- Email: [email protected]
Environmentally-enhanced cavity growth in nickel and nickel-based alloys
Pergamon 1359-6454(95)00405-X
Acta mater. Vol. 44, No. 8, pp. 3259-3266, 1996 Copyright 0 1996 Acta Metallurgica Inc. Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 1359-6454/96 $15.00 + 0.00
ENVIRONMENTALLY-ENHANCED CAVITY GROWTH NICKEL AND NICKEL-BASED ALLOYS
IN
H.-M. LU’, T. J. DELPHI, D. J. DWYER’, M. GAO’ and R. P. WEI’ ‘Department of Mechanical Engineering and Mechanics, Lehigh University, Bethelehem, PA 18015 and 2Department of Chemistry, University of Maine, Orono, ME 04469, U.S.A. (Received
12 June 1995; in revised form
20 October
1995)
Abstract-Environmental factors have a strong effect on the elevated-temperature failure behavior of nickel-based alloys. It has been proposed that this effect is due to the reactions of oxygen with carbon in the interior of creep cavities. Such reactions can lead to quite high internal gas pressures, sufficient to result in substantial increases in the cavity growth rates. This hypothesis is investigated by carrying out detailed calculations for a simple system which take into account the coupled effects of oxygen diffusion into the cavity and concurrent cavity growth. The results show that creep cavity growth may or may not be affected by internal, gas-producing reactions, depending upon the nature of the carbon-containing particle, the ratio of the grain boundary oxygen diffusivity to the self-diffusivity of nickel, and upon other factors as well. Copyright 0 1996 Acta Metallurgica Inc.
1. INTRODUCTION
It is well-known that environmental factors have a strong effect on the high-temperature failure behavior of nickel-based alloys [l-3]. A number of explanations have been put forward for this phenomenon, most of which have been summarized in the review article of Floreen and Raj [4]. One explanation which has a good deal of support, both experimental and analytical, is that environmental effects act to enhance the growth of intergranular cavitation. According to this hypothesis, a reacting species, typically oxygen, diffuses from the external environment into the interior of the creep cavities. Here it reacts with carbon, either in solid solution or in the form of grain boundary carbides, to produce high-pressure gases. The internal gas pressure thus generated acts to enlarge the cavities at a much greater rate than would be the case under inert conditions, accelerating the failure process. This hypothesis was first advanced by Bricknell and Woodford [5], who demonstrated experimentally that exposure of carbon-containing nickel to oxygen at 1000°C led to the formation of numerous intergranular cavities near the surface of the specimen. Mass spectroscopy indicated the presence of CO, within the cavities. Similar cavities were not, however, observed in decarburized nickel specimens exposed to the same environment. Subsequently, the chemistry of oxygen/carbon reactions in nickel alloys was analyzed by Raj [6], and by Dyson [7], in terms of equilibrium chemical thermodynamics. Both analyses indicated that, under the proper conditions, the gases produced by these reactions are capable
of generating quite high internal cavity pressures in static, non-growing cavities. These analyses, however, did not explore in depth the complications embodied in Bricknell and Woodford’s hypothesis, namely the coupled effects of oxygen diffusion into the cavity and diffusive cavity growth. The outstanding question, which bears directly upon the validity of BricknellWoodford’s hypothesis, is the extent to which pressure generated by oxygen/carbon reactions can act to enhance the cavity growth rate. The high pressures predicted by the Raj and Dyson analyses will inevitably result in cavity growth. However, the increase in volume associated with cavity growth will immediately act to slow the rate at which the internal pressure increases, or perhaps to decrease the pressure. In order to make a substantial contribution to the cavity growth rate, gas must be generated at a rate sufficient to maintain high internal pressure in the face of increasing cavity volume. Whether or not this will occur depends upon the rate at which oxygen diffuses into the cavity and the rate at which the cavity is able to grow under the combined influences of internal pressure and external stress. The purpose of the present paper is to investigate this question by considering a situation in which the combined effects of oxygen diffusion, internal chemical reactions and cavity growth kinetics are all taken into account. A somewhat idealized geometry is considered, which allows an analytically tractable problem to be posed. The resulting problem, however, is not at all simple, with diffusion, chemical reactions and cavity growth all strongly coupled to each other. The results of the analysis have qualitative implications for more complex situations.
3259
3260
LU et al.:
ENVIRONMENTALLY-ENHANCED
2. ANALYSIS
Metal Matrix (Ni)
GROWTH
2.1. Cavity growth
The situation shown in Fig. 1 is considered, in which a cylindrical bicrystal of radius b is subjected to a remote tensile stress un. The grain boundary of the bicrystal, which is oriented normal to the tensile axis, contains at its center a cavity of radius a. Oxygen from the external environment diffuses into the cavity, where it reacts with a carbon-containing particle, either in pure form or in the form of a carbidic particle, to form CO and COZ. Cavity growth is assumed to occur by self-diffusion of atoms from the cavity surface onto the grain boundary, and depends upon the rate of self-diffusion, the applied stress and the internal cavity pressure generated by the gas-producing reactions. The process of internal pressurization involves three sequential steps: the transport of oxygen through an exterior oxide layer, the diffusion of oxygen from the metal/oxide interface to the cavity, and the reaction of oxygen with the carboncontaining particle within the cavity. It is assumed that the rate at which oxygen is supplied from the exterior environment is fast enough to maintain the oxygen at the metal/oxide interface in chemical thermodynamic equilibrium with metal and the surface oxide. It is assumed also that the relevant chemical reactions within the cavity occur at a sufficiently rapid rate to ensure chemical thermodynamic equilibrium at all times. Under these assumptions, the rate of formation of the internal gases (e.g. CO and CO,) will be essentially controlled by the rate of oxygen diffusion from the metal/oxide interface to the cavity. The problem is then reduced to the analysis of three separate, but coupled, processes; growth of the cavity under the influence of stress and internal pressure; diffusion of oxygen from the metal/oxide interface into the cavity; and equilibrium chemical reactions occurring in the cavity interior. Each of these will be considered in turn.
Oxide
CAVITY
For simplicity, only diffusional cavity growth is considered and the effects of inelastic deformation upon cavity growth are neglected. Because only relatively small cavities are to be considered, the cavity is also assumed to grow purely in a quasi-equilibrium mode. Thus, the possibility of crack-like cavity growth described by Chuang and Rice [8], which is typically of importance for larger cavities, is neglected. The development here basically parallels earlier work in this area, e.g. Refs [9, lo], but is modified to take account of the effects of internal cavity pressure [l 11. Here the grain boundary chemical potential L(~ is shown to be
/LL~ = 2R
crab’+ y,(b + 2a sin c()
b* - a* + 2 ln(b/a)
2y, sin c( yS ’ - 7 - b
+ [b4 - a4 - (b2 - a*)*/ln(b/a)]
xrZ+Aln
$
+B
0 where
-
$$$
and the product Db& is the effective grain boundary diffusivity. The chemical potential is related to the grain boundary normal stress (TVby pb = -o&I. The cavity volume growth rate is given by the sum of the mass volume flux from the cavity, which is proportional to V.uLbrand the grain boundary [ 121. This yields [ 1 l] “jacking” contribution
Oxide
dt
Fig. 1, Axisymmetric
bicrystal
containing
in Boundary
central
1 (2)
dv= _2&3!% arbon-Containing
(6’ - a’)
cavity.
kT
(3)
where A is given by the first of equations (2). The growth rate of the cavity da/dt in the plane of the grain boundary may be found by noting that V = 2rca’/sin3 c((; - cos c( + i co? a), where the cavity tip angle CIis given by a = coss’(yb/2y,) and yb is the grain boundary surface energy. Here the effects of oxygen concentration upon the tip angle [13] have been neglected.
LU et al.:
2.2. Oxygen
ENVIRONMENTALLY-ENHANCED
d@ision
Now consider the process by which oxygen diffuses from the metal/oxide interface into the cavity. In general, oxygen diffusion along the grain boundary is assumed to occur at a much more rapid rate than diffusion through the bulk material [14, 151 so that bulk diffusion will be neglected. From Stevens et al. [16, 171 the mass flux of oxygen along the grain boundary may be expressed in terms of the chemical potential of oxygen on the grain boundary PWI
by
CAVITY
3261
GROWTH
of oxygen on where C&, is the concentration the cavity surface. This condition can also be shown to lead to continuity of oxygen concentration at r = a. At r = b, the situation is somewhat more complicated. Here the formation of an exterior layer of nickel oxide is anticipated. As noted previously, the oxygen at the Ni/NiO interface is assumed to be in chemical thermodynamic equilibrium with Ni and the NiO layer. Assuming that the NiO layer is stress-free, this leads to
J where CbLOlis the concentration of oxygen on the grain boundary, expressed as the number of atoms per unit area, and D,tUIis the grain boundary diffusion coefficient for oxygen. Conservation of mass considerations lead directly to
$
V.(C,,,,Vp)
=
G,o,(b) = Cwo
dC
2.
From Stevens et al. [16], the chemical potential atom of the diffusing species is given by p,,,,,, = pa + kT In F
”
- a,,!&
per
(6)
where oh is the grain boundary normal stress, determined from the analysis in the preceding section, /*” is the standard potential of oxygen, QtoI is the atomic volume of oxygen, and Co is the number of available lattice sites per unit area of grain boundary. Substitution into equation (5) and taking account of axisymmetry, yields the governing equation for grain boundary oxygen diffusion as
To this equation, appropriate initial and boundary conditions must be appended. To account for the amount of oxygen dissolved in the alloy before it is exposed to the high-temperature environment, it is assumed that residual oxygen is uniformly distributed along the grain boundary before the cavity is nucleated. Thus, the initial condition is C,,&,
0) = cb”.
(8)
The boundary conditions at r = a and r = b can be derived from continuity of chemical potential at these points. At r = a, this yields
Cbpj(a) = C,Wexp g [
the oxygen concentration where CN,,N,~ denotes at the Ni/NiO interface, but by the boundary condition at r = b for self-diffusion [1 11,oh(b) = - rs/ b, so that
11
2y, sin c( p + oh(a) - ~ a (
(9)
(11)
From thermodynamic data given by Park and Altstetter [15], the value of CN,,N,Ois approximately = 1.544 x lOI8 e-ss.ooo’n’, where the units of C NI.‘NIO CN,,‘N,O are atoms/m2 and those of RT are J/mol. From equation (7) and the initial and boundary conditions of equations (8))( 11), one may obtain the CbLOl(rrt). grain boundary oxygen concentration Then, from equations (4) and (6), the total oxygen flux Fro1into the cavity, in terms of number of atoms per unit time, may be given by
-_--
R,~,a6 kT
2.3.
& cb’o’ ,=8’ (12)
Chemical reactions
In this section, the reactions of oxygen diffusing into the cavity with pure carbon will be considered. Following Raj [6] and Dyson [7], the relevant chemical reactions occurring in the cavity interior will be analyzed in terms of equilibrium chemical thermodynamics. For the case of a pure carbon particle inside the cavity, the possible chemical reactions, along with the standard change in free-energy AGO (in J/mol, with T in degrees Kelvin) associated with each reaction [15, 18, 191, are listed below. [O](at.%)efOz,,,; (Ni)
+ ;O Z(gIe(NiO); (C)#Z];
[C] + fOz&CO,,,; CO<,, + ;O~&CO~(gI;
AGp = 189,550 - 80.9T AG$’= -244,550
+ 98.57
AC: = 39,600 + 5.76T AG: = - 151,300 - 93.41T AG: = -282,400
+ 86.81 T. (13)
The notation ( ) represents the pure solid, [] represents a species dissolved in Ni and (g) indicates
3262
LU et al.:
ENVIRONMENTALLY-ENHANCED
the gaseous state. Here the Henrian standard state and the mole fraction composition coordinate have been utilized for those reactions involving species in solution, with the exception of the first reaction, where the atomic percent composition coordinate has been adopted. Note that the second of these reactions will occur only when the oxygen concentration on the cavity surface is sufficiently high to allow the formation of NiO. Following Dyson [7] all gaseous species in the above reactions are assumed to obey the perfect gas law. This implies that f; = p,, where p, is the partial pressure of the ith gaseous species (expressed in atmospheres) and f; is its fugacity. Furthermore, for the ith pure solid constituent, the activity a, is unity. Finally, by expressing the concentration of carbon dissolved in the nickel in terms of mole fractions Xrcl, the activity of the dissolved carbon is given by aIcl = Xrcl for the Henrian standard state. With reference to equation (13), chemical thermodynamic considerations lead to
equations algebraic
CAVITY GROWTH (14), the following equations
Nro,=
=
PO, =
of nonlinear
dt i‘40,(t)
Npo, +
(
system
JO
~P:O+PCO+~p;o c
c
>
j$Pi0 c
&I
=
K
pco, =
5 P60
‘@I -
lOOK,
KC
CNi Kc
PC0
(15)
k,
(14) where K, = exp(-AGy/RT). Here, [at.%01 is the atomic percentage of oxygen atoms on the cavity surface. This quantity may be related to the oxygen concentration on the cavity surface, C,tol, by C,, = ([at.%O]/lOO)C~,, where CN, is the total number of lattice sites on the cavity surface per unit area. CN, can be estimated from the lattice parameters for Ni by assuming that the cavity surface is the most closely-packed plane of the nickel f.c.c. structure. This yields a value of CNI = 1.86 x lOI9 atoms/m2. The second of equations (14) is valid only for the case in which NiO forms in the interior of the cavity. However, given that the grain boundary oxygen is assumed to be in chemical equilibrium with Ni and NiO at the outer surface of the bicrystal, this would be expected to occur only if the partial pressure of O2 in the interior of the cavity exceeded that at the outer surface. These conditions would occur only when oxygen diffusion had essentially come to a halt. Hence it will be assumed that NiO does not form in the cavity interior, in which case the second of equations (14) would read -‘*’ > Kz. PO, In addition, conservation of carbon and oxygen atoms must be satisfied in the cavity interior. Assuming that the carbon particle is not completely oxidized (Nc > 0), these yield, in conjunction with
where Np is the initial number of atoms of species i in the interior of the cavity, N, the number at time t, and Ffo,(t) is the diffusive oxygen flux from the grain boundary into the cavity, given by equation (12). Furthermore, S = 4na*/(l + cos a) is the current cavity surface area, and V,,, = V - Nc/pC the net cavity volume, where pc is the number of carbon atoms/unit volume. Finally, NA is Avogadro’s number and Ru is the universal gas constant. Note that the units of partial pressure, in conjunction with equation (14), are atmospheres, and those of Ru Tare atom.m3/mol. 2.4. Solution procedure As seen in the preceding sections, the equations for the cavity growth, oxygen diffusion and internal gas production reactions are strongly coupled, and must be solved simultaneously using standard time-integration procedures. In practice, it was found that the quasi-steady state solution for grain boundary oxygen diffusion was reached rather rapidly. Hence, to speed the calculations, diffusion was assumed to proceed under quasi-steady state conditions for all times. As was discussed in Section 2, internal gas production is controlled by the rate of grain boundary oxygen diffusion, and cavity growth depends upon the rate of nickel self-diffusion. Hence, the extent to which the internal pressurization affects the cavity growth will depend directly upon the relative values of the grain boundary diffusivity of oxygen in nickel Dbrol and the grain boundary
LU er al.:
ENVIRONMENTALLY-ENHANCED
‘I- (“C) 12001100 1000 900 I’,~I~I~I’
..
600
700
800
1
1 :
Mean Scatterband
CAVITY
3263
GROWTH
(a) ::I.1--:--_/ 0.0006 -
D-
0.0004 -
Carbon Particle D,6, = 1.09x10-‘5 cm3/s D boo,= 5.51x10-” cm2/s
0.0002 -
0.0000 lo-‘21
6
’
7
’
’ ’ ’ j
8
9
’ I
10
11
-7 7.7
Tf
5
0
’
_ _._ ’ -..
12
r
.._ ’ -.-
’ -
10
r,-i-‘-‘15
20
t (W
104/T (l/OK) Fig. 2. Reported
40
values of I&cl vs T
I
”
7
I
”
Carbon Particle
self-diffusivity of nickel D&,. Values for Dbte] are very limited in the literature, and tend to be based upon indirect evidence, e.g. the extent of observable oxidation after a fixed period of time. The only available data appear to be those of Bricknell and Woodford [20] on pure nickel and Pedron and Pineau [14] on the nickel-based alloy Inconel 718. Figure 2 shows an Arhennius-type plot of these data with a scatterband constructed about the mean trend line. Data on Db& for pure nickel are more numerous, and have been obtained by more direct methods, e.g. radioisotope tracer techniques. These data are summarized in Table 1. The values shown here were obtained by interpolating or extrapolating the minimum and maximum reported values to 650°C using the values of the activation energy cited in each reference, and assuming a grain boundary thickness of &, = 5 x lo-* cm. It can be seen that the reported values exhibit about five orders of magnitude in scatter. 3. RESULTS
Given the large scatter in the diffusivity data, the approach will be to examine the least and most favorable conditions for environmentallyenhanced cavity growth. From physical considerations, the cavity growth rate is expected to be least influenced by the effects of internal gas pressure for small values of the grain boundary oxygen diffusivity and for large values of the self-diffusivity. Low grain boundary oxygen diffusivities will result in relatively
25 -
0
5
10
15
t (W
Fig. 3. Internal least favorable
cavity pressure vs time under conditions to environmental influence: (a) growing cavity; (b) nongrowing cavity.
slow rates of internal gas production, while high values of self-diffusivity will lead to relatively large cavity volume growth rates, so that the internal pressure will be unlikely to attain appreciable values. Conversely, a large value of grain boundary oxygen diffusivity, combined with a small value of selfdiffusivity, will be most favorable. Thus the lower and upper limit values of Db&, given in Table 1 at 650°C will be considered for the bounding cases, namely, 5.52 x 10~” and 1.09 x lo-l5 cm)/s. From the scatterband in the grain boundary oxygen diffusivity data, at 650°C (see Fig. 2), the minimum and maximum values of Dbrol are 1.22 x 1O-9 and
Table 1. Estimated maximum and minnnum values of LA& at 65O’C Temperature range of actual data (“C) 85&l 100 70&l 100 600-970 475450
(DbSb),“.X(cm’is) T = 650°C 1.21 3.47 9.24 1.09
x x x x
lo-‘5 lo-‘6 1O-‘8 10-15
20
(Db&),,. (CrnW T = 650°C 2.65 5.85 5.52 1.09
x x x x
10-l’ lo-” 1O-2” lo-‘1
Ref.
Pll [22,231 [241 125. 261
Specimen type polycrystal
bicrystal bicrystal Dolvcrvstal
3264
LU et al.:
ENVIRONMENTALLY-ENHANCED
5.51 x 10-l’ cm2/s, respectively. Additional parameter values were Q = 50 MPa, b = 50 pm, and the initial cavity radius a0 = 0.1 pm. The temperature was taken to be T = 650°C. Figures 3(a) and 3(b) show the computed cavity internal pressure, as well as the partial pressures of CO and CO,, as a function of time for the conditions least favorable to environmentally-enhanced cavity growth, i.e. Dh& = 1.09 x 10~‘5cm3/s and Dbrol= 5.51 x lo-” cm’js. Plotted on Fig. 3(a)]is the variation of pressure in a growing cavity. Here, an initial rapid rise in pressure is followed by an almost immediate decrease, so that the total gas pressure is limited to fairly small values. These values are so much smaller than the applied external stress (- 10m3 vs 50 MPa) as to have practically no effect upon the kinetics of cavity growth. At these low pressures, almost all of the internal pressure is due to CO. By contrast, when the cavity is constrained against growth [Fig. 3(b)], the internal pressure reaches fairly substantial values. At these pressures, chemical equilibrium favors the formation of coz. The situation in which the values of the diffusivities are most favorable to environmentallyenhanced cavity growth (DbSb = 5.52 x 10~“cm3/s, Dbrol= 1.22 x 10m9cm’/s) is quite different. Here the rate of cavity growth is considerably lower, and the rate of oxygen diffusion considerably higher than in the previous case. Consequently the internal gas pressure for the growing cavity, as shown in Fig. 4, reaches substantially higher values. In this case, almost all of the contribution to the total pressure comes from CO>, in agreement with the results of Dyson [7]. Figure 5 shows the growth in cavity radius versus time with and without the effects of internal pressure. After 20 h of growth, the pressurized cavity has reached a size approximately 25% greater than
-P
CAVITY GROWTH
I __
0.005
1,
7
7
I
’
0.004 Carbon Particle
0.002 -
0
10
5
15
20
t (hr) Fig. 5. Cavity growth in inert and oxygen environments under conditions most favorable to environmental influence.
that of the unpressurized cavity, with a growth rate about 65% larger than that of the unpressurized cavity. Figure 6 illustrates the combined effects of oxygen and self-diffusivity in more detail. Here the time required for a change in cavity radius of Aa = 0.01 pm, starting from an initial radius of u0 = 0.1 pm, is plotted as a function of Db& and Dbrol. For the parameters used here, environmental effects upon cavity growth are noticeable when the ratio DbtOl/Db is greater than about 10 (assuming a value of b,, = 5 x lo-* cm). Of interest is the fact that environmental effects are present over only a relatively small portion of the domain of Db& and Dbrolvalues. Other parameters in the problem, of course, have an influence upon the environmentally-enhanced cavity growth behavior. The most obvious is the applied stress 0”. For fixed values of the diffusion coefficients, a smaller value of go would lead to less rapid cavity growth and the effects of internal gas
I
0
Carbon
I
oxygenenvironment inert environment
inert environment oxygen environment
Particle
D&$,= 5.52~10-~~ cm3/s
Fig. 4. Internal cavity most favorable
pressure vs time under conditions to environmental influence.
Fig. 6. Time required
for Aa = 0.01 pm vs DbSb and Dbcol.
LU et al.:
ENVIRONMENTALLY-ENHANCED
production would be more marked. For example, separate simulations have been conducted for Q = 0 which yield values of internal cavity pressure about 30% greater than those shown in Fig. 4 after 20 h, and which are large enough to lead to cavity growth under the influence of internal pressure alone. 4. DISCUSSION
The foregoing analysis considered only the case of a pure carbon particle inside a nickel matrix. Many nickel-based alloys, however, contain significant amounts of chromium, and Dyson [7] has considered in some detail the case of a static (non-growing) cavity containing a Cr6C2, particle. He concluded that the presence of chromium would almost entirely suppress the growth of internal pressure within the cavity, because the reaction 4[Cr] + 302(,, + 2(Crz0?) is thermodynamically favored over the gas-producing oxygen/carbon reactions. In spite of this prediction, cavities, apparently of environmental origin, were observed by Pandey et al. [27] in a chromium-containing nickel-based superalloy. These authors speculated that these cavities could result from the formation of COZ as a metastable phase if the value of the oxygen activity was allowed to reach the formation pressure of NiO, instead of being limited by the formation pressure of Cr203 as assumed by Dyson [7]. To further examine this issue, an order of magnitude estimate of cavity pressure is made by replacing the pure carbon particle with a CrzzCs particle. For simplicity, it is assumed that chromium oxide does not form on the external surface of the bicrystal, and the diffusional results derived for pure nickel may be used. With this assumption, oxygen activity within the cavity will be allowed to reach the formation pressure of NiO. The analysis can be treated in the same fashion as the case of the pure
0.0010~
t 0.0008 -
1
”
1 7
1
’
”
9
7
j
A
P,PC0
~
”
cr,,c, Particle D&J =
0.0006-
D
0.0000~ 0
5
10
15
fh9
Fig. 7. Internal
cavity
pressure
vs time.
20
CAVITY
GROWTH
carbon particle [11] with the following chromium reactions
i
z$ g[Cr] + [Cl;
3265
inclusion
of
the
AG,O= 132,870 + 18.95T
2[Crl + iOZ(E)ti (CrZOl>; AG; = - 1,133,200 + 256.57.
(16)
Figure 7 shows the results of this analysis, in terms of the internal cavity pressure as a function of time, for the diffusivity coefficients most favorable to environmentally-enhanced cavity growth and o0 = 50 MPa. It can be seen that, even allowing for a higher oxygen activity than that considered by Dyson [7], the pressures are far too small to influence cavity growth. Several simplifying assumptions and approximations have been used in the preceding analyses. In the interests of analytical tractability. a geometrically simple configuration has been analyzed. However, the results obtained here have qualitative implications for more complex situations. Likewise, following Dyson [7] ideal gas behavior has been assumed. When internal cavity pressures exceed about 0.5 MPa, this assumption becomes increasingly in error. Raj [6] considered real gas effects and concluded that these act to reduce the cavity pressure as compared to ideal gas behavior. Additionally, a rather simplified model of the cavity growth which is entirely due to a self-diffusive mass transport mechanism has been employed. More complicated models, in which matrix creep contributes to cavity growth [28, 291 or in which a polycrystalline matrix acts to constrain the rate of cavity growth [30]. would alter the effect of internal gas production on cavity growth because of the changes in cavity growth rates relative to that for oxygen diffusion. The effects of these assumptions would need to be examined for each case, but they are not expected to alter the overall conclusions of this study. Finally, limiting values on the grain boundary self-diffusivity have been used to bound the range of expected responses. It is worth noting that the minimum value of the self-diffusivity resulted from data taken on bicrystalline specimens, while the maximum value was taken from tests performed on polycrystalline specimens. Because polycrystalline self-diffusivities might be expected to represent in some sense an average value for bicrystalline self-diffusivities, it is not clear that the range of self-diffusivities to be expected in a polycrystalline specimen are as wide as those indicated here for a bicrystal. Given the higher values reported for polycrystals, it is quite possible that the internal pressurization mechanism would not play a significant role in the growth of cavities in polycrystalline materials, at least in the temperature range 600700°C. In addition to their effect upon chemical equilibrium, alloying elements may affect the efficacy of the internal pressurization mechanism through shifts in the ratio of self-diffusivity to grain boundary
3266
LU et al.:
ENVIRONMENTALLY-ENHANCED
oxygen diffusivity. However, there are insufficient data in the literature to assess this effect at present. It should be noted, however, that the analysis presented here considered only the effect of gas-production upon cavity growth, not upon their formation. It is quite possible that high initial internal gas pressures might be important in the nucleation of cavities of very small size. If high initial pressures resulted in the nucleation of substantially more cavities than would be nucleated in an inert environment, then this would in itself constitute a mechanism for environmentally-enhanced creep damage, even if the internal pressure was reduced to negligible levels by subsequent cavity growth. Finally, it should be noted that reported observations of environmentally-induced cavitation, e.g. Refs [5] and [27], were carried out at temperatures considerably higher than that considered here and without external load. Increasing temperature favors oxygen diffusion over self-diffusion, which would tend to favor the enhanced cavity growth mechanism discussed here. For example, at 650°C a ratio of Db&/Dblol of 1.05, which results in negligible enhancement of cavity growth, translates into a value of 11.9 at lOOO”C, a value for which enhanced cavity growth would be expected to occur. There is an additional increase in oxygen concentration at the free surface with increasing temperature, which would likewise serve to promote enhanced cavity growth. However, even at these elevated temperatures, the presence of chromium should still effectively suppress gas production. Hence, the suggestion of Pandey et al. [27] that metastable COz might lead to cavities of environmental origin in nickel-based superalloys, does not seem to be a tenable one. SUMMARY
The hypothesis of Bricknell and Woodford that internal gas pressure resulting from oxygen carbon reactions could result in enhanced creep cavity growth, and hence might explain the marked environmental sensitivity of the failure behavior of nickel-based alloys, has been investigated in some detail. It is concluded that, at typical operating temperatures and stress levels for polycrystalline nickel-based alloys, this hypothesis is viable if two conditions are met. The first is that the ratio of the grain boundary oxygen diffusivity to the grain boundary self-diffusivity of nickel is sufficiently high so that cavity growth does not outpace the increase in internal pressure. For the parameters used herein, this condition is met over only a relatively small domain of Db& and Dbrol values at 650°C. At lOOO”C, however, this domain appears to be somewhat larger. The second condition is that the
GROWTH
alloy does not contain significant amounts of chromium (and possibly other alloying elements, as well) which would act to suppress CO/CO, production. Acknowledgement-This work was supported by the Materials Research Group (MRG) Program of the Division of Materials Research, National Science Foundation under Grant No. DMR-9102093.
REFERENCES
5. 6. 7. 8. 9. 10.
11. 12. 13. 14. 15. 16. 17. 18.
5.
CAVITY
P. N. Chaku and C. J. McMahon, Jr, Met&l. Trans. 5, 441 (1974). D. A. Woodford, MetaN. Trans. A 12A, 299 (1981). E. Andrieu and J. P. Henon, Muter. Sci. Engng 88, 191 (1987). S. Floreen and R. Raj, in Flow and Fracture at Elevated Temperatures (edited by R. Raj), p. 383. American Society for Metals, Metals Park, OH (1985). R. H. Bricknell and D. A. Woodford, Acta metall. 30, 257 (1982). R. Raj, Acta metall. 30, 1259 (1982). B. F. Dyson, Acta metall. 30, 1639 (1982). T.-J. Chuane and J. R. Rice. Acta metall. 21. 1625 (1973). D. Hull and D. E. Rimmer, Phil. Mug. 4, 673 (1959). M. V. Speight and J. E. Harris, MetaN. Sot. J. 1, 83 (1967). H.-M. Lu, Ph.D. Dissertation, Lehigh University (1995). M. V. Speight and W. Beere, Metal. Sci. 9, 190 (1975). J. Kameda, Acta metall. 34, 867 (1986). J. P. Pedron and A. Pineau, Mem. Etudes Sci. M&all. 80, 665 (1983). J.-W. Park and C. J. Altstetter. Metall. Trans. 18A. 43 (1987). R. N. Stevens, R. Dutton and M. P. Puls, Acta metall. 22, 629 (1974). R. N. Stevens, R. Dutton and M. P. Puls, Acta metall. 22, 639 (1974). M. Hansen, Constitution of Binary Alloys. McGraw-
Hill, New York (1958). 19. D. R. Gaskell, Introduction 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
to Metallurgical Thermodynamics, 2nd edition. Hemisphere Publishing Co. (1981). R. H. Bricknell and D. A. Woodford, Scripta metall. 16, 761 (1982). W. Lange, A. Hassner and G. Mixcher, Physica status solidi 5, 63 (1964). W. R. Upthegrove and M. J. Sinnott, Trans. Am. Sot. Metals 50, 1031 (1958). R. F. Cannon and J. P. Stark, J. appl. Phys. 40, 4361 (1969). R. F. Cannon and J. P. Stark, J. appl. Phys. 40, 4366 (1969). A. R. Wazzan, J. appl. Phys. 36, 3596 (1965). H. J. Frost and M. F. Ashby, Deformation-Mechanism Maps for Metals and Alloys. Pergamon Press, Oxford (1983). M. C. Pandey, B. F. Dyson and D. M. R. Taplin, Proc. R. Sot. London A 393, 117 (1984). A. Needleman and J. R. Rice, Acta metall. 28, 1315 (1980). H.-M. Lu and T. J. Delph, Scripta metall. mater. 29,281 (1993). J. R. Rice, Acta metall. 29, 675 (1981).
Acta mater. Vol. 44, No. 8, pp. 3259-3266, 1996 Copyright 0 1996 Acta Metallurgica Inc. Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 1359-6454/96 $15.00 + 0.00
ENVIRONMENTALLY-ENHANCED CAVITY GROWTH NICKEL AND NICKEL-BASED ALLOYS
IN
H.-M. LU’, T. J. DELPHI, D. J. DWYER’, M. GAO’ and R. P. WEI’ ‘Department of Mechanical Engineering and Mechanics, Lehigh University, Bethelehem, PA 18015 and 2Department of Chemistry, University of Maine, Orono, ME 04469, U.S.A. (Received
12 June 1995; in revised form
20 October
1995)
Abstract-Environmental factors have a strong effect on the elevated-temperature failure behavior of nickel-based alloys. It has been proposed that this effect is due to the reactions of oxygen with carbon in the interior of creep cavities. Such reactions can lead to quite high internal gas pressures, sufficient to result in substantial increases in the cavity growth rates. This hypothesis is investigated by carrying out detailed calculations for a simple system which take into account the coupled effects of oxygen diffusion into the cavity and concurrent cavity growth. The results show that creep cavity growth may or may not be affected by internal, gas-producing reactions, depending upon the nature of the carbon-containing particle, the ratio of the grain boundary oxygen diffusivity to the self-diffusivity of nickel, and upon other factors as well. Copyright 0 1996 Acta Metallurgica Inc.
1. INTRODUCTION
It is well-known that environmental factors have a strong effect on the high-temperature failure behavior of nickel-based alloys [l-3]. A number of explanations have been put forward for this phenomenon, most of which have been summarized in the review article of Floreen and Raj [4]. One explanation which has a good deal of support, both experimental and analytical, is that environmental effects act to enhance the growth of intergranular cavitation. According to this hypothesis, a reacting species, typically oxygen, diffuses from the external environment into the interior of the creep cavities. Here it reacts with carbon, either in solid solution or in the form of grain boundary carbides, to produce high-pressure gases. The internal gas pressure thus generated acts to enlarge the cavities at a much greater rate than would be the case under inert conditions, accelerating the failure process. This hypothesis was first advanced by Bricknell and Woodford [5], who demonstrated experimentally that exposure of carbon-containing nickel to oxygen at 1000°C led to the formation of numerous intergranular cavities near the surface of the specimen. Mass spectroscopy indicated the presence of CO, within the cavities. Similar cavities were not, however, observed in decarburized nickel specimens exposed to the same environment. Subsequently, the chemistry of oxygen/carbon reactions in nickel alloys was analyzed by Raj [6], and by Dyson [7], in terms of equilibrium chemical thermodynamics. Both analyses indicated that, under the proper conditions, the gases produced by these reactions are capable
of generating quite high internal cavity pressures in static, non-growing cavities. These analyses, however, did not explore in depth the complications embodied in Bricknell and Woodford’s hypothesis, namely the coupled effects of oxygen diffusion into the cavity and diffusive cavity growth. The outstanding question, which bears directly upon the validity of BricknellWoodford’s hypothesis, is the extent to which pressure generated by oxygen/carbon reactions can act to enhance the cavity growth rate. The high pressures predicted by the Raj and Dyson analyses will inevitably result in cavity growth. However, the increase in volume associated with cavity growth will immediately act to slow the rate at which the internal pressure increases, or perhaps to decrease the pressure. In order to make a substantial contribution to the cavity growth rate, gas must be generated at a rate sufficient to maintain high internal pressure in the face of increasing cavity volume. Whether or not this will occur depends upon the rate at which oxygen diffuses into the cavity and the rate at which the cavity is able to grow under the combined influences of internal pressure and external stress. The purpose of the present paper is to investigate this question by considering a situation in which the combined effects of oxygen diffusion, internal chemical reactions and cavity growth kinetics are all taken into account. A somewhat idealized geometry is considered, which allows an analytically tractable problem to be posed. The resulting problem, however, is not at all simple, with diffusion, chemical reactions and cavity growth all strongly coupled to each other. The results of the analysis have qualitative implications for more complex situations.
3259
3260
LU et al.:
ENVIRONMENTALLY-ENHANCED
2. ANALYSIS
Metal Matrix (Ni)
GROWTH
2.1. Cavity growth
The situation shown in Fig. 1 is considered, in which a cylindrical bicrystal of radius b is subjected to a remote tensile stress un. The grain boundary of the bicrystal, which is oriented normal to the tensile axis, contains at its center a cavity of radius a. Oxygen from the external environment diffuses into the cavity, where it reacts with a carbon-containing particle, either in pure form or in the form of a carbidic particle, to form CO and COZ. Cavity growth is assumed to occur by self-diffusion of atoms from the cavity surface onto the grain boundary, and depends upon the rate of self-diffusion, the applied stress and the internal cavity pressure generated by the gas-producing reactions. The process of internal pressurization involves three sequential steps: the transport of oxygen through an exterior oxide layer, the diffusion of oxygen from the metal/oxide interface to the cavity, and the reaction of oxygen with the carboncontaining particle within the cavity. It is assumed that the rate at which oxygen is supplied from the exterior environment is fast enough to maintain the oxygen at the metal/oxide interface in chemical thermodynamic equilibrium with metal and the surface oxide. It is assumed also that the relevant chemical reactions within the cavity occur at a sufficiently rapid rate to ensure chemical thermodynamic equilibrium at all times. Under these assumptions, the rate of formation of the internal gases (e.g. CO and CO,) will be essentially controlled by the rate of oxygen diffusion from the metal/oxide interface to the cavity. The problem is then reduced to the analysis of three separate, but coupled, processes; growth of the cavity under the influence of stress and internal pressure; diffusion of oxygen from the metal/oxide interface into the cavity; and equilibrium chemical reactions occurring in the cavity interior. Each of these will be considered in turn.
Oxide
CAVITY
For simplicity, only diffusional cavity growth is considered and the effects of inelastic deformation upon cavity growth are neglected. Because only relatively small cavities are to be considered, the cavity is also assumed to grow purely in a quasi-equilibrium mode. Thus, the possibility of crack-like cavity growth described by Chuang and Rice [8], which is typically of importance for larger cavities, is neglected. The development here basically parallels earlier work in this area, e.g. Refs [9, lo], but is modified to take account of the effects of internal cavity pressure [l 11. Here the grain boundary chemical potential L(~ is shown to be
/LL~ = 2R
crab’+ y,(b + 2a sin c()
b* - a* + 2 ln(b/a)
2y, sin c( yS ’ - 7 - b
+ [b4 - a4 - (b2 - a*)*/ln(b/a)]
xrZ+Aln
$
+B
0 where
-
$$$
and the product Db& is the effective grain boundary diffusivity. The chemical potential is related to the grain boundary normal stress (TVby pb = -o&I. The cavity volume growth rate is given by the sum of the mass volume flux from the cavity, which is proportional to V.uLbrand the grain boundary [ 121. This yields [ 1 l] “jacking” contribution
Oxide
dt
Fig. 1, Axisymmetric
bicrystal
containing
in Boundary
central
1 (2)
dv= _2&3!% arbon-Containing
(6’ - a’)
cavity.
kT
(3)
where A is given by the first of equations (2). The growth rate of the cavity da/dt in the plane of the grain boundary may be found by noting that V = 2rca’/sin3 c((; - cos c( + i co? a), where the cavity tip angle CIis given by a = coss’(yb/2y,) and yb is the grain boundary surface energy. Here the effects of oxygen concentration upon the tip angle [13] have been neglected.
LU et al.:
2.2. Oxygen
ENVIRONMENTALLY-ENHANCED
d@ision
Now consider the process by which oxygen diffuses from the metal/oxide interface into the cavity. In general, oxygen diffusion along the grain boundary is assumed to occur at a much more rapid rate than diffusion through the bulk material [14, 151 so that bulk diffusion will be neglected. From Stevens et al. [16, 171 the mass flux of oxygen along the grain boundary may be expressed in terms of the chemical potential of oxygen on the grain boundary PWI
by
CAVITY
3261
GROWTH
of oxygen on where C&, is the concentration the cavity surface. This condition can also be shown to lead to continuity of oxygen concentration at r = a. At r = b, the situation is somewhat more complicated. Here the formation of an exterior layer of nickel oxide is anticipated. As noted previously, the oxygen at the Ni/NiO interface is assumed to be in chemical thermodynamic equilibrium with Ni and the NiO layer. Assuming that the NiO layer is stress-free, this leads to
J where CbLOlis the concentration of oxygen on the grain boundary, expressed as the number of atoms per unit area, and D,tUIis the grain boundary diffusion coefficient for oxygen. Conservation of mass considerations lead directly to
$
V.(C,,,,Vp)
=
G,o,(b) = Cwo
dC
2.
From Stevens et al. [16], the chemical potential atom of the diffusing species is given by p,,,,,, = pa + kT In F
”
- a,,!&
per
(6)
where oh is the grain boundary normal stress, determined from the analysis in the preceding section, /*” is the standard potential of oxygen, QtoI is the atomic volume of oxygen, and Co is the number of available lattice sites per unit area of grain boundary. Substitution into equation (5) and taking account of axisymmetry, yields the governing equation for grain boundary oxygen diffusion as
To this equation, appropriate initial and boundary conditions must be appended. To account for the amount of oxygen dissolved in the alloy before it is exposed to the high-temperature environment, it is assumed that residual oxygen is uniformly distributed along the grain boundary before the cavity is nucleated. Thus, the initial condition is C,,&,
0) = cb”.
(8)
The boundary conditions at r = a and r = b can be derived from continuity of chemical potential at these points. At r = a, this yields
Cbpj(a) = C,Wexp g [
the oxygen concentration where CN,,N,~ denotes at the Ni/NiO interface, but by the boundary condition at r = b for self-diffusion [1 11,oh(b) = - rs/ b, so that
11
2y, sin c( p + oh(a) - ~ a (
(9)
(11)
From thermodynamic data given by Park and Altstetter [15], the value of CN,,N,Ois approximately = 1.544 x lOI8 e-ss.ooo’n’, where the units of C NI.‘NIO CN,,‘N,O are atoms/m2 and those of RT are J/mol. From equation (7) and the initial and boundary conditions of equations (8))( 11), one may obtain the CbLOl(rrt). grain boundary oxygen concentration Then, from equations (4) and (6), the total oxygen flux Fro1into the cavity, in terms of number of atoms per unit time, may be given by
-_--
R,~,a6 kT
2.3.
& cb’o’ ,=8’ (12)
Chemical reactions
In this section, the reactions of oxygen diffusing into the cavity with pure carbon will be considered. Following Raj [6] and Dyson [7], the relevant chemical reactions occurring in the cavity interior will be analyzed in terms of equilibrium chemical thermodynamics. For the case of a pure carbon particle inside the cavity, the possible chemical reactions, along with the standard change in free-energy AGO (in J/mol, with T in degrees Kelvin) associated with each reaction [15, 18, 191, are listed below. [O](at.%)efOz,,,; (Ni)
+ ;O Z(gIe(NiO); (C)#Z];
[C] + fOz&CO,,,; CO<,, + ;O~&CO~(gI;
AGp = 189,550 - 80.9T AG$’= -244,550
+ 98.57
AC: = 39,600 + 5.76T AG: = - 151,300 - 93.41T AG: = -282,400
+ 86.81 T. (13)
The notation ( ) represents the pure solid, [] represents a species dissolved in Ni and (g) indicates
3262
LU et al.:
ENVIRONMENTALLY-ENHANCED
the gaseous state. Here the Henrian standard state and the mole fraction composition coordinate have been utilized for those reactions involving species in solution, with the exception of the first reaction, where the atomic percent composition coordinate has been adopted. Note that the second of these reactions will occur only when the oxygen concentration on the cavity surface is sufficiently high to allow the formation of NiO. Following Dyson [7] all gaseous species in the above reactions are assumed to obey the perfect gas law. This implies that f; = p,, where p, is the partial pressure of the ith gaseous species (expressed in atmospheres) and f; is its fugacity. Furthermore, for the ith pure solid constituent, the activity a, is unity. Finally, by expressing the concentration of carbon dissolved in the nickel in terms of mole fractions Xrcl, the activity of the dissolved carbon is given by aIcl = Xrcl for the Henrian standard state. With reference to equation (13), chemical thermodynamic considerations lead to
equations algebraic
CAVITY GROWTH (14), the following equations
Nro,=
=
PO, =
of nonlinear
dt i‘40,(t)
Npo, +
(
system
JO
~P:O+PCO+~p;o c
c
>
j$Pi0 c
&I
=
K
pco, =
5 P60
‘@I -
lOOK,
KC
CNi Kc
PC0
(15)
k,
(14) where K, = exp(-AGy/RT). Here, [at.%01 is the atomic percentage of oxygen atoms on the cavity surface. This quantity may be related to the oxygen concentration on the cavity surface, C,tol, by C,, = ([at.%O]/lOO)C~,, where CN, is the total number of lattice sites on the cavity surface per unit area. CN, can be estimated from the lattice parameters for Ni by assuming that the cavity surface is the most closely-packed plane of the nickel f.c.c. structure. This yields a value of CNI = 1.86 x lOI9 atoms/m2. The second of equations (14) is valid only for the case in which NiO forms in the interior of the cavity. However, given that the grain boundary oxygen is assumed to be in chemical equilibrium with Ni and NiO at the outer surface of the bicrystal, this would be expected to occur only if the partial pressure of O2 in the interior of the cavity exceeded that at the outer surface. These conditions would occur only when oxygen diffusion had essentially come to a halt. Hence it will be assumed that NiO does not form in the cavity interior, in which case the second of equations (14) would read -‘*’ > Kz. PO, In addition, conservation of carbon and oxygen atoms must be satisfied in the cavity interior. Assuming that the carbon particle is not completely oxidized (Nc > 0), these yield, in conjunction with
where Np is the initial number of atoms of species i in the interior of the cavity, N, the number at time t, and Ffo,(t) is the diffusive oxygen flux from the grain boundary into the cavity, given by equation (12). Furthermore, S = 4na*/(l + cos a) is the current cavity surface area, and V,,, = V - Nc/pC the net cavity volume, where pc is the number of carbon atoms/unit volume. Finally, NA is Avogadro’s number and Ru is the universal gas constant. Note that the units of partial pressure, in conjunction with equation (14), are atmospheres, and those of Ru Tare atom.m3/mol. 2.4. Solution procedure As seen in the preceding sections, the equations for the cavity growth, oxygen diffusion and internal gas production reactions are strongly coupled, and must be solved simultaneously using standard time-integration procedures. In practice, it was found that the quasi-steady state solution for grain boundary oxygen diffusion was reached rather rapidly. Hence, to speed the calculations, diffusion was assumed to proceed under quasi-steady state conditions for all times. As was discussed in Section 2, internal gas production is controlled by the rate of grain boundary oxygen diffusion, and cavity growth depends upon the rate of nickel self-diffusion. Hence, the extent to which the internal pressurization affects the cavity growth will depend directly upon the relative values of the grain boundary diffusivity of oxygen in nickel Dbrol and the grain boundary
LU er al.:
ENVIRONMENTALLY-ENHANCED
‘I- (“C) 12001100 1000 900 I’,~I~I~I’
..
600
700
800
1
1 :
Mean Scatterband
CAVITY
3263
GROWTH
(a) ::I.1--:--_/ 0.0006 -
D-
0.0004 -
Carbon Particle D,6, = 1.09x10-‘5 cm3/s D boo,= 5.51x10-” cm2/s
0.0002 -
0.0000 lo-‘21
6
’
7
’
’ ’ ’ j
8
9
’ I
10
11
-7 7.7
Tf
5
0
’
_ _._ ’ -..
12
r
.._ ’ -.-
’ -
10
r,-i-‘-‘15
20
t (W
104/T (l/OK) Fig. 2. Reported
40
values of I&cl vs T
I
”
7
I
”
Carbon Particle
self-diffusivity of nickel D&,. Values for Dbte] are very limited in the literature, and tend to be based upon indirect evidence, e.g. the extent of observable oxidation after a fixed period of time. The only available data appear to be those of Bricknell and Woodford [20] on pure nickel and Pedron and Pineau [14] on the nickel-based alloy Inconel 718. Figure 2 shows an Arhennius-type plot of these data with a scatterband constructed about the mean trend line. Data on Db& for pure nickel are more numerous, and have been obtained by more direct methods, e.g. radioisotope tracer techniques. These data are summarized in Table 1. The values shown here were obtained by interpolating or extrapolating the minimum and maximum reported values to 650°C using the values of the activation energy cited in each reference, and assuming a grain boundary thickness of &, = 5 x lo-* cm. It can be seen that the reported values exhibit about five orders of magnitude in scatter. 3. RESULTS
Given the large scatter in the diffusivity data, the approach will be to examine the least and most favorable conditions for environmentallyenhanced cavity growth. From physical considerations, the cavity growth rate is expected to be least influenced by the effects of internal gas pressure for small values of the grain boundary oxygen diffusivity and for large values of the self-diffusivity. Low grain boundary oxygen diffusivities will result in relatively
25 -
0
5
10
15
t (W
Fig. 3. Internal least favorable
cavity pressure vs time under conditions to environmental influence: (a) growing cavity; (b) nongrowing cavity.
slow rates of internal gas production, while high values of self-diffusivity will lead to relatively large cavity volume growth rates, so that the internal pressure will be unlikely to attain appreciable values. Conversely, a large value of grain boundary oxygen diffusivity, combined with a small value of selfdiffusivity, will be most favorable. Thus the lower and upper limit values of Db&, given in Table 1 at 650°C will be considered for the bounding cases, namely, 5.52 x 10~” and 1.09 x lo-l5 cm)/s. From the scatterband in the grain boundary oxygen diffusivity data, at 650°C (see Fig. 2), the minimum and maximum values of Dbrol are 1.22 x 1O-9 and
Table 1. Estimated maximum and minnnum values of LA& at 65O’C Temperature range of actual data (“C) 85&l 100 70&l 100 600-970 475450
(DbSb),“.X(cm’is) T = 650°C 1.21 3.47 9.24 1.09
x x x x
lo-‘5 lo-‘6 1O-‘8 10-15
20
(Db&),,. (CrnW T = 650°C 2.65 5.85 5.52 1.09
x x x x
10-l’ lo-” 1O-2” lo-‘1
Ref.
Pll [22,231 [241 125. 261
Specimen type polycrystal
bicrystal bicrystal Dolvcrvstal
3264
LU et al.:
ENVIRONMENTALLY-ENHANCED
5.51 x 10-l’ cm2/s, respectively. Additional parameter values were Q = 50 MPa, b = 50 pm, and the initial cavity radius a0 = 0.1 pm. The temperature was taken to be T = 650°C. Figures 3(a) and 3(b) show the computed cavity internal pressure, as well as the partial pressures of CO and CO,, as a function of time for the conditions least favorable to environmentally-enhanced cavity growth, i.e. Dh& = 1.09 x 10~‘5cm3/s and Dbrol= 5.51 x lo-” cm’js. Plotted on Fig. 3(a)]is the variation of pressure in a growing cavity. Here, an initial rapid rise in pressure is followed by an almost immediate decrease, so that the total gas pressure is limited to fairly small values. These values are so much smaller than the applied external stress (- 10m3 vs 50 MPa) as to have practically no effect upon the kinetics of cavity growth. At these low pressures, almost all of the internal pressure is due to CO. By contrast, when the cavity is constrained against growth [Fig. 3(b)], the internal pressure reaches fairly substantial values. At these pressures, chemical equilibrium favors the formation of coz. The situation in which the values of the diffusivities are most favorable to environmentallyenhanced cavity growth (DbSb = 5.52 x 10~“cm3/s, Dbrol= 1.22 x 10m9cm’/s) is quite different. Here the rate of cavity growth is considerably lower, and the rate of oxygen diffusion considerably higher than in the previous case. Consequently the internal gas pressure for the growing cavity, as shown in Fig. 4, reaches substantially higher values. In this case, almost all of the contribution to the total pressure comes from CO>, in agreement with the results of Dyson [7]. Figure 5 shows the growth in cavity radius versus time with and without the effects of internal pressure. After 20 h of growth, the pressurized cavity has reached a size approximately 25% greater than
-P
CAVITY GROWTH
I __
0.005
1,
7
7
I
’
0.004 Carbon Particle
0.002 -
0
10
5
15
20
t (hr) Fig. 5. Cavity growth in inert and oxygen environments under conditions most favorable to environmental influence.
that of the unpressurized cavity, with a growth rate about 65% larger than that of the unpressurized cavity. Figure 6 illustrates the combined effects of oxygen and self-diffusivity in more detail. Here the time required for a change in cavity radius of Aa = 0.01 pm, starting from an initial radius of u0 = 0.1 pm, is plotted as a function of Db& and Dbrol. For the parameters used here, environmental effects upon cavity growth are noticeable when the ratio DbtOl/Db is greater than about 10 (assuming a value of b,, = 5 x lo-* cm). Of interest is the fact that environmental effects are present over only a relatively small portion of the domain of Db& and Dbrolvalues. Other parameters in the problem, of course, have an influence upon the environmentally-enhanced cavity growth behavior. The most obvious is the applied stress 0”. For fixed values of the diffusion coefficients, a smaller value of go would lead to less rapid cavity growth and the effects of internal gas
I
0
Carbon
I
oxygenenvironment inert environment
inert environment oxygen environment
Particle
D&$,= 5.52~10-~~ cm3/s
Fig. 4. Internal cavity most favorable
pressure vs time under conditions to environmental influence.
Fig. 6. Time required
for Aa = 0.01 pm vs DbSb and Dbcol.
LU et al.:
ENVIRONMENTALLY-ENHANCED
production would be more marked. For example, separate simulations have been conducted for Q = 0 which yield values of internal cavity pressure about 30% greater than those shown in Fig. 4 after 20 h, and which are large enough to lead to cavity growth under the influence of internal pressure alone. 4. DISCUSSION
The foregoing analysis considered only the case of a pure carbon particle inside a nickel matrix. Many nickel-based alloys, however, contain significant amounts of chromium, and Dyson [7] has considered in some detail the case of a static (non-growing) cavity containing a Cr6C2, particle. He concluded that the presence of chromium would almost entirely suppress the growth of internal pressure within the cavity, because the reaction 4[Cr] + 302(,, + 2(Crz0?) is thermodynamically favored over the gas-producing oxygen/carbon reactions. In spite of this prediction, cavities, apparently of environmental origin, were observed by Pandey et al. [27] in a chromium-containing nickel-based superalloy. These authors speculated that these cavities could result from the formation of COZ as a metastable phase if the value of the oxygen activity was allowed to reach the formation pressure of NiO, instead of being limited by the formation pressure of Cr203 as assumed by Dyson [7]. To further examine this issue, an order of magnitude estimate of cavity pressure is made by replacing the pure carbon particle with a CrzzCs particle. For simplicity, it is assumed that chromium oxide does not form on the external surface of the bicrystal, and the diffusional results derived for pure nickel may be used. With this assumption, oxygen activity within the cavity will be allowed to reach the formation pressure of NiO. The analysis can be treated in the same fashion as the case of the pure
0.0010~
t 0.0008 -
1
”
1 7
1
’
”
9
7
j
A
P,PC0
~
”
cr,,c, Particle D&J =
0.0006-
D
0.0000~ 0
5
10
15
fh9
Fig. 7. Internal
cavity
pressure
vs time.
20
CAVITY
GROWTH
carbon particle [11] with the following chromium reactions
i
z$ g[Cr] + [Cl;
3265
inclusion
of
the
AG,O= 132,870 + 18.95T
2[Crl + iOZ(E)ti (CrZOl>; AG; = - 1,133,200 + 256.57.
(16)
Figure 7 shows the results of this analysis, in terms of the internal cavity pressure as a function of time, for the diffusivity coefficients most favorable to environmentally-enhanced cavity growth and o0 = 50 MPa. It can be seen that, even allowing for a higher oxygen activity than that considered by Dyson [7], the pressures are far too small to influence cavity growth. Several simplifying assumptions and approximations have been used in the preceding analyses. In the interests of analytical tractability. a geometrically simple configuration has been analyzed. However, the results obtained here have qualitative implications for more complex situations. Likewise, following Dyson [7] ideal gas behavior has been assumed. When internal cavity pressures exceed about 0.5 MPa, this assumption becomes increasingly in error. Raj [6] considered real gas effects and concluded that these act to reduce the cavity pressure as compared to ideal gas behavior. Additionally, a rather simplified model of the cavity growth which is entirely due to a self-diffusive mass transport mechanism has been employed. More complicated models, in which matrix creep contributes to cavity growth [28, 291 or in which a polycrystalline matrix acts to constrain the rate of cavity growth [30]. would alter the effect of internal gas production on cavity growth because of the changes in cavity growth rates relative to that for oxygen diffusion. The effects of these assumptions would need to be examined for each case, but they are not expected to alter the overall conclusions of this study. Finally, limiting values on the grain boundary self-diffusivity have been used to bound the range of expected responses. It is worth noting that the minimum value of the self-diffusivity resulted from data taken on bicrystalline specimens, while the maximum value was taken from tests performed on polycrystalline specimens. Because polycrystalline self-diffusivities might be expected to represent in some sense an average value for bicrystalline self-diffusivities, it is not clear that the range of self-diffusivities to be expected in a polycrystalline specimen are as wide as those indicated here for a bicrystal. Given the higher values reported for polycrystals, it is quite possible that the internal pressurization mechanism would not play a significant role in the growth of cavities in polycrystalline materials, at least in the temperature range 600700°C. In addition to their effect upon chemical equilibrium, alloying elements may affect the efficacy of the internal pressurization mechanism through shifts in the ratio of self-diffusivity to grain boundary
3266
LU et al.:
ENVIRONMENTALLY-ENHANCED
oxygen diffusivity. However, there are insufficient data in the literature to assess this effect at present. It should be noted, however, that the analysis presented here considered only the effect of gas-production upon cavity growth, not upon their formation. It is quite possible that high initial internal gas pressures might be important in the nucleation of cavities of very small size. If high initial pressures resulted in the nucleation of substantially more cavities than would be nucleated in an inert environment, then this would in itself constitute a mechanism for environmentally-enhanced creep damage, even if the internal pressure was reduced to negligible levels by subsequent cavity growth. Finally, it should be noted that reported observations of environmentally-induced cavitation, e.g. Refs [5] and [27], were carried out at temperatures considerably higher than that considered here and without external load. Increasing temperature favors oxygen diffusion over self-diffusion, which would tend to favor the enhanced cavity growth mechanism discussed here. For example, at 650°C a ratio of Db&/Dblol of 1.05, which results in negligible enhancement of cavity growth, translates into a value of 11.9 at lOOO”C, a value for which enhanced cavity growth would be expected to occur. There is an additional increase in oxygen concentration at the free surface with increasing temperature, which would likewise serve to promote enhanced cavity growth. However, even at these elevated temperatures, the presence of chromium should still effectively suppress gas production. Hence, the suggestion of Pandey et al. [27] that metastable COz might lead to cavities of environmental origin in nickel-based superalloys, does not seem to be a tenable one. SUMMARY
The hypothesis of Bricknell and Woodford that internal gas pressure resulting from oxygen carbon reactions could result in enhanced creep cavity growth, and hence might explain the marked environmental sensitivity of the failure behavior of nickel-based alloys, has been investigated in some detail. It is concluded that, at typical operating temperatures and stress levels for polycrystalline nickel-based alloys, this hypothesis is viable if two conditions are met. The first is that the ratio of the grain boundary oxygen diffusivity to the grain boundary self-diffusivity of nickel is sufficiently high so that cavity growth does not outpace the increase in internal pressure. For the parameters used herein, this condition is met over only a relatively small domain of Db& and Dbrol values at 650°C. At lOOO”C, however, this domain appears to be somewhat larger. The second condition is that the
GROWTH
alloy does not contain significant amounts of chromium (and possibly other alloying elements, as well) which would act to suppress CO/CO, production. Acknowledgement-This work was supported by the Materials Research Group (MRG) Program of the Division of Materials Research, National Science Foundation under Grant No. DMR-9102093.
REFERENCES
5. 6. 7. 8. 9. 10.
11. 12. 13. 14. 15. 16. 17. 18.
5.
CAVITY
P. N. Chaku and C. J. McMahon, Jr, Met&l. Trans. 5, 441 (1974). D. A. Woodford, MetaN. Trans. A 12A, 299 (1981). E. Andrieu and J. P. Henon, Muter. Sci. Engng 88, 191 (1987). S. Floreen and R. Raj, in Flow and Fracture at Elevated Temperatures (edited by R. Raj), p. 383. American Society for Metals, Metals Park, OH (1985). R. H. Bricknell and D. A. Woodford, Acta metall. 30, 257 (1982). R. Raj, Acta metall. 30, 1259 (1982). B. F. Dyson, Acta metall. 30, 1639 (1982). T.-J. Chuane and J. R. Rice. Acta metall. 21. 1625 (1973). D. Hull and D. E. Rimmer, Phil. Mug. 4, 673 (1959). M. V. Speight and J. E. Harris, MetaN. Sot. J. 1, 83 (1967). H.-M. Lu, Ph.D. Dissertation, Lehigh University (1995). M. V. Speight and W. Beere, Metal. Sci. 9, 190 (1975). J. Kameda, Acta metall. 34, 867 (1986). J. P. Pedron and A. Pineau, Mem. Etudes Sci. M&all. 80, 665 (1983). J.-W. Park and C. J. Altstetter. Metall. Trans. 18A. 43 (1987). R. N. Stevens, R. Dutton and M. P. Puls, Acta metall. 22, 629 (1974). R. N. Stevens, R. Dutton and M. P. Puls, Acta metall. 22, 639 (1974). M. Hansen, Constitution of Binary Alloys. McGraw-
Hill, New York (1958). 19. D. R. Gaskell, Introduction 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
to Metallurgical Thermodynamics, 2nd edition. Hemisphere Publishing Co. (1981). R. H. Bricknell and D. A. Woodford, Scripta metall. 16, 761 (1982). W. Lange, A. Hassner and G. Mixcher, Physica status solidi 5, 63 (1964). W. R. Upthegrove and M. J. Sinnott, Trans. Am. Sot. Metals 50, 1031 (1958). R. F. Cannon and J. P. Stark, J. appl. Phys. 40, 4361 (1969). R. F. Cannon and J. P. Stark, J. appl. Phys. 40, 4366 (1969). A. R. Wazzan, J. appl. Phys. 36, 3596 (1965). H. J. Frost and M. F. Ashby, Deformation-Mechanism Maps for Metals and Alloys. Pergamon Press, Oxford (1983). M. C. Pandey, B. F. Dyson and D. M. R. Taplin, Proc. R. Sot. London A 393, 117 (1984). A. Needleman and J. R. Rice, Acta metall. 28, 1315 (1980). H.-M. Lu and T. J. Delph, Scripta metall. mater. 29,281 (1993). J. R. Rice, Acta metall. 29, 675 (1981).