site approximation to fcc phases in the Ni–Al–Cr–Re system

Acta Materialia 55 (2007) 4545–4551 www.elsevier.com/locate/actamat

Application of the cluster/site approximation to fcc phases in the Ni–Al–Cr–Re system J. Zhu a, W. Cao b, Y. Yang b, F. Zhang b, S. Chen b, W.A. Oates c, Y.A. Chang a

a,*

Department of Materials Science and Engineering, University of Wisconsin, Madison, WI 53706, USA b CompuTherm LLC, 437 South Yellowstone Drive, Madison, WI 53719, USA c Institute for Materials Research, University of Salford, Salford M5 4WT, UK Received 5 December 2006; received in revised form 8 April 2007; accepted 11 April 2007 Available online 7 June 2007

Abstract The cluster/site approximation (CSA) has been used to model the face-centered cubic (fcc) phases (disordered c with the A1 structure and ordered c 0 with the L12 structure) in the Ni–Al–Cr–Re quaternary system. The CSA takes into account short-range order (SRO), which is vital for describing order/disorder transitions. The approximation possesses computational advantages over the cluster variation method while offering comparable accuracy in the calculation of multi-component phase diagrams. The CSA-calculated phase diagrams are in good accord with the experimental data. In addition, the fcc metastable phase diagrams calculated by using the CSA show the expected ordering/disordering behavior and phase relationships. Ó 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nickel alloys; Cluster/site approximation; Order–disorder phenomena; Phase diagram; Thermodynamics

1. Introduction The fundamental point of interest in Ni-based superalloys is that they consist of mainly two phases: the solid solution c, which is a disordered face-centered cubic (fcc) phase with the A1 structure, and the intermetallic compound c 0 , which is an ordered fcc phase with the L12 structure. The Ni–Al–Cr ternary system is the basis of Ni-based superalloys and has been studied extensively. It is known that the mechanical properties of Ni-based superalloys can be improved by the addition of Re as an alloying element [1]. Thus a thermodynamic description which can correctly describe the order (c 0 )/disorder (c) transformation in the Ni–Al–Cr–Re quaternary system is therefore needed. Previous published work on this system [2] does not satisfy this requirement due to the use of compound energy formalism, which is based on the point or Bragg– Williams approximation. Since the point approximation *

Corresponding author. E-mail address: [email protected] (Y.A. Chang).

does not account for the existence of short-range order (SRO) in alloys at high temperatures [3–5], it has difficulties in describing the thermodynamics of phases with order/disorder transformation. Accordingly, descriptions based on this approximation are often unreliable when extrapolating the thermodynamic properties to metastable regions or extrapolating the thermodynamic descriptions from lower order systems to multi-component systems. This point has been extensively discussed by Oates et al. [6], Zhang et al. [7], Chang et al. [8,9] and Cao [10]. The cluster variation method (CVM) [11] has the advantage over point approximation in taking SRO into consideration, and is able to account for the thermodynamics of order/disorder transformations in a much more satisfactory manner. CVM-calculated fcc prototype systems are in good agreement with the results from Monte Carlo simulation based on the same energy parameters. The CVM is, however, computationally too intensive for use in multicomponent systems. On the other hand, the Cluster/Site Approximation (CSA) [6,12] also considers the existence of SRO but is computationally much less demanding.

1359-6454/$30.00 Ó 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2007.04.019

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J. Zhu et al. / Acta Materialia 55 (2007) 4545–4551

To date, the CSA has been successfully applied to the fcc phases in the calculation of the prototype ternary Cu–Ag– Au system [13] and the real alloy Ni–Al binary [7] and Ni– Al–Cr ternary systems [14]. It has been demonstrated that the CSA can achieve comparable accuracy with the CVM in the tetrahedron approximation for multi-component systems whilst remaining computationally much more efficient. In this spirit, we have explored the possibility of applying the CSA to the fcc phases in the Ni–Al–Cr–Re quaternary system while retaining the earlier descriptions for the other phases. The new description may be used not only for calculating the stable phase diagram but also for calculating reliable metastable phase diagrams. Moreover, we will have more confidence in using this type of thermodynamic description for calculating thermodynamic driving forces for phase transformation, which are necessary for the prediction of microstructure and, ultimately, the mechanical behavior of the materials. 2. The CSA model The formalism used in the CSA takes both long-range order (LRO) and SRO into consideration. The technique used was first introduced by Fowler [15] for treating atom/molecule equilibria in gases and subsequently applied to cluster/site equilibria in solid solutions by Yang and Li [16–19]. In the CSA, the clusters are energetically noninterfering, so that they are permitted to share only corners and not faces or edges. The result of this assumption is that there are only two terms in the configurational entropy expression, the cluster entropy and the single-site entropy. The entropy per site (per atom) Ssite is given by S site ¼ fS n  ðnf  1ÞS 1 ;

ð1Þ

where f is the number of energetically non-interfering clusters per site. In the nearest neighbor pairwise approximation, as was used by Yang and Li, f = z/2p, z being the nearest-neighbor coordination number and p the number of nearest-neighbor pairs in the cluster of size n sites. The cluster and P site entropies Sn and P S1 are expressed as S n ¼ k i Z i ln Z i and S 1 ¼ k i X i ln X i , respectively, where Zi is the cluster probability, Xi is the site probability and k is the Boltzmann constant. The molar equilibrium Helmholtz energy (in the case of constant volume, it is equivalent to the Gibbs energy) Fm for a C-component system is given by [6]: ! ! C1 X n C X n X X Fm i i i i ¼f y P lP  lnk  ðnf  1Þ fi y P ln y P ; RT P P i¼1 i¼1 ð2Þ where fi is the fraction of sublattice of type i and y iP is the sublattice mole fraction of component P on sublattice i. The liP are Lagrangian multipliers arising from the mass balance constraints and are related to the virtual or species chemical potentials of the atoms on the sublattice i. The

cluster partition function, k, is expressed in terms of the cluster energies ej and the liP : 2 3 ! Cn C 1 X n X X ð3Þ exp 4 liP  ej 5: k¼ j¼1

P

i¼1

j

The coherent (no atomic size mismatch) fcc phase diagram calculated from the original Yang–Li formalism represented by the above equations with f = z/2p is at variance with that obtained from Monte Carlo simulations and CVM calculations. Although the separation of the three critical regions for the L10 and the two L12 phases is correctly found, the disordered A1 phase is incorrectly predicted to be stable down to 0 K at compositions between the ordered L12 and L10 phases. By permitting f to deviate by a small amount from the value given by z/2p, it is possible to overcome the problems of the original Yang–Li formulation. By using f as an adjustable parameter, good agreement with the binary alloy fcc prototype phase diagrams derived from either MC simulations or from using the CVM in the tetrahedron approximation can be achieved [6,10]. Note that there is a substantial difference between the MC results and those from the CSA in the tetrahedron approximation, so that different values of f are required depending on which phase diagram is being considered, i.e. f = 1.42 for the Cu–Au binary system [6] and f = 1.35 for the Ni–Al binary system [7]. The application of the CSA to real alloy systems also requires taking into account both atomic size mismatch and excitational contributions in the expression for the free energy of mixing. The excitational contributions are coarse-grained into the configurational free energy which, in the high temperature approximation, leads to a linear temperature variation of the cluster energies. For the atomic size mismatch, the method suggested by Ferreira et al. [20] is used in which the elastic energy is assumed to be configuration independent (volume changes between ordered and disordered phases are very small) so that the free energy consists of two different parts [14]: F m ¼ F CD þ F CI ;

ð4Þ

where FCD and FCI represent the configuration-dependent and configuration-independent contributions, respectively. The first term FCD is determined by Eq. (2), while FCI can be expressed as a Redlich–Kister polynomial of the mole fractions x and temperature T, with coefficients L: X F CI ¼ xP xQ ðL0 þ L1 ðxP  xQ Þ . . .Þ ð5Þ P ;Q

in which P and Q are components. Eq. (4) is then seen to be a function of the temperaturedependent cluster energies, the geometrical factor f, the temperature T, the Lagrangian multipliers liP and the coefficients L. Both the f and L parameters and the temperature-dependent cluster energies are then used as optimizing parameters in obtaining a satisfactory description of the experimental data. With these modifications

J. Zhu et al. / Acta Materialia 55 (2007) 4545–4551

to the original CSA, as illustrated by Cao in Ref. [14], it is possible to obtain good descriptions for the high temperature free energies of mixing for real alloy systems. In the Ni–Al–Cr–Re system, there are five principal phases: liquid, fcc, B2 (b), hexagonal close-packed (hcp) (d), and body-centered cubic (bcc) (a), where the fcc phase has two different stable states, i.e. the disordered state (A1), represented by c in the stable phase diagram, and the ordered state (L12), represented by c 0 . Since the objective of this study was to explore the possibility of applying CSA to the Ni–Al–Cr–Re alloys, only the fcc phase has been modeled using the CSA in the nearest neighbor tetrahedron cluster with four sublattices. 3. Results and discussions Optimization of the model parameters and the subsequent phase diagram calculations were carried out using the Pandat software [10,21]. Since the fcc phase in the Ni– Al binary system has already been modeled using the CSA by Zhang et al. [7], and the Ni–Al–Cr ternary by Cao et al. [14], their descriptions were adopted in the present study. The model parameters for the fcc phase in the other binaries, Ni–Re, Al–Re and Cr–Re, and in the other ternaries, Ni–Al– Re, Ni–Cr–Re and Al–Cr–Re, were optimized using the same experimental data as used in the assessment of Huang

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and Chang [2]. The model parameters for the fcc phase obtained from the optimization are listed in Table 1. 3.1. The Ni–Re, Al–Re and Cr–Re binary systems The experimental data for the Ni–Re binary were reviewed by Nash and Nash [22] and the phase diagram suggested by them was based on the experimental work of Savitskii et al. [23]. Fig. 1 shows the CSA-calculated Ni–Re phase diagram in which the fcc phase was modeled using the CSA and the other phases were modeled using substitutional solution model. The phase relations in this binary system are relatively simple compared with the Ni–Al system. There are only three stable phases in this system: liquid phase at high temperatures, fcc (c) phase in the Ni-rich side and hcp (d) phase in the Re-rich side. The three phases form a peritectic reaction L + d = c at about 1895 K. The calculation is in good agreement with the experimental data. For the Al–Re system, the fcc phase is only stable on the Al side and has a very limited solubility of Re. For the Cr– Re system, the fcc phase is not stable over the whole composition range. The model parameters for the fcc phase in the binary Al–Re and Cr–Re systems have been obtained mainly by considering the ternary phase equilibrium information.

Table 1 Model parameters for the fcc phase of the Ni–Al–Cr–Re system developed in the present study (pair exchange energies x are given in J mol1) Al–Re binary

xAlRe = 4360, f = 1.35 fcc eAlRe3 ¼ 0:25  Gfcc Al þ 0:75  GRe þ 3  xAlRe fcc eAl2 Re2 ¼ 0:5  Gfcc þ 0:5  G Al Re þ 4  xAlRe fcc eAl3 Re ¼ 0:75  Gfcc Al þ 0:25  GRe þ 3  xAlRe

Cr–Re binary

xCrRe = 1500, f = 1.35 fcc eCrRe3 ¼ 0:25  Gfcc Cr þ 0:75  GRe þ 3  xCrRe fcc eCr2 Re2 ¼ 0:5  Gfcc þ 0:5  G Cr Re þ 4  xCrRe fcc eCr3 Re ¼ 0:75  Gfcc Cr þ 0:25  GRe þ 3  xCrRe

Ni–Re binary

xNiRe ¼ 3302:6, xNi3 Re ¼ 611:9, f ¼ 1:35 fcc eNiRe3 ¼ 0:25  Gfcc Ni þ 0:75  G Re þ 3  xNiRe fcc eNi2 Re2 ¼ 0:5  Gfcc þ 0:5  G Ni Re þ 4  xNiRe fcc eNi3 Re ¼ 0:75  Gfcc þ 0:25  G Ni Re þ 3  xNi3 Re

Al–Cr–Re ternary

f = 1.35 fcc fcc eAl2 CrRe ¼ 0:5  Gfcc Al þ 0:25  G Cr þ 0:25  GRe þ 2  xAlCr þ 2  xAlRe þ xCrRe fcc fcc fcc eAlCr2 Re ¼ 0:25  GAl þ 0:5  GCr þ 0:25  GRe þ 2  xAlCr þ xAlRe þ 2  xCrRe fcc fcc eAlCrRe2 ¼ 0:25  Gfcc Al þ 0:25  GCr þ 0:5  GRe þ xAlCr þ 2  xAlRe þ 2  xCrRe

Al–Ni–Re ternary

f = 1.35 fcc fcc eAl2 NiRe ¼ 0:5  Gfcc Al þ 0:25  GNi þ 0:25  GRe þ 2  xAlNi þ 2  xAlRe þ xNiRe fcc fcc eAlNi2 Re ¼ 0:25  Gfcc þ 0:5  G þ 0:25  G Al Ni Re þ 2  xAlNi þ x AlRe þ 2  xNi3 Re fcc fcc eAlNiRe2 ¼ 0:25  Gfcc þ 0:25  G þ 0:5  G Al Ni Re þ xAlNi þ 2  xAlRe þ 2  xNiRe

Cr–Ni–Re ternary

f = 1.35 fcc fcc eCr2 NiRe ¼ 0:5  Gfcc Cr þ 0:25  GNi þ 0:25  GRe þ 2  xCrNi þ 2  xCrRe þ xNiRe fcc fcc eCrNi2 Re ¼ 0:25  Gfcc þ 0:5  G þ 0:25  G Cr Ni Re þ 2  xCrNi þ xCrRe þ 2  xNi3 Re  1000 fcc fcc eCrNiRe2 ¼ 0:25  GCr þ 0:25  GNi þ 0:5  Gfcc Re þ xCrNi þ 2  xCrRe þ 2  xNiRe

Al–Cr–Ni–Re quaternary

f = 1.35 fcc fcc fcc eAlCrNiRe ¼ 0:25  Gfcc Al þ 0:25  GCr þ 0:25  GNi þ 0:25  GRe þ xAlCr þ xAlNi þ xCrNi þ xAlRe þ xCrRe þ xNiRe

The parameters for the binary Al–Cr, Al–Ni, Cr–Ni and ternary Al–Cr–Ni systems are available in Ref. [14]. From the pair exchange energies x, we can compute all the cluster energies e. The cluster partition function, k, can be obtained from Eq. (3). Eventually, the free energy is determined by Eq. (4). The reader is referred to Ref. [10] for more details about the model.

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J. Zhu et al. / Acta Materialia 55 (2007) 4545–4551 1

3500

Savitskii 1970 melting one-phase two-phase

Liquid

Mol. Fracn. Element

Temperature(K)

3000

2500

δ

2000

1500

1000

0.1

0.01

Miyazaki 1994 Ni, Al, Re

1E-3 0.00

500 0.0

Ni

γ' phase

γ phase

γ 0.2

0.4

0.6

Mol. Fracn. Re

0.8

0.01

1.0

0.02

Ni,

0.03

0.00

Miyazaki 1994 Al, Re

0.01

0.02

0.03

Mol. Fracn. Re

Mol. Fracn. Re

Re

Fig. 1. The CSA-calculated Ni–Re binary phase diagram with experimental data [23].

Fig. 2. The CSA-calculated partition of alloy elements between c and c 0 of the Ni–Al–Re alloy with xNi  xAl = 0.63 at 1313 K compared with experimental data [24]. (a) The composition of c in equilibrium with c 0 . (b) The composition of c 0 in equilibrium with c.

3.2. The Ni–Al–Re ternary system

3.3. The Ni–Cr–Re ternary system The sigma phase in the Cr–Re system is an important constituent in Ni-based superalloys. Huang and Chang [2] discussed the thermodynamic model which they used for this phase in detail and their description was used in the current study. Slyusarenko et al. [25] published an isothermal section at 1425 K for the Ni–Cr–Re system. Fig. 5 shows that their experimental results are represented well by our CSA calculations.

l. F

r ac

n.

Ni

0.7

γ'

0.8

0.4

0.6

T = 1313 K

β+γ'+ δ

0.3

γ'+δ

0.2

γ+γ'+ δ

Mo

There are many intermetallic compounds in the Ni–Al and Al–Re binaries. For the Ni-based superalloys, the important phases are: liquid, B2 (b), hcp (d), fcc-A1 (c) and fcc-L12 (c 0 ). Therefore, the experimental phase-equilibrium studies focused on these phases only. In the current study, Al3Ni, Al3Ni2 and Al3Ni5 in the Ni–Al binary and all the intermetallic compounds in the Al–Re binary were treated as binary phases without ternary solubilities due to the lack of any experimental data suggesting otherwise. The fcc-A1 (c) and fcc-L12 (c 0 ) are described by the CSA model simultaneously and ternary interaction parameters for the fcc phases were extrapolated directly from the constituent binaries without any correction terms. Miyazaki et al. [24] studied the phase equilibria at 1313 K between c and c 0 in the Ni–Al–Re system using electron probe microanalysis. As shown in Fig. 2, the calculated partition of each alloy element agrees well with the experimental data [24]. Fig. 3 shows the calculated isothermal section at 1313 K, which is also in accord with the experimental data. The solubility of Re in the ordered c 0 phase is up to 2 at.%. The liquidus projection is shown in Fig. 4. It can be seen that the d phase is the dominating phase and that the c 0 phase region is very narrow.

0.1

0.9

γ+δ

γ

Miyazaki 1994 0.0

1.0 0.0

Ni

0.1

0.2

0.3

0.4

Mol. Fracn. Re

Fig. 3. The CSA-calculated Ni-rich region of the Ni–Al–Re isothermal section at 1313 K with experimental data [24].

3.4. The Al–Cr–Re ternary system There are no experimental data available for the Al–Cr– Re system. Because of this lack of experimental information, we made two assumptions. Firstly, we neglected the Al solubility in the sigma phase of the Cr–Re system. We regard this assumption as plausible because the Al solubility in the sigma phase of the proxy systems Cr–Fe and Cr– Mn is less than 2% and 3%, respectively [26]. Secondly, we neglected third element solubilities of Re and Cr in the other intermetallic phases of the systems Al–Cr and Al– Re, respectively, since these intermetallic compounds are not important for Ni-based superalloy. The fcc phase has almost no solubility in both the Al–Cr and Al–Re systems and is not stable in the Cr–Re system. The ternary interac-

J. Zhu et al. / Acta Materialia 55 (2007) 4545–4551

0.0

Liquidus Projection

Al Al Re 4 Al11Re4

0.2

0.4

l. F

ra c

n.

Ni

AlRe2

Mo

β 0.6

δ

γ'

0.8

γ 1.0 0.0

0.2

0.4

0.6

0.8

1.0

Re

Mol. Fracn. Re

Ni

Fig. 4. The liquidus projection of the Ni–Al–Re system with c and c 0 described by the CSA model.

tion parameters for the fcc phase were extrapolated from the constituent binaries without introducing any ternary interaction parameters.

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Miyazaki et al. [24] investigated the partition of alloying elements between c and c 0 in the Ni–Al–Cr–Re system at 1313 K. As shown in Fig. 6, the calculated results are in good agreement with these experimental data. Huang [28] measured the partition coefficient of the alloying elements between the c and liquid phase in a Ni–8Cr–6Al–3Re (at.%) alloy. The partition coefficients were obtained as the ratio of the composition of the c dendrite center and that of the bulk alloy. Considering the back diffusion of the alloying elements during solidification and the difficulty in locating the dendrite center, the measured composition is an average composition of the c phase formed in a certain temperature range. The calculated values are compared with experimental ones in Table 2. With increasing undercooling of the liquid phase from 0 to 30 K, the calculation under the Scheil condition showed that the Al and Cr contents of the c phase increase while the Re content decreases. The result from this calculation is in agreement with experimental data. Fig. 7a shows the Ni-corner of an isothermal section for the Ni–Al–Cr–Re system with 1 at.% of Re at 1473 K. Cao et al. [14] calculated an isothermal section for the Ni–Al–Cr ternary system under similar conditions as shown in Fig. 7b. With a very small Re addition, the phase equilibria

3.5. The Ni–Al–Cr–Re quaternary system

0.0

T = 1425K

Cr

0.1

0.01

γ' phase Miyazaki 1994 Ni Al Cr Re 1E-3 0.00

1.0

α

0.02

0.04

0.06

0.08

γ phase Miyazaki 1994 Ni Al Cr Re 0.10 0.00

0.02

0.04

0.06

0.08

0.10

Mol. Fracn. Cr

Mol. Fracn. Cr

Slyusarenko 1998

Fig. 6. The CSA-calculated compositions of c and c 0 in equilibrium for the Ni–Al–Cr–Re alloys with 1.5 at.% Re and xNi  xAl = 0.625 at 1313 K compared with experimental data [24]. (a) The compositions of c in equilibrium with c 0 . (b) The compositions of c 0 in equilibrium with c.

0.8

0.6

0.4

Mo

l. F

ra c

n.

Ni

0.2

1

Mol. Fracn. Element

No quaternary phases have been reported in the literature for the Ni–Al–Cr–Re system. In the current study, therefore, it was assumed that no quaternary phases are formed. The Gibbs energies for the liquid, bcc (a), hcp (d), B2 (b), sigma (r) and fcc phases (c and c 0 ) were obtained by extrapolation from those in the corresponding subsystems following the Muggianu geometric model [27]. No quaternary parameters were introduced. All the other intermetallic compounds were considered to be binary phases with no ternary solubility.

0.6

Table 2 The calculated and experimental partition coefficients [28] (the temperatures in brackets indicate the undercooling temperatures of the liquid phase)

0.4

σ

0.2

0.8

γ δ

1.0 0.0

Ni

0.2

0.4

0.6

Mol. Fracn. Re

0.8

0.0 1.0

Re

Fig. 5. Comparison between the experimental [25] and the CSA-calculated isothermal section at 1425 K of the Ni–Cr–Re system.

Xfcc/Xliq

Ni

Cr

Al

Re

Experimental Calculation (0 K) Calculation (5 K) Calculation (10 K) Calculation (15 K) Calculation (20 K) Calculation (30 K)

0.98 1.00 1.00 1.00 1.00 1.00 1.01

0.95 0.84 0.86 0.87 0.89 0.90 0.94

0.77 0.68 0.71 0.74 0.77 0.80 0.88

1.90 2.15 1.99 1.84 1.69 1.52 1.18

J. Zhu et al. / Acta Materialia 55 (2007) 4545–4551

a

a

1

γ

0.0

0.0

0.1

0.2

0.3

0.4

0.5

T = 1473K

1

+γ +γ

L1

0.0

0.0

0.5

0.1

0.2

0.3

0.5

0.5

0.5

T = 1473K 0.4

0.6

Ni n. α+β+γ

β +γ

L10+γ 0.3

0.7

ra c

0.3

L10+γ'+γ

γ'

l. F

ra c γ'

L10+γ'

Mo

Ni n.

α+β β+γ'

l. F Mo

0.4

β

0.7

0.8

0.2

γ'+γ

0.2

γ'+γ

0.1

0.9

β+γ'+γ

0.9

0.5

L10 0.6

0.8

0.4

Mol. Fracn. Cr

Ni

b

b

0.1

1.0

Mol. Fracn. Cr

Ni

L1 +

0

Ni n. Mo

β+ γ'+ δ

Ni n. r ac l. F 1.0

γ': L12 γ : Ni-rich A1 γ1: Re-rich A1

γ

γ

0.1 α+γ

γ

1.0 1.0

0.0

0.0

Ni

0.1

Re

0.9

0.1

0.2

γ'+γ

A l+

α+γ

γ

0.8

n.

0.9

0.3

γ'

Re

0.2

β +γ

L10+γ

ra c

α+β+γ

β+γ'+γ

γ'+γ

Al +

3

γ+γ1

0.7

0.4

l. F

n.

γ'

0.8

4

0.3

2

ra c

5

1 β+γ+σ

l. F

0.7

α+β

L10+γ'+γ1 γ'+γ1

Mo

β +σ

0.6

Mo

Mo

0.4

β +δ

rac

0.6

0.5

0.5

T=1473K

l. F

T=1473K

1: β+γ'+γ+σ 2: β+γ+α+σ 3: β+γ' 4: β+γ'+σ 5: γ'+δ

0.5

0.5

0

4550

0.2

0.3

0.4

0.5

0.0

0.0

Ni

0.1

0.2

0.3

0.4

0.5

Mol. Fracn. Cr

Mol. Fracn. Cr

Fig. 7. (a) The CSA-calculated Ni–Al–Cr–Re isothermal section with 1 at.% Re at 1473 K. (b) The CSA-calculated Ni–Al–Cr isothermal section at 1473 K using the description of Cao et al. [14].

among the b, c and c 0 phases change markedly. Two more Re-rich phases d and r are stable and the phase equilibria are much more complicated than that found in the Ni–Al– Cr ternary system. The CSA-calculated metastable fcc isothermal section with 1 at.% of Re at 1473 K is also shown in Fig. 8a. Comparing this with the Ni–Al–Cr ternary system shown in Fig. 8b, one more Re-rich disordered A1 phase is involved, which will form a miscibility gap together with the Ni-rich A1 phase in the Ni–Re binary when only fcc phases are considered. Zhang et al. [7] and Cao et al. [14] have demonstrated that the CSA-calculated metastable phase diagrams show correct topologies due to the consideration of SRO in the calculation of entropy of mixing. This is not the case for those models based on the point approximation. Even though the topological features can be reproduced by adjusting the values of the reciprocal L parameters in the point approximation types of models, the entropy of mix-

Fig. 8. (a) The CSA-calculated metastable fcc Ni–Al–Cr–Re isothermal section with 1 at.% Re at 1473 K. (b) The CSA-calculated metastable fcc Ni–Al–Cr isothermal section at 1473 K using the description of Cao et al. [14].

ing does not include the existence of SRO. A detailed discussion of this point can be found in Refs. [9,10]. 4. Conclusions The CSA considers SRO in calculating the configurational free energy, which is a great advantage over models based on the point approximation. On the other hand, the CSA is computationally more efficient when compared with the CVM, particularly for multi-component alloy systems. The present study has focused on using the CSA to describe the fcc phases in the real Ni–Al–Cr–Re quaternary system. The calculated stable phase diagrams from the current study show good agreement with the experimental data available in the literature and the total number of model parameters used is less than that when the model is based on the point approximation [2]. Moreover, the current thermodynamic description is believed to be reliable

J. Zhu et al. / Acta Materialia 55 (2007) 4545–4551

for calculations outside the composition and temperature regions where experimental measurements are not available. Lastly, the calculated thermodynamic properties of phases in the metastable states, such as undercooling, needed for microstructural evolution calculations are more realistic than those obtained using models based on point approximations. The resulting microstructure of a material ultimately governs its performance. Acknowledgment Financial support from the AFOSR Grant No. F4962003-1-0083 is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7]

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