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γ equilibrium phase compositions in nickel-base superalloys by cluster variation method
CMHAD
Vol. 15, No. 2, pp. 143-158, 1991
0364-5916/91 $3.00 + .OO (c) 1991 Pergamon Press plc
PWed in the USA.
CALCULATION OF r'/'7 EQUILIBRIUM PHASE ~O~OS~TIO~S IN NICKEL-BASE SUPERALLOYS BY CLUSTER VARIATION METHOD
M. Enomoto", H. Harada** and M. Yamazaki** (r)Materials Physics Division and (**) Materials Design Division, National Research Institute for Metals, Tokyo 153, JAPAN Uhis paper was presented at CALPHAD XXJamshedpur,India Februaxy1991) ABSTRACT
They',/ 7 equflibrium phase boundaries in ternary, quaternary and even higher order nickel-base superalloys are calculated by the cluster variation method (CW) which utilizes the tetrahedron approximation and the phenomenolo~ical Lennard-Jones pair interaction potential. The potential parameter@ are first estimated from thermochemical data and /or semi-empirical formulae on cohesive energies and lattice parameters of binary alloys or compounds. Some of these values are adjusted by a small amount to obtain better agreement with the phase bpundaries in Lower order alloys reported in the literature. These paramter values are shown to reproduce y'/ 7 equilibrium phase compositions, 7’ volume fraction and misfit between the two lattices etc. in higher order alloys close to experimental observation in the temperature range from 750 to 125o'C. Hence, this method may be applicable to the designing of engineering nickel base superalloys in which the improvement of high temperature performance is effected by multiple alloying additions.
I. Introduction The superior properties of nickel-base superalloys at high temperatures can be attributed to the fine dispersion of coherent 7' precipitates in the ductile 7 matrix. Hence, the necessity has long been recognized of developing a reliable method to evaluate the effects of multiple alloying additions on the characteristics of 7’/ 7 phase equilibrium in the alloys. A multiple regression analysis has been utilized for the design of engineering superalloys to predict the alloy constitution and mechanical properties such as the creep rupture life at high temperatures etc.(l-3). Although this method is in princfple authentic, it needs a large amount of experimental data to assure its reliability in mult~eomponent alloys. Often the choice of a base formula for the regression analysis imposes a serious limftation to the accuracy of analysis. To avoid these difficulties, it is preferred to use a thermodynamic model which can describe adequately thermodynamic Properties and phase equilibria in multicomponent superalloys. So far, various modifications of the regular solution model have been utilized to analyze the constitution of binary or at most ternary nickel alloys(4-7). In the earliest treatments(4.5) compound phases were treated as a line compound or as a solid solution when compounds have a sizable stability composition range. Recently, a sublattice regular solution model was used to analyze the equilibrium involving Ll2 ordered r* and B2 ordered fiNiAl. As long as the random mixing is assumed in these models, the atom configuration in the nearest neighbor shell which encompasses different sublattices in compounds may be quite different from reality and thus, calculated cohesive energies of ----_*______________ Received 13August1990 143
M. ENOMOTO et
144
al.
compounds or alloys may contain a considerable amount of error. For instance, the preferential substitution site of alloying elements is likely to have an important influence on thermodynamic properties (and mechanical properties as well) of compounds. An extreme pattern of substitution behavior, e.g. entire substitution in the Ni or Al site, is often assumed in the above models. The interaction parameters necessary for subsequent analyses are evaluated under such assumptions. The cluster variation method (CVM) can calculate the probabilities of atom configurations affected by the interactions between atoms in the same and different sublattices in a most reliable manner (8.9). By using the interaction potential such as the Lennard-Jones (L-J) pair potential. the 7’/ 7 phase boundaries were reproduced very well in Ni-Al binary systems (10). This method has been applied to ternary alloys (11-14) and it has been demonstrated that the substitution behavior of some alloying elements, partition behavior thereof between 7’ transformation temperature and lattice paraand 7 , the 7’ to 7 order-disorder meters in ternary 7’ (mostly containing transition elements). can be treated in a unified manner. However, extensive comparisons with experiment is necessary to establish the reliability of the method in higher order alloys for applying it to engineering superalloys. In the first part of this report, comparisons of 7’/ 7 phase boundaries calculated by CVMwith the L-J pair potential and with the tetrahedron approximation (10,lZ) are made with experimentally determined phase boundaries in eight ternary systems. Subsequently, calculations are extended in quaternary, sexinary and septenary systems and results are compared with experimental data. Before entering the phase boundary calculations in quaternary systems, the basic characteristics of the effects of interaction between two alloying elements on their substitution and partition behavior are investigated.
2.
lattice
Method of Calculations
Under the tetrahedron approximation, point of 7’ phase is written as: ABBB r, 3L(zijkl) - 3% { L(Y:;l+L(Y$ ijkl ij
the configurational
>
entropy
(L(xiA)+3L(xlB1)
per each
1Ill
where, z~~~~ is the probability of finding atoms i,j,k and 1 in a tetrahedron nearest neighbor atom cluster. The summation is taken with all component atom species. The Al and Ni sublattices are denoted by superscripts A and B, respectively. yqfi and yaj are the probabilities of finding atoms i and j in a nearest neighbor atom pair between the two sublattices and in the Ni sublattice. x~A and xi% are the probability of finding atom i at a lattice point belonging to the ~1 and Nf sublattices, respectively. k is the Boltzmann constant, and L(c )= 1: lnc variables of clusters of different sizes. the following c * Among the probability relationships hold: ABBB = % xijkl kl
12Al
BB ABBB yij = % skkij kl
tSB1
AB
Yij
CALCULATION OF~‘/~EQUiLl6~lUM
xiA = z j
XiP = 2 j
PHASE COMPOSITIONS IN NICKEL-BASE SUPERALLOYS
AB yij
145
[3A1
y;; = c j
BB yij
[3B1
Following procedures were first proposed by Sanchez et al.(lO). Only the nearest neighbor pair interaction is taken into account in the enthalpy. namely:
' h,‘ = 7
AB BB I: eij(r) (Yij + Yijf LJ
141
where z = 12 is the nearest neighbor coordination number in the fee lattice. The atom configuration which minimizes the grand potential: 1 Q 7' = H,' - TS,' + PV,' +--x 4i
,Ui(xiA + 3xiB)
151
is sought under the restrictive condition: ABBB 2 zijkl ijkl
= 1
I81
Here, T, P, VT' and .ai are the temperature, pressure, atomic volume and chemical potential of the i-th component. respectively. By means of the Lagrangian multiplier method, the most probable atom configuration is expressed as:
171
where .I is the Lagrange multiplier and:
Uijkl = eij(rJ + eik(r) + eil(r) + ejk(r) + ejl(r) + ekl(rE
[31
On the other hand, the equilibrium distance between atoms fn rt is calculated from the equatzon:
-+p=o 6 v;
r91
All corresponding equations for the 7 phase are obtained by removing the subscripts A and B from the above equations.
146
M. ENOMOTO etal.
The Lennard-Jones
= eij .
eijtr)
pair
[yi”
potential
(2c)
is
written
as:
(g]
[lOI
The average values of the exponents, mij and nij, of pure metals are close to 8 and 4, respectively (10.15). The potential parameters, eij’ and rij are determined from thermochemical data on the heat of mixing of binary solid solutions (161, semi-empirical formula for the heat of formation of compounds (17-191, the lattice parameter of binary alloys or compounds (20). The procedures of the determination of their values have been described in detail previously (12). The potential parameter values are shown in TABLE 1.
the
In the cohesive
calculations all potential parameters energy and the lattice parameter of
are pure
normalized
by ell’
and rll,
Ni (i.j=l and 2 for Ni and Al, respectively). Performing differentiation, eq.[9] is written explicitly in terms of the normalized potential parameters as:
3Pr113 + z x
E ij’
AB BB P ij+Yij+Yij)P
4
ij
2+e11*
AB BB - z fj E ij'P ij8(Yij+Yij)P
8 = 0
[ill
Eq.[ll] is incorporated in / ell’ , P =r/ r11. and P ij=rij/ rll. the scheme of natural iteration: given an atom configuration characterized by y$j” the most probable atom distance is calculated. This distance is used to etc., in every iteration cycle. calculate uijkl by means of eq.[8]
where E ij* =eij’
In analyzing the phase component atoms are usually
c
equilibrium, set equal to
the sum of chemical zero for convenience:
potentials
of
all
[121
/Ai=0
i
Under this condition it is difficult to adjust all fii values to yield concentrations of all component species in the desired range because the alteration of a certain fli inevitably incurs the alteration of another IIi in eq.[ll]. This poses a serious problem in the calculation in higher order alloys. Accordingly, the above condition is applied only to the chemical potential of Ni and Al: [131
fl1+LL2=0 and fli for alloying elements (i = 3,4;...)
are changed freely.
It is difficult to take into account completely the temperature ,dependence of parameter values. For instance, to incorporate the temperature dependence of rij the lattice parameter data at some fixed alloy compositions are required in a significantly wide range of temperature. Hence, calculations are made primarily
CALCULATION OF y’ly
Al Ti Cr Fe co Nb MO Ta W
1.000 0.766 __ ___ -_ -_ __
1.384 0.909 1.095 --_-
1.000 --------__
--_
0.982 0.760 1.089 0.902 ---_-
1.057 0.790 N.D. 0.911 0.924 -__ __
1.002 0.910 1.248 0.968 N.D. 0.998 ---
__
-_
-_
-_
-_
Ni
Al
Ti
Cr
Fe
CO
Nb
1.000 ----------
1.053 1.149 -_ __ ___ __
1.035 1.122 1.174 --__ _-
1.018 1.073 1.080 1.032 __ -_ -_
1.017 1.031 N.D. 1.037 1.014 __ __
1.007 1.009 1.062 1.021 N.D. 1.006 --
1.038 1.171 1.169 1.075 N.D. 1.063 1.181 __ __
TABLE 2
_I
__
--
_-
1.488 1.298 1.374 1.490 N.D. 1.805 1.656 __
f Ta(lI
1.275 1.035 1.336 1.150 N.D. 1.365 1.685 1.356 __
--
1
1.543 1.262 1.095 1.353 N.D. 1.600 1.664 1.787 1.737
1.560 1.238 1.549 1.659 N.D. 1.509 1.990 1.731 1.901 1.960
Ta
W
1.04s 1.165 1.186 1.073 N.D. 1.074 1.187 1.137 1.180 --
1.029 1.150 1.151 1.041 N.D. 1.030 1.155 1.126 1.155 1.132
I_
MO
1.028 1.147 1.156 1.070 N.D. 1.062 1.157 1.095
1 W(If
Values of Chemical Potential of Alloy Elements to Initiate Iteration.
Ni tc1
CotI 1 Nb(g 1 MofII
Al
(b) Ni
Ti(lI 1 Crflll)Fe(W)
Ni
(a)
Al Ti Cr Fe co Nb MO Ta W
147
Lennard-Jones Potential Parameters used in Calculations. (a) E ij” , (b) p 13. For I ,Ii and Ill,See Text.
TABLE 1
Ni
EQUILIBRIUM PHASE COMPOSITIONS IN NICKEL-BASE SUPERALLOYS
O-3
Al -0.3
Ti
Cr
CO
Nb
MO
Ta
W
5.0
0.5
1.0
5.0
4.5
7.0
6.5
assuming the same temperature dependence as the parameters for pure Ni. namely, keeping the E ij' and p ij constant. When necessary, the temperature dependence of particular parameter values can be incorporated by fitting the phase boundary of published phase diagram.
3. Results 3-l. 7'/ 7
equilibrium
phase boundaries in ternary alloys
To begin with, calculated phase boundaries are compared with those In the literature in a number of Ni-Al-X ternary systems (21-27). Figures l-8 show the results. The potential parameters which were evaluated from thermochemical data do not always give phase boundaries close to reported ones. However, usually agreement is improved remarkably by the alteration of eij' by less than 15%.
M. ENOMOTO
81al.
FIG.1 Calculated 7'/ 7 phase boundaries in the Ni-Al-Ti system compared with experiment (21).
a)
b)
FIG.2 Calculated 7’~’ 7 phase boundaries in the Ni-Al-Cr system compared with experiment (22).
CALCULATION OF y’ / y EQUILIBRIUM PHASE COMPOSITIONS
INNICKEL-BASE SUPERALLOYS
Fe, at%
1150°C
___
BMdlW
30
-.i.=.
10
NJ
20
30
Al.at!4
b)
a)
Calculated r'/ 7 experiment (23).
Ni
10
20
30
a)
Calculated r'/ y experiment (24).
FIG.3 phase boundaries in the Ni-Al-Fe system compared with
40
50
60
Al, at%
b)
FIG.4 phase boundaries in the Ni-Al-Co system compared with
149
150
etal.
M. ENOMOTO
Nb. at% \ 21A :
750°C
/ Nf
10
20
25
AI.
*tt
b)
a)
Calculated 7’/ 7 experiment (25).
FIG.5 phase boundaries in the NI-Al-Nb system compared with
---
Lsh. ma
A
WI
I
10
20
Flelalng west
Chlk~AVOrtY ma
nest ll25o'C~
3C
Al, iitt
FIG.6 Calculated 7’/ 7 phase boundaries in the Ni-Al-MO system compared with experiment (26,271.
Nl
10
20
30
Al. At%
FIG.7 Calculated 7’/ 7 phase boundaries in the Ni-Al-Ta system compared with experiment (28).
CALCULATION OF y’ I y EQUILIBRIUM PHASE COMPOSITIONS IN NICKEL-BASE SUPERALLOYS
FIG.8 Calculated 7'/ 7 phase boundaries in the Ni-Al-W system compared with experiment (26).
151
A few basic rules of alteration of the parameters to improve agreement are recognized. Alloying elements which substitute preferentially for Ni and Al are termed Type I and X , respectively (12). Elements which change their substitution sites on both sides of the stoichiometry COInpOSitiOn (x~i=o.%) are termed Type Iii (see Table la). Firstly, for Type I and Hi elements. such as Co and Fe, the phase boundaries lean to a greater extent toward the NiX side as e23' is increased in relative to e13' . This is because X atoms in the Ni sublattice are surrounded by eight Ni and four Al atoms. Accordingly, the number of Al-X pairs is usually larger in 7' than in 7 . This implies that the number of Ni-X pairs is smaller in 7 ’ than in 7 . Consequently, if e23' is increased in relative to el3' , the 7’ phase becomes more stable than 7 , resulting in the contraction of the 7 phase field.
Secondly, for some Type II and m elements, such as MO and Cr, the f7’+7 ) phase region leans to a greater extent toward the Ni-X side as el3' is increased. In this case, the magnitude of e23' in relative to e13' is not very important. This is because these elements occupy the Al sublattice and are surrounded in twelve Ni atoms in the 7 ’ lattice. Considering that the E 13‘ value becomes larger in the order of Cr, MO and W (TABLE la). the experimental observation (28-30) that the two phase region in these ternary systems tends to lean progressively toward the Ni-X side as the atomic number of the elements (all in the VIa group) is increased may be interpreted along the same line. two
Thirdly, elements which substitute preferentially for Al and are segregated the curvature of the 7 / (7 +7’) boundary is in 7 , again such as MO and W etc., increased as e3j is increased; the boundary becomes more convex toward the Ni-X side. This is because the number of X-X pairs rapidly increases to form an energetically more favorable atom configuration in 7 as eS3’ is increased, whereas X atoms occupying the Al sublattice are unable to form a nearest neighbor atom pair in 7’. It is seen that the direction in the isothermal section of both calculated and experimental phase boundaries at small concentrations show a good correlation with the substitution behavior of X atoms in 7’; the phase boundaries are nearly parallel with the Ni-X and Al-X sides for Type I and H elements, respectively. For Type illelements the (7’+7 ) two phase region takes a path in the intermediate direction at small alloy element concentrations and is often curved to the Al-X side as the concentration is increased. 3-2. Characteristics of 7'/ 7
equilibrium in Ni-Al-Xl-X2 quaternary alloys
Preferential substitution site of alloying elements Before calculating the equilibrium phase boundaries the effects of interaction of alloying elements on various characteristics of 7’/ 7 equilibrium in
152
M. ENOMOTO et al.
quaternary alloys are investigated. Figures Sshows the variation of alloying element concentrations in the Al sublattice, XgA and x~A with the Ni (or Al) concentration in ternary and quaternary 7'. The bulk concentration of alloying elements are all 2 at.%. The calculation was made at 1OOo'C. It was demonstrated (12) that the present method predicted the site occupancy of a number of elements in good agreement with experimental observation. Figure 9a shows the results of calculation for Co(I 1 and Ta(ll ). It is seen that both XTaA and %oA do not alter significantly from those in ternary 7’ in which these elements are added separately. As seen in FIG.Sb, it is also the case with the combination of Ta(g ) and Cr(lTl ). Figure 9c is the result for the combination of CrfIU f and Fe(lU 1. Cr atoms have a slightly stronger tendency to substitute for Al than Fe atoms. It is seen that Fe exhibits a very similar substitution behavior in ternary and quaternary 7'. whereas the Cr curve for quaternary 7' lies significantly to the left of the curve for ternary 7’. However, this is an artifact caused by the choice of the Ni or Al concentration for the abscissa. If one chooses Al for the abscissa, the Fe curves in quaternary and ternary 7' are separated horizontally instead of Cr. This implies that Cr atoms regard Fe atoms as Ni as far as the substitution is eoncerned. Also, Fe atoms regard Cr atoms as Al when they search for their substitution sites in the 7’ lattice.
at%
Al,
a)
25
I
0
10.0
*
2. Ternary
# I
a I
I
,
,
,
25 20 I g ) ’ t , 1 I a~aternary
I
I 75
70
-
-
-
Ternary
eiix
0
70
80
Ta
75
,o,op Cr
FIG.3 Variation of alloy element concentration in the Al-sublattice with Ni(or Al) concentration in quaternary 7’ (thick curves). The bulk concentration of alloy element is 2 at.%. The concentrations in in ternary 7' are also shown for comparison (thin curves). Ni, at%
80
CALCULATION OF y’ I y EQUILIBRIUM PHASE COMPOSITIONS IN NICKEL-BASE SUPERALLOYS
a) t 2.0
-
Ouaternary
-
Ternary
153
bf
/
.$ !z 0
0.2
I 0.1 0
Chromium
Concentration
in Y.
Quaternary
-
Ternary
10
at%
FIG.10 Variation of equilibrium partition coefficient, kr'/ 7 , with the alloy element concentration in quaternary alloys (thick curves). For comparison. k7'/ 7' of Cr in ternary alloys is included (thin curves).
-
20
Chromium Concentration in Y.
2.0
-
~aternary
-
Ternary
at%
Cl
Partition of Alloying Elements between 7' and 7 Figures 10 show the variation of partition coefficient, kr'/ 7 , of Cr and the other alloying element between quaternary 7' and 7 with the Cr concentration. The bulk concentration of the second alloying elements is invariably 4 at.%. The calculation was again conducted at 1OOO'C. For the sake of comparison the partition coefficient of Cr in ternary alloys is included.
Cr 0.2
t 0.11 0 Chromium
I
20
10 concentration
In
Y.
at%
It is seen in FIG.lOa that at small Cr concentrations kr'/ 7 of Cr is somewhat decreased by the addition of Co( I 1, but the difference is diminished as the Cr concentration is increased. On the other hand, k7'/ 7 of Co is increased moderately from the value in the ternary alloy (the intercept of the Co curve with the ordinate) with the Cr concentration. Because the Cr-Co interaction is
M. ENOMOTO et al.
154
relatively weak, the coexistence of these elements i,sunlikely to affect their respective partition coefficients to each other significantly. When the second alloying element is Tat5 1, k7'/ 7 of Cr is decreased in quaternary alloys (FIG.lOb), whereas k7'/ 7 of Ta is slightly increased as the Cr concentration is increased. However, the differences are again small. W(W ) belongs to the same group as Ta in terms of the substitution behavior. However, the addition of W have a quite large influence on the Cr partition; k7'/ 7 of Cr is considerably decreased (more enriched in 7 ) in the presence of W (FIG.lOc). Concommittantly. k7'/ 7 of W is decreased remarkably as the Cr concentration is increased. This may be interpreted as follows. In contrast to Ta, a significant amount of W is present in 7 ; k7'/ 7 of W in the ternary alloy is only slightly larger than unity. From the evaluated E CrW' value (TABLE I), W atoms are likely to have a strong (attractive) interaction with Cr to form an energetically stable Cr-W pairs in the 7 lattice. Since both Cr and W substitute preferentially for Al (Cr does so in the Ni excess side of the stoichiometric composition), these elements are unable to form a nearest neighbor atom pair in the 7' lattice. Accordingly, more Cr and W atoms are segregated in 7 to form Cr-W pairs and hence, k7'/ 7 of these elements is decreased as the concentration of either element is increased. 3-3.
7'~'7
Equilibrium Phase Boundaries in Quaternary Systems
Experimentally determined 7’/ 7 phase boundaries in the xNi = 0.75 section are reported in some quaternary systems (27,29). Comparisons of calculated phase boundaries with them are made in FIGs.11 and 12. Usually, the equilibrium tielines were not in these isoconcentration sections. Accordingly, the 7 / (7 +7’) and 7'/ (7 +7') phase boundaries were calculated separately so that each boundary is exactly on the isoconcentration section. The deviation of the other end of the tie-line from the xNi = 0.75 plane is less than 0.03. The agreement between the calculated and reported phase boundaries is satisfactory in these systems. 3-4.
Calculation in Higher Order Alloys
In order to calculate the 7’/ 7 phase boundary composition etc. in alloys containing more than two alloying elements, a tie-line passing through a specified bulk alloy composition has to be searched in the multi-dimensional space. The search was initiated with the values of chemical potential shown in TABLE 2. In each cycle, these values are increased or decreased by a small amount depending on the sign of the difference between the two slopes, xi 7’
-
xi7
~27’
-
x27
xi7*
-
xib
mi =
113Al
and
m'i =
[13Bl x27' - x2b
where i= 3,4,..,* and x17', xi7 and xib are the compositions of alloying elements in 7' and 7 at equilibrium and in the bulk. These procedures are repeated until mi and m'i become nearly identical. The tie-line search was conducted in a sexinaryandaseptenary alloys. The results are compared with experimental data in TABLE 3. These alloys were prepared by arc-melting under
CALCULATION OF y ’ / y EQUILIBRIUM PHASE CoMPOSfTiONS
IN NICKEL-BASE SUPERALLOYS
155
argon. The compositions in each phase and 7’ volume fraction of these alloys were measured by EPMA and the conventinal metallographical method, respectively (31). In Alloy A excellent agreement is obtained between calculation and measurement. In particular, calculation yields very small k7'/ 7 of Cr at 909Cf=O.l7). As mentioned previously, such an extensive enrichment of Cr in 7 may be due to a strong interaction of nearest neighbor Cr-W pairs which are likely to exist in r. The lattice misfit, 6 , is a7’
6
defined as:
- ay
=
[141 a7
The calculated 6 is shown in the last column of TABLE 3. The "experimental" 6 was obtained by multiple regression analysis which incorporates the results of Xray diffractometry in a number of alloys (31). The X-ray analysis was made at room temperature. Grose and Ansell(32) reported that the thermal expansion coefficient of the lattice parameter of 7’ and 7 are (1.22-1.44)x10-5 and (1.48-
FIG. 11 Calculated 7’/ 7 phase boundaries in the xNi = 0.75 isoconcentration section of the Ni-Al-Ta-Mo systems.
FIG.12 Calculated 7’/ 7 phase boundaries in the xNi = 0.75 isoconcentration section of the Ni-Al-Cr-Ta systems.
Ni@
20
15
Ni3Ta Ni3Ta
Nt$r
a)
20
t5
IO
5
NIjAl
b)
-
Ni3Ta
CVM talc.
10
5
Nf3Al
-
Cvbi talc.
- - -
Chakravorty and West (29)
156
M. ENOMOTO
81al.
1.74)x10-5, respectively, in a number of Ni-15Cr-Al-Ti-Mo alloys. Taking the average value of these thermal expansion coefficients, 6 at high temperatures was back calculated from 6 measured at room temperature. It is not clear whether equilibrium atom configurations were obtained, or an atom configuration at some high temperatures were retained during the measurement of 6 at room temperature. However, it is seen that the temperature-corrected 6 agrees well with the calculated misfit, albeit the temperature dependence of all potential parameters are assumed to be the same as those of pure nickel.
TABLE 3
Comparison of Calculated and Experimentally Determined 7' and 7 Compositions, 7’ Volume Fraction and Lattice Misfit.
(a) Alloy A Composition in at.% Temp. C Bulk
Exp.
900
7’ 7
1240
7’ 7
Calc. 900
7’ 7
1240
7’ 7
Ni
Al
Cr
Co
Ta
bal.
12.3
6.75
8.12
1.83
5.76
bal. bal.
16.2
2.6
5.5
15.5
13.6
2.4 0.6
5.0 7.4
68
3.9
bal. bal.
15.5 lo:2
2.7 7.4
6.1 9.0
3.1 1.7
2.7 5.1
N.D.
bal. bal.
16.0 2.8 3.2 16.3
6.5 12.1
2.4 0.4
4.6 8.5
70
-0.15
bal. bal.
16.2 10.4
7.9 8.8
3.3 1.0
3.4 5.4
36
+0.08
3.1 8.5
W
7’ volume fraction,%
Lattice Misfit,%
+0.16 (-0.12') N.D.
(b) Alloy B at 9OO'C Composition in at.%
Exp.
7’ 7
Calc.
7’ 7
Ni
Al
Ti
Cr
Co
Ta
bal.
9.1
1.9
W
11.8
5.22
4.08
1.34
7’ volume fraction,%
bal. 13.4 bal. 2.6
2.7 0.7
3.1 23.6
2.9 8.2
5.3 1.4
0.8 2.0
59
bal. 13.5 bal. 2.8
3.0 0.4
2.7 25.0
4.0 7.3
6.0 0.9
0.4 2.8
61
I) Temperature-corrected values.
Lattice Misfit,%
+0.58 (+0.29') +0.21
CALCULATION OF ~‘/yEQUlLll3RlUM
PHASE COMPOSITIONS IN NICKEL-BASE SUPERALLOYS
157
4.Discussion This method may have the following advantages when used for the practical design of superalloys. Firstly, calculations are made without an assumption on the substitution sites of alloying elements. It may be of course possible to analyze the substitution behaviour by the sublattice regular solution model (33). However, the procedures of determining the potential parameters separately in each sublattice may be quite cumbersome. In this method, the interaction between atoms in the same and different sublattices is expressed by common potential parameters. Thus, the number of parameters necessary for calculation is significantly less than that in previous models. Secondly, the use of an atom cluster larger than the tetrahedron is unlikely to be advantageous unless potential parameters related to the 2nd and 3rd nearest neighbor interactions etc. are obtained very accurately. It took about one minute to search a tie-line which passed a specified bulk composition in a septenary alloy with the supercomputer. Accordingly, the computation time needed when a larger atom cluster is employed in alloys of practical significance, say, in a ten-component alloy may become enormous, even if a supercomputer is available. Thirdly, the lattice misfit is calculated simultaneously with the equilibrium phase composition. This is practically very useful because the lattice misfit is an important parameter affecting the high temperature performance of superalloys. An improved treatment of misfit at coherent y'/ 7 interface which incorporates the segregation of alloying elements at interfaces might be possible in the framework of this method (34).
4.Su~ary The 7'/ 7 equilibrium phase boundaries, 7’ volume fraction, and lattice misfit between 7' and 7 in ternary, quaternary and up to septenary Ni-base superalloys were calculated using the cluster variation method with the tetrahedron approximation. The potential parameters first evaluated from the cohesive energy and the lattice parameter of binary alloys or compunds were adjusted to bring the calculated phase boundaries into close agreement with the reported ones in ternary or quaternary systems. Coexistence of strongly interactive alloying elements can affect the characteristics of 7’/ 7 phase equilibrium, e.g. partition between 7’ and 7 , due to the difference in the atomic arrangement in the two lattices. The 7’/ 7 equilibrium phase compositions, etc. were reproduced reasonably well also in multicomponent alloys in this method. In view of these advantages and others concerning the execution of computation, this method may be competent for the practical design of a class of superalloys in which fine coherent compound precipitates are dispersed as a strengthener.
S.Acknowledgement The authors express their thanks to the Numerical Simulator Center of the National Aeronautical Laboratory. Tokyo, for permitting them to use a FACOM-~200 supercomputer.
References 1. H.Harada, M.Yamazaki. Y.Koizumi, N.Sakuma and N.Furuya: High Temperature Alloys for Gas Turbines. Conf Proc., D.Re idel Publishing Company, Boston, MA, 721(1982). 2. H.Harada. K.Ohno, T.Yamagata, T.Yokokawa, and M.Yamazaki: Proc.Gth Int. SYmp. Superalloys, Seven Springs, PA, 733 (1988).
156
M. ENOMOTO et
al.
H.Harada, T.Yokokawa, K.Ohno, T.Yamagata and M.Yamazaki: Int. Conf. Superalloys, Liege, Belgium, (1990). 4. L.Kaufman and H.Nesor:CALPHAD, 2, 325(1976). 5. L.Kaufman and H.Bernstein: Computer Calculations of Phase Diagrams, Academic Press, New York, 1970. 6. I.Ansara. B.Sundman and P.Willemin: Acta Metall., 36, 977(1966). 7. C.-C.Jia, K.Ishida, T.Nishizawa and R.Kainuma: Unpublished work at Tohoku University. 6. R.Kikuchi and D.de Fontaine: NBS Special Publications SP-496. 967(1977). 9. J.M.Sanchez and D.de Fontaine: Phys Rev.B, 17, 2926(1976). 10. J.M.Sanchez, J.R.Barefoot. R.N.Jarrett and J.K.Tien: Acta Metall., 32, 1519(1964). 11. J.K.Tien: Proc.Japan-US Seminar on Superalloys, Japan Institute for Metals, Fuji Susono, Japan, 333(1964). 12. M.Enomoto and H.Harada: Metall.Trans., m. 649(1969). 13. H.Harada. K.Ohno and M.Enomoto: Unpublished work at National Research Institute for Metals, 1969. 14. Y.P.Wu, N.C.Tso. J.M.Sanchez and J.K.Tien: Acta Metall., 37. 2635(1969). 15. R.Furth: Proc.Roy.Soc., m, 67(1944). 16. R.Hultgren, P.D.Desai, D.T.Hawkins. M.Gleiser, K.K.Kelly, and D.D.Wagman: Selected Values of the Thermodynamic Properties of the Alloys, ASM, Metals Park, OH, 1973. 17. A.R.Miedema, F.R.de Boer and R.Room: CALPHAD, 1, 341(1977). 16. A.R.Miedema and P.F.de ChAtel and F.R.de Boer: Physica, m, l(1926). 19. A.K.Niessen, F.R.de Boer, R.Room, P.F.de Chatel. W.C.M.Mattens and A.R.Miedema: CALPHAD, 1, 51(1963). 20. H.W.King: J.Mater.Sci.. 1, 79(1966). 21. A.Taylor and R.W.Floyd: J.Inst.Metals, a, 25(1952-53). 22. A.Taylor and R.W.Floyd: J.Inst.Metals. a, 451(1952-53). 23. A.J.Bradley: J.Iron Steel Inst., 163, 1930(1949); 168, 233(1951). 24. J.Schramm: Z.Metallkde, 33, 403(1941). 25. D.S.Duvall and M.J.Donachie,Jr.: J.Inst.Metals. 98. 162(1970). 26. P.Nash, S.Fielding, and D.R.F.West: Metal Sci.. 17. 192(1963). 27. S.Chakravorty and D.R.F.West: Metal Sci.. Is. 207(1964). 26. P.Nash and D.R.F.West: Metal Sci., 13. 670(1979). 29. S.Chakravorty and D.R.F.West: Metal Sci. Tech., 1, 976(1965). 30. T.Suzuki: unpublished research at Tokyo Institute of Technology, 1990. 31. H.Harada, T.Yamagata, S.Nakazawa. K.Ohno and M.Yamazaki: Report of the 123rd Committee on Heat-Resistent Metals and Alloys, Japan Society for the Promotion of Science, Vo1.30. No.3. 251(1969). 32. D.A.Grose and G.S.Ansell: Metall.Trans., m. 1631(1961). 33. M.Hillert: Hardenability Concepts with Applications to Steel, eds.D.V.Doane and J.S.Kirkaldy, 5(1976), TMS-AIME. 34. R.Kikuchi and J.W.Cahn: Acta Metall.. 27. 1337(1979). 3.
Vol. 15, No. 2, pp. 143-158, 1991
0364-5916/91 $3.00 + .OO (c) 1991 Pergamon Press plc
PWed in the USA.
CALCULATION OF r'/'7 EQUILIBRIUM PHASE ~O~OS~TIO~S IN NICKEL-BASE SUPERALLOYS BY CLUSTER VARIATION METHOD
M. Enomoto", H. Harada** and M. Yamazaki** (r)Materials Physics Division and (**) Materials Design Division, National Research Institute for Metals, Tokyo 153, JAPAN Uhis paper was presented at CALPHAD XXJamshedpur,India Februaxy1991) ABSTRACT
They',/ 7 equflibrium phase boundaries in ternary, quaternary and even higher order nickel-base superalloys are calculated by the cluster variation method (CW) which utilizes the tetrahedron approximation and the phenomenolo~ical Lennard-Jones pair interaction potential. The potential parameter@ are first estimated from thermochemical data and /or semi-empirical formulae on cohesive energies and lattice parameters of binary alloys or compounds. Some of these values are adjusted by a small amount to obtain better agreement with the phase bpundaries in Lower order alloys reported in the literature. These paramter values are shown to reproduce y'/ 7 equilibrium phase compositions, 7’ volume fraction and misfit between the two lattices etc. in higher order alloys close to experimental observation in the temperature range from 750 to 125o'C. Hence, this method may be applicable to the designing of engineering nickel base superalloys in which the improvement of high temperature performance is effected by multiple alloying additions.
I. Introduction The superior properties of nickel-base superalloys at high temperatures can be attributed to the fine dispersion of coherent 7' precipitates in the ductile 7 matrix. Hence, the necessity has long been recognized of developing a reliable method to evaluate the effects of multiple alloying additions on the characteristics of 7’/ 7 phase equilibrium in the alloys. A multiple regression analysis has been utilized for the design of engineering superalloys to predict the alloy constitution and mechanical properties such as the creep rupture life at high temperatures etc.(l-3). Although this method is in princfple authentic, it needs a large amount of experimental data to assure its reliability in mult~eomponent alloys. Often the choice of a base formula for the regression analysis imposes a serious limftation to the accuracy of analysis. To avoid these difficulties, it is preferred to use a thermodynamic model which can describe adequately thermodynamic Properties and phase equilibria in multicomponent superalloys. So far, various modifications of the regular solution model have been utilized to analyze the constitution of binary or at most ternary nickel alloys(4-7). In the earliest treatments(4.5) compound phases were treated as a line compound or as a solid solution when compounds have a sizable stability composition range. Recently, a sublattice regular solution model was used to analyze the equilibrium involving Ll2 ordered r* and B2 ordered fiNiAl. As long as the random mixing is assumed in these models, the atom configuration in the nearest neighbor shell which encompasses different sublattices in compounds may be quite different from reality and thus, calculated cohesive energies of ----_*______________ Received 13August1990 143
M. ENOMOTO et
144
al.
compounds or alloys may contain a considerable amount of error. For instance, the preferential substitution site of alloying elements is likely to have an important influence on thermodynamic properties (and mechanical properties as well) of compounds. An extreme pattern of substitution behavior, e.g. entire substitution in the Ni or Al site, is often assumed in the above models. The interaction parameters necessary for subsequent analyses are evaluated under such assumptions. The cluster variation method (CVM) can calculate the probabilities of atom configurations affected by the interactions between atoms in the same and different sublattices in a most reliable manner (8.9). By using the interaction potential such as the Lennard-Jones (L-J) pair potential. the 7’/ 7 phase boundaries were reproduced very well in Ni-Al binary systems (10). This method has been applied to ternary alloys (11-14) and it has been demonstrated that the substitution behavior of some alloying elements, partition behavior thereof between 7’ transformation temperature and lattice paraand 7 , the 7’ to 7 order-disorder meters in ternary 7’ (mostly containing transition elements). can be treated in a unified manner. However, extensive comparisons with experiment is necessary to establish the reliability of the method in higher order alloys for applying it to engineering superalloys. In the first part of this report, comparisons of 7’/ 7 phase boundaries calculated by CVMwith the L-J pair potential and with the tetrahedron approximation (10,lZ) are made with experimentally determined phase boundaries in eight ternary systems. Subsequently, calculations are extended in quaternary, sexinary and septenary systems and results are compared with experimental data. Before entering the phase boundary calculations in quaternary systems, the basic characteristics of the effects of interaction between two alloying elements on their substitution and partition behavior are investigated.
2.
lattice
Method of Calculations
Under the tetrahedron approximation, point of 7’ phase is written as: ABBB r, 3L(zijkl) - 3% { L(Y:;l+L(Y$ ijkl ij
the configurational
>
entropy
(L(xiA)+3L(xlB1)
per each
1Ill
where, z~~~~ is the probability of finding atoms i,j,k and 1 in a tetrahedron nearest neighbor atom cluster. The summation is taken with all component atom species. The Al and Ni sublattices are denoted by superscripts A and B, respectively. yqfi and yaj are the probabilities of finding atoms i and j in a nearest neighbor atom pair between the two sublattices and in the Ni sublattice. x~A and xi% are the probability of finding atom i at a lattice point belonging to the ~1 and Nf sublattices, respectively. k is the Boltzmann constant, and L(c )= 1: lnc variables of clusters of different sizes. the following c * Among the probability relationships hold: ABBB = % xijkl kl
12Al
BB ABBB yij = % skkij kl
tSB1
AB
Yij
CALCULATION OF~‘/~EQUiLl6~lUM
xiA = z j
XiP = 2 j
PHASE COMPOSITIONS IN NICKEL-BASE SUPERALLOYS
AB yij
145
[3A1
y;; = c j
BB yij
[3B1
Following procedures were first proposed by Sanchez et al.(lO). Only the nearest neighbor pair interaction is taken into account in the enthalpy. namely:
' h,‘ = 7
AB BB I: eij(r) (Yij + Yijf LJ
141
where z = 12 is the nearest neighbor coordination number in the fee lattice. The atom configuration which minimizes the grand potential: 1 Q 7' = H,' - TS,' + PV,' +--x 4i
,Ui(xiA + 3xiB)
151
is sought under the restrictive condition: ABBB 2 zijkl ijkl
= 1
I81
Here, T, P, VT' and .ai are the temperature, pressure, atomic volume and chemical potential of the i-th component. respectively. By means of the Lagrangian multiplier method, the most probable atom configuration is expressed as:
171
where .I is the Lagrange multiplier and:
Uijkl = eij(rJ + eik(r) + eil(r) + ejk(r) + ejl(r) + ekl(rE
[31
On the other hand, the equilibrium distance between atoms fn rt is calculated from the equatzon:
-+p=o 6 v;
r91
All corresponding equations for the 7 phase are obtained by removing the subscripts A and B from the above equations.
146
M. ENOMOTO etal.
The Lennard-Jones
= eij .
eijtr)
pair
[yi”
potential
(2c)
is
written
as:
(g]
[lOI
The average values of the exponents, mij and nij, of pure metals are close to 8 and 4, respectively (10.15). The potential parameters, eij’ and rij are determined from thermochemical data on the heat of mixing of binary solid solutions (161, semi-empirical formula for the heat of formation of compounds (17-191, the lattice parameter of binary alloys or compounds (20). The procedures of the determination of their values have been described in detail previously (12). The potential parameter values are shown in TABLE 1.
the
In the cohesive
calculations all potential parameters energy and the lattice parameter of
are pure
normalized
by ell’
and rll,
Ni (i.j=l and 2 for Ni and Al, respectively). Performing differentiation, eq.[9] is written explicitly in terms of the normalized potential parameters as:
3Pr113 + z x
E ij’
AB BB P ij+Yij+Yij)P
4
ij
2+e11*
AB BB - z fj E ij'P ij8(Yij+Yij)P
8 = 0
[ill
Eq.[ll] is incorporated in / ell’ , P =r/ r11. and P ij=rij/ rll. the scheme of natural iteration: given an atom configuration characterized by y$j” the most probable atom distance is calculated. This distance is used to etc., in every iteration cycle. calculate uijkl by means of eq.[8]
where E ij* =eij’
In analyzing the phase component atoms are usually
c
equilibrium, set equal to
the sum of chemical zero for convenience:
potentials
of
all
[121
/Ai=0
i
Under this condition it is difficult to adjust all fii values to yield concentrations of all component species in the desired range because the alteration of a certain fli inevitably incurs the alteration of another IIi in eq.[ll]. This poses a serious problem in the calculation in higher order alloys. Accordingly, the above condition is applied only to the chemical potential of Ni and Al: [131
fl1+LL2=0 and fli for alloying elements (i = 3,4;...)
are changed freely.
It is difficult to take into account completely the temperature ,dependence of parameter values. For instance, to incorporate the temperature dependence of rij the lattice parameter data at some fixed alloy compositions are required in a significantly wide range of temperature. Hence, calculations are made primarily
CALCULATION OF y’ly
Al Ti Cr Fe co Nb MO Ta W
1.000 0.766 __ ___ -_ -_ __
1.384 0.909 1.095 --_-
1.000 --------__
--_
0.982 0.760 1.089 0.902 ---_-
1.057 0.790 N.D. 0.911 0.924 -__ __
1.002 0.910 1.248 0.968 N.D. 0.998 ---
__
-_
-_
-_
-_
Ni
Al
Ti
Cr
Fe
CO
Nb
1.000 ----------
1.053 1.149 -_ __ ___ __
1.035 1.122 1.174 --__ _-
1.018 1.073 1.080 1.032 __ -_ -_
1.017 1.031 N.D. 1.037 1.014 __ __
1.007 1.009 1.062 1.021 N.D. 1.006 --
1.038 1.171 1.169 1.075 N.D. 1.063 1.181 __ __
TABLE 2
_I
__
--
_-
1.488 1.298 1.374 1.490 N.D. 1.805 1.656 __
f Ta(lI
1.275 1.035 1.336 1.150 N.D. 1.365 1.685 1.356 __
--
1
1.543 1.262 1.095 1.353 N.D. 1.600 1.664 1.787 1.737
1.560 1.238 1.549 1.659 N.D. 1.509 1.990 1.731 1.901 1.960
Ta
W
1.04s 1.165 1.186 1.073 N.D. 1.074 1.187 1.137 1.180 --
1.029 1.150 1.151 1.041 N.D. 1.030 1.155 1.126 1.155 1.132
I_
MO
1.028 1.147 1.156 1.070 N.D. 1.062 1.157 1.095
1 W(If
Values of Chemical Potential of Alloy Elements to Initiate Iteration.
Ni tc1
CotI 1 Nb(g 1 MofII
Al
(b) Ni
Ti(lI 1 Crflll)Fe(W)
Ni
(a)
Al Ti Cr Fe co Nb MO Ta W
147
Lennard-Jones Potential Parameters used in Calculations. (a) E ij” , (b) p 13. For I ,Ii and Ill,See Text.
TABLE 1
Ni
EQUILIBRIUM PHASE COMPOSITIONS IN NICKEL-BASE SUPERALLOYS
O-3
Al -0.3
Ti
Cr
CO
Nb
MO
Ta
W
5.0
0.5
1.0
5.0
4.5
7.0
6.5
assuming the same temperature dependence as the parameters for pure Ni. namely, keeping the E ij' and p ij constant. When necessary, the temperature dependence of particular parameter values can be incorporated by fitting the phase boundary of published phase diagram.
3. Results 3-l. 7'/ 7
equilibrium
phase boundaries in ternary alloys
To begin with, calculated phase boundaries are compared with those In the literature in a number of Ni-Al-X ternary systems (21-27). Figures l-8 show the results. The potential parameters which were evaluated from thermochemical data do not always give phase boundaries close to reported ones. However, usually agreement is improved remarkably by the alteration of eij' by less than 15%.
M. ENOMOTO
81al.
FIG.1 Calculated 7'/ 7 phase boundaries in the Ni-Al-Ti system compared with experiment (21).
a)
b)
FIG.2 Calculated 7’~’ 7 phase boundaries in the Ni-Al-Cr system compared with experiment (22).
CALCULATION OF y’ / y EQUILIBRIUM PHASE COMPOSITIONS
INNICKEL-BASE SUPERALLOYS
Fe, at%
1150°C
___
BMdlW
30
-.i.=.
10
NJ
20
30
Al.at!4
b)
a)
Calculated r'/ 7 experiment (23).
Ni
10
20
30
a)
Calculated r'/ y experiment (24).
FIG.3 phase boundaries in the Ni-Al-Fe system compared with
40
50
60
Al, at%
b)
FIG.4 phase boundaries in the Ni-Al-Co system compared with
149
150
etal.
M. ENOMOTO
Nb. at% \ 21A :
750°C
/ Nf
10
20
25
AI.
*tt
b)
a)
Calculated 7’/ 7 experiment (25).
FIG.5 phase boundaries in the NI-Al-Nb system compared with
---
Lsh. ma
A
WI
I
10
20
Flelalng west
Chlk~AVOrtY ma
nest ll25o'C~
3C
Al, iitt
FIG.6 Calculated 7’/ 7 phase boundaries in the Ni-Al-MO system compared with experiment (26,271.
Nl
10
20
30
Al. At%
FIG.7 Calculated 7’/ 7 phase boundaries in the Ni-Al-Ta system compared with experiment (28).
CALCULATION OF y’ I y EQUILIBRIUM PHASE COMPOSITIONS IN NICKEL-BASE SUPERALLOYS
FIG.8 Calculated 7'/ 7 phase boundaries in the Ni-Al-W system compared with experiment (26).
151
A few basic rules of alteration of the parameters to improve agreement are recognized. Alloying elements which substitute preferentially for Ni and Al are termed Type I and X , respectively (12). Elements which change their substitution sites on both sides of the stoichiometry COInpOSitiOn (x~i=o.%) are termed Type Iii (see Table la). Firstly, for Type I and Hi elements. such as Co and Fe, the phase boundaries lean to a greater extent toward the NiX side as e23' is increased in relative to e13' . This is because X atoms in the Ni sublattice are surrounded by eight Ni and four Al atoms. Accordingly, the number of Al-X pairs is usually larger in 7' than in 7 . This implies that the number of Ni-X pairs is smaller in 7 ’ than in 7 . Consequently, if e23' is increased in relative to el3' , the 7’ phase becomes more stable than 7 , resulting in the contraction of the 7 phase field.
Secondly, for some Type II and m elements, such as MO and Cr, the f7’+7 ) phase region leans to a greater extent toward the Ni-X side as el3' is increased. In this case, the magnitude of e23' in relative to e13' is not very important. This is because these elements occupy the Al sublattice and are surrounded in twelve Ni atoms in the 7 ’ lattice. Considering that the E 13‘ value becomes larger in the order of Cr, MO and W (TABLE la). the experimental observation (28-30) that the two phase region in these ternary systems tends to lean progressively toward the Ni-X side as the atomic number of the elements (all in the VIa group) is increased may be interpreted along the same line. two
Thirdly, elements which substitute preferentially for Al and are segregated the curvature of the 7 / (7 +7’) boundary is in 7 , again such as MO and W etc., increased as e3j is increased; the boundary becomes more convex toward the Ni-X side. This is because the number of X-X pairs rapidly increases to form an energetically more favorable atom configuration in 7 as eS3’ is increased, whereas X atoms occupying the Al sublattice are unable to form a nearest neighbor atom pair in 7’. It is seen that the direction in the isothermal section of both calculated and experimental phase boundaries at small concentrations show a good correlation with the substitution behavior of X atoms in 7’; the phase boundaries are nearly parallel with the Ni-X and Al-X sides for Type I and H elements, respectively. For Type illelements the (7’+7 ) two phase region takes a path in the intermediate direction at small alloy element concentrations and is often curved to the Al-X side as the concentration is increased. 3-2. Characteristics of 7'/ 7
equilibrium in Ni-Al-Xl-X2 quaternary alloys
Preferential substitution site of alloying elements Before calculating the equilibrium phase boundaries the effects of interaction of alloying elements on various characteristics of 7’/ 7 equilibrium in
152
M. ENOMOTO et al.
quaternary alloys are investigated. Figures Sshows the variation of alloying element concentrations in the Al sublattice, XgA and x~A with the Ni (or Al) concentration in ternary and quaternary 7'. The bulk concentration of alloying elements are all 2 at.%. The calculation was made at 1OOo'C. It was demonstrated (12) that the present method predicted the site occupancy of a number of elements in good agreement with experimental observation. Figure 9a shows the results of calculation for Co(I 1 and Ta(ll ). It is seen that both XTaA and %oA do not alter significantly from those in ternary 7’ in which these elements are added separately. As seen in FIG.Sb, it is also the case with the combination of Ta(g ) and Cr(lTl ). Figure 9c is the result for the combination of CrfIU f and Fe(lU 1. Cr atoms have a slightly stronger tendency to substitute for Al than Fe atoms. It is seen that Fe exhibits a very similar substitution behavior in ternary and quaternary 7'. whereas the Cr curve for quaternary 7' lies significantly to the left of the curve for ternary 7’. However, this is an artifact caused by the choice of the Ni or Al concentration for the abscissa. If one chooses Al for the abscissa, the Fe curves in quaternary and ternary 7' are separated horizontally instead of Cr. This implies that Cr atoms regard Fe atoms as Ni as far as the substitution is eoncerned. Also, Fe atoms regard Cr atoms as Al when they search for their substitution sites in the 7’ lattice.
at%
Al,
a)
25
I
0
10.0
*
2. Ternary
# I
a I
I
,
,
,
25 20 I g ) ’ t , 1 I a~aternary
I
I 75
70
-
-
-
Ternary
eiix
0
70
80
Ta
75
,o,op Cr
FIG.3 Variation of alloy element concentration in the Al-sublattice with Ni(or Al) concentration in quaternary 7’ (thick curves). The bulk concentration of alloy element is 2 at.%. The concentrations in in ternary 7' are also shown for comparison (thin curves). Ni, at%
80
CALCULATION OF y’ I y EQUILIBRIUM PHASE COMPOSITIONS IN NICKEL-BASE SUPERALLOYS
a) t 2.0
-
Ouaternary
-
Ternary
153
bf
/
.$ !z 0
0.2
I 0.1 0
Chromium
Concentration
in Y.
Quaternary
-
Ternary
10
at%
FIG.10 Variation of equilibrium partition coefficient, kr'/ 7 , with the alloy element concentration in quaternary alloys (thick curves). For comparison. k7'/ 7' of Cr in ternary alloys is included (thin curves).
-
20
Chromium Concentration in Y.
2.0
-
~aternary
-
Ternary
at%
Cl
Partition of Alloying Elements between 7' and 7 Figures 10 show the variation of partition coefficient, kr'/ 7 , of Cr and the other alloying element between quaternary 7' and 7 with the Cr concentration. The bulk concentration of the second alloying elements is invariably 4 at.%. The calculation was again conducted at 1OOO'C. For the sake of comparison the partition coefficient of Cr in ternary alloys is included.
Cr 0.2
t 0.11 0 Chromium
I
20
10 concentration
In
Y.
at%
It is seen in FIG.lOa that at small Cr concentrations kr'/ 7 of Cr is somewhat decreased by the addition of Co( I 1, but the difference is diminished as the Cr concentration is increased. On the other hand, k7'/ 7 of Co is increased moderately from the value in the ternary alloy (the intercept of the Co curve with the ordinate) with the Cr concentration. Because the Cr-Co interaction is
M. ENOMOTO et al.
154
relatively weak, the coexistence of these elements i,sunlikely to affect their respective partition coefficients to each other significantly. When the second alloying element is Tat5 1, k7'/ 7 of Cr is decreased in quaternary alloys (FIG.lOb), whereas k7'/ 7 of Ta is slightly increased as the Cr concentration is increased. However, the differences are again small. W(W ) belongs to the same group as Ta in terms of the substitution behavior. However, the addition of W have a quite large influence on the Cr partition; k7'/ 7 of Cr is considerably decreased (more enriched in 7 ) in the presence of W (FIG.lOc). Concommittantly. k7'/ 7 of W is decreased remarkably as the Cr concentration is increased. This may be interpreted as follows. In contrast to Ta, a significant amount of W is present in 7 ; k7'/ 7 of W in the ternary alloy is only slightly larger than unity. From the evaluated E CrW' value (TABLE I), W atoms are likely to have a strong (attractive) interaction with Cr to form an energetically stable Cr-W pairs in the 7 lattice. Since both Cr and W substitute preferentially for Al (Cr does so in the Ni excess side of the stoichiometric composition), these elements are unable to form a nearest neighbor atom pair in the 7' lattice. Accordingly, more Cr and W atoms are segregated in 7 to form Cr-W pairs and hence, k7'/ 7 of these elements is decreased as the concentration of either element is increased. 3-3.
7'~'7
Equilibrium Phase Boundaries in Quaternary Systems
Experimentally determined 7’/ 7 phase boundaries in the xNi = 0.75 section are reported in some quaternary systems (27,29). Comparisons of calculated phase boundaries with them are made in FIGs.11 and 12. Usually, the equilibrium tielines were not in these isoconcentration sections. Accordingly, the 7 / (7 +7’) and 7'/ (7 +7') phase boundaries were calculated separately so that each boundary is exactly on the isoconcentration section. The deviation of the other end of the tie-line from the xNi = 0.75 plane is less than 0.03. The agreement between the calculated and reported phase boundaries is satisfactory in these systems. 3-4.
Calculation in Higher Order Alloys
In order to calculate the 7’/ 7 phase boundary composition etc. in alloys containing more than two alloying elements, a tie-line passing through a specified bulk alloy composition has to be searched in the multi-dimensional space. The search was initiated with the values of chemical potential shown in TABLE 2. In each cycle, these values are increased or decreased by a small amount depending on the sign of the difference between the two slopes, xi 7’
-
xi7
~27’
-
x27
xi7*
-
xib
mi =
113Al
and
m'i =
[13Bl x27' - x2b
where i= 3,4,..,* and x17', xi7 and xib are the compositions of alloying elements in 7' and 7 at equilibrium and in the bulk. These procedures are repeated until mi and m'i become nearly identical. The tie-line search was conducted in a sexinaryandaseptenary alloys. The results are compared with experimental data in TABLE 3. These alloys were prepared by arc-melting under
CALCULATION OF y ’ / y EQUILIBRIUM PHASE CoMPOSfTiONS
IN NICKEL-BASE SUPERALLOYS
155
argon. The compositions in each phase and 7’ volume fraction of these alloys were measured by EPMA and the conventinal metallographical method, respectively (31). In Alloy A excellent agreement is obtained between calculation and measurement. In particular, calculation yields very small k7'/ 7 of Cr at 909Cf=O.l7). As mentioned previously, such an extensive enrichment of Cr in 7 may be due to a strong interaction of nearest neighbor Cr-W pairs which are likely to exist in r. The lattice misfit, 6 , is a7’
6
defined as:
- ay
=
[141 a7
The calculated 6 is shown in the last column of TABLE 3. The "experimental" 6 was obtained by multiple regression analysis which incorporates the results of Xray diffractometry in a number of alloys (31). The X-ray analysis was made at room temperature. Grose and Ansell(32) reported that the thermal expansion coefficient of the lattice parameter of 7’ and 7 are (1.22-1.44)x10-5 and (1.48-
FIG. 11 Calculated 7’/ 7 phase boundaries in the xNi = 0.75 isoconcentration section of the Ni-Al-Ta-Mo systems.
FIG.12 Calculated 7’/ 7 phase boundaries in the xNi = 0.75 isoconcentration section of the Ni-Al-Cr-Ta systems.
Ni@
20
15
Ni3Ta Ni3Ta
Nt$r
a)
20
t5
IO
5
NIjAl
b)
-
Ni3Ta
CVM talc.
10
5
Nf3Al
-
Cvbi talc.
- - -
Chakravorty and West (29)
156
M. ENOMOTO
81al.
1.74)x10-5, respectively, in a number of Ni-15Cr-Al-Ti-Mo alloys. Taking the average value of these thermal expansion coefficients, 6 at high temperatures was back calculated from 6 measured at room temperature. It is not clear whether equilibrium atom configurations were obtained, or an atom configuration at some high temperatures were retained during the measurement of 6 at room temperature. However, it is seen that the temperature-corrected 6 agrees well with the calculated misfit, albeit the temperature dependence of all potential parameters are assumed to be the same as those of pure nickel.
TABLE 3
Comparison of Calculated and Experimentally Determined 7' and 7 Compositions, 7’ Volume Fraction and Lattice Misfit.
(a) Alloy A Composition in at.% Temp. C Bulk
Exp.
900
7’ 7
1240
7’ 7
Calc. 900
7’ 7
1240
7’ 7
Ni
Al
Cr
Co
Ta
bal.
12.3
6.75
8.12
1.83
5.76
bal. bal.
16.2
2.6
5.5
15.5
13.6
2.4 0.6
5.0 7.4
68
3.9
bal. bal.
15.5 lo:2
2.7 7.4
6.1 9.0
3.1 1.7
2.7 5.1
N.D.
bal. bal.
16.0 2.8 3.2 16.3
6.5 12.1
2.4 0.4
4.6 8.5
70
-0.15
bal. bal.
16.2 10.4
7.9 8.8
3.3 1.0
3.4 5.4
36
+0.08
3.1 8.5
W
7’ volume fraction,%
Lattice Misfit,%
+0.16 (-0.12') N.D.
(b) Alloy B at 9OO'C Composition in at.%
Exp.
7’ 7
Calc.
7’ 7
Ni
Al
Ti
Cr
Co
Ta
bal.
9.1
1.9
W
11.8
5.22
4.08
1.34
7’ volume fraction,%
bal. 13.4 bal. 2.6
2.7 0.7
3.1 23.6
2.9 8.2
5.3 1.4
0.8 2.0
59
bal. 13.5 bal. 2.8
3.0 0.4
2.7 25.0
4.0 7.3
6.0 0.9
0.4 2.8
61
I) Temperature-corrected values.
Lattice Misfit,%
+0.58 (+0.29') +0.21
CALCULATION OF ~‘/yEQUlLll3RlUM
PHASE COMPOSITIONS IN NICKEL-BASE SUPERALLOYS
157
4.Discussion This method may have the following advantages when used for the practical design of superalloys. Firstly, calculations are made without an assumption on the substitution sites of alloying elements. It may be of course possible to analyze the substitution behaviour by the sublattice regular solution model (33). However, the procedures of determining the potential parameters separately in each sublattice may be quite cumbersome. In this method, the interaction between atoms in the same and different sublattices is expressed by common potential parameters. Thus, the number of parameters necessary for calculation is significantly less than that in previous models. Secondly, the use of an atom cluster larger than the tetrahedron is unlikely to be advantageous unless potential parameters related to the 2nd and 3rd nearest neighbor interactions etc. are obtained very accurately. It took about one minute to search a tie-line which passed a specified bulk composition in a septenary alloy with the supercomputer. Accordingly, the computation time needed when a larger atom cluster is employed in alloys of practical significance, say, in a ten-component alloy may become enormous, even if a supercomputer is available. Thirdly, the lattice misfit is calculated simultaneously with the equilibrium phase composition. This is practically very useful because the lattice misfit is an important parameter affecting the high temperature performance of superalloys. An improved treatment of misfit at coherent y'/ 7 interface which incorporates the segregation of alloying elements at interfaces might be possible in the framework of this method (34).
4.Su~ary The 7'/ 7 equilibrium phase boundaries, 7’ volume fraction, and lattice misfit between 7' and 7 in ternary, quaternary and up to septenary Ni-base superalloys were calculated using the cluster variation method with the tetrahedron approximation. The potential parameters first evaluated from the cohesive energy and the lattice parameter of binary alloys or compunds were adjusted to bring the calculated phase boundaries into close agreement with the reported ones in ternary or quaternary systems. Coexistence of strongly interactive alloying elements can affect the characteristics of 7’/ 7 phase equilibrium, e.g. partition between 7’ and 7 , due to the difference in the atomic arrangement in the two lattices. The 7’/ 7 equilibrium phase compositions, etc. were reproduced reasonably well also in multicomponent alloys in this method. In view of these advantages and others concerning the execution of computation, this method may be competent for the practical design of a class of superalloys in which fine coherent compound precipitates are dispersed as a strengthener.
S.Acknowledgement The authors express their thanks to the Numerical Simulator Center of the National Aeronautical Laboratory. Tokyo, for permitting them to use a FACOM-~200 supercomputer.
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