As part of Wahl's treatment, the following expression for the depletionzone thickness, Xd, was obtained
x~ = ~ - - ° o ~/
Ng
'
(1)
where DE is the interdiffusion coefficient in the dissolution zone, t is time, and N~B and Ng are the atom fractions of B in the matrix and bulk alloy, respectively.
4033
4034
CARTER et al.:
PRECIPITATE DISSOLUTION ZONE KINETICS
A limitation of Wahl's treatment is that it fails to take into account the contribution of solute from the alloy matrix phase when establishing the mass balance between BO scale formation and consumption of solute metal B. Gesmundo and Gleeson [17] presented a modification of Wahl's treatment to account for the solute concentration in the matrix and found f 4D,N~Lt g0 = q 2 N ~ - N~B"
(2)
It is seen that if N~Bis negligible, equation (2) reduces to equation (1). Gesmundo et al. [21-23] considered the more realistic situation of a non-linear concentration gradient of the reactive solute component. However, this treatment did not lead to an expression for Xd. From a practical standpoint, an accurate expression for Xd is desirable since the service life of a component made from a multiphase material is likely to be dependent on the precipitate-dissolution zone thickness. This is because the dissolution of the precipitates will correspond to a reduction in the supply of solute for the formation of its protective oxide. Further, precipitates often provide strength to a multiphase alloy, and the formation of a deep precipitate-dissolution zone would weaken the alloy. In the following treatment, a diffusional analysis to predict Xd is presented. No assumption is made as to the shape of the solute-concentration gradient and both stationary and mobile alloy/scale interfaces are considered. The predicted values of X~ derived from both the present and previous treatments [equations (1) and 2)] are then compared with new experimental results for the oxidation behaviour of a Ni-Ni3Si alloy [25].
fraction of component B in the alloy, N~. It is assumed that the alloy contains B-rich precipitates of the fl-phase in equilibrium with an s-phase matrix of composition N~ (i,e. N~ > N~). As shown in Fig. 1, it is further assumed that the alloy oxidises to form an exclusive BO scale together with a near-surface, precipitate-free zone of width X~. As indicated in the figure, the composition of the alloy at the alloy/scale interface is N'B, which is assumed to remain constant with respect to time. This assumption is only precisely correct for parabolic kinetics and an immobile interface. Two cases are considered. In the first case, it is assumed that the alloy/scale interface is stationary with respect to time, there is a non-linear concentration profile in the precipitatefree zone, precipitate dissolution kinetics are rapid and, consequently, the rate at which the precipitatedissolution zone widens is diffusion controlled. This last assumption leads to the condition that the boundary compositions are time-invariant. The second case is identical to the first with the single exception that the alloy/scale interface is considered to be mobile with respect to time. It is recognised that the assumption of a stationary alloy/scale interface does not allow for a proper mass balance calculation at this interface, since the growth of a scale results in metal consumption. However, in instances where the rate o f scale growth is low, the amount of metal consumed is small and, hence, this error in the mass balance calculation may be considered negligible. This can be of value in simplifying the numerical analysis.
2.1. Stationary alloy~scale interface model The concentration profile of B in the single-phase region of the precipitate-dissolution zone is obtained by solving Fick's second law 3NB O2NB 0t - D ~ at x > 0
2. PROPOSED MODELS
We consider a two-phase binary alloy, in which components A and B form immiscible oxides AO and BO. It is assumed that no ternary oxides form, and that BO is much more stable than AO. The bulk composition of the alloy is defined by the atom
s~!e
alloy stanonary~
;if :
~
~ I I
/
/
I
I
',~J~
interface
(3)
for the boundary conditions Ns=N'B
at
x=0
NB=N',
at
x=Xd
for for
t>0 t~>0
where x = 0 corresponds to the original alloy/gas interface (Fig. 1). Gesmundo et al. [22, 26] have solved equation (3) for similar boundary conditions, except it was assumed that the concentration of solute B at the interface was zero. Modified for a finite interface composition, the solution for the concentration profile of B in the single phase dissolution-zone is of the form
I
erf(~x~
\ 2x/ Dst J N, = (N~ - N'a) 0 Fig.
X I. Schematic defining boundary conditions for diffusional analysis models.
eft(V)
+ Nh,
(4)
where Xd - 2~',,/OBt
(5)
CARTER et al.: PRECIPITATE DISSOLUTION ZONE KINETICS In these expressions, DB is the interdiffusion coefficient. The flux of B at the alloy/scale interface, JB, is given as (6)
J B - D B d N B ~=0 Vm dx '
where V,, is the molar volume of the alloy. Differentiating equation (41)with respect to x at x = 0 and substituting the solution into equation (6) gives D~ '-~ (N~ - N~) JB = - - 1,2 12 - Vm~
~/~ _ _
(7)
eft(y)
The instantaneous flux of B diffusing through the oxide away from the alloy/scale interface, Ja~o~), is given by (8)
where V~~ is the molar volume of the alloy at the alloy/scale interface and k~ is the corrosion constant of the alloy, a measure of its surface recession rate. For an oxide M.Ob it is related to the parabolic scaling rate constant kp by
4035
2.2. Moving alloy~scale interface model Maak [27] has solved Fick's second law [equation (3)] for the case of internal oxidation accompanied by the formation of an external scale, where both the alloy/scale interface and the internal-oxide/alloy interface are mobile. M a a k ' s solution was recently modified by Gesmundo et al. [22, 26] for the converse case of two-phase alloy oxidation with the formation of an exclusive external scale and a precipitate-dissolution zone. Both Gesmundo and M a a k assumed the concentration of solute B at one of the interfaces to be zero. In the present analysis the boundary conditions for Fick's second law [equation (3)] are NB=N~B
for
x=X
and
t>0
NB=N~
for
X=Xd
and
t>0,
where x = X corresponds to the alloy/scale interface and x = 0 is again the original alloy/gas interface (Fig. 1). The modified solution for the concentration profile of B in the single phase region is then
Ierf(
\ 2 \ / D , tJ
NB = (m~ - N~B)
[erf(~.) -- eft(u)]
+ APe(15)
kp = k J b M o y " \aV,.,]
(9)
where Mo is the atomic weight of oxygen. At steady-state JB = J~ox) so that, equating equations (7) and (8) yields, after rearrangement, N~ = ~'-'uerf(7) + APE,
(10)
where
It can be shown that this concentration profile reduces to the expression presented by Whittle and W o o d [28] for a single-phase alloy, by considering the limit as the volume fraction of B-rich phase tends toward zero, and the depletion zone becomes large. Substituting Ng for N~ in equation (15) and noting that for large Xo, erf(~,) --* 1, gives the expression for the concentration profile obtained by Whittle and Wood, i.e.
(11)
U ~
~ e r f ( - - - - . ~ x "] - e r f ( u ) 1 The mass balance at x = Xd, i.e. the a/~ +/~ interface, is
UB -- APB= L Ng - AP,
\ 2x/ D,t,/ erfc(u)
(16)
The flux at x = X = x/~ct, is given by
DB (dNB~ Fro ~,~ d x / v = Xd
_ (N~ - N~) dXd FI11
dt
(12)
DB(dNB~ J"=-~ dx }~.x
"
Hence, obtaining expressions for (dNB/dX) ..... , from equation (4) and dXd/dt from equation (5), one finds N~ - N~ N~ - N-~ - ~'~'~ exp(y2)erf(~')'
(13)
Substituting the expression for N~ from equation (10) into (13) and rearranging shows how the bulk alloy concentration of B and the scale/alloy interface value are related through the kinetic parameters ~, and u, i.e.
______L____ u N~° = y exp(~:) + n~"2uerf(~/) + APB.
(14)
(17)
= D~'2(N~ - APE) 1 Vmn'/2t ':2 exp(u2)[erf(7) -- err(u)] and the flux through the oxide, J~ox, is still given by equation (8). At steady state, JB = JB~ox),and therefore N~" = x','2u exp(u2)[erf(y) - eft(u) + APE. (18) AS before, the mass balance at x = Xd (the ~/u + fl interface) is given by equation (12) and therefore N~ -- APB= hE,27 exp(y2)[erf(?) _ eft(u)]. N.° - N~
(19)
4036
CARTER et al.:
PRECIPITATE DISSOLUTION ZONE KINETICS 2O i
-
-
-
Slatlonar/
•
~mnn~rli 1251
o4
Xd
•
•
/
•
•
4
/
•
•
•
pl
i
/ / i
s~"
0
~ o
,
~
,
I
t
L
,,
I, 100
5o x
Fig. 2. Cross sectional optical image of the Ni-16.2 at.%Si alloy oxidised for 8 h at l l 3 0 C in a 3% CO~ + Ar mixture with a total pressure of 1 atm [25].
Substituting for rearranging gives
N~
from
equation
(18)
and
+ n ~:u exp(u2)[erf(7) -- eft(u)] + N~.
(20)
N~ = u exp(u'-) "¢ exp(72)
3. APPLICATION OF THE MODELS The values of X~ predicted by both the present and previous [17, 18] treatments were compared with recent experimental results for the oxidation behaviour of a Ni-Ni3Si alloy of measured composition N i - 1 6 . 2 a t . % S i [25]. The alloy was oxidised in a thermogravimetric apparatus for up to 10h at 1130°C in a 3% C O . ~ + A r mixture with a total pressure of 1 atm. The specimens were found to form a scale which was identified by electron microprobe analysis ( E P M A ) as SiO~. Figure 2 shows a cross-section of the Ni-16.2 at.%Si alloy after 8 h oxidation. It is seen that a Ni3Si precipitate-dissolution zone of uniform width formed in the subsurface region of the alloy as a result of depletion of silicon to form the SiO: scale. Composition profiles within this dissolution zone were determined by E P M A , and interdiffusion coefficients calculated via the M a t a n o [29] technique. An example of a concentration profile is shown in Fig. 3, which was obtained after oxidation for 8h. Figure 3 shows that the silicon concentration varies from about 5.5 at.% at the alloy/scale interface to about 14 at.% at the y-Ni/),-Ni + fl-Ni3Si interface. These compositions were found to be time-invariant for the oxidation times studied, thus indicating that
, ,,
I, 150
,,, 200
(~)
Fig. 3. Experimental concentration profile of the Ni16.2 at.%Si oxidised for 8 h at 1130:'C in a 3% CO., + Ar mixture with a total pressure of 1 atm. O:. The two linear zones used for determining D~n are superimposed.
the steady-state condition existed in the precipitatedissolution zone. Inspection of the concentration profile in Fig. 3 shows that it consists essentially of two regions: one with a very steep gradient for about 30% of the total zone thickness and the other with a relatively shallow gradient for the remaining 70%. Using the M a t a n o [29] technique, the average interdiffusion coefficient was found [30] to be about 3.5 x 10 -~° and 1.2 x 10 -9 cm2/s in the regions of the steep and shallow gradient regions, respectively. Owing to the relatively constant value of D in these two regions, it was possible to estimate an effective interdiffusion coefficient, De~, using a weighted average. Values of Xd as a function of oxidation time were calculated for the Ni-16.2 at.%Si alloy, by solving equations (14) and (20), and using the experimentally determined parameters given in Table 1. The equations were simplified by substituting five-term Taylor series expansions for both the exponential and error function terms involving Xd. This resulted in a univariant polynomial in Xd and t, which was solved for Xd for values of t between 0 and l0 h, using a symbolic mathematics program ( M A P L E , Waterloo Software, Waterloo, Canada). The values of Xd calculated from the present and previous models are compared with the experimentally determined values in the parabolic plots in Fig. 4. The predicted and experimentally determined rate constants for dissolution-zone widening are summarised in Table 2. It is seen that the present treatment agrees well with the Table 1. Experimentally determined parameters used to solve equations (14) and (20) Vo, = 23.11 g/cm3 k~ = 1.55 x 10-~ cmZ/s Ng = 16.17 at.% V= = 6.63 g/cm3 D~e= 9.40 x 10-L°cmZ/s Nh = 5.5 at.%
CARTER
et al.:
4037
PRECIPITATE DISSOLUTION ZONE KINETICS
2rio I I
'
- --
Geamundoi~IGM111on[17J
~IO"
•
2OO
lOO
111 p
~o
Two-pb $ t~gJOn
Fig. 5. Schematic representation of the linear and error function concentration profiles. 0 40
j
I 8o
,
I ~2o
I
I ~eo
t
aoo
Time I/2 (s)
Fig. 4. Parabolic plots for the experimentally and theoretically determined Xn for Ni-16.2 at.%Si oxidised at ll3ffC in 7 x 10-6atm O.,. experimental results, whereas the previous models produce significant underestimates. 4. DISCUSSION In contrast with the models of Wahl [18] and Gesmundo and Gleeson [17], the present description provided very close agreement with Xd values obtained experimentally for the Ni-Nir Si system. The significant underestimations of Xd produced by the former models is explained by comparing their assumed linear concentration profile in the dissolution zone with the error-function profile of the present model, as shown in Fig. 5. The shaded area in this figure is equivalent to the mass of B which has diffused from the alloy to form the surface oxide BO. It is clear that the corresponding area for the linear profile is greater than that for the error function profile. It follows therefore that the depth of an actual dissolution zone, in which the concentration follows an error function profile, must be greater than would be predicted for a linear profile corresponding to the same amount of B consumption for the growth of a given scale thickness. The X~ values calculated from Gesmundo and Gleeson's model are greater than those from Wahl's model because the former model takes into account the rather high solubility of silicon in the 7-Ni matrix ( ~ 1 4 a t . % at 1130°C [31]). Notwithstanding this improvement, the assumption of a linear concentration profile is invalid and Table
2. Experimental and calculated parabolic dissolution-zone widening
Model Experimental [25] Stationary interface model Mobile interface model G e s m u n d o and Gleeson [17] Wahl [18]
constants
Parabolic constant k~, (cm"/s) 1.3 1.2 1.2 3.5 1.8
x x × x ×
10 -s 10-" 10 -s 10-' 10 -9
for
significantly underestimates the depth of solute consumption from the alloy. The present model's high degree of accuracy is somewhat surprising given it's simplifying assumptions of a constant diffusion coefficient and rapid precipitate dissolution. The validity and limitations of these assumptions are now discussed. The interdiffusion coefficient in fact varies by about an order of magnitude in the composition range of the precipitate-dissolution zone. In spite of this, the use of a constant (weighted average) interdiffusion coefficient gave excellent agreement between the experimental and calculated Xd values (Fig. 4). Figure 3 compares the calculated (based on D,g) and experimentally determined silicon concentration profiles in the depletion zone. The error function curve is seen to deviate from the experimental concentration profile; however, it does provide a reasonable approximation of the experimental profile at both the alloy/scale and the ";/? + // interfaces. It is for this reason that the calculations were reasonably accurate, since the model is constructed on the basis of mass balances at these interfaces. The compositions at these interfaces were time-invariant. Thus, changing the value of Den has the effect of changing the slopes of the error-function curve at the boundaries in order to provide the requisite flux of silicon as determined by the rate of scale growth. The simple weighting method adopted in this analysis to determine D~ was adequate. A more rigorous method does exist [32], but was not used in the present analysis due to the reasonable results obtained. In applying the model to other systems which show a more marked concentration dependence on D, the assumption of a constant Dc~ may not be appropriate. Due to the lack of appropriate experimental data for the oxidation or sulphidation of multiphase alloys, however, the applicability of the models to other systems cannot be assessed. The second assumption of rapid precipitate dissolution kinetics was appropriate for the Ni-Ni3Si system. A TEM study [33] of the Ni-Ni~Si system has shown that the Ni/Ni~Si interface is incoherent, a fact reflected in the globular shape of the Ni3Si
4038
CARTER et al.: PRECIPITATE DISSOLUTION ZONE KINETICS
precipitates shown in Fig. 2. The incoherency of the precipitate interfaces would allow for their rapid dissolution, and this is reflected in the fact that the silicon content of the matrix phase at the bulk alloy/dissolution-zone boundary was time-invariant [25]. Thus the rate of Ni3Si dissolution was sufficient to maintain local equilibrium, and did not contribute to the overall rate control. The planarity of the Ni/Ni + Ni3Si interface shows that this was uniformly the case, and the process was controlled by diffusion, not dissolution. The assumption of rapid precipitate dissolution kinetics will not always be valid in systems containing semicoherent or coherent precipitates [e.g. F e (Fe, Cr)23C6and Ni-Ni~AI]. If precipitate dissolution is too slow to sustain the exclusive growth of the solute-metal scale, formation of the nonprotective, solvent-metal oxide may occur. In the case where solvent-metal oxidation occurs because the precipitates fail to dissolve, the solute-rich precipitates are engulfed in the metal consumption zone so that no noticeable depletion zone exists. Alternatively, internal oxidation of the precipitates may result. In both cases the precipitates fail to dissolve sufficiently fast, thus rendering the model inapplicable. F r o m Fig. 4 it can be seen that both the stationary and the mobile-interface models are in close agreement, with the stationary interface model giving a slightly smaller value of Xd. The difference is small because of the relatively small kc value used. Only if the kr values are high, as in sulphidation, is the approximation of a stationary alloy/scale interface likely to be invalid.
5. CONCLUSIONS (1) A diffusional analysis can be used to accurately predict the growth kinetics of a precipitate-dissolution zone formed in a two-phase alloy during thermal oxidation. In this situation, the precipitate phase acts as a reservoir, providing a supply of the more reactive metal which leads to the exclusive formation of solute-metal oxide scale. (2) The rate o f Ni3Si precipitate-dissolution in ),-Ni matrix alloys is enough to maintain local equilibrium at the precipitate-dissolution front, and not contribute to overall rate control. (3) The assumption of a stationary alloy/scale interface is a reasonable approximation in the Ni-Ni3Si system where the rate of scale growth and, hence, metal consumption is sufficiently low. (4) Even though D varies over an order of magnitude in the precipitate-dissolution zone, an appropriate weighted average value is successful in predicting zone widening kinetics. Acknowledgements--This research was carried out with the assistance of a grant from the Australian Research Council. One of the authors, P. Carter, also gratefully acknowledges the receipt of an Australian Postgraduate Research Award
during the course of this research. Assistance in obtaining some of the experimental results was provided by Dr Mark Harper.
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