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A thermodynamic description of the TiAl system
PII:
SO966-9795(97)00030-7
Intermetallics 5 (1997) 471482 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0966-9795/97/%17.00 + 0.00
ELSEVIER
A thermodynamic description of the Ti-Al system F. Zhang,” S. L. Chen,” Y. A. Chang= & U. R. Kattner” aDepartment of Materials Science and Engineering, University of Wisconsin at Madison, Madison, Wisconsin 53706, USA ‘National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
(Received 27 November 1996; accepted 5 March 1997)
A thermodynamic description of the Ti-AI system was developed in this study. Nine phases were considered and can be classified as three types - disordered solution phases: liquid, (crTi, hcp), (#?Ti, bee), (Al, fee); ordered intermetallic phases: 02-T&Al (Dots), y-TiAl (Lls), TiAl3 (DO&; and stoichiometric phases: TiAl2 and Ti2A15. While the Redlich-Kister equation was used to describe the excess Gibbs energy of the disordered solution phases, a generalized bond-energy model recently developed by us at the University of Wisconsin at Madison was used to describe the Gibbs energy of the ordered intermetallic phases. The model parameters were optimized using the experimental phase equilibrium and thermodynamic data available in the literature. The calculated phase diagram as well as the thermodynamic functions are in good agreement with experimental data. The intrinsic defect concentrations calculated from the model parameters for y-TiAl were found to be in accord with those obtained from a semiempirical relationship in terms of its enthalpy of formation and the available experimental data. The generalized bond-energy model parameters for the ordered intermetallic phases were converted to those of the compound energy model for the convenience of the users of Thermo-Calc. 0 1997 Elsevier Science Limited Key words: A, Ti-Al
thermodynamic
system, D. point defects, E. phase diagram calculation, modelling, thermodynamic properties,
INTRODUCTION
concentrations of the y-TiAl calculated are not physically realistic. Ever since then, we have been making a continuous effort to develop a better description for this system. At the 1995 international conference on v-TiAl held at the TMS annual meeting, we presented a description for this system. i4 In this description, a significant improvement was made in describing the stability of y-TiAl. For instance, the calculated defect concentrations of y-TiAl from the model parameters are in accord with those obtained from a semiempirical relationship for this type of phase.” However, hypothetical standard states were used for the ordered phases az-Ti3Al (D0i9), y-TiAl (L&J, and TiAls (DO& in that study. Unfortunately, this choice of standard states precludes the use of this description by other researchers in developing thermodynamic descriptions for higher order systems consisting of Ti and Al. The objective of the present study is to present a description of the Ti-Al system using Ti (fee) and Al (fee) as the standard state for the y-TiAl and TiA13 phases and Ti (hcp) and Al (hcp) for the a2Ti3Al phase.
The development of ga.mma titanium aluminides as a structural material for use at high temperatures in the aerospace industries has in recent years stimulated intensive experimental investigations of the Ti-AI phase equilibria in the mid composition range.‘-’ Simultaneously, several attempts*-I4 have been made to develop thermodynamic descriptions for the Ti-Al system in addition to the earlier ones made by Kaufman and Nesor.‘5*16 The newer descriptions have been developed using more realistic models for the ordered intermetallic pjhases in this system and incorporating newer experimental data. The most satisfactory description prior to 1993 is that by Kattner et a1.i3 Not only the calculated phase diagram of the Ti-Al system but also the calculated thermodynamic values for most of the phases are in accord with the experimental data. However, a compromise was made in describing the thermodynamic stability of y-TiAl. In other words, although the model-calculated thermodynamic values are quite reasonable, the defect 471
F. Zhang
472
THERMODYNAMIC
et al. G = XTi ‘GTi + XA~‘GA,
MODELS
2
The nine phases considered in the present study for the Ti-Al system are liquid, (aTi, hcp), @Ti, bee), (Al, fee), a*-T&Al (DOis), y-TiAl (Lie), TiAls (DO&, TiA12, and Ti2A15.The first four phases are modeled as disordered solution phases and the next three c+-TisAl, y-TiAl, and TiAls as ordered intermetallic phases. The last two phases are treated as stoichiometric phases. The thermodynamic models for these three types of phases are described below. Stoichiometric compounds The Gibbs energy for a stoichiometric compound is described as G = XTiOGTi+ XAI‘GA, + AGr
(1)
where xi is the mole fraction of component i, and ‘Gi represents the Gibbs energy of a component in its standard state. AGr is the Gibbs energy of formation per mole of atoms of the stoichiometric compound. Disordered solution phases The Gibbs energy of a disordered solution phase is described by the following equation: G = XTiOGTi+ + XTi XAl
XAl
‘GAG-i- RT(xTi h XTi i-
[GO + G
(XTi
-
XAl hl XAl)
XAI )]
2
+ c j=l
C( _j=l
+ RT
C
If,*@
+
v,*zij)x’y;iy’,, (3)
2 x’(y~i
In $ri + _vA,In yi,)
i=l
where x’ (i = 1,2) is the site fraction of sublattice i relative to the total lattice sites; J$ is the concentration of the component p on sublattice i; Zq and z’j are the numbers of the first nearest neighbors and the second nearest neighbors of an atom on sublattice i with its neighbors on sublattice j(i,j = 1,2); I’12 represents the interchange energy between the first nearest neighbors, defined as Vi2 = N[Ei2 - 4 (El 1 + l&2)] with Ev representing the bond energy between i and j, and N the Avogardro constant; pi2 represents the interchange energy between the second nearest neighbors, taken to be n. V,2 with n being a constant in the present study. The assumption of constant Vi2 is reasonable to describe the thermodynamic behavior of an ordered intermetallic phase with a rather narrow range of homogeneity and varying symmetrically with respect to its stoichiometric composition. If this is not the case, it becomes necessary to make this term composition dependent. It is analogous to the extension of a regular solution model for a disordered phase to a subregular (or a subsubregular) solution model.
(2)
where the first two terms on the right hand side represent the Gibbs energy of the mechanical mixture of the components, the third term the ideal Gibbs energy of mixing, and the fourth term the excess Gibbs energy. The quantities Go and G1 are the parameters of the model, whose values are obtained by optimization. When Gi = 0 and Go = constant, it becomes the familiar regular solution model. Ordered intermetallic compound phases In contrast to stoichiometric compounds, many ordered intermetallic phases exist over a range of homogeneity. In the present study, a generalized bond-energy model’* is used to describe the thermodynamic properties of these phases. This model can be applied to any multicomponent, multisublattice intermetallic phase with antistructure defects. For a binary Ti-Al intermetallic phase with two sublattices, it has the following form:
OPTIMIZATION
AND CALCULATION
Optimization of the model parameters was carried out using PANDA, a program being developed by Chang’s group at the University of Wisconsin at Madison.i9 The thermodynamic and phase equilibrium values used in the optimization are essentially those obtained from the same set of dataid 2s29 used by Kattner et ~1.‘~ and Zhang.i4 However, two different sets of data were used to describe the Gibbs energies of pure Ti and Al. The first set is that given by Dinsdale, which is frequently referred to in the literature as the SGTE data. The second set is that used by Kattner et aLI3 and Zhang. I4 While a complex equation, i.e. a+bT+cTlnT+dT2++
+f T3
+ gT4 + hT7 + iT9
Thermodynamic description of Ti-Al system
with a, b, c, etc. as constants, was used by Dinsdale30 to represent the Gibbs energy of the different states of Ti and Al, a linear relationship in terms of T was used to represent the Gibbs energy differences between the various states of the elements by Kattner et a1.,13 Zhang14 and many others in the materials field. However, since many researchers in the field are currently developing thermodynamic descriptions of alloy systems using the SGTE data, we adopt this practice in the present study primarily for the purpose of compatibility when one extends the phase diagram calculations from binary Ti-Al to higher order systems involving these two elements. On the other hand, the linear relationships for the Gibbs energy differences are much simpler and can be used more conveniently by materials scientists doing research in related fields such as in phase transformation. Moreover, within the experimental uncertainties of the data for the elements, there is no difference between the two sets of data. Nevertheless, since many of the phase equilibria are governed by the rather small Gibbs energy differences between the various phases, the optimized higher order parameters are different
473
depending on which set of the data for pure Ti and Al are used. The two sets of optimized model parameters for the phases in the Ti-Al system obtained in the present study are given in Table 1, one using the SGTE data30 and the other using the linear relationship in terms of T for the pure elements.13,‘4 An examination of the two sets of model parameters given in Table 1 shows that the leading terms for all the phases are essentially the same. The higher order parameters which have rather small effects on the Gibbs energy values of the phases are different. Although the thermodynamic equation according to the generalized bond-energy model for ordered phases formulated by Chen et a1.18 retains the original Bragg-Williams form in terms of VAB, an equivalent form of equation, referred to in the literature as the compound energy model, has been developed. 31 This form of equation has been adopted to describe the thermodynamic properties of an ordered intermetallic phase in Thermo-Calc, a major computer program used to calculate phase diagrams. Consequently, many researchers in the field use this model to describe the thermodynamic properties of ordered intermetallic phases to
Table 1. Model parameters of the Ti-AI system obtained in this study Phases
Standard
A. SGTE data
state
B. Linear relationship to represent the Gibbs energy differences
Liquid
Liquid Ti and Al
Go=-111811~4+34~19962T G, = -9746.9-7.69422T
Go=--111811~4+35~08807T G, = -9746.9-7~18705T
B
Bee Ti and Al
Go=-127431.5+32.39035T G, = 82.2-8.24268T
Go=--132014.3+35.73744T G, =968.0-8.39094T
ff
Hcp Ti and Al
Go = - 127552.5 + 29.77509T G, =-14831.6+3.679267
Go=-123476.4+27.38338T G,-16484.1+3.74935T
(Al)
FCC Ti and Al
Go = - 114784.7+ 42.06532T
Go = - 114784.7 + 42.968543
TiAlz
FCC Ti and Al
AGf = -37730.1+6.701967
AGf = -37730.1+
6.706883
T&AI5
FCC Ti and Al
AGr = -34400.4+
AC‘ = -34400.4
+ 5.50811 T
arz-T&Al
Hcp Ti and All
V,2 = -10688.1 +2.25383T + (2239.7-2.302327J.x~ -(9958.8-10.406087) .xi, Z”=8, 212~4, Z21=12,J22=0
V,2 = -10511.5+2.24598T +(902.4-2.262197’)~~ -(7435.1-10.359487)x& Z”=8, Z12=4, Z2’=12,
V,2=2900.3-4.26077T -(26795.717.774063). x,z,l +(21896.2-14.38504T).xz,, Zll=4,Z’2=8,Z21=8,Z22=4 z”=6,2’2=16,221=16,~22=6,
V,2 =2900.3-4.25469T -(26871.2-17.79261T).xA] +(22047.2-14.446467,)x2,, Z” =4, Z’2=8, Z2’=8, 222 =4 z”=6,212=16,~221=16,~22=6
9=0.6
r]=O+6
V,2 = 234000.0 -(669443.8-13.21837).xAl +(453720.1-13.21837).x2,, Z’l=O,Z’2=12,Z21=4,Z22=8 211 =4, g12=2, $1=2, $2,4
V,2 = 234000.0 -(669348.9-13.146OZ). xAl + (453625.2-13~1462Z-)xz, Z’l=O,Z’2=12,_21=4,Z22=8 ~ll=4,~l2=2,~2l=2,~22=4
7]= -0.7
?J= -0.7
y-TiAl
TiA13
FCC Ti and Al
FCC Ti and Al
550773T
@2=0
F. Zhang et al.
474
calculate phases diagrams. Zhang et ~1.~~ have recently converted the model parameters of the generalized bond-energy model’* to those of the compound energy model.31 As a convenience for materials scientists in calculating multicomponent phase diagrams involving Ti and Al, the model parameters of the compound energy model for the three ordered intermetallic phases in the Ti-Al system are given in the Appendix. These parameters are converted directly from the values of VAB. It is noteworthy that the parameter VAB is a function of composition, in contrast to it being constant as in the original Bragg-Williams model. Using the optimized parameters given in column A of Table 1 and the SGTE data for Ti and Al, the phase diagram of the Ti-Al system is calculated as shown in Fig. 1 together with the experimental data points. The calculated phase diagram using the model parameters in column B of Table 1 and the linear relationship to represent the Gibbs energy difference between different states of the pure elements is essentially the same as that shown in Fig. 1. As shown in this figure, good agreement is obtained between the calculated phase diagram and the experimental data. A detailed comparison is presented below.
DISCUSSION
In this section, we first discuss the model-calculated thermodynamic values of the phases in the Ti-Al system in relation to the known experimental data. Second, we extend the discussion to the defect concentrations of y-TiAl. Third, we discuss the agreement between calculated phase boundaries and experimental phase boundary data. Lastly, we suggest additional improvements needed from both experimental and modelling points of view. Enthalpy of formation data are available for the liquid alloys at 2000K,33 shown in Fig. 2, and for solid alloys at 298 K, 34 shown in Fig. 3. As shown in these figures, there is reasonable agreement between the model-calculated values and experimental data considering the uncertainties of the measurements. In their paper, Esin et ~1.~~ did not mention such details as possible contamination of the alloys with impurities and evaporation of Al during their experiments. In view of the many intrinsic difficulties of high temperature experimental measurements, we estimate the uncertainty to be *4 kJmol-‘. Although Kubaschewski et a1.34 gave an uncertainty of about f 1 kJmol-’ for their data, a more realistic value for their data may be twice as large. As shown in Fig. 3, agreement
0” M c-4
* [Ref. I]
g_ N
n [Ref. 21
0 mef. 31
A [Ref. 41 0 Bef. 51 + Pef. 201
A [Ref 211 0 Ref. 221 0 [Ref. 231
Q [Ref. 241 t?s[Ref. 271 * [Ref. 251 v [Ref. 281 x [Ref. 261 V [Ref. 291
Liquid
L--. (Al
“0.0
Ti Fig. 1. Comparison
0.1
0.2
0.3
0.4
0.5
0.6
mole C-action of AI
0.7
0.8
0.9
)-
1
.o
Al
of the calculated Ti-Al phase diagram using the parameters in column A of Table 1 with the experimental data.
Thermodynamic description of Ti-Al system I
0
I
I
-
-4
I
-10
This work
J 0
[Ref.
I
331
F 1
-50
b
-60
A
O
/_I
0
El. 4
0.2
0
+
between the model-calculated values and experimental data is quite reasonable except for the compositions in the az-T&Al phase field. In their original investigation, Kubaschewski et a1.34 identified these alloys as disordered hcp rather than ordered D0i9. If this is the case, it is reasonable to expect that their measured values would be less exothermic than those for a*-Ti3Al, which is ordered. Figure 4 shows a comparison of the model-calculated partial Gibbs energy values of Al for (aTi), &-Ti3Al, and y-TiAl at 940 K with the experimental values of Samokhval et a1.35 who used a solid electrolyte cell technique. It is evident from this figure that significant deviations
4
I
-5
0
-10 E 3 3
-
-0.5
O/ I
This work [Ref. 341
0.2
0.3
0.4
0.5
+
0.6
-
Thisstudy
O,O,A
IRef.
361
t
-1.0-1.5-
e -2.0z a -2.5-
-30
0.1
occur between the model-calculated values and experimental data. In view of the fact that Samokhval et al. 35 did not give such experiment details as the level of impurities in their alloys, a low weight factor was given to their data in the optimization. After the optimization of the model parameters was completed, some activity data became available.36,37 Figure 5 shows comparisons between the model-calculated values and the experimental data of log,, a(A1) in the az-T&Al+ y-TiAl two-phase field as a function of the reciprocal temperature. Both Jacobson et a1.36 and Eckert et a1.37 used a
-20
Lj
a
Fig. 4. Comparison of the calculated partial Gibbs energy of Al in cz, az-TisAl, y-TiAl phase fields at 940 K with the experimental data.
,”
-25
A
mole fraction of Al
-15
2
A
-110
I
I
_“,, , A
0
Fig. 2. Comparison of the calculated enthalpy of mixing of the liquid phase at 2000 K with the experimental data.
A
A
-100 1.0
i’
AAA
AAAA
-90
0
0.8
Y
A
-80
0.6
This work [Ref. 351
a2
2
mole fraction of Al
0
I
i
-70
0
0
-28
I
-40
0
0
-24
I
t
-30
0
3
I
-20
0
$ -20
475
s*
I-
-3.0-35
1
-4E I-45 0
0.2
0.4
0.6
O.E
mole fraction of Al Fig. 3. Comparison of the calculated enthalpies of formation of the solid phases at 298 K with the experimental data.
Fig. 5. Comparison of the calculated activity of Al in the TisAl + TiAl two-phase field with the experimental data.
F.Zhang et al.
476
0
AH; = $t -... RT
00
-- AHOf-RT *
.-----. ---.-~--..-..:.yy-%
0
i
...-..--...
hHOf= RT
f
ln@x*) ln(2&)
...” ...”
0
“......_
.-“.. ....._.
‘Y..
-1.5 1
-2.0 I 6
I 7
1 8
....._
I 9
-lna*
10000/T(K) Fig. 6. Comparison of the calculated activity of Al in the TiAl + TiAlz two-phase field with the experimental data.
Knudsen cell mass spectrometric technique, while a solid electrolyte emf cell method was also used by Jacobson et ~1.~~ As shown in Fig. 5, excellent agreement is obtained between the model-calculated values and experimental data of Jacobson et a1.36The activity data of Eckert et ~1.~~are notably more negative than our calculated values. As shown in both Figs 4 and 5, the activity data of Samokhval et ~1.~~are lower than the modelcalculated values and the experimental data of Jacobson et ~1.~~ in the az-TisAl+ y-TiAl twophase field. The agreement between the modelcalculated and experimental data of Jacobson et ~1.~~ in the y -TiAl+ TiA12 two-phase field as shown in Fig. 6 is not as good as that for the cz2-Ti3Al + y-TiAl two-phase field given in Fig. 5.
10Ti
Fig. 8. Relationship between Ai$ and a* for antistructure phases at stoichiometry.
B2
However, as discussed later, there remain rather large uncertainties concerning the phase equilibria in this part of the phase diagram. Until these equilibria are better established, we cannot give a definitive rationale for the existence of this disagreement. The activity data of Eckert et ~1.~~in this two-phase field are also more negative, similar to that shown in Fig. 5. The y-TiAl phase exhibits the LIO structure; a unit cell for a perfectly ordered structure is shown in Fig. 7. The defect structure of this phase was determined by Elliot and Rostoker38 to be antistructure defects by measuring the intensities of the diffraction lines as a function of composition. In other words, at any finite temperature, lattice disordering of stoichiometric y-TiAl occurs by some of the Al atoms
2@Al
-8-9-10 0.5
\ I 1.0
I 1.5
I 2.0
1 2.5
I 3.0
I 3.5
4.0
1000/T(K) Fig. 7. Perfect unit cell of y-TiAl, in which the Ti and Al atoms occupy sublattice 1 and 2, respectively.
Fig. 9. Comparison of the calculated defect concentrations of y-TiAl at its stoichiometric composition with other studies.
Thermodynamic description of Ti-AI system 0.15
I
I
I
477
I
1::;:; -
‘......’
This work 0
[Ref. 431
-22.5
Ref : Bee Ti, Liq Al
0.10 “N
I /
by parameters of column B.
0
0.05
0
0
/
0 0 .5 0
I
I
0.52
1
0.54
0.56
mole
fraction of Al
,
0.58
+
-23.1
0.60
I 1400
I 1420
/ 1460
I 1440
I 1480
I 1500
T(K)
Fig. 10. Comparison of the calculated defect concentrations y-TiAl at 600°C with the experimental data.
of
on the Al sites being relplaced by Ti atoms. At the same time, equal numblers of the Ti atoms on the Ti sites are replaced with Al atoms. Inthe present paper, the model-calculalted defect concentration of y-TiAl is first presented to be reasonable by showing the agreement between the calculated data and the predicted values of a semiempirical relationship. This conclusion is based on the fact that the defect
Fig. 12. Comparison of the calculated Gibbs energy curves of the az-T&Al phase at x,&t= 0.3 from the two sets of parameters of Table 3.
concentrations obtained using a similar approach are in accord with experimental data available in the literature for antistructure B2 inter-metallic phases. In their review paper, Chang and Neumanni showed that an approximate relationship exists between the enthalpy of formation, Aq, and the intrinsic disorder parameter, (Y*, for antistructure B2 phases as given here:
A 0
o pef. 61 0 [Ref. 221
0
“-I Liquid
5
Ti Fig. 11. Comparison
0.6
mole fraction of Al
I.7
0.8
0.9
1 .o Al
of the calculated phase boundaries using the model parameters in column B of Table 3 for the az-T&Al phase with the experimental data.
F. Zhang
478
Ref : Bee Ti, Liq Al 24.02 % 3 -24.50 -25.0-
-25.5
1 0.30
I 0.32
I 0.34
I 0.36
1 0.38
I 0.40
mole fraction of Al Fig. 13. Comparison of the calculated Gibbs energy curves of the czz-T&Al phase at T= 1400K from the two sets of parameters of Table 3.
In a* M -ln2+3
(4)
where a*, the intrinsic disorder parameter, is the concentration of antistructure defects at stoichiometry a!* = 0 . 5y:, where y,2 is the mole fraction of the first component in the second sublattice. This relationship was derived using the same assumptions as those for deriving the generalized bondenergy equations used in the present study. As shown in Fig. 8, the experimental results for five B2 phases are slightly higher than those calculated from eqn (4). The agreement is improved when a factor of 4/3 is introduced, obtained from Monte Carlo calculations.39”0 Nevertheless, eqn (4) certainly gives an approximate description of the experimental data. This implies that we might be able to derive a similar relationship for antistructure L 1o phases:41
In CP M -ln2+-
AH,0
The definition of this intrinsic disorder parameter is the same as that for the antistructure B2 phases. Figure 9 shows the values of a* as a function of temperature obtained from the model parameters given in Table 1, as well as those obtained by Fu and Yoo4* using first principle calculations and by the approximate relationship given by eqn (5). As shown in Fig. 9, the calculated defect concentra-
et al. tions of y-TiAl from the present model parameters are in reasonable accord with those calculated from eqn (5). On the other hand, the values obtained by Fu and Yoo are about two orders of magnitude lower. Since the completion of this study, experimental data on the antistructure defects in y-TiAl determined by Collins and his students became available.43 As shown in Fig. 9, our calculated defect concentrations at stoichiometry are in reasonable accord with their experimental data at 1000 and 1400°C. In addition, Collins43 also determined the antistructure defects for several Al-rich alloys with Al contents varying from 50 to 55 at%. These samples were annealed at 1200°C for 1 h and then slowly cooled down to room temperature for measurements. Although the exact heat-treated temperature for these specimens is not known, we estimate it to be about half of. the melting temperature of y-TiAl, which is approximately 600°C. Following Chang and Neumann’s definition,” the defect concentration z*, defined as the number of Al atoms in the Ti sublattice divided by the total number of atoms, is calculated for y-TiAl at 600°C and compared with Collins’s data.43 As shown in Fig. 10, excellent agreement is obtained between the model-calculated values and the experimental data of Collins.43 As shown in this figure, the intrinsic antistructure defect concentration at stoichiometry is extremely small and those at nonstoichiometric compositions are primarily due to the constitutionally generated defects.” As mentioned in the last section, good agreement is obtained between the calculated phase diagram and the experimental data as shown in Fig. 1. Let us now discuss in detail the calculated invariant equilibria with experimental data as summarized in Table 2. It is evident from this table that the calculated invariant temperatures for most of the invariant reactions are in agreement with the experimental values within 1 to 2K. The exceptions are for the peritectic formation of o from /?, i.e. L + /? = a; the peritectoid formation of TiA12 from y and Ti2A15, i.e. y + Ti2A15= TiAlz;, and the eutectoid decomposition of a to form a2 and y, i.e. (Y= o2 + y, with discrepancies varying from 5 to 7K. The calculated phase boundaries are also in good agreement with the experimental data. An issue that is not yet completely resolved, however, is the type of phase transformation from a! to a2-Ti3Al. In other words, is it a congruent transformation as adopted in the present study and shown in Fig. l? Or the transformation may occur via other routes as shown in Fig. 11, which involve two peritectoid transformations, i.e. B + a! = (~2
Thermodynamic description of Ti-Al system
479
Table 2. Calculated and experimental invariant equilibria of the Ti-AI system Reaction
Calculated
results
Composition of the respective phase (~~1)
Experimental T(K)
0.497 0.549 0.719
0.460 0.517 0.676
0.479 0.547 0.714
1763.0 1735.9 1689.0
L + T&Al5 = TiA13
0.786
0.714
0.740
1666.0
y + Ti2AIS = TiA12
0.612 0.392 0.714 0.998 0.310
0.714 0.377 0.667 0.751 0.310
0.667 0.478 0.736 0.992
1472.5 1387.1 1265.0 938.2 1438.7
TizAls = TiAlz + TiA13 L + TiA& = (Al) (Y= cq
and B + Q = Q as suggested by Kainuma et al6 Since the congruent transformation point as shown in Fig. 1 is so close to the (a! + /I)/a phase boundary, any small increase in the stability of the az-TisAl phase at high temperatures would result in the peritectoid reactions. In order to demonstrate this effect, the model parameters for the Gibbs energy of aI-TiJA1 were reoptimized using the experimental data of Kainuma et aL6 The optimized parameters are listed in column B of Table 3. By using this set of model parameters for the a*-T&Al phase and the model parameters in column A of Table 1 for all the other phases, the Ti-AI phase diagram -is calculated as shown in Fig. 11. The model parameters of the az-TisAl phase used to calculate Fig. 1 are also listed in column A of Table 3. It is evident from Table 3 that the leading terms in Vi2 are exactly the same for both sets; only small changes in the higher order parameters cause changes of the reaction type. This is even mor’e clearly demonstrated by comparing the Gibbs energy curves of this phase calculated using the two sets of parameters. Figure 12 shows the Gibbs energy curve of crz-TijAl at XAi= 0.3, between 1400 and 1500 K, while Fig. 13 shows a similar curve at T= 1400 K from x.41= 0.3 to XAr= 0.4. From these two figures, we can see that a small decrease of less than 50 J mol-’ in the Gibbs energy of az-Ti,JAl changes the transformation from a congruent type to a peritectoid type. Table 3. The two sets of model parameters of the a2-TiJAl phase A. crz-Ti3Al forms congruently
of the
T(K)
Ref.
1768.2 1735.2 1688.0 1689.0 1698.0 1668.0 1669.0 1478.0 1380.2 1263.2 938.0 1442.0
5 5 2 4 5 2 5 5 5 2 24 29
reSpeCtiVephase @Al)
L+/3=a L+cr=y L + y = T&Al5
a=a2+y
Composition
results
B. cz2-Ti3AI forms peritectoidly
V,2= -10688~1+2~25383T V,*= -10688.1+2.25383T + (4152.2-3.680187).~~l + (2239.7-2.302329.~~~ -(11879.3-11.754167’)~x~, -(9958.8- 10~40608T)~&
0.500 -0.56 -0.68 -0.76 -0.72 -0.79 -0.78 -0.60 0.390 0.702 0.999 0.325
0.465 -0.51 -0.64 -0.63 0.714 0.714 0.714 0.370 0.667 0.75 0.325
0.475 -0.53 0.714 -0.75 -0.75 0.667 0.485 0.751 0.993
These results demonstrate the experimental difficulty in determining the correct type of phase transformation from a! to a*-TisAl. Until additional experimental evidence become available, the congruent transformation a! to aZ-Ti3Al is adopted in the present study. On the basis of the experimental data available in the literature, we believe the phase equilibria are reasonably established from Ti to about 60at% Al. However, this is not the case for phase equilibria at higher Al concentrations. Additional experimental investigations are needed. On the other hand, there is also a need to improve the models used to describe the thermodynamic properties of ordered intermetallic phases in the Ti-Al system. Even though the Bragg-Williams model works reasonably well at low temperatures, it is clearly inadequate when the temperatures approach the disordering temperature. Although Asta et a1.44 have successfully computed a metastable phase diagram of Ti-Al based on an fee lattice using first principle calculations coupled with the cluster variation method (CVM), it is more challenging with that approach to calculate the stable phase diagram for related materials research and engineering applications. Perhaps the cluster site approximation (CSA) offers in the near future a viable alternative to CVM for describing the thermodynamic properties of ordered intermetallic phases such as those in the Ti-Al system. 45 The great advantage of CSA over CVM is for applications to phase diagram calculations of multicomponent systems.
CONCLUSIONS A thermodynamic description for the Ti-Al system was obtained in the present study. The Redlich-
480
F.Zhang et al.
Kister equation was used to describe the excess Gibbs energy of the disordered solution phases while a generalized bond-energy model recently developed by Chang’s group at the University of Wisconsin at Madison18 was used for the ordered intermetallic phases. Optimization of the model parameters was carried out using PANDA, a program being developed by Chang’s group.” The calculated Ti-Al phase diagram is in good agreement with a large set of selected experimental phase equilibrium data, while agreement was also obtained for some of the thermodynamic data.33,34,36 Discrepancies between the model-calculated thermodynamic properties and other experimental data can be rationalized. The calculated defect concentrations of y-TiAl .are in accord with the predicted values of a semiempirical relationship,17 which has been successfully applied to ordered phases with the B2 structure, and the experimental data. 43 The generalized bond-energy model parameters for the three ordered intermetallic phases were converted to those of the compound energy model. While the phase equilibrium data from Ti to about 60at% Al are believed to be reliable, this is not true for data at higher Al concentrations. Additional experimental data are needed. Moreover, an improved model, such as cluster site approximation rather than that based on the basic Bragg-Williams assumptions, is needed to describe the thermodynamic properties of the ordered intermetallic phases.
ACKNOWLEDGEMENTS
The first three authors from the University of Wisconsin at Madison wish to acknowledge the financial support received from NASA’s Office of Space Access and Technology under Grant NAGW-1192 and Dr Tony Overfelt of Auburn for his interest in this work. The first three authors also wish to acknowledge Dr N. S. Jacobson and Dr G. S. Collins for providing their experimental data of this system. The authors also thank Professor W. A. Oates for his valuable comments on the manuscript. REFERENCES
4. Schuster, J. C. and Ipser, H., Z. Metallkd., 1990, 81, 389. 5. Mishurda, J. C. and Perepezko, J. H. MicrostructureProperty Relationships in Titanium Aluminides and Alloys,
6. 7. 8.
9.
10. 11.
12. 13. 14. 15.
16. 17. 18. 19.
20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
30. 31. 32. 33. 34.
1. Huang, S. C. and Siemers, P. A., Metall. Trans. A, 1989, 2OA, 1899. 2. Kaltenbach, K., Gama, S., Pinatti, D. G. and Schulze, K., Z. Metallkd., 1989, 80, 511. 3. McCullough, C., Valencia, J. J., Levi, C. G. and Mehrabian, R., Acta Metall., 1989, 37, 1321.
35. 36.
ed. Y. W. Kim and R. R. Boyer. TMS, Warrendale, PA, 1991, p. 3. Kainuma, R., Pa‘lhr, M. and Inden, G., Intermetalhcs, 1994,2(4), 321. Ding, J. J., Qin, G. W., Hao, S. M., Wang, X. T. and Chen, G. L., Journal of Phase Equilibria, 1996, 17, 117. Hsieh, K.-C., Jewett, T., Chang, Y. A. and Perepezko, J. H., First Annual URI Report, supported by DARPA through ONR Contract No. 0014-86-K-0753, 1987. Murray, J. L., Phase Diagrams of Binary Titanium Alloys, American Society for Metals, Metals Park, OH, 1987, p. 12. Gros, J. P., Sundman, B. and Ansana, I., Scri. Metail., 1988,22, 1587. Lin, J.-C., Jewett, T., Mishurda, J. C., Chang, Y. A. and Perepezko, J. H., Second Annual URF Report, supported by DARPA through ONR Contract No. 0014-86-K-0753, 1988. Murray, J. L., Metall. Trans., 1988, 19A, 243. Kattner, U. R., Lin, J.-C. and Chang, Y. A., Metall. Trans., 1992,23A, 2081. Zhang, F., Gamma Titanium Aluminides, ed. Y.-W. Kim, R. Wagner and M. Yamaguchi. TMS, Las Vegas, 1995, 131. Kaufman, L. and Nesor, H., Titanium Sci. Tech., Vol. 2, ed. R. I. Jaffe and H. M. Burte. Plenum Press, New York, 1973, 773. Kaufman, L. and Nesor, H., CALPHAD, 1978,4, 325. Chang, Y. A. and Neumann, J. P., Progress in Solid State Chemistry, 1982, 14, 221. Chen, S.-L., Kao, C. R. and Chang, Y. A., Intermetallics, 1995, 3(3), 233 Chen, S.-L. and Chang, Y. A., “PANDA” - Binary Phase Diagram Calculation and Optimization Program. University of Wisconsin at Madison, Madison, 1996. Fink, W. L., Van Horn, K. R. and Budge, P. M., Trans. AIME, 1931,93,421. Ogden, H. R., Maykuth, D. J., Finlay, W. L. and Jaffe, R. I., Trans. AIME, 1951, 191, 1150. Bumps, E. S., Kessler, H. D. and Hansen, M., Trans. AIME, 1952,194,609. Blackburn, M. J., Trans. AIME, 1967, 239, 1200. Cisse, J., Kerr, H. W. and Bolling, G. F., Metall. Trans., 1974, 5, 633. Heckler, M., Aluminum, 1974, 50, 405. Shibata, K., Sato, T. and Ohira, G., Journal of Crystal Growth, 1978,44, 435. Collings, E. W., Metall. Trans., 1979, lOA, 463. Abdel-Hamid, A., Allibert, C. H. and Durand, F., Z. Metallkd., 1984, 75, 455. Shull, R. D., McAlister, A. J. and Reno, R. C., Titanium Sci. Tech., Vol. 3, ed. G. Luetjering, U. Zwicker and W. Bunk. Deutsche Gesellschaft Fuer Metallkunde, Oberusel, Germany, 1985, p. 1459. Dinsdale, A. T., CALPHAD, 1991, 15, 317. Andersson, J.-O., Guillermet, A. F., Hillert, M., Jansson, B. and Sundman, B., Acta Metall., 1986, 34, 437. Zhang, F., Huang, W. and Chang, Y. A,, CALPHAD, in press. Esin, Yu. O., Bobrov, N. P., Petrushevskii, M. S. and Gel’d, P. V., Russian Metallurgy, 1974, 5, 86. Kubaschewski, 0. and Dench, W. A., Acta Metall., 1955, 3, 339. Samokhval, V. V., Poleshchuk, P. A. and Vecher, A. A., Russian Journal of Physical Chemistry, 1971,45(g), 1174. Jacobson, N. S., Brady, M. P. and Mehrotra, G., private communication, National Aeronautics and Space Administration, Lewis Research Center, Cleveland, 1996.
Thermodynamic description of Ti-AI system 31. Eckert, M., Bencze, L., Kath, D., Nickel, H. and Hilpert, K., Ber. Bunsenges. Phys. Chem., 1996, 100, 418. 38. Elliott, R. P. and Rostoker, W., Acta Met&., 1954,2, 884.
39. Guttman, L., Journalof IChem.Phys., 1961, 34, 1024. 40. Moscinski, J. and Rycerz, Z., Journal of Phys. F. (Metal Phys.), 1974, 4, 1853.
41. Chen, S. L., PhD thesis, University Madison, Madison, 1994.
of Wisconsin
at
APPENDIX: In Thermo-Calc, the Gibbs energy of a disordered solution phase is described by the Redlich-Kister equation.. The Gibbs energy of an ordered intermetallic phase is described by the compound energy modeL3’ which has the following form: GS = Y~,_Y~,GAI:AI + Y,&_YtiGAI:ri
481
42. Fu, C. L. and Yoo, M. H., Intermetallics, 1993, 1, 59. 43. Collins, G., private communication, Washington State
University, WA, 1995. 44. Asta, M., de Fontaine, D., van Schilfgaarde, M., Sluiter, M. and Methfessel, M., Physics Review, 1992, 46B,
5055. 45. Oates, W. A. and Wenzl, H., Scripta Materialia, 1996, 35, 623.
(Al) phase oLW)
AI,Ti =
-114784.7
+ 42.06532.
T
TiA12 oGTj!;-“G~-2.0G~ = -113190.3+20.10588.T
+ .J&_J$,,Gri:A* + YiiJ&%i:Ti TizAls +
R T[X’
<.A,
+
~‘(A
ln_h
+
Y~$+iY~l
In
YA,
+
--
J-4
Y$
Vii
)
+‘LAI(AI,Ti)(Y&
+lLTi:(A1,Ti)
(Y’,, - Y&l
+ YalY:iY~l[OL~AI,Ti):AI
+2LTi:(*1,Ti)
.&)
+‘L(*I.Ti):AIO.‘~~
-
Y?.i)2l + Y$..,Y+iY+i [oLTi:(*l.Ti)
+2L*l:(*l.Ti)
+‘L(*I,Ti):Ti(YJjl
“Gfj~$ _2.“@c_5.0@=
1nYti)l
[‘*l:(*l,Ti)
+2L(*I,Ti):Al
In
Y+i)
+2L(*l.Ti):Ti(Y,&
- Vii) Ie(A1,Ti):Ti -
Y&J21
+ Y~,Y~iY~lY:ioL(*l,l-i):(*l,Ti)
The model parameters of the Ti-Al system listed in column A of Table 1 are converted to the ThermoCalc format, as listed below:
= -240802.8 +;8.55411
TiSAl Two sublattices, sites 0.75:0*25, constituents (Ti,Al):(Al,Ti) oe;:A, - 0.75 .OG;;* - 0.25 .OG;,c* = -32251.8 + 6.98589 . T oGk,lTi - 0.25 . OG;;* - 0.75 . OG;;*
= -43830.4
+ 19.1415.
OLU2
(AI,Ti):AI =
1L’y2
(AI,Ti):AI =
Liquid
2LU2
OLliquid _ AI,Ti -
‘Lzq
(Al,Ti):AI
-111811.4+34.19962.T
= 9746.9 + 7.69422-T
‘L;,
Ti = -127431.5
‘LBAI,Ti = -82.2
$- 32.39035.T
+ 8.24268.T
OLa A,,Ti = -127552.5
i- 29.77509.T
‘L”A,,Ti = 14831.6 - 3’67926.T
-93637.7
+ 44.5853 . T
-31372.5
+ 32.867.
-8402.74
+ 8.78013.
T T
+ 10.9557 . T
1La2
+ 0.97557.
T
=
-933.637
OLZ(~~,~i)
=
1121.14 - 1.199967.
’ L;:(Al,Ti)
=
933.637 - 0.97557 ’ T
(Al,Ti):Ti
=
-47323.1
- 4.03728 . T
(AI,Ti):Ti
=
-3363.41
+ 3.59991 - T
2L(r2 (A1,Ti):Ti =
-8402.74
+ 8.78013.
OLU2
1L”2
(Y phase
=
T
oL~,:CA,:Tij= -10457.5 Al:(Al,Ti)
p phase
.;
0L”2
(Al,Ti):(Al,Ti)
=
26141.8 - 27.316.
T
T T
482
F. Zhang
TiAl
Two sublattices, sites O.kO.5, constituents (Ti,Al): (A1,Ti)
OGI;I:Ti 4e”Gp
-
05.OG;F
OLY (AI,Ti):Al - oL!h:(Al,Ti) ‘LY(Al,Ti):Al -
. lLLl*(Al
=
,Ti) =
= -44206-g+
9.064 - T
36409.2 - 33.4572 ’ T
‘L~i.(,1 Ti) = “~A*,Tij:Ti . 1
= -3914’
2~i:(A,,,)
=
f
25.876 . T
TiA13
Two sublattices, sites 0.25:0#75,constituents (Ti,Al): (Al,Ti)
=
0.75. OGE
-;4093 + 6.56787 - T
OGz;Ir =
-
-
0.75 -OG;p - 0.25. OGE
185243 + 4.83296 - T
26674.1 - 0.22719 . T
2LTiA13 (AI,Ti):AI
=
Al: (AI,Ti) =
-
4962.56 + 0.144576 a T
-426397 + 14.4989 ’ T 99232.6 - 12.0824 - T
=
248837 - 7.24944 . T
OL;$,
Ti) = .
498405 + 15.3663 - T
’ LTi:il
Q) =
819551 + 14.6848. T
2L~~~~,,i)
5200.35 - 3.41645 ’ T
-234289 + 153.92 - T
OG;!; - 0-25.OG;y
CAI,Ti):AI =
1LTiA13
I LTiA13
=
=
48020.3 - 6.48525 . T
29832 - 19.4369 . T
= OLY OLY Ti:(AI,Ti)
OLY (AI,Ti):(Al,Ti)
=
OLTiA13 Al: (AI.Ti) =
5200.35 - 3.41645. T
= 2’l,~,Ti):Ti
0LTiAl2 (AI,Ti):AI
16015.5 + 2.80527 - T
2Ly(Al,Ti):Al 2L:, :(Al ,Ti) = (AI,Ti):Ti
et al.
2 LTiA13 Ti:(AI,Ti)
=
248837 - 7.24944. T
0LTiA13 (A,,Tj):Ti
=
133222 - 5 ~03949 - T
=
21002-6 - 0.061961 - T
1LTiA13 (Al,Ti):Ti
2 LTiAl3 CAI,Ti):Ti = 0LTiA13 (AL,Ti):(AI,Ti)
-4962-56 + O-144576 - T =
-714609
f
20.8 189 . T
SO966-9795(97)00030-7
Intermetallics 5 (1997) 471482 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0966-9795/97/%17.00 + 0.00
ELSEVIER
A thermodynamic description of the Ti-Al system F. Zhang,” S. L. Chen,” Y. A. Chang= & U. R. Kattner” aDepartment of Materials Science and Engineering, University of Wisconsin at Madison, Madison, Wisconsin 53706, USA ‘National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
(Received 27 November 1996; accepted 5 March 1997)
A thermodynamic description of the Ti-AI system was developed in this study. Nine phases were considered and can be classified as three types - disordered solution phases: liquid, (crTi, hcp), (#?Ti, bee), (Al, fee); ordered intermetallic phases: 02-T&Al (Dots), y-TiAl (Lls), TiAl3 (DO&; and stoichiometric phases: TiAl2 and Ti2A15. While the Redlich-Kister equation was used to describe the excess Gibbs energy of the disordered solution phases, a generalized bond-energy model recently developed by us at the University of Wisconsin at Madison was used to describe the Gibbs energy of the ordered intermetallic phases. The model parameters were optimized using the experimental phase equilibrium and thermodynamic data available in the literature. The calculated phase diagram as well as the thermodynamic functions are in good agreement with experimental data. The intrinsic defect concentrations calculated from the model parameters for y-TiAl were found to be in accord with those obtained from a semiempirical relationship in terms of its enthalpy of formation and the available experimental data. The generalized bond-energy model parameters for the ordered intermetallic phases were converted to those of the compound energy model for the convenience of the users of Thermo-Calc. 0 1997 Elsevier Science Limited Key words: A, Ti-Al
thermodynamic
system, D. point defects, E. phase diagram calculation, modelling, thermodynamic properties,
INTRODUCTION
concentrations of the y-TiAl calculated are not physically realistic. Ever since then, we have been making a continuous effort to develop a better description for this system. At the 1995 international conference on v-TiAl held at the TMS annual meeting, we presented a description for this system. i4 In this description, a significant improvement was made in describing the stability of y-TiAl. For instance, the calculated defect concentrations of y-TiAl from the model parameters are in accord with those obtained from a semiempirical relationship for this type of phase.” However, hypothetical standard states were used for the ordered phases az-Ti3Al (D0i9), y-TiAl (L&J, and TiAls (DO& in that study. Unfortunately, this choice of standard states precludes the use of this description by other researchers in developing thermodynamic descriptions for higher order systems consisting of Ti and Al. The objective of the present study is to present a description of the Ti-Al system using Ti (fee) and Al (fee) as the standard state for the y-TiAl and TiA13 phases and Ti (hcp) and Al (hcp) for the a2Ti3Al phase.
The development of ga.mma titanium aluminides as a structural material for use at high temperatures in the aerospace industries has in recent years stimulated intensive experimental investigations of the Ti-AI phase equilibria in the mid composition range.‘-’ Simultaneously, several attempts*-I4 have been made to develop thermodynamic descriptions for the Ti-Al system in addition to the earlier ones made by Kaufman and Nesor.‘5*16 The newer descriptions have been developed using more realistic models for the ordered intermetallic pjhases in this system and incorporating newer experimental data. The most satisfactory description prior to 1993 is that by Kattner et a1.i3 Not only the calculated phase diagram of the Ti-Al system but also the calculated thermodynamic values for most of the phases are in accord with the experimental data. However, a compromise was made in describing the thermodynamic stability of y-TiAl. In other words, although the model-calculated thermodynamic values are quite reasonable, the defect 471
F. Zhang
472
THERMODYNAMIC
et al. G = XTi ‘GTi + XA~‘GA,
MODELS
2
The nine phases considered in the present study for the Ti-Al system are liquid, (aTi, hcp), @Ti, bee), (Al, fee), a*-T&Al (DOis), y-TiAl (Lie), TiAls (DO&, TiA12, and Ti2A15.The first four phases are modeled as disordered solution phases and the next three c+-TisAl, y-TiAl, and TiAls as ordered intermetallic phases. The last two phases are treated as stoichiometric phases. The thermodynamic models for these three types of phases are described below. Stoichiometric compounds The Gibbs energy for a stoichiometric compound is described as G = XTiOGTi+ XAI‘GA, + AGr
(1)
where xi is the mole fraction of component i, and ‘Gi represents the Gibbs energy of a component in its standard state. AGr is the Gibbs energy of formation per mole of atoms of the stoichiometric compound. Disordered solution phases The Gibbs energy of a disordered solution phase is described by the following equation: G = XTiOGTi+ + XTi XAl
XAl
‘GAG-i- RT(xTi h XTi i-
[GO + G
(XTi
-
XAl hl XAl)
XAI )]
2
+ c j=l
C( _j=l
+ RT
C
If,*@
+
v,*zij)x’y;iy’,, (3)
2 x’(y~i
In $ri + _vA,In yi,)
i=l
where x’ (i = 1,2) is the site fraction of sublattice i relative to the total lattice sites; J$ is the concentration of the component p on sublattice i; Zq and z’j are the numbers of the first nearest neighbors and the second nearest neighbors of an atom on sublattice i with its neighbors on sublattice j(i,j = 1,2); I’12 represents the interchange energy between the first nearest neighbors, defined as Vi2 = N[Ei2 - 4 (El 1 + l&2)] with Ev representing the bond energy between i and j, and N the Avogardro constant; pi2 represents the interchange energy between the second nearest neighbors, taken to be n. V,2 with n being a constant in the present study. The assumption of constant Vi2 is reasonable to describe the thermodynamic behavior of an ordered intermetallic phase with a rather narrow range of homogeneity and varying symmetrically with respect to its stoichiometric composition. If this is not the case, it becomes necessary to make this term composition dependent. It is analogous to the extension of a regular solution model for a disordered phase to a subregular (or a subsubregular) solution model.
(2)
where the first two terms on the right hand side represent the Gibbs energy of the mechanical mixture of the components, the third term the ideal Gibbs energy of mixing, and the fourth term the excess Gibbs energy. The quantities Go and G1 are the parameters of the model, whose values are obtained by optimization. When Gi = 0 and Go = constant, it becomes the familiar regular solution model. Ordered intermetallic compound phases In contrast to stoichiometric compounds, many ordered intermetallic phases exist over a range of homogeneity. In the present study, a generalized bond-energy model’* is used to describe the thermodynamic properties of these phases. This model can be applied to any multicomponent, multisublattice intermetallic phase with antistructure defects. For a binary Ti-Al intermetallic phase with two sublattices, it has the following form:
OPTIMIZATION
AND CALCULATION
Optimization of the model parameters was carried out using PANDA, a program being developed by Chang’s group at the University of Wisconsin at Madison.i9 The thermodynamic and phase equilibrium values used in the optimization are essentially those obtained from the same set of dataid 2s29 used by Kattner et ~1.‘~ and Zhang.i4 However, two different sets of data were used to describe the Gibbs energies of pure Ti and Al. The first set is that given by Dinsdale, which is frequently referred to in the literature as the SGTE data. The second set is that used by Kattner et aLI3 and Zhang. I4 While a complex equation, i.e. a+bT+cTlnT+dT2++
+f T3
+ gT4 + hT7 + iT9
Thermodynamic description of Ti-Al system
with a, b, c, etc. as constants, was used by Dinsdale30 to represent the Gibbs energy of the different states of Ti and Al, a linear relationship in terms of T was used to represent the Gibbs energy differences between the various states of the elements by Kattner et a1.,13 Zhang14 and many others in the materials field. However, since many researchers in the field are currently developing thermodynamic descriptions of alloy systems using the SGTE data, we adopt this practice in the present study primarily for the purpose of compatibility when one extends the phase diagram calculations from binary Ti-Al to higher order systems involving these two elements. On the other hand, the linear relationships for the Gibbs energy differences are much simpler and can be used more conveniently by materials scientists doing research in related fields such as in phase transformation. Moreover, within the experimental uncertainties of the data for the elements, there is no difference between the two sets of data. Nevertheless, since many of the phase equilibria are governed by the rather small Gibbs energy differences between the various phases, the optimized higher order parameters are different
473
depending on which set of the data for pure Ti and Al are used. The two sets of optimized model parameters for the phases in the Ti-Al system obtained in the present study are given in Table 1, one using the SGTE data30 and the other using the linear relationship in terms of T for the pure elements.13,‘4 An examination of the two sets of model parameters given in Table 1 shows that the leading terms for all the phases are essentially the same. The higher order parameters which have rather small effects on the Gibbs energy values of the phases are different. Although the thermodynamic equation according to the generalized bond-energy model for ordered phases formulated by Chen et a1.18 retains the original Bragg-Williams form in terms of VAB, an equivalent form of equation, referred to in the literature as the compound energy model, has been developed. 31 This form of equation has been adopted to describe the thermodynamic properties of an ordered intermetallic phase in Thermo-Calc, a major computer program used to calculate phase diagrams. Consequently, many researchers in the field use this model to describe the thermodynamic properties of ordered intermetallic phases to
Table 1. Model parameters of the Ti-AI system obtained in this study Phases
Standard
A. SGTE data
state
B. Linear relationship to represent the Gibbs energy differences
Liquid
Liquid Ti and Al
Go=-111811~4+34~19962T G, = -9746.9-7.69422T
Go=--111811~4+35~08807T G, = -9746.9-7~18705T
B
Bee Ti and Al
Go=-127431.5+32.39035T G, = 82.2-8.24268T
Go=--132014.3+35.73744T G, =968.0-8.39094T
ff
Hcp Ti and Al
Go = - 127552.5 + 29.77509T G, =-14831.6+3.679267
Go=-123476.4+27.38338T G,-16484.1+3.74935T
(Al)
FCC Ti and Al
Go = - 114784.7+ 42.06532T
Go = - 114784.7 + 42.968543
TiAlz
FCC Ti and Al
AGf = -37730.1+6.701967
AGf = -37730.1+
6.706883
T&AI5
FCC Ti and Al
AGr = -34400.4+
AC‘ = -34400.4
+ 5.50811 T
arz-T&Al
Hcp Ti and All
V,2 = -10688.1 +2.25383T + (2239.7-2.302327J.x~ -(9958.8-10.406087) .xi, Z”=8, 212~4, Z21=12,J22=0
V,2 = -10511.5+2.24598T +(902.4-2.262197’)~~ -(7435.1-10.359487)x& Z”=8, Z12=4, Z2’=12,
V,2=2900.3-4.26077T -(26795.717.774063). x,z,l +(21896.2-14.38504T).xz,, Zll=4,Z’2=8,Z21=8,Z22=4 z”=6,2’2=16,221=16,~22=6,
V,2 =2900.3-4.25469T -(26871.2-17.79261T).xA] +(22047.2-14.446467,)x2,, Z” =4, Z’2=8, Z2’=8, 222 =4 z”=6,212=16,~221=16,~22=6
9=0.6
r]=O+6
V,2 = 234000.0 -(669443.8-13.21837).xAl +(453720.1-13.21837).x2,, Z’l=O,Z’2=12,Z21=4,Z22=8 211 =4, g12=2, $1=2, $2,4
V,2 = 234000.0 -(669348.9-13.146OZ). xAl + (453625.2-13~1462Z-)xz, Z’l=O,Z’2=12,_21=4,Z22=8 ~ll=4,~l2=2,~2l=2,~22=4
7]= -0.7
?J= -0.7
y-TiAl
TiA13
FCC Ti and Al
FCC Ti and Al
550773T
@2=0
F. Zhang et al.
474
calculate phases diagrams. Zhang et ~1.~~ have recently converted the model parameters of the generalized bond-energy model’* to those of the compound energy model.31 As a convenience for materials scientists in calculating multicomponent phase diagrams involving Ti and Al, the model parameters of the compound energy model for the three ordered intermetallic phases in the Ti-Al system are given in the Appendix. These parameters are converted directly from the values of VAB. It is noteworthy that the parameter VAB is a function of composition, in contrast to it being constant as in the original Bragg-Williams model. Using the optimized parameters given in column A of Table 1 and the SGTE data for Ti and Al, the phase diagram of the Ti-Al system is calculated as shown in Fig. 1 together with the experimental data points. The calculated phase diagram using the model parameters in column B of Table 1 and the linear relationship to represent the Gibbs energy difference between different states of the pure elements is essentially the same as that shown in Fig. 1. As shown in this figure, good agreement is obtained between the calculated phase diagram and the experimental data. A detailed comparison is presented below.
DISCUSSION
In this section, we first discuss the model-calculated thermodynamic values of the phases in the Ti-Al system in relation to the known experimental data. Second, we extend the discussion to the defect concentrations of y-TiAl. Third, we discuss the agreement between calculated phase boundaries and experimental phase boundary data. Lastly, we suggest additional improvements needed from both experimental and modelling points of view. Enthalpy of formation data are available for the liquid alloys at 2000K,33 shown in Fig. 2, and for solid alloys at 298 K, 34 shown in Fig. 3. As shown in these figures, there is reasonable agreement between the model-calculated values and experimental data considering the uncertainties of the measurements. In their paper, Esin et ~1.~~ did not mention such details as possible contamination of the alloys with impurities and evaporation of Al during their experiments. In view of the many intrinsic difficulties of high temperature experimental measurements, we estimate the uncertainty to be *4 kJmol-‘. Although Kubaschewski et a1.34 gave an uncertainty of about f 1 kJmol-’ for their data, a more realistic value for their data may be twice as large. As shown in Fig. 3, agreement
0” M c-4
* [Ref. I]
g_ N
n [Ref. 21
0 mef. 31
A [Ref. 41 0 Bef. 51 + Pef. 201
A [Ref 211 0 Ref. 221 0 [Ref. 231
Q [Ref. 241 t?s[Ref. 271 * [Ref. 251 v [Ref. 281 x [Ref. 261 V [Ref. 291
Liquid
L--. (Al
“0.0
Ti Fig. 1. Comparison
0.1
0.2
0.3
0.4
0.5
0.6
mole C-action of AI
0.7
0.8
0.9
)-
1
.o
Al
of the calculated Ti-Al phase diagram using the parameters in column A of Table 1 with the experimental data.
Thermodynamic description of Ti-Al system I
0
I
I
-
-4
I
-10
This work
J 0
[Ref.
I
331
F 1
-50
b
-60
A
O
/_I
0
El. 4
0.2
0
+
between the model-calculated values and experimental data is quite reasonable except for the compositions in the az-T&Al phase field. In their original investigation, Kubaschewski et a1.34 identified these alloys as disordered hcp rather than ordered D0i9. If this is the case, it is reasonable to expect that their measured values would be less exothermic than those for a*-Ti3Al, which is ordered. Figure 4 shows a comparison of the model-calculated partial Gibbs energy values of Al for (aTi), &-Ti3Al, and y-TiAl at 940 K with the experimental values of Samokhval et a1.35 who used a solid electrolyte cell technique. It is evident from this figure that significant deviations
4
I
-5
0
-10 E 3 3
-
-0.5
O/ I
This work [Ref. 341
0.2
0.3
0.4
0.5
+
0.6
-
Thisstudy
O,O,A
IRef.
361
t
-1.0-1.5-
e -2.0z a -2.5-
-30
0.1
occur between the model-calculated values and experimental data. In view of the fact that Samokhval et al. 35 did not give such experiment details as the level of impurities in their alloys, a low weight factor was given to their data in the optimization. After the optimization of the model parameters was completed, some activity data became available.36,37 Figure 5 shows comparisons between the model-calculated values and the experimental data of log,, a(A1) in the az-T&Al+ y-TiAl two-phase field as a function of the reciprocal temperature. Both Jacobson et a1.36 and Eckert et a1.37 used a
-20
Lj
a
Fig. 4. Comparison of the calculated partial Gibbs energy of Al in cz, az-TisAl, y-TiAl phase fields at 940 K with the experimental data.
,”
-25
A
mole fraction of Al
-15
2
A
-110
I
I
_“,, , A
0
Fig. 2. Comparison of the calculated enthalpy of mixing of the liquid phase at 2000 K with the experimental data.
A
A
-100 1.0
i’
AAA
AAAA
-90
0
0.8
Y
A
-80
0.6
This work [Ref. 351
a2
2
mole fraction of Al
0
I
i
-70
0
0
-28
I
-40
0
0
-24
I
t
-30
0
3
I
-20
0
$ -20
475
s*
I-
-3.0-35
1
-4E I-45 0
0.2
0.4
0.6
O.E
mole fraction of Al Fig. 3. Comparison of the calculated enthalpies of formation of the solid phases at 298 K with the experimental data.
Fig. 5. Comparison of the calculated activity of Al in the TisAl + TiAl two-phase field with the experimental data.
F.Zhang et al.
476
0
AH; = $t -... RT
00
-- AHOf-RT *
.-----. ---.-~--..-..:.yy-%
0
i
...-..--...
hHOf= RT
f
ln@x*) ln(2&)
...” ...”
0
“......_
.-“.. ....._.
‘Y..
-1.5 1
-2.0 I 6
I 7
1 8
....._
I 9
-lna*
10000/T(K) Fig. 6. Comparison of the calculated activity of Al in the TiAl + TiAlz two-phase field with the experimental data.
Knudsen cell mass spectrometric technique, while a solid electrolyte emf cell method was also used by Jacobson et ~1.~~ As shown in Fig. 5, excellent agreement is obtained between the model-calculated values and experimental data of Jacobson et a1.36The activity data of Eckert et ~1.~~are notably more negative than our calculated values. As shown in both Figs 4 and 5, the activity data of Samokhval et ~1.~~are lower than the modelcalculated values and the experimental data of Jacobson et ~1.~~ in the az-TisAl+ y-TiAl twophase field. The agreement between the modelcalculated and experimental data of Jacobson et ~1.~~ in the y -TiAl+ TiA12 two-phase field as shown in Fig. 6 is not as good as that for the cz2-Ti3Al + y-TiAl two-phase field given in Fig. 5.
10Ti
Fig. 8. Relationship between Ai$ and a* for antistructure phases at stoichiometry.
B2
However, as discussed later, there remain rather large uncertainties concerning the phase equilibria in this part of the phase diagram. Until these equilibria are better established, we cannot give a definitive rationale for the existence of this disagreement. The activity data of Eckert et ~1.~~in this two-phase field are also more negative, similar to that shown in Fig. 5. The y-TiAl phase exhibits the LIO structure; a unit cell for a perfectly ordered structure is shown in Fig. 7. The defect structure of this phase was determined by Elliot and Rostoker38 to be antistructure defects by measuring the intensities of the diffraction lines as a function of composition. In other words, at any finite temperature, lattice disordering of stoichiometric y-TiAl occurs by some of the Al atoms
2@Al
-8-9-10 0.5
\ I 1.0
I 1.5
I 2.0
1 2.5
I 3.0
I 3.5
4.0
1000/T(K) Fig. 7. Perfect unit cell of y-TiAl, in which the Ti and Al atoms occupy sublattice 1 and 2, respectively.
Fig. 9. Comparison of the calculated defect concentrations of y-TiAl at its stoichiometric composition with other studies.
Thermodynamic description of Ti-AI system 0.15
I
I
I
477
I
1::;:; -
‘......’
This work 0
[Ref. 431
-22.5
Ref : Bee Ti, Liq Al
0.10 “N
I /
by parameters of column B.
0
0.05
0
0
/
0 0 .5 0
I
I
0.52
1
0.54
0.56
mole
fraction of Al
,
0.58
+
-23.1
0.60
I 1400
I 1420
/ 1460
I 1440
I 1480
I 1500
T(K)
Fig. 10. Comparison of the calculated defect concentrations y-TiAl at 600°C with the experimental data.
of
on the Al sites being relplaced by Ti atoms. At the same time, equal numblers of the Ti atoms on the Ti sites are replaced with Al atoms. Inthe present paper, the model-calculalted defect concentration of y-TiAl is first presented to be reasonable by showing the agreement between the calculated data and the predicted values of a semiempirical relationship. This conclusion is based on the fact that the defect
Fig. 12. Comparison of the calculated Gibbs energy curves of the az-T&Al phase at x,&t= 0.3 from the two sets of parameters of Table 3.
concentrations obtained using a similar approach are in accord with experimental data available in the literature for antistructure B2 inter-metallic phases. In their review paper, Chang and Neumanni showed that an approximate relationship exists between the enthalpy of formation, Aq, and the intrinsic disorder parameter, (Y*, for antistructure B2 phases as given here:
A 0
o pef. 61 0 [Ref. 221
0
“-I Liquid
5
Ti Fig. 11. Comparison
0.6
mole fraction of Al
I.7
0.8
0.9
1 .o Al
of the calculated phase boundaries using the model parameters in column B of Table 3 for the az-T&Al phase with the experimental data.
F. Zhang
478
Ref : Bee Ti, Liq Al 24.02 % 3 -24.50 -25.0-
-25.5
1 0.30
I 0.32
I 0.34
I 0.36
1 0.38
I 0.40
mole fraction of Al Fig. 13. Comparison of the calculated Gibbs energy curves of the czz-T&Al phase at T= 1400K from the two sets of parameters of Table 3.
In a* M -ln2+3
(4)
where a*, the intrinsic disorder parameter, is the concentration of antistructure defects at stoichiometry a!* = 0 . 5y:, where y,2 is the mole fraction of the first component in the second sublattice. This relationship was derived using the same assumptions as those for deriving the generalized bondenergy equations used in the present study. As shown in Fig. 8, the experimental results for five B2 phases are slightly higher than those calculated from eqn (4). The agreement is improved when a factor of 4/3 is introduced, obtained from Monte Carlo calculations.39”0 Nevertheless, eqn (4) certainly gives an approximate description of the experimental data. This implies that we might be able to derive a similar relationship for antistructure L 1o phases:41
In CP M -ln2+-
AH,0
The definition of this intrinsic disorder parameter is the same as that for the antistructure B2 phases. Figure 9 shows the values of a* as a function of temperature obtained from the model parameters given in Table 1, as well as those obtained by Fu and Yoo4* using first principle calculations and by the approximate relationship given by eqn (5). As shown in Fig. 9, the calculated defect concentra-
et al. tions of y-TiAl from the present model parameters are in reasonable accord with those calculated from eqn (5). On the other hand, the values obtained by Fu and Yoo are about two orders of magnitude lower. Since the completion of this study, experimental data on the antistructure defects in y-TiAl determined by Collins and his students became available.43 As shown in Fig. 9, our calculated defect concentrations at stoichiometry are in reasonable accord with their experimental data at 1000 and 1400°C. In addition, Collins43 also determined the antistructure defects for several Al-rich alloys with Al contents varying from 50 to 55 at%. These samples were annealed at 1200°C for 1 h and then slowly cooled down to room temperature for measurements. Although the exact heat-treated temperature for these specimens is not known, we estimate it to be about half of. the melting temperature of y-TiAl, which is approximately 600°C. Following Chang and Neumann’s definition,” the defect concentration z*, defined as the number of Al atoms in the Ti sublattice divided by the total number of atoms, is calculated for y-TiAl at 600°C and compared with Collins’s data.43 As shown in Fig. 10, excellent agreement is obtained between the model-calculated values and the experimental data of Collins.43 As shown in this figure, the intrinsic antistructure defect concentration at stoichiometry is extremely small and those at nonstoichiometric compositions are primarily due to the constitutionally generated defects.” As mentioned in the last section, good agreement is obtained between the calculated phase diagram and the experimental data as shown in Fig. 1. Let us now discuss in detail the calculated invariant equilibria with experimental data as summarized in Table 2. It is evident from this table that the calculated invariant temperatures for most of the invariant reactions are in agreement with the experimental values within 1 to 2K. The exceptions are for the peritectic formation of o from /?, i.e. L + /? = a; the peritectoid formation of TiA12 from y and Ti2A15, i.e. y + Ti2A15= TiAlz;, and the eutectoid decomposition of a to form a2 and y, i.e. (Y= o2 + y, with discrepancies varying from 5 to 7K. The calculated phase boundaries are also in good agreement with the experimental data. An issue that is not yet completely resolved, however, is the type of phase transformation from a! to a2-Ti3Al. In other words, is it a congruent transformation as adopted in the present study and shown in Fig. l? Or the transformation may occur via other routes as shown in Fig. 11, which involve two peritectoid transformations, i.e. B + a! = (~2
Thermodynamic description of Ti-Al system
479
Table 2. Calculated and experimental invariant equilibria of the Ti-AI system Reaction
Calculated
results
Composition of the respective phase (~~1)
Experimental T(K)
0.497 0.549 0.719
0.460 0.517 0.676
0.479 0.547 0.714
1763.0 1735.9 1689.0
L + T&Al5 = TiA13
0.786
0.714
0.740
1666.0
y + Ti2AIS = TiA12
0.612 0.392 0.714 0.998 0.310
0.714 0.377 0.667 0.751 0.310
0.667 0.478 0.736 0.992
1472.5 1387.1 1265.0 938.2 1438.7
TizAls = TiAlz + TiA13 L + TiA& = (Al) (Y= cq
and B + Q = Q as suggested by Kainuma et al6 Since the congruent transformation point as shown in Fig. 1 is so close to the (a! + /I)/a phase boundary, any small increase in the stability of the az-TisAl phase at high temperatures would result in the peritectoid reactions. In order to demonstrate this effect, the model parameters for the Gibbs energy of aI-TiJA1 were reoptimized using the experimental data of Kainuma et aL6 The optimized parameters are listed in column B of Table 3. By using this set of model parameters for the a*-T&Al phase and the model parameters in column A of Table 1 for all the other phases, the Ti-AI phase diagram -is calculated as shown in Fig. 11. The model parameters of the az-TisAl phase used to calculate Fig. 1 are also listed in column A of Table 3. It is evident from Table 3 that the leading terms in Vi2 are exactly the same for both sets; only small changes in the higher order parameters cause changes of the reaction type. This is even mor’e clearly demonstrated by comparing the Gibbs energy curves of this phase calculated using the two sets of parameters. Figure 12 shows the Gibbs energy curve of crz-TijAl at XAi= 0.3, between 1400 and 1500 K, while Fig. 13 shows a similar curve at T= 1400 K from x.41= 0.3 to XAr= 0.4. From these two figures, we can see that a small decrease of less than 50 J mol-’ in the Gibbs energy of az-Ti,JAl changes the transformation from a congruent type to a peritectoid type. Table 3. The two sets of model parameters of the a2-TiJAl phase A. crz-Ti3Al forms congruently
of the
T(K)
Ref.
1768.2 1735.2 1688.0 1689.0 1698.0 1668.0 1669.0 1478.0 1380.2 1263.2 938.0 1442.0
5 5 2 4 5 2 5 5 5 2 24 29
reSpeCtiVephase @Al)
L+/3=a L+cr=y L + y = T&Al5
a=a2+y
Composition
results
B. cz2-Ti3AI forms peritectoidly
V,2= -10688~1+2~25383T V,*= -10688.1+2.25383T + (4152.2-3.680187).~~l + (2239.7-2.302329.~~~ -(11879.3-11.754167’)~x~, -(9958.8- 10~40608T)~&
0.500 -0.56 -0.68 -0.76 -0.72 -0.79 -0.78 -0.60 0.390 0.702 0.999 0.325
0.465 -0.51 -0.64 -0.63 0.714 0.714 0.714 0.370 0.667 0.75 0.325
0.475 -0.53 0.714 -0.75 -0.75 0.667 0.485 0.751 0.993
These results demonstrate the experimental difficulty in determining the correct type of phase transformation from a! to a*-TisAl. Until additional experimental evidence become available, the congruent transformation a! to aZ-Ti3Al is adopted in the present study. On the basis of the experimental data available in the literature, we believe the phase equilibria are reasonably established from Ti to about 60at% Al. However, this is not the case for phase equilibria at higher Al concentrations. Additional experimental investigations are needed. On the other hand, there is also a need to improve the models used to describe the thermodynamic properties of ordered intermetallic phases in the Ti-Al system. Even though the Bragg-Williams model works reasonably well at low temperatures, it is clearly inadequate when the temperatures approach the disordering temperature. Although Asta et a1.44 have successfully computed a metastable phase diagram of Ti-Al based on an fee lattice using first principle calculations coupled with the cluster variation method (CVM), it is more challenging with that approach to calculate the stable phase diagram for related materials research and engineering applications. Perhaps the cluster site approximation (CSA) offers in the near future a viable alternative to CVM for describing the thermodynamic properties of ordered intermetallic phases such as those in the Ti-Al system. 45 The great advantage of CSA over CVM is for applications to phase diagram calculations of multicomponent systems.
CONCLUSIONS A thermodynamic description for the Ti-Al system was obtained in the present study. The Redlich-
480
F.Zhang et al.
Kister equation was used to describe the excess Gibbs energy of the disordered solution phases while a generalized bond-energy model recently developed by Chang’s group at the University of Wisconsin at Madison18 was used for the ordered intermetallic phases. Optimization of the model parameters was carried out using PANDA, a program being developed by Chang’s group.” The calculated Ti-Al phase diagram is in good agreement with a large set of selected experimental phase equilibrium data, while agreement was also obtained for some of the thermodynamic data.33,34,36 Discrepancies between the model-calculated thermodynamic properties and other experimental data can be rationalized. The calculated defect concentrations of y-TiAl .are in accord with the predicted values of a semiempirical relationship,17 which has been successfully applied to ordered phases with the B2 structure, and the experimental data. 43 The generalized bond-energy model parameters for the three ordered intermetallic phases were converted to those of the compound energy model. While the phase equilibrium data from Ti to about 60at% Al are believed to be reliable, this is not true for data at higher Al concentrations. Additional experimental data are needed. Moreover, an improved model, such as cluster site approximation rather than that based on the basic Bragg-Williams assumptions, is needed to describe the thermodynamic properties of the ordered intermetallic phases.
ACKNOWLEDGEMENTS
The first three authors from the University of Wisconsin at Madison wish to acknowledge the financial support received from NASA’s Office of Space Access and Technology under Grant NAGW-1192 and Dr Tony Overfelt of Auburn for his interest in this work. The first three authors also wish to acknowledge Dr N. S. Jacobson and Dr G. S. Collins for providing their experimental data of this system. The authors also thank Professor W. A. Oates for his valuable comments on the manuscript. REFERENCES
4. Schuster, J. C. and Ipser, H., Z. Metallkd., 1990, 81, 389. 5. Mishurda, J. C. and Perepezko, J. H. MicrostructureProperty Relationships in Titanium Aluminides and Alloys,
6. 7. 8.
9.
10. 11.
12. 13. 14. 15.
16. 17. 18. 19.
20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
30. 31. 32. 33. 34.
1. Huang, S. C. and Siemers, P. A., Metall. Trans. A, 1989, 2OA, 1899. 2. Kaltenbach, K., Gama, S., Pinatti, D. G. and Schulze, K., Z. Metallkd., 1989, 80, 511. 3. McCullough, C., Valencia, J. J., Levi, C. G. and Mehrabian, R., Acta Metall., 1989, 37, 1321.
35. 36.
ed. Y. W. Kim and R. R. Boyer. TMS, Warrendale, PA, 1991, p. 3. Kainuma, R., Pa‘lhr, M. and Inden, G., Intermetalhcs, 1994,2(4), 321. Ding, J. J., Qin, G. W., Hao, S. M., Wang, X. T. and Chen, G. L., Journal of Phase Equilibria, 1996, 17, 117. Hsieh, K.-C., Jewett, T., Chang, Y. A. and Perepezko, J. H., First Annual URI Report, supported by DARPA through ONR Contract No. 0014-86-K-0753, 1987. Murray, J. L., Phase Diagrams of Binary Titanium Alloys, American Society for Metals, Metals Park, OH, 1987, p. 12. Gros, J. P., Sundman, B. and Ansana, I., Scri. Metail., 1988,22, 1587. Lin, J.-C., Jewett, T., Mishurda, J. C., Chang, Y. A. and Perepezko, J. H., Second Annual URF Report, supported by DARPA through ONR Contract No. 0014-86-K-0753, 1988. Murray, J. L., Metall. Trans., 1988, 19A, 243. Kattner, U. R., Lin, J.-C. and Chang, Y. A., Metall. Trans., 1992,23A, 2081. Zhang, F., Gamma Titanium Aluminides, ed. Y.-W. Kim, R. Wagner and M. Yamaguchi. TMS, Las Vegas, 1995, 131. Kaufman, L. and Nesor, H., Titanium Sci. Tech., Vol. 2, ed. R. I. Jaffe and H. M. Burte. Plenum Press, New York, 1973, 773. Kaufman, L. and Nesor, H., CALPHAD, 1978,4, 325. Chang, Y. A. and Neumann, J. P., Progress in Solid State Chemistry, 1982, 14, 221. Chen, S.-L., Kao, C. R. and Chang, Y. A., Intermetallics, 1995, 3(3), 233 Chen, S.-L. and Chang, Y. A., “PANDA” - Binary Phase Diagram Calculation and Optimization Program. University of Wisconsin at Madison, Madison, 1996. Fink, W. L., Van Horn, K. R. and Budge, P. M., Trans. AIME, 1931,93,421. Ogden, H. R., Maykuth, D. J., Finlay, W. L. and Jaffe, R. I., Trans. AIME, 1951, 191, 1150. Bumps, E. S., Kessler, H. D. and Hansen, M., Trans. AIME, 1952,194,609. Blackburn, M. J., Trans. AIME, 1967, 239, 1200. Cisse, J., Kerr, H. W. and Bolling, G. F., Metall. Trans., 1974, 5, 633. Heckler, M., Aluminum, 1974, 50, 405. Shibata, K., Sato, T. and Ohira, G., Journal of Crystal Growth, 1978,44, 435. Collings, E. W., Metall. Trans., 1979, lOA, 463. Abdel-Hamid, A., Allibert, C. H. and Durand, F., Z. Metallkd., 1984, 75, 455. Shull, R. D., McAlister, A. J. and Reno, R. C., Titanium Sci. Tech., Vol. 3, ed. G. Luetjering, U. Zwicker and W. Bunk. Deutsche Gesellschaft Fuer Metallkunde, Oberusel, Germany, 1985, p. 1459. Dinsdale, A. T., CALPHAD, 1991, 15, 317. Andersson, J.-O., Guillermet, A. F., Hillert, M., Jansson, B. and Sundman, B., Acta Metall., 1986, 34, 437. Zhang, F., Huang, W. and Chang, Y. A,, CALPHAD, in press. Esin, Yu. O., Bobrov, N. P., Petrushevskii, M. S. and Gel’d, P. V., Russian Metallurgy, 1974, 5, 86. Kubaschewski, 0. and Dench, W. A., Acta Metall., 1955, 3, 339. Samokhval, V. V., Poleshchuk, P. A. and Vecher, A. A., Russian Journal of Physical Chemistry, 1971,45(g), 1174. Jacobson, N. S., Brady, M. P. and Mehrotra, G., private communication, National Aeronautics and Space Administration, Lewis Research Center, Cleveland, 1996.
Thermodynamic description of Ti-AI system 31. Eckert, M., Bencze, L., Kath, D., Nickel, H. and Hilpert, K., Ber. Bunsenges. Phys. Chem., 1996, 100, 418. 38. Elliott, R. P. and Rostoker, W., Acta Met&., 1954,2, 884.
39. Guttman, L., Journalof IChem.Phys., 1961, 34, 1024. 40. Moscinski, J. and Rycerz, Z., Journal of Phys. F. (Metal Phys.), 1974, 4, 1853.
41. Chen, S. L., PhD thesis, University Madison, Madison, 1994.
of Wisconsin
at
APPENDIX: In Thermo-Calc, the Gibbs energy of a disordered solution phase is described by the Redlich-Kister equation.. The Gibbs energy of an ordered intermetallic phase is described by the compound energy modeL3’ which has the following form: GS = Y~,_Y~,GAI:AI + Y,&_YtiGAI:ri
481
42. Fu, C. L. and Yoo, M. H., Intermetallics, 1993, 1, 59. 43. Collins, G., private communication, Washington State
University, WA, 1995. 44. Asta, M., de Fontaine, D., van Schilfgaarde, M., Sluiter, M. and Methfessel, M., Physics Review, 1992, 46B,
5055. 45. Oates, W. A. and Wenzl, H., Scripta Materialia, 1996, 35, 623.
(Al) phase oLW)
AI,Ti =
-114784.7
+ 42.06532.
T
TiA12 oGTj!;-“G~-2.0G~ = -113190.3+20.10588.T
+ .J&_J$,,Gri:A* + YiiJ&%i:Ti TizAls +
R T[X’
<.A,
+
~‘(A
ln_h
+
Y~$+iY~l
In
YA,
+
--
J-4
Y$
Vii
)
+‘LAI(AI,Ti)(Y&
+lLTi:(A1,Ti)
(Y’,, - Y&l
+ YalY:iY~l[OL~AI,Ti):AI
+2LTi:(*1,Ti)
.&)
+‘L(*I.Ti):AIO.‘~~
-
Y?.i)2l + Y$..,Y+iY+i [oLTi:(*l.Ti)
+2L*l:(*l.Ti)
+‘L(*I,Ti):Ti(YJjl
“Gfj~$ _2.“@c_5.0@=
1nYti)l
[‘*l:(*l,Ti)
+2L(*I,Ti):Al
In
Y+i)
+2L(*l.Ti):Ti(Y,&
- Vii) Ie(A1,Ti):Ti -
Y&J21
+ Y~,Y~iY~lY:ioL(*l,l-i):(*l,Ti)
The model parameters of the Ti-Al system listed in column A of Table 1 are converted to the ThermoCalc format, as listed below:
= -240802.8 +;8.55411
TiSAl Two sublattices, sites 0.75:0*25, constituents (Ti,Al):(Al,Ti) oe;:A, - 0.75 .OG;;* - 0.25 .OG;,c* = -32251.8 + 6.98589 . T oGk,lTi - 0.25 . OG;;* - 0.75 . OG;;*
= -43830.4
+ 19.1415.
OLU2
(AI,Ti):AI =
1L’y2
(AI,Ti):AI =
Liquid
2LU2
OLliquid _ AI,Ti -
‘Lzq
(Al,Ti):AI
-111811.4+34.19962.T
= 9746.9 + 7.69422-T
‘L;,
Ti = -127431.5
‘LBAI,Ti = -82.2
$- 32.39035.T
+ 8.24268.T
OLa A,,Ti = -127552.5
i- 29.77509.T
‘L”A,,Ti = 14831.6 - 3’67926.T
-93637.7
+ 44.5853 . T
-31372.5
+ 32.867.
-8402.74
+ 8.78013.
T T
+ 10.9557 . T
1La2
+ 0.97557.
T
=
-933.637
OLZ(~~,~i)
=
1121.14 - 1.199967.
’ L;:(Al,Ti)
=
933.637 - 0.97557 ’ T
(Al,Ti):Ti
=
-47323.1
- 4.03728 . T
(AI,Ti):Ti
=
-3363.41
+ 3.59991 - T
2L(r2 (A1,Ti):Ti =
-8402.74
+ 8.78013.
OLU2
1L”2
(Y phase
=
T
oL~,:CA,:Tij= -10457.5 Al:(Al,Ti)
p phase
.;
0L”2
(Al,Ti):(Al,Ti)
=
26141.8 - 27.316.
T
T T
482
F. Zhang
TiAl
Two sublattices, sites O.kO.5, constituents (Ti,Al): (A1,Ti)
OGI;I:Ti 4e”Gp
-
05.OG;F
OLY (AI,Ti):Al - oL!h:(Al,Ti) ‘LY(Al,Ti):Al -
. lLLl*(Al
=
,Ti) =
= -44206-g+
9.064 - T
36409.2 - 33.4572 ’ T
‘L~i.(,1 Ti) = “~A*,Tij:Ti . 1
= -3914’
2~i:(A,,,)
=
f
25.876 . T
TiA13
Two sublattices, sites 0.25:0#75,constituents (Ti,Al): (Al,Ti)
=
0.75. OGE
-;4093 + 6.56787 - T
OGz;Ir =
-
-
0.75 -OG;p - 0.25. OGE
185243 + 4.83296 - T
26674.1 - 0.22719 . T
2LTiA13 (AI,Ti):AI
=
Al: (AI,Ti) =
-
4962.56 + 0.144576 a T
-426397 + 14.4989 ’ T 99232.6 - 12.0824 - T
=
248837 - 7.24944 . T
OL;$,
Ti) = .
498405 + 15.3663 - T
’ LTi:il
Q) =
819551 + 14.6848. T
2L~~~~,,i)
5200.35 - 3.41645 ’ T
-234289 + 153.92 - T
OG;!; - 0-25.OG;y
CAI,Ti):AI =
1LTiA13
I LTiA13
=
=
48020.3 - 6.48525 . T
29832 - 19.4369 . T
= OLY OLY Ti:(AI,Ti)
OLY (AI,Ti):(Al,Ti)
=
OLTiA13 Al: (AI.Ti) =
5200.35 - 3.41645. T
= 2’l,~,Ti):Ti
0LTiAl2 (AI,Ti):AI
16015.5 + 2.80527 - T
2Ly(Al,Ti):Al 2L:, :(Al ,Ti) = (AI,Ti):Ti
et al.
2 LTiA13 Ti:(AI,Ti)
=
248837 - 7.24944. T
0LTiA13 (A,,Tj):Ti
=
133222 - 5 ~03949 - T
=
21002-6 - 0.061961 - T
1LTiA13 (Al,Ti):Ti
2 LTiAl3 CAI,Ti):Ti = 0LTiA13 (AL,Ti):(AI,Ti)
-4962-56 + O-144576 - T =
-714609
f
20.8 189 . T