Thermodynamic analysis of the Co–Al–C and Ni–Al–C systems by incorporating ab initio energetic calculations into the CALPHAD approach

Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 177–190 www.elsevier.com/locate/calphad

Thermodynamic analysis of the Co–Al–C and Ni–Al–C systems by incorporating ab initio energetic calculations into the CALPHAD approach Hiroshi Ohtania,c,∗, Maki Yamanob, Mitsuhiro Hasebea,c a Department of Materials Science and Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan b Graduate School, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan c CREST, Japan Science and Technology Agency, Japan

Received 9 June 2004; received in revised form 13 August 2004; accepted 13 August 2004 Available online 16 September 2004

Abstract The Co–Al–C ternary phase diagram has been constructed by combining ab initio energetic calculations with the CALPHAD approach, and the results have been compared with the Ni–Al–C ternary phase diagram obtained using the same procedure. In the thermodynamic analysis, special care was taken evaluating the expression of the free energy for the L12 and E21 structures. To treat these two structures as a continuous solution, the free energy was expressed using the (M, Al)3 (Al, M)1 (C, Va)1 -type sublattice model. Because of the lack of experimental data, the thermodynamic properties of the E21 structure were evaluated using the Full Potential Linearized Augmented Plane Wave method. The calculated results show that the E21 phase in the Co–Al–C system is in equilibrium with fcc Co, the B2-type intermetallic compound, and the graphite phase. This finding is in good agreement with previous experimental results. On the other hand, the E21 phase does not exist in the Ni–Al–C ternary system. The phase equilibrium of the Co–Ni–Al–C quaternary system is discussed, taking into account the two-phase separation between the E21 phase and the L12 phase. © 2004 Elsevier Ltd. All rights reserved. Keywords: Phase diagram; Thermodynamic analysis; Ab initio energetic calculations; Ternary carbide; Phase separation

1. Introduction The microstructures of Ni-base superalloys contain the Ni3 Al–L12 phase, which shows an anomalous flow-stress dependence on temperature. In addition, because they exhibit very high melting temperatures and have good resistance to oxidation, alloys with complex phase structures containing the NiAl–B2, fcc-Ni, and Ni3 Al–L12 phases have been investigated for technological applications [1]. It is difficult to produce Co–Al-based superalloys with a microstructure consisting of phases isotypic with fcc-Ni and ∗ Corresponding author at: Materials Science and Engineering, Kyushu Institute of Technology, Sensui-cho 1-1, Tobata-ku, 8048550 Kitakyushu, Japan. Tel.: +81 93 884 3359; fax: +81 93 884 3359. E-mail address: [email protected] (H. Ohtani).

0364-5916/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2004.08.003

Ni3 Al–L12 , because of the absence of a stable strengthening L12 phase in the Co–Al binary system. However, the addition of carbon to this alloy stabilizes the formation of the perovskite type carbide (E21) with the composition M3 AlC (κ phase). This carbide has a carbon atom at the body centre site in the L12 structure, and from its crystallographic similarity, we anticipate the formation of a fine coherent microstructure in a Co-based solid solution. The Co–Al–C ternary phase diagram was investigated by Hütter et al. [2] and Kimura et al. [3] for the Co corner of the system. Based on these works, the outline of the projected liquidus surface, as well as the isothermal sections at several temperatures, was revealed. However, the thermodynamic aspects of the ternary system are still unknown, and in particular, information on the thermodynamic properties of

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the perovskite type carbide is lacking. Thus, we attempted to calculate the formation energy of Co3 AlC using a firstprinciple band energetic calculation method. The objective of the present study was to clarify the entire phase equilibria of the Co–Al–C ternary system, by introducing estimated values into a CALPHAD-type thermodynamic analysis. The same procedure was then applied to the Ni–Al–C ternary system, and the results were compared to the Co–Al–C system. 2. Computational procedure This section discusses the evaluation of the thermodynamic properties of each phase by first-principle calculations, and expression of the Gibbs free energy. 2.1. Ab initio energetic calculations The formation energy of the κ phase was calculated using the Full Potential Linearized Augmented Plane Wave (FLAPW) method. The FLAPW method, as embodied in the WIEN2k software package [4], is one of the most accurate schemes for electronic calculations, and allows for precise calculations of the total energies in a solid. This package was therefore employed in our energetic calculations. The FLAPW method uses a scheme to solve many-electron problems based on the local spin density approximation (LSDA) approach. In this framework, the unit cell is divided into two regions: nonoverlapping atomic spheres and an interstitial region. Inside the atomic spheres, the wave functions of the valence states are expanded by a linear combination of radial functions and spherical harmonics, while a plane wave expansion is used in the interstitial region. Because the LSDA approach includes an approximation for both the exchange and correlation energies, it has been recently expanded on by adding gradient terms for the electron density to the exchange–correlation energy. This has led to the generalized gradient approximation (GGA) method suggested by Perdew et al. [5], and we used this improved method over the conventional LSDA approach. In our computations, muffin-tin radii of 2.0 au (0.106 nm) for Co, Ni, and Al were assumed, and a muffin-tin radius of 1.4 au (0.0742 nm) was assumed for C. The value of RKmax was fixed at 9.0, which almost corresponds to the 20 Ry (270 eV) cut-off energy used in our work. Spin-polarized electronic structure calculations were carried out for the κ phase. 2.2. Thermodynamic modelling of the solution phases A description of the Gibbs energy for each phase appearing in the Co–Al–C and Ni–Al–C ternary systems will be presented in this section. The symbol “M” in the following equations denotes either Co or Ni hereafter.

2.2.1. Liquid (L) The regular solution approximation was applied to the liquid phase. The molar Gibbs energy, G Lm , was calculated using the following equation. G Lm = x Al oG LAl + x C oG LC + x M oG LM + RT (x Al ln x Al + x C ln x C + x M ln x M ) + x Al x C L LAl,C + x Al x M L LAl,M + x C x M L LC,M + x Al x C x M L LAl,C,M

(1)

where oG Li denotes the molar Gibbs energy of element i in the liquid state. This quantity is called the lattice stability parameter, and is described by the formula oG φ i

− o H iref = A + BT + C T ln T + DT 2 + E T 3 + F T 7 + I T −1 + J T −9

(2)

where o H iref denotes the molar enthalpy of the pure element i in its stable state at T = 298 K. The parameter L Li, j denotes the interaction energy between i and j in the liquid phase, and has a compositional dependency following the Redlich–Kister polynomial as L Li, j = 0 L Li, j + 1 L Li, j (x i − x j ) + 2 L Li, j (x i − x j )2 + · · · + n L Li, j (x i − x j )n

(3)

where n LL i, j

= a + bT + cT ln T + d T 2 + · · · .

(4)

The term L LAl,C,M is the ternary interaction parameter between elements Al, C, and M. The composition dependency of the interaction parameters is expressed as follows L LAl,C,M = x Al 0 L LAl,C,M + x C 1 L LAl,C,M + x M 2 L LAl,C,M. (5) 2.2.2. Fcc solid solution (γ ) The fcc solid solution, which exhibits a range of nonstoichiometric alloys, was modelled using a two-sublattice model [6], which had a general formula described by 

C y (2) Va y (2) (6) M y (1) Al y (1) M

Al

u1

C

Va

u2

where u1 and u2 are the numbers of the sites of the sublattice in parentheses, and u1 = u2 = 1 for the fcc phase. The quantities yi(1) and y (2) are the site fractions of the j elements i and j in their respective sublattice, which are designated by the numerals 1 and 2. In the Ni–Al system, the fcc phase is the disordered state of the L12 phase, and the thermodynamic behaviour of these two states can be described by a single formula of the sublattice model. However, as described in Section 2.2.4, such a treatment was restricted to the description for the κ and L12 phases to avoid the complexity of phase diagram computations. The Gibbs energy of the phase per mole of formula unit is expressed by

H. Ohtani et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 177–190 γ

(1) (2)

(1) (2)

179

(1) (2)

G m = yAl yVa oG γAl:Va + yM yVa o G γM:Va + yAl yC o G γAl:C
(1) (2) o γ (1) (1) (1) (1) + yM yC G M:C + RT yAl ln yAl + yM ln yM  (2) (2) (2) (2) + yC ln yC + yVa ln yVa
  (1) (1) (2) (1) (1) v vLγ y + yAl yM yVa − y M Al Al,M:Va v

+

(1) (1) (2) yAl yM yC

+

(1) (2) (2) yAl yC yVa

+

(1) (2) (2) yM yC yVa

 v

 v

 v




vLγ Al,M:C vLγ Al:C,Va vLγ M:C,Va





 (1) v

(1)

yAl − yM

(2) yC(2) − yVa

(2) yC(2) − yVa

v

v

(7)

where oG i: j denotes the Gibbs energy of a compound i j . The parameter L i, j :k (or L i: j,k ) is the interaction parameter between unlike atoms on the same sublattice. 2.2.3. Bcc phase (β) In the thermodynamic modelling of the bcc structure, the three-sublattice model denoted by (M y (1) Al y (1) )u1 M Al (M y (2) Al y (2) )u2 (C y (3) Va y (3) )u3 was applied to the β phase. In M Va Al C the above formula, the first and second sublattices represent the substitution sites occupied by the metallic elements, while the third sublattice shows the interstitial sites in which the C and the vacancies are located. The values of the numbers of the sublattice sites for the β phase are u1 = u2 = (l) 0.5 and u3 = 3. The quantity yi has the same meaning as in Eq. (6). The Gibbs energy of the phase is expressed as (1) (2) (3) o

G βm = yAl yAl yC

(1) (2) (3) o

+ yAl yM yC

(1) (2) (3)

G βAl:Al:C + yAl yAl yVa oG βAl:Al:Va (1) (2) (3)

G βAl:M:C + yAl yM yVa oG βAl:M:Va β

(1) (2) (3) o β (1) (2) (3) + yM yAl yC G M:Al:C + yM yAl yVa G M:Al:Va (1) (2) (3)

(1) (2) (3)

+ yM yM yC oG βM:M:C + yM yM yVa oG βM:M:Va 
(1) (1) (1) (1) + 0.5RT yAl ln yAl + yM ln yM
 (2) (2) (2) (2) + 0.5RT yAl ln yAl + yM ln yM (3) (3) + 3RT (yC(3) ln yC(3) + yVa ln yVa ) β

β

(1) (1) (2) (3) β

(1) (1) (2) (3) β

(1) (2) (2) (3) β

(1) (2) (2) (3) β

(1) (2) (2) (3) β

(1) (2) (2) (3) β

β

β

β

β

(1) (1) (2) (3) (1) (1) (2) (3) + yAl yM yAl yC L Al,M:Al:C + yAl yM yAl yVa L Al,M:Al:Va

+ yAl yM yM yC L Al,M:M:C + yAl yM yM yVa L Al,M:M:Va + yAl yAl yM yC L Al:Al,M:C + yAl yAl yM yVa L Al:Al,M:Va + yM yAl yM yC L M:Al,M:C + yM yAl yM yVa L M:Al,M:Va (1) (2) (3) (3) (1) (2) (3) (3) + yAl yAl yC yVa L Al:Al:C,Va + yAl yM yC yVa L Al:M:C,Va (1) (2) (3) (3) (1) (2) (3) (3) + yM yAl yC yVa L M:Al:C,Va + yM yM yC yVa L M:M:C,Va

(8) where oG i: j :k represents the Gibbs energy of a hypothetical compound i 0.5 j0.5 k3 , and terms relative to the same stoichiometry are identical. The parameter L i, j :k:l , for

Fig. 1. Crystal structures of: (a) M3 Al–L12 and (b) M3 AlC–E21 .

example, denotes the interaction energies between unlike atoms on the first sublattice, varying with composition as a polynomial expansion as follows 
1L (9) L i, j :k:l = 0 L i, j :k:l + yi(1) − y (1) j i, j :k:l . A similar compositional dependency was introduced to L i:k, j :l and L i:k:l, j . 2.2.4. Perovskite-type carbide (κ) and L12 phase For the perovskite M3 AlC (κ) structure, M and Al form an L12 -type superlattice in which the C atoms occupy interstitial sites, resulting in the so-called E21 superstructure. Fig. 1(a) and (b) show schematic drawings of the M3 Al–L12 and M3 AlC–E21 structures, respectively. If the C atom occupies only the body-centred sites, surrounded by the six M atoms, then this will be preferable from an energetic point of view. Occupation of the other interstitial sites for the fcc phase enhances the tetragonality in the L12 crystal structure, producing larger strain energy. The only difference between these two structures is the existence of a C atom in the octahedral interstitial site. Therefore, the Gibbs free energies for these two ordered structures can be described by the same thermodynamic model. The two-sublattice model denoted by the formula (M y (1) Al y (1) )3 M Al (M y (2) Al y (2) )1 has been applied by the present authors to the M Al L12 structure [7], and the three-sublattice model described by (M y (1) Al y (1) )3 (M y (2) Al y (2) )1 (C y (3) Va y (3) )1 was applied to M M Va Al Al C the κ phase. This model is essentially that used for the β phase in Eq. (8), assuming u1 = 3, u2 = 1, and u3 = 1. The Gibbs free energy for the κ phase was evaluated using Eq. (10). (1) (2) (3) o κ (1) (2) (3) o κ yAl yC G Al:Al:C + yAl yAl yVa G Al:Al:Va G κm = yAl (1) (2) (3) o

G κAl:M:C + yAl yM yVa oG κAl:M:Va

(1) (2) (3) o

G κM:Al:C + yM yAl yVa oG κM:Al;Va

+ yAl yM yC + yM yAl yC

(1) (2) (3) (1) (2) (3)

180

H. Ohtani et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 177–190 (1) (2) (3)

(1) (2) (3)

+ yM yM yC oG κM:M:C + yM yM yVa oG κM:M:Va 
(1) (1) (1) (1) +3RT yAl ln yAl + yM ln yM
 (2) (2) (2) (2) + RT yAl ln yAl + yM ln yM 
(3) (3) (3) (3) + RT yC ln yC + yVa ln yVa (1) (1) (2) (3)

where

   474 1 1 79τ −1 + −1 f (τ ) = 1 − A 140 p 497 p   τ3 τ9 τ 15 × + + for τ < 1 6 135 600

(1) (1) (2) (3)

+ yAl yM yAl yC L κAl,M:Al:C + yAl yM yAl yVa L κAl,M:Al:Va + + + + +

(1) (1) (2) (3) (1) (1) (2) (3) yAl yM yM yC L κAl,M:M:C + yAl yM yM yVa L κAl,M:M:Va (1) (2) (2) (3) (1) (2) (2) (3) yAl yAl yM yC L κAl:Al,M:C + yAl yAl yM yVa L κAl:Al,M:Va (1) (2) (2) (3) (1) (2) (2) (3) yM yAl yM yC L κM:Al,M:C + yM yAl yM yVa L κM:Al,M:Va (1) (2) (3) (3) κ (1) (2) (3) (3) κ yAl yAl yC yVa L Al:Al:C,Va + yAl yM yC yVa L Al:M:C,Va (1) (2) (3) (3) κ (1) (2) (3) (3) κ yM yAl yC yVa L M:Al:C,Va + yM yM yC yVa L M:M:C,Va .

(10) 2.2.5. Ni2 Al3 , Co2 Al5 , and Co4 Al13 phases Ni2 Al3 is an intermetallic compound that extends into non-stoichiometry on the Al-rich side. The two-sublattice model denoted by (Al y (1) )3 (Al y (2) Co y (2) Ni y (2) )2 was applied Al Al Co Ni to this phase. (2)

(2)

(2)

o Ni2 Al3 o Ni2 Al3 o Ni2 Al3 2 Al3 = y G Ni m Al G Al:Al + yCo G Al:Co + yNi G Al:Ni 
(2) (2) (2) (2) (2) (2) + 2RT yAl ln yAl + yCo ln yCo + yNi ln yNi (2) (2) 2 3 yNi L Al:Al,Ni . + yAl Ni Al

(11)

In the same manner, Co2 Al5 and Co4 Al13 phases were described by the two-sublattice formula as (Al y (1) )0.714 Al (Co y (2) Ni y (2) )0.286 and (Al y (1) )0.765(Co y (2) Ni y (2) )0.235 , reCo Ni Al Co Ni spectively. 2.2.6. Stoichiometric compounds The CoAl3 , Co2 Al9 , Ni5 Al3 , and the NiAl3 phases with zero homogeneity ranges were treated as stoichiometric compounds. The Gibbs energy of the Co2 Al5 phase, for example, is expressed by the following equation, adopting the stable structure for the elements as the thermodynamic standard state. o G Co2 Al5 Co:Al

− 0.714 oG γAl − 0.286 oG εCo

= a + bT + cT ln T + d T 2 .

(12)

2.2.7. Contribution of magnetic transition to Gibbs energy The contribution to the Gibbs free energy due to the magnetic ordering was added to the non-magnetic part of the free energy as G φ = G φm + G φmag .

(13) φ

The magnetic Gibbs energy, G mag , is given by the expression G φmag = RT · f (τ ) ln(β + 1)

(14)

and 1 f (τ ) = − A



τ −5 τ −15 τ −25 + + 10 315 1500

where   11 692 1 518 + −1 . A= 1125 15 975 p

(15)

 for τ ≥ 1

(16)

(17)

The variable τ is defined as T /TC , where TC is the Curie temperature, and β is the mean atomic moment expressed in Bohr magnetons, µB . The parameter p depends on the structure, with p = 0.4 for the bcc phase and p = 0.28 for the fcc phase [8,9]. 3. Results and discussion 3.1. Calculation of the thermodynamic properties of the κ phase The thermodynamic parameters required by the (M, Al)3 (M, Al)1 (C, Va)1 model were evaluated using first-principle calculations. The calculated results are listed in Table 1, together with the lattice constants. The calculated values denote the formation enthalpies based on the stable structure of the pure element in the ground state. The entropy terms of the formation energies were estimated from the experimental phase boundaries. 3.2. The electronic structure and phase stability of the κ phase Figs. 2 and 3 show the angular-momentum-resolved density of state for each element (PDOS) and the total density of states (DOS) for the Co3 AlC and Ni3 AlC structures, respectively. The term E f denotes the Fermi energy, and no electrons occupy the electronic states above this energy level. In both structures, the DOS mainly consists of the contribution from the Co d or Ni d electrons, and the two distributions of the DOS look similar. However, when observing these diagrams, one can see a difference between these two structures, in that the Fermi level is located near the small trapezoid region of the DOS for the Ni3 AlC structure, but decreases in a region with a very low DOS in the Co3 AlC structure. This fact indicates that, from an energetic point of view, the Co3 AlC structure is more stable compared to the Ni3 AlC phase. This difference in the DOS at the Fermi level may be considered to be the result of the extra d electron of Ni versus Co [7]. From the point of view of the rigid

H. Ohtani et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 177–190

181

Table 1 The calculated thermodynamic parameters required by the (M, Al)3 (M, Al)1 (C, Va)1 -type three-sublattice model Structure

Strukturbericht symbol

Calculated lattice parameter (nm)

Observed lattice parameter (nm)

Calculated formation enthalpy in the ground state (kJ/mol of compound)

Co3 AlC Al3 CoC Co3 Al CoAl3 Co4 C Al4 C Ni3 AlC Al3 NiC Ni3 Al NiAl3 Ni4 C

E21 E21 L12 L12 – – E21 E21 L12 L12 –

0.3675 0.3909 0.3515 0.3726 0.3621 0.4057 0.3713 0.3905 0.3504 0.3774 0.3645

0.3700 – – – – – – – 0.3572 – –

−179.0 +113.5 −79.6 −90.4 +98.0 +160.5 −141.0 +64.5 −169.5 −91.6 +61.0

Fig. 2. (a) The total density of states, and the angular-momentum-resolved density of states for: (b) Co, (c) Al, and (d) C for the Co3 AlC–E21 (κ) structure.

band approximation [10], this difference in the number of electrons of two neighbouring elements shifts the Fermi level towards the higher energy side in Ni3 AlC versus Co3 AlC. This raises the different relative positions of the Fermi level in the DOS curve, which consequently leads to the different structural stabilities.

3.3. Thermodynamic analysis A brief outline of the thermodynamic analysis for each binary system and the ternary system will be presented in this section. Most of the descriptions of the lattice stability

parameters for each pure element were obtained from the Scientific Group Thermodata Europe (SGTE) data [11], and are shown in Table 2. 3.3.1. The Co–Al binary system The Co–Al binary system is composed of the liquid (L), fcc (γ Co and γ Al), B2 (β), hcp (εCo), Co2 Al5 , CoAl3 , Co4 Al13 , and Co2 Al9 phases. According to the critical assessment by McAlister [12], one eutectic reaction involving the liquid phase and six invariant reactions between the solid phases have been confirmed. The peritectic reactions of Co2 Al5 , CoAl3 , Co4 Al13 , and Co2 Al9 have been determined by thermal

182

H. Ohtani et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 177–190

Fig. 3. (a) Total density of states, and the angular-momentum-resolved density of states for: (b) Ni, (c) Al, and (d) C for the Ni3 AlC–E21 (κ) structure.

analysis and metallographic studies [13], and there were only small discrepancies between the values, which were within experimental error. Several liquidus studies employing melt analysis as well as thermal analysis have been reported, and the data given in Refs. [13–18] have been adopted in our work. The B2 phase, which is formed congruently from the melt at almost the equiatomic composition, has a homogeneity range extending from 22% Al to almost 50% Al at higher temperatures, but this range decreases markedly in the lower temperature region. The phase boundary on the Co-rich side has been studied using metallography and X-ray diffraction [16], EPMA [19], electromotive force [20], lattice constant measurements [21], and EPMA and lattice parameter measurements [22]. In a previous work [7], we have carried out thermodynamic analysis of the Co–Al and Ni–Al systems by applying the four-sublattice model for the bcc phase to clarify the phase separation of the B2 and D03 structures. However, in this work, alternative descriptions for these systems were obtained using the twosublattice model for the bcc phase. The main reason for this choice is to maintain consistency of thermodynamic descriptions concerning this phase with previous works on binary and ternary systems containing Fe, Mn and other elements. The evaluated thermodynamic parameters based on the experimental data are shown in Table 3. The calculated Co–Al binary phase diagram is shown in Fig. 4(a).

3.3.2. The Ni–Al binary system The Ni–Al binary system is composed of the liquid, fcc (γ Ni and γ Al), L12 (κ), B2(β), Ni5 Al3 , Ni2 Al3 , and NiAl3 phases. A large volume of experimental data has been reported for this binary system, and a critical assessment was carried out by Nash et al. [23]. Nash et al. reported that the liquidus lines were determined using thermal analysis and microstructural observations [14,16,24,25], and these experimental data show good agreement with each other. The B2 phase has an ordered structure, with a wide homogeneity range extending from about 31% Al to 58% Al, and it melts congruently at T = 1911 K. The B2/L12 phase boundary has been established using electron probe microanalysis [16,24,26]. A small difference in the experimental results can be seen for the B2/Ni2 Al3 phase boundary data, and the results obtained using X-ray diffraction and microstructural observation [24] are usually referred to. Although the Ni5 Al3 phase boundary has not been established, an approximate position was indicated by the work reported in Refs. [27] and [28]. The fcc-Ni phase dissolves Al up to 21.2% at the Ni-rich eutectic temperature of T = 1658 K [24]. Since the L12 phase is an ordered structure that is based on the fcc structure, then there is some scatter in the experimental data, which originates in the coherency with the fcc-Ni matrix. Nash et al. [23] adopted the experimental data in Refs. [24] and [29] among the several values reported, and we adopted the same strategy in this work.

H. Ohtani et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 177–190

183

Table 2 Lattice stability parameters for Co, Ni, Al, and C Element

Phase

Co

β

γ

κ ε

Lattice stability parameters (J/mol) o β

G

o ε

− G

o ε

Co:Va

− G

−4 G

Co:Co:Va

o ε

G

298 < T < 5000

= 427.59 − 0.615248T

298 < T < 6000

Co γ γ TC(Co) = 1396, βCo = 1.35 o ε o γ

o κ

G

= 2938 − 0.7138T

Co:Co:Va Co β β TC(Co) = 1450, βCo = 1.35

o γ

G

Temperature (K)

o

Co:Va

− H

Co

ε

= 4( G

Co

o ε

− G

)

Co

= 310.241 + 133.36601T − 25.0861T ln T − 0.002654739 × T 2

Co

−1.7348 × 10−7 T 3 + 72 527T −1 = −17 197.666 + 253.28374T − 40.5T ln T + 9.3488 × 1030 T −9 ε ε = 1.35 = 1396, βCo TC(Co)

L

Ni

β γ

o L

G

Co

o ε

− G

o β

G

o γ

− G

Ni:Ni:Va

o γ

G

Co

o

Ni:Va

− H

Ni

γ Ni

γ

TC(Ni) = 633, κ L

Al

β γ

κ Ni2 Al3 ε L

C

C

L

o κ

G

Ni:Ni:Va

o L

G

o

Ni

− H

Al:Al

o ε

G G

Al

o C

γ Al

−4 H

γ Al

o γ

−5 G

Al

− G

o γ

− G

o

G − H

Al

C

C

C

o L

o C

C

C

G − G

AR

1768 < T < 6000 298 < T < 6000 298 < T < 1768

= 16 531.056 − 9.683796T − 9.3488 × 1030 T −9

1768 < T < 6000

= 8715.084 − 3.556T

298 < T < 5000

= −5179.159 + 117.854T − 22.096T ln T − 0.0048407T 2

298 < T < 1728

= −27 840.655 + 279.135T − 43.1T ln T + 1.12754 × 1031 T −9 γ βNi = 0.52

1728 < T < 3000 298 < T < 6000

o γ

= 4( G

Ni:Va

o

γ

− H ) Ni

298 < T < 1728

−3.82318 × 10−21 T 7 = −9549.775 + 268.598T − 43.1T ln T

1728 < T < 3000

= 10 083 − 4.813T

298 < T < 2900

= −7976.15 + 137.071542T − 24.3671976T ln T − 0.001884662T 2

298 < T < 700

o γ Al:Va

o

γ

− H ) Al

[11]

700 < T < 933.5 933.5 < T < 6000 298 < T < 6000

= 27 405 − 8.995T

298 < T < 6000

= 5481 − 1.799T

298 < T < 6000

= 11 005.553 − 11.840873T + 7.9401 × 10−20 T 7

298 < T < 933.5

= 10 481.974 − 11.252014T + 1.234264 × 1028 T −9

933.5 < T < 6000

= −17 368.441 + 170.73T − 24.3T ln T − 4.723 × 10−4 T 2

298 < T < 6000

+2562 600T −1 − 2.643 × 108 T −2 + 1.2 × 1010 T −3 = 117 369 − 24.63T

[11]

298 < T < 3000

= 11 235.527 + 108.457T − 22.096T ln T − 0.0048407T 2

= 4( G

[11]

298 < T < 1768

= 15 085.037 − 8.931932T − 2.19801 × 10−21 T 7

−8.77664 × 10−7 T 3 + 74 092T −1 = −11 276.24 + 223.02695T − 38.5844296T ln T + 0.018531982T 2 −5.764227 × 10−6 T 3 + 74 092T −1 = −11 277.683 + 188.661987T −31.748192T ln T − 1.234264 × 1028 T −9

o γ

AR:Va

o L

Al

o

Al:Al:Va

o Ni2 Al3

G

− G o

Al:Va

o κ

G

Ni

o γ

Al:Al:Va

o γ

G

Ni

o γ

− H

o β

G

−4 H

γ

298 < T < 6000

Reference

Present work

[11]

[11] 298 < T < 6000

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Table 3 Optimized thermodynamic parameters for the binary and ternary systems System

Phase

Co–Al

β

γ

Co4 Al13 CoAl3 Co2 Al9 Co2 Al5 Ni2 Al3 κ

L

Thermodynamic parameters (J/mol) o β o γ o ε G − 0.5 G − 0.5 G = −73 900 + 16.8T Al:Co:Va Al Co o β o γ o ε G − 0.5 G − 0.5 G = −73 900 + 16.8T Co:Al:Va Al Co 0 β L = 42 600 − 31.5T Co:Al,Co:Va 0 β L = 42 600 − 31.5T Al,Co:Co:Va 0 β L = 3600 Al,Co:Al:Va 0 β L = 3600 Al:Al,Co:Va 0 γ L = −105 000 Al,Co:Va

Reference Present work

0 γ T = −2300 C(Al,Co:Va) o γ o ε o Co4 Al13 G − 0.765 G − 0.235 G = −40 200 + 7.1T Al:Co Al Co o γ o ε o CoAl3 G − 0.75 G − 0.25 G = −42 500 + 7.78T + 5.14 × 10−4 T ln T Al:Co Al Co o γ o ε o Co2 Al9 G − 0.818 G − 0.182 G = −32 100 + 5.346T + 9.4 × 10−4 T ln T Al:Co Al Co o γ o ε o Co2 Al5 G − 0.714 G − 0.286 G = −47 000 + 8.914T Al:Co Al Co o γ o ε o Ni2 Al3 G − 3 G − 2 G = −180 000 Al:Co Al Co o κ o γ o ε G − G − 3 G = −79 600 Co:Al:Va Al Co o κ o γ o ε G − 3 G − G = −90 400 Al:Co:Va Al Co 0 L L = −157 700 + 30.4T Al,Co 1 L L = −2600 Al,Co 2 L L = 11 600 Al,Co 3 L L = 3250 + 1.735T Al,Co

Ni–Al

β

γ

o β o γ o γ G − 0.5 G − 0.5 G = −56 500 − 10.7T + 1.4975T ln T Ni:Al:Va Al Ni o β o γ o γ G − 0.5 G − 0.5 G = −56 500 − 10.7T + 1.4975T ln T Al:Ni:Va Al Ni 0 β L = −14 225 − 5.625T Al,Ni:Al:Va 0 β L = −14 225 − 5.625T Al:Al,Ni:Va 0 β L = −22 050 Ni:Al,Ni:Va 1 β L = 1115 Ni:Al,Ni:Va 0 β L = −22 050 Al,Ni:Ni:Va 1 β L = 1115 Al,Ni:Ni:Va 0 γ L = −169 600 + 12.65T Al,Ni:Va 1 γ L = −27 800 + 26.185T Al,Ni:Va 2 γ L = −1270 + 1.188T Al,Ni:Va

Present work

H. Ohtani et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 177–190

185

Table 3 (continued) System

Phase κ

NiAl3 Ni2 Al3

Ni5 Al3 Co2 Al5 Co4 Al13 L Al–C

Thermodynamic parameters (J/mol) o κ o γ o γ G − 3 G − G = 1180 Al:Ni:Va Al Ni o κ o γ o γ G − G − 3 G = −147 800 + 1.489T Ni:Al:Va Al Ni 0 κ L = −343 100 Al,Ni:Al:Va 0 κ L = 64 660 − 51.915T Ni:Al,Ni;Va o γ o γ o NiAl3 G − 3 G − G = −341 500 + 168T Al:Ni Al Ni o γ o γ o Ni2 Al3 G − 3 G − 2 G = −336 700 + 57.67T Al Ni Al:Ni 0 Ni2 Al3 L = −27 650 Al:Al,Ni o Ni5 Al3 o γ o γ G − 0.375 G − 0.625 G = −53 300 + 5.5T Al:Ni Al Ni o γ o γ o Co2 Al5 G − 0.714 G − 0.286 G = −39 000 Al:Ni Al Ni o γ o γ o Co4 Al13 G − 0.765 G − 0.235 G = −35 900 Al:Ni Al Ni 0 L 1 L 2 L L = −154 400 + 0.168T L = −12 550 L = 10 650 − 14.08T Al,Ni Al,Ni Al,Ni

Al4 C3

o β o γ o C G − G − 3 G = 1040 500 Al:Al:C Al C o γ o γ o C G − G − G = 80T Al:C Al C 0 γ L = 80T Al:C,Va o γ o C o Al4 C3 G − 4 H − 3 H = −265 237.816 + 938.200313T − 148.7408T ln T Al:C Al C

κ

o κ o γ o C G − 4 G − G = 160 500 Al:Al:C Al C

L

0 L L = 29 910 − 25.586T Al,C

β γ

Reference

Present work [38]

− 0.01672941T 2 − 6.53485 × 10−11 T 3 + 1863 975.5T −1

Co–C

β γ ε κ L

Ni–C

β γ

κ L

o β o ε o C G − G − 3 G = 1039 500 Co:Co:C Co C o γ o ε o C G − G − G = 50 463.8 − 6.849T Co:C Co C o ε o ε o C G − G − 0.5 G = 22 916.5 − 2.855T Co:C Co C o κ o ε o C G − 4 G − G = 98 000 Co:Co:C Co C 0 L L = −107 940.6 + 24.956T C,Co 1 L L = −9805.5 C,Co o β o γ o C G − G − 3 G = 1092 000 Ni:Ni:C Ni C o γ o γ o C G − G − G = 62 000 − 7.6T Ni:C Ni C 0 γ L = −14 902 + 7.5T Ni:C,Va 0 γ T = 633 C(Ni:C) 0 γ β = 0.52 Ni:C o κ o γ o C G − 4 G − G = 61 000 Ni:Ni:C Ni C 0 L L = −111 479 + 35.2685T C,Ni

Present work

Present work [35]

Present work [35]

Present work [37]

Present work [37]

(continued on next page)

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Table 3 (continued) System

Phase

Thermodynamic parameters, (J/mol)

Reference

Co–Al–C

β

o β o γ o ε o C G − 0.5 G − 0.5 G − 3 G = 1046 500 Al:Co:C Al Co C o β o ε o γ o C G − 0.5 G − 0.5 G − 3 G = 1046 500 Co:Al:C Co Al C o κ o ε o γ o C G − 3 G − G − G = −178 750 + 31.5T Co:Al:C Co Al C o κ o γ o ε o C G − 3 G − G − G = 113 500 Al:Co:C Al Co C 2 L L = −260 000 Al,C,Co

Present work

κ

L

Ni–Al–C

β

κ

L

o β o γ o γ o C G − 0.5 G − 0.5 G − 3 G = 1077 500 Al:Ni:C Al Ni C o β o γ o γ o C G − 0.5 G − 0.5 G − 3 G = 1077 500 Ni:Al:C Ni Al C o κ o γ o γ o C G − 3 G − G − G = −141 000 Ni:Al:C Ni Al C

Present work

o κ o γ o γ o C G − 3 G − G − G = 64 500 Al:Ni:C Al Ni C 2 L L = −200 000 Al,C,Ni

To obtain the thermodynamic properties, the enthalpy of formation of each solid phase [30], and the mixing enthalpy of the liquid phase have been reported [31]. In addition to the above information, the enthalpies of formation of the L12 , Ni2 Al3 , and NiAl3 phases have been published [32]. The enthalpy of formation for the L12 phase was given as −37 660 kJ/mol of atoms [32], which is slightly less stable than the calculated value as listed in Table 1. A larger weight was put on the experimental data in the optimization process. The evaluated thermodynamic parameters based on these experimental data are given in Table 3. The calculated Ni–Al binary phase diagram is shown in Fig. 4(b). 3.3.3. The Co–C binary system The solubility of C in fcc-Co was determined using the diffusion couple method, and the data thermodynamically analysed by taking into account the effect of the magnetic transition [33]. According to a previously assessed diagram [34], the equilibrium phase diagram is a simple eutectic-type with a liquid and two terminal solid solutions: fcc Co (γ ) and graphite. The adopted thermodynamic description [35] is shown in Table 3. 3.3.4. The Ni–C binary system The Ni–C system also consists of a simple eutectic, with two terminal phases: graphite and fcc-Ni (γ ). The assessed phase diagram was reported by Singleton and Nash [36], and the thermodynamic parameters of this system have been analysed [37], and are shown in Table 3. 3.3.5. The Al–C binary system Three phases, liquid, graphite, and Al4 C3 constitute the Al–C binary system. A thermodynamic analysis of this binary system was carried out in the present work, since the results of Gröbner et al. [38] showed a slight deviation from

the experimental data on the solubility of C in the liquid phase [39–43], as shown in Fig. 5. The assessed parameters are listed in Table 3. 3.3.6. The Co–Al–C and Ni–Al–C ternary systems Grieb and Stadelmaier [44] reviewed the experimental data on the Co–Al–C ternary system. The phase diagram at 1173 K was studied by Hütter et al. [2], using metallographic observations on samples annealed for 35 h. More recently, Kimura et al. [3] have examined isothermal sections and isopleths for this ternary system using optical microscopy, differential thermal analysis, X-ray diffraction, and energy dispersive X-ray spectroscopy. Kimura et al. reported that the κ phase in the fcc-Co phase exhibited a fine cuboidal shape, closely resembling the morphology of L12 -type Ni3 Al precipitates in fcc-Ni. The liquidus surface in the Co-rich region was composed of the γ , κ, β, and graphite primary crystalline phases, and these phases form the three types of four-phase invariant reactions. In contrast to the abundant information on the phase equilibria, knowledge on the thermodynamic properties of the Co–Al–C system is lacking. The phase equilibria in the Ni–Al–C ternary system have been determined using metallography, X-ray diffraction, and Vickers hardness measurements [45]. These show that there is no ternary phase in the isothermal sections at 1273 K. However, C is soluble in the Ni3 Al phase at concentrations of up to 7–8 mol%. In regard to the liquidus surface, only the monovariant curves have been reported [45]. Our thermodynamic analysis was performed based on the experimental phase diagrams of Kimura et al. [3] of the Co–Al–C system, and the experimental phase diagrams by Schuster and Nowotny [45] of the Ni–Al–C system. No data concerning the thermodynamic properties of the ternary carbide were available in the literature. Thus, the formation energies of this phase in both systems were determined using the results of the first-principle calculations listed in Table 1.

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187

Fig. 5. The calculated binary phase diagram for the Al–C system.

Fig. 4. The calculated binary phase diagrams for: (a) the Co–Al and (b) the Ni–Al systems.

The optimized thermodynamic parameters are summarized in Table 3. 3.4. Comparison of the calculated phase equilibria with the experimental data The Co–Al–C ternary phase diagrams were calculated using the thermodynamic parameters described in the previous section. These are shown in Fig. 6 for temperatures of 1173, 1373, and 1573 K. The enlarged portion of the isothermal section diagrams at 1373 K was compared with the experimental data [3]. The calculated values agree well with the experimental results, and hence, the present approach based on the incorporation of the CALPHAD method into ab initio calculations has proven to be applicable to phase diagram calculations for higher-order systems.

The κ phase appears only at the stoichiometric composition in our results, while a small homogeneity range was exhibited by this phase in the experimental phase diagrams. This aspect is closely related to the large, positive formation energies in the metastable ordered structures, such as Al3 CoC and Al4 C. As the formation energy of Co3 AlC shows an extremely large negative value when compared to these structures, the κ phase forms only at the stoichiometric position. The homogeneity could be expressed by introducing ternary interaction parameters, however, this treatment was not applied in the present analysis, because the experimental phase boundaries of the κ phase are still uncertain. The calculated vertical section diagrams at constant 10 mol% C and 30 mol% Al are shown in Fig. 7. Compared to the experimental phase fields determined by differential thermal analysis [3], the correspondence with the entire phase diagram is fairly acceptable. The calculated liquidus projection on the Co-rich side is shown in Fig. 8. The liquidus isotherms obtained at the temperatures indicated in the figure are also presented. The invariant reactions are denoted by the filled circles. The following four phases are in equilibrium at these points E 1 : L ⇔ γ + κ + (C) at 1461 K, 8.23 mol% Al, and 9.73 mol% C. E 2 : L ⇔ γ (Co) + β + κ at 1471 K, 16.92 mol% Al, and 4.32 mol% C. E 3 : L ⇔ β + (C) + Al4 C3 at 1782 K, 56.63 mol% Al, and 4.78 mol% C. and U : L + (C) ⇔ β + κ at 1658 K, 25.32 mol% Al, and 9.35 mol% C.

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Fig. 7. Calculated vertical section diagrams at constant (a) 10 mol% C and (b) 30 mol% Al.

phase, as evaluated by the ab initio calculations, resulted in reasonable agreement between the calculated results and the observed data [45].

Fig. 6. Isothermal section diagrams of the Co–Al–C system calculated at: (a) 1173 K, (b) 1373 K, and (c) 1573 K.

In the Ni–Al–C system on the other hand, it has been experimentally verified that the κ phase does not appear in the vicinity of the stoichiometric composition. Fig. 9(a) through (c) show the calculated isothermal section diagrams at 1173, 1273 and 1573 K, respectively. As seen in the Co–Al–C ternary system, the formation energies for the κ

3.5. Phase separation of the κ phase in the Co–Ni–Al–C quaternary system According to our first-principle calculations, there was a large difference in the phase stability between the Co3 Al–L12 and Ni3 Al–L12 phases, compared to the difference between the Ni3 AlC–E21 and Co3 AlC–E21 phases. In such an energetic situation, a two-phase separation should occur, dependent on the difference in the formation energies of the compounds, given by the

H. Ohtani et al. / Computer Coupling of Phase Diagrams and Thermochemistry 28 (2004) 177–190

189

Fig. 8. Calculated liquidus projection on the Co-rich side.

following expression G = oG Ni3 Al + oG Co3 AlC − oG Co3 Al − o G Ni3 AlC.

(18)

This miscibility gap originates in the energy difference between the terminal compounds, and was studied in Ref. [46], which points out that as the absolute value of G increases, then the critical temperature of the miscibility gap increases. Such a phase separation is often observed in some complex carbonitrides or alloy semiconductor systems. Fig. 10 shows the calculation of a miscibility gap in the Co3 AlC–Ni3 AlC–Co3 Al–Ni3 Al pseudo-quaternary system at T = 1273 K. In this model calculation, only the formation enthalpies of the stoichiometric compounds were used, designated by the four vertices of the composition square. One can see that a two-phase separation forms in the direction of the Co3 AlC and Ni3 Al diagonal. Then, a twophase separation between these terminal compounds will be involved in the phase equilibria of the quaternary system. To our knowledge, very little is known about the microstructural evolution of the phase separation of coherent precipitates in these superalloys. Therefore, the development of a new heat resistant alloy may be possible, based on the findings revealed by the present study. The detailed discussion on the respect will be carried out in a separate publication. 4. Conclusions The phase equilibria in the Co–Al–C and Ni–Al–C ternary systems were investigated by incorporating ab initio energetic calculations into the CALPHAD approach, yielding the following results. 1. The Gibbs free energy for the ternary perovskite carbide, M3 AlC, was expressed using the (M, Al)3 (M, Al)1 (C, Va)1 -type three-sublattice model, and treating this phase as a continuous solution with a M3 Al-type L12

Fig. 9. Isothermal section diagrams of the Ni–Al–C system calculated at: (a) 1173 K, (b) 1273 K, and (c) 1573 K.

structure. An attempt was carried out to obtain the formation enthalpies for the stoichiometric compounds using this model employing the Full Potential Linearized Augmented Plane Wave method. No interaction parameter for this perovskite carbide was introduced in the phase diagram calculation. However, the entropy terms of the

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Fig. 10. Calculated miscibility gap in the Co3 AlC–Ni3 AlC–Co3 Al–Ni3 Al pseudo-quaternary system at 1273 K.

formation energies for the terminal compounds were estimated, based on the experimental phase boundaries. 2. According to the electronic structure calculations, the Fermi level is located near the small trapezoid region of the DOS for the Ni3 AlC structure, but decreases in a region with a very low DOS in the Co3 AlC structure. This makes the Co3 AlC structure more stable than the Ni3 AlC structure. This difference in the stability of these two structures corresponds to the different number of d electrons between Co and Ni. 3. The calculated phase diagrams are in good agreement with previous experimental results. Furthermore, the phase separation of the κ phase in the Co–Ni–Al–C quaternary system was predicted by the simple model calculation. This phase separation may play a key role in the development of new types of heat resistant alloys. Acknowledgements The authors gratefully acknowledge discussions with Dr. Y. Kimura and Prof. Y Mishima at Tokyo Institute of Technology. The authors are also grateful to Mr. H. Minooka for helpful work on the Al–C system. This work was partially supported by a Grant-in-Aid for Scientific Research (No. 14550720) from the Ministry of Education, Science, Sports and Culture of Japan. References [1] R. Kainuma, S. Imano, H. Ohtani, K. Ishida, Intermetallics 4 (1996) 37–45. [2] L.J. Hütter, H.H. Stadelmaier, A.C. Fraker, Metall. 14 (1960) 113–115. [3] Y. Kimura, M. Takahashi, S. Miura, T. Suzuki, Y. Mishima, Intermetallics 3 (1995) 413–425.

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