CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133
Contents lists available at ScienceDirect
CALPHAD: Computer Coupling of Phase Diagrams and
Thermochemistry
journal homepage: www.elsevier.com/locate/calphad
Solution-based thermodynamic modeling of the Ni–Al–Mo system
using first-principles calculations
S.H. Zhou a,b,n, Y. Wang a, L.-Q. Chen a, Z.-K. Liu a, R.E. Napolitano b,c
a
Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA
Materials & Engineering Physics Program, Ames Laboratory, USDOE, USA
c
Department of Materials Science and Engineering, Iowa State University, Ames, IA 50011, USA
b
art ic l e i nf o
a b s t r a c t
Article history:
Received 31 October 2013
Received in revised form
9 January 2014
Accepted 6 March 2014
Available online 22 March 2014
A solution-based thermodynamic description of the ternary Ni–Al–Mo system is developed here,
incorporating first-principles calculations and reported modeling of the binary Ni–Al, Ni–Mo and
Al–Mo systems. To search for the configurations with the lowest energies of the N phase, the Alloy Theoretic
Automated Toolkit (ATAT) was employed and combined with VASP. The liquid, bcc and γ-fcc phases are
modeled as random atomic solutions, and the γʹ-Ni3Al phase is modeled by describing the ordering
within the fcc structure using two sublattices, summarized as (Al,Mo,Ni)0.75(Al,Mo,Ni)0.25. Thus, γ-fcc
and γʹ-Ni3Al are modeled with a single Gibbs free energy function with appropriate treatment of the
chemical ordering contribution. In addition, notable improvements are the following: first, the ternary
effects of Mo and Al in the B2-NiAl and D0a-Ni3Mo phases, respectively, are considered; second, the
N-NiAl8Mo3 phase is described as a solid solution using a three-sublattice model; third, the X-Ni14Al75Mo11
phase is treated as a stoichiometric compound. Model parameters are evaluated using first-principles
calculations of zero-Kelvin formation enthalpies and reported experimental data. In comparison with the
enthalpies of formation for the compounds ψ-AlMo, θ-Al8Mo3 and B2-NiAl, the first-principles results
indicate that the N-NiAl8Mo3 phase, which is stable at high temperatures, decomposes into other phases
at low temperature. Resulting phase equilibria are summarized in the form of isothermal sections and
liquidus projections. To clearly identify the relationship between the γ-fcc and γʹ-Ni3Al phases in the
ternary Ni–Al–Mo system, the specific γ-fcc and γʹ-Ni3Al phase fields are plotted in x(Al)–x(Mo)–T space
for a temperature range 1200–1800 K.
& 2014 Elsevier Ltd. All rights reserved.
Keywords:
CALPHAD
First-principles calculation
Ni–Al–Mo phase diagram
ATAT
1. Introduction
The high temperature performance of superalloys based on the
Ni–Al binary system is a critical factor in current and future power
generation and transportation technologies. Moreover, the refractory metal Mo is well established as a key contributor to the high
temperature stability of many multicomponent Ni-base superalloys [1–6]. Despite its widespread use, quantitative guidelines
for future alloy development and processing design call for more
complete descriptions of the effects of Mo additions on the phase
stability and phase transformation in the Ni–Al–Mo system. This is
the focus of the present work.
Thermodynamic modeling for the ternary Ni–Al–Mo system
was first performed by Kaufman and Nesor [7], in which all
compounds were described as stoichiometric compounds. Subsequent to treatments of the Ni–Al [8], Al–Mo [9], and Ni–Mo [10]
systems, Lu et al. [11] reassessed the ternary Ni–Al–Mo system
n
Corresponding author.
http://dx.doi.org/10.1016/j.calphad.2014.03.002
0364-5916/& 2014 Elsevier Ltd. All rights reserved.
which represented a great improvement over the modeling by
Kaufman and Nesor [7], and, despite several limitations, has been
used with reasonable success for quite some time. As increased
demands for high temperature materials drive the exploration of
new multicomponent alloy regimes, however, these limitations
become problematic. First, the binary Ni–Mo phase diagram
resulting from the model by Lu et al. [11] is different from those
by Frisk [10] in Fig. 1(a) although Lu et al. [11] use the database
developed by Frisk [10]. Lu’s model (dashed curves) indicates that
(i) the B2 phase becomes stable around 1900 K indicating the B2
phase can be a primary phase in a ternary Ni–Al–Mo system and
(ii) the γʹ phase becomes stable at low temperature, which does
not agree with Frisk’s model [10] and the reported experiments
[12–14]. In addition, Lu’s model was found to be inconsistent with
experimental phase equilibrium data [15–18], limiting its utility in
phase-field modeling [19]. These problems arise from inappropriate descriptions of the thermodynamic effects of Mo in the ternary
system [11], and several important improvements are called here
for. First, recent advances in modeling the Ni–Al [20] and Ni–Mo
[21] binary systems should be incorporated into the ternary
S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133
125
Fig. 1. The binary phase diagrams of the (a) Ni–Mo system calculated using the parameters by Frist [10] in solid-line and Lu et al. [11] in dash-line; (b) Ni–Mo system
calculated using the parameters by Zhou et al. [21] in solid-line and Lu et al. [11] in dash-line; (c) Al–Mo system [9] and (d) Al–Ni system [8,20].
Table 1
A listing of phases modeled in the current treatment.
Formula unit
Sym.
Phases extending from the
Ni, Ta, Mo
–
Ni
γ
Al
γ
Mo
–
Phases extending from the
Al3Ni
–
Al3Ni2
–
AlNi
–
Al3Ni5
ε
AlNi3
γ’
Phases extending from the
NiMo
δ
Ni2Mo
ρ
Ni3Mo
–
Ni4Mo
–
Ni8Mo
ζ
Phases extending from the
Al12Mo
ϕ
Al5Mo
η
Al4Mo
φ
Al8Mo3
s
Al63Mo37
τ
AlMo
ψ
AlMo3
–
Ternary compounds
NiAl8Mo3
N
Ni14Al75Mo11
X
Lattice
pure component states:
–
Cub
Cub
Cub
Ni–Al binary system:
Ort
Tri
Cub
Ort
Cub
Ni–Mo binary system:
Orth
Tet
Tet
Tet
Tet
Al–Mo binary system:
Cub
Hex
Mon
Mon
–
Cub
Cub
Tet
–
Struk. des.
Prototype
–
A1
A1
A2
Liquid
fcc
fcc
bcc
D011
D513
B2
–
L12
Fe3C
Al3Ni2
ClCs
Ga5Pt3
AuCu3
–
–
D0a
D1a
–
NiMo
Pt2Mo
Cu3Ti
Ni4Mo
Ni8Nb
–
–
–
–
–
A2
A15
Al12W
Al5W
Al4W
Al8W3
–
W
Cr3Si
D022
–
Al3Ti
–
description. Second, the ternary N and X phases listed in Table 1,
which were not described in previous reports [7,11], should be
considered.
In the present work, the thermodynamic descriptions of the binary
Ni–Al [8,20], Al–Mo [9], and Ni–Mo [21] systems with the associated
phase diagrams shown in Fig. 1(b–d) are employed within a general
solution-based approach, using first-principles calculations and experimental data [11,15–18,22–39], which are appropriate to determine the
ternary model parameters. The ternary N and X phases are included
and the full Ni–Al–Mo phase diagram is computed. The equilibrium
phase diagram is presented with a series of isothermal sections and a
3D plot for the γ and γʹ phase fields. The resulting phase diagrams of
the ternary Ni–Al–Mo system are compared with the experimental
data [11,15–18,22–39] and the prior report by Lu [11].
2. First-principles calculations
To determinate the Gibbs free energy of formation for the intermediate phases, the zero-Kelvin energies of the N phases and the
unstable end-members (described in Section 3) for the B2 and D0aNi3Mo phases as well as fcc-Al, fcc-Ni, bcc-Mo were calculated by
means of VASP [40] employing Vanderbilt ultrasoft pseudopotential
[41] and the generalized gradient approximation (GGA) [42] with the
high precision choice. Monkhost 15 15 15 k points were used for
the pure elements Ni, Al and Mo, 11 11 11 k points for the
compounds B2 and D0a-Ni3Mo, and 6 6 6 k points for the compound N. For the stable structures, we fully relaxed the volume, the
126
S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133
cell shape, and the internal atomic coordinates while for the unstable
structures of the end-members we only relaxed the cell volume to
maintain the unit cell structures to their stable phases.
The enthalpy of formation, ΔH Φ
f , for a given compound
Φ is
calculated as the difference between the energy EΦ
TOT of the
compound and the linear combination of the pure element
reference state energies, EfAlcc , EfNicc and Ebcc
Mo
Φ
Φ f cc
Φ f cc
Φ bcc
ΔH Φ
f ¼ ETOT xNi ENi xAl EAl xMo E Mo
ð1Þ
where xΦ
i is the mole fraction of component i in Φ. The calculated
enthalpies of formation of the compounds are listed in Table 2a.
The compositions of the N phase were reported as
Ni12.5Al62.5Mo25 by Raman and Schubert [23,24] and as
Ni3Al72Mo25 by Markiv et al. [22] as the dashed-lines in Fig. 2,
respectively. To search for the configurations with the lowest
energies, ATAT developed by van de Walle [43] was employed
and combined with VASP. The starting geometry was limited to
the D022 structure with the fixed composition (25 at% Mo) and Mo
atom positions. Gamma centered k points were used and more
than 65 configurations were studied. The calculated enthalpies of
formation for some of the structures found by ATAT are plotted in
Fig. 2 indicating that the structure at the lowest point along the
concave curve is at composition NiAl8Mo3 and gives a formation
energy of 38.17 kJ/mol, which is lower than the those values at
both Ni12.5Al62.5Mo25 and Ni3Al72Mo25 by Raman and Schubert
[23,24] and Markiv et al. [22], respectively. This ternary compound
is a tripled superstructure of that of DO22 along b direction,
resulting in a body centered orthorhombic structure with the
lattice parameters being a ¼3.6904 Å, b¼11.1050 Å, and
Table 2a
A summary of results from the first-principles calculations.
Phase
Ni
Al
Mo
Symbol
Formula
ΔH, kJ/mol of atoms
Ni
Al
Mo
NiMo
AlMo
Ni3Al
Ni3Mo
Al2NiMo
Al8NiMo3
Al3Mo
0
0
0
27.84
5.88
38.12
6.15
28.01
38.17
27.19
A2
NiAl
B2
CsCl
Ni3Mo
D0a
Cu3Ti
NiAl8Mo3
N
Al3Ti
Position type
Atomic position
Al1
Al1
Al2
Mo
Ni1
Ni1
Ni2
Ni2
Ni3
Ni3
Ni4
Ni5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0.9938
0.4938
0.0194
0.7525
0.4821
0.9821
0.2486
0.7486
0.7503
0.2503
0.5210
0.2575
0.3284
0.1716
0
0
0.3399
0.1601
0.1704
0.3296
0.1518
0.3482
0
0
c¼8.5710 Å. The primitive unit cell of this ternary compound
contains 12 atoms which are arranged in the symmetry. The
atomic positions of this ternary compound are listed in Table 2b.
In this work, we designated this ternary compound as N-NiAl8Mo3
and describe it as a solution phase with small homogeneous
composition by considering the experimental observations by
Raman and Schubert [23,24] and Markiv et al. [22].
As listed in Table 2a, the first-principles calculated enthalpy of
formation for the N-NiAl8Mo3 phase is 38.17 kJ/mol, which is
less negative than the enthalpy of mixing, i.e. 40.19 kJ/mol, for
the ψ-AlMo, θ- and B2-NiAl phases from the binary systems,
indicating that the N phase decomposes into other phases at low
temperature. Due to the lack of structural information, the firstprinciples calculation was not performed for the compound X.
3. Thermodynamic models
Prototype
fcc
fcc
bcc
γ
Table 2b
The crystal structure of the lowest energy structure at composition NiAl8Mo3.
The thermodynamic properties of pure Al, Mo, and Ni in
various structures were computed using the parameters from
the SGTE database [44] as listed in Table 3. In addition to the
compounds in the Ni–Al [8,20], Al–Mo [9] and Ni–Mo [21]
systems, the liquid, bcc, γ-fcc, B2-NiAl, γʹ-Ni3Al, D0a-Ni3Mo, and
N were modeled as ternary solution phases and X phase as a
stoichiometric compound. The associated Gibbs free energy functions are defined in Table 4, where the total Gibbs free energy for
the phase, Φ, is generally given by the sum of three contributions:
ref Φ
xs Φ
Gm þ id GΦ
GΦ
m ¼
m þ Gm ;
ð2Þ
where the subscript m denotes that all terms are molar quantities.
The first term in Eq. (2) is the sum of occupancy-weighted
sublattice end-member contributions. The second and third terms
are the ideal and excess parts of the Gibbs free energy of mixing,
respectively. The excess Gibbs free energy in all Gibbs free energy
equations is expressed in terms of the Redlich–Kister polynomial
[45]. The specific treatment of each phase is discussed briefly here.
The liquid, bcc and γ-fcc are described using the model given in
Table 4, where 1GΦ
i is the molar Gibbs free energy of the pure
element i with the structure Φ (Φ ¼ liquid, bcc or γ-fcc) as listed in
Table 3 and xi is the mole fraction of the indicated component
Φ
i. j LΦ
i;k is the binary interaction parameter, and LAl;Mo;Ni is a composition dependent ternary interaction parameter evaluated with the
experimental data. Magnetic property of the Ni is considered for
its fcc structure. mg GΦ
m is the magnetic contribution [46] to Gibbs
free energy in solution phases, and expressed as the following:
mg
Fig. 2. Enthalpy of formation for the N phase calculated using ATAT and VASP with
the fixed Mo atomic positions xMo ¼ 0.25 and current thermodynamic parameters in
Tables 6 and 7. Reference states: fcc-Ni, fcc-Al and bcc-Mo.
Φ
Φ
GΦ
m ¼ RT lnðβ þ1Þf ðτ Þ;
Φ
ð3Þ
where β is the quantity related to the total magnetic entropy and
is set equal to the Bohr magnetic moment per mole of atom. τΦ is
S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133
127
Table 3
Thermodynamic parameters for pure elements [44].
Al phases
0
Gliq
Al
0
GfAlcc
0
Gbcc
Al
Tmin, K
Tmax, K
298
933
933
6000
298
700
700
933
933
6000
298
3200
0
0 f cc
GAl
11,005.029
11.841867
–
–
–
–
7.934 10 20
–
0 f cc
GAl
10,482.282
11.253974
–
–
–
–
–
1.231 1028
–
–
–
7976.15
137.093038
24.3671976
1.884662 10 3
74,092
8.77664 10 7
–
–
11,276.24
223.048446
38.5844296
0.018531982
74,092
5.764227 10 6
–
–
11,278.378
188.684153
31.748192
0 f cc
GAl
10,083
4.813
–
–
–
–
–
–
GREF
A
B
C
D
E
F
G
H
1.234 1028
Mo phases
0
Gliq
Mo
0
cc
GfMo
Tmin
Tmax
298
2896
2896
6000
298
6000
0
0
0
0 bcc
GMo
15,200
0.63
–
–
–
–
–
–
–
Gbcc
Mo
41,831.347
14.694912
–
–
–
–
4.24519 10 22
–
–
GREF
A
B
C
D
E
F
G
H
I
Gbcc
Mo
34,095.373
11.890046
–
–
–
–
4.849315 1033
–
–
0
Gbcc
Mo
298
2896
0
7746.302
131.9197
23.56414
3.443396 10 3
65,812
5.66283 10 7
–
–
1.30927 10 10
2896
3000
0
30,556.41
283.559746
42.63829
–
–
–
–
4.849315 1033
–
Ni phases
0
Gliq
Ni
0
Gbcc
Ni
0
GfNicc
Tmin
Tmax
298
1728
1728
6000
298
6000
298
1728
1728
3000
0
GREF
A
B
C
D
E
F
G
H
Tc
β
0 f cc
GNi
16,414.686
9.397
–
–
–
–
3.82318 10 21
–
–
–
0 f cc
GNi
18,290.88
10.537
–
–
–
–
–
1.12754 1031
–
–
0 f cc
GNi
8715.084
3.556
–
–
–
–
–
–
575
0.85
0
0
5179.159
117.854
22.096
4.8407 10 3
–
–
–
–
633
0.52
27,840.655
279.135
43.1
–
–
–
–
1.12754 1031
–
–
Note: O Gθi ¼ O GRef þ a þ bT þ cT ln T þ dT 2 þ eT 1 þ f T 3 þ gT 7 þ hT 9 þ iT 4 (J/mol) and Tc (K) is the Curie temperature and β is the average magnetic monent per atom (Bohr
magnetons).
Φ
defined as T=T Φ
c and T c is the Curie temperature. For the solution
Φ
phase, T c and β
Φ
are described as the following:
n
0 Φ
j Φ
0 Φ
j
TΦ
c ¼ xA T cA þ xB T cB þ xA xB ∑ LTðA;BÞ ðxA xB Þ
j¼0
n
j Φ
j
0 Φ
βΦ ¼ xA 0 βΦ
A þ xB β B þ xA xB ∑ Lβ ðA;BÞ ðxA xB Þ
j¼0
Φ
0 Φ
T ci and 0 i are from the pure
j Φ
j Φ
3. LTðA;BÞ and LβðA;BÞ are the magnetic
where
Table
β
ð4Þ
ð5Þ
et al. [20] in the thermodynamic evaluation of the Ni–Al, and Ni–Al–Cr
systems with a special emphasis on the relationship among the
parameters for the ordered γʹ phase. The γʹ phase in the binary
Ni–Al system was modeled with the two-sublattice model (Al,Ni)0.75
(Al,Ni)0.25 [8]. Considering the thermodynamic effect of the component
Mo, the two-sublattice model for γʹ was modified as (Al,Mo,Ni)0.75
(Al,Mo,Ni)0.25. Adopting the descriptions by Ansara et al. [8,47] and
'
Dupin et al. [20], the Gibbs free energy, Gγm , for γʹ is expressed as the
following:
element i as listed in
ðaÞ
interaction parameters
Gγm' ¼ Gγm ðxi Þ þ ΔGord
m ði : jÞ;
ðbÞ
ord
ord
ΔGord
m ði : jÞ ¼ Gm ði : jÞyIi ;yIIi Gm ði : jÞxi ¼ yIi ¼ yIIi ;
between the elements A and B. In this work, the magnetic
parameters from the binary Al–Ni system [8] are employed as
listed in Table 7.
The thermodynamic modeling of the ordered γʹ-Ni3Al and disordered γ-fcc phases was discussed by Ansara et al. [8,47] and Dupin
ð6Þ
where Gγm ðxi Þ is the Gibbs free energy of the disordered γ-fcc phase as
described in Table 4. ΔGord
m ði; jÞ in Eq. (6) corresponds to the ordering
energy, being described with Eq. (6b) [20,48]. Gord
m ði : jÞ in Eq. (6b) is
128
S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133
Table 4
ref Φ
xs Φ 0 ref
0 f cc
Gm þ id GΦ
¼ 0 GfAlcc , 0 Gbcc
Summary of the thermodynamic models used for the Al–Mo-Ni ternary system with the total Gibbs free energy GΦ
m ¼
m þ Gm ( Gi
Mo or GNi ).
Phase
Formulation
Model
Liquid
γ-fcc
Bcc
(Al,Mo,Ni)1
ref
GΦ
m ¼
id
xs
GΦ
m
xi 1GΦ
i
∑
i ¼ Al;Mo;Ni
¼ RT
∑
xi ln xi
i ¼ Al;Mo;Ni
n
n
j¼0
j¼0
j Φ
j Φ
j
j
GΦ
m ¼ xAl xMo ∑ LAl;Mo ðxAl xMo Þ þ xAl xNi ∑ LAl;Ni ðxAl xNi Þ
n
Φ
j
þ xMo xNi ∑ j LΦ
Mo;Ni ðxMo xNi Þ þ xAl xMo xNi LAl;Mo;Ni ;
j¼0
2 Φ
0 Φ
1 Φ
LΦ
Al;Mo;Ni ¼ xAl LAl;Mo;Ni þ xMo LAl;Mo;Ni þ xNi LAl;Mo;Ni
ψ-AlMo
B2-NiAl
D0a-Ni3Mo
(Al,Mo)1(Al,Mo)1
(Al,Mo,Ni)1(Al,Mo,Ni,Va)1
(Al,Mo,Ni)0.75(Al,Mo,Ni)0.25
ref
D0a
D0a
a
GD0
Al:Al ¼ GAl:Mo ¼ GAl:Ni ¼ 0
GΦ
m ¼
∑
yIi
∑
yIIj 1GΦ
i:j
i ¼ Al;Mo;Ni j ¼ Al;Mo;Ni
ref
ref
Φ
1GΦ
¼
p1G
þ
q1G
þ
ΔG
¼
p1Gref
i:j
i:j
i
j
i
id
GΦ
m
¼
xs
GΦ
m
I
I k
¼ ∑ ∑ yIi yIl ∑ ∑ k LΦ
i;l:j ðyi yl Þ
i l4i
j k¼0
RT
∑
ðpyIi
i ¼ Al;Mo;Ni
ln
yIi þ qyIIi
Φ
Φ
þ q1Gref
j þ ai:j þ bi:j T
ln
yIIi Þ
II
II k
þ ∑ ∑ ∑ yIIi yIIl ∑ k LΦ
j:i;l ðyi yl Þ
i l4i
j
k¼0
(p and q are the subscript numbers of sublattices, respectively)
γ’-Ni3Al
(Al,Mo,Ni)0.75(Al,Mo,Ni)0.25
0
0 ord
1
Gord
i:j ¼ Gj:i ¼ 3uij
0 ord
Li;j:i
1
¼ 0 Lord
i;j:j ¼ 6uij
1 ord
Li;j:i
4
¼ 1 Lord
i;j:j ¼ 3uij
1 ord
Li:i;j
4
¼ 1 Lord
j:i;j ¼ uij
0 ord
Li;j:k
¼ 6u1ij þ ðuiijk þ ujijk Þ3=2
1 ord
Li;j:k
¼ 3u4ij þ ðuiijk ujijk Þ=2
0 ord
Li;j;k:i
N
(Al,Ni)2(Al,Ni)1(Mo)1
D513-Al3Ni2
δ-NiMo
GN
Ni:Al:Mo ¼ 0
(Al)3(Al,Ni)2(Ni,Va)1
(Ni)24(Mo,Ni)20(Mo)12
ref
¼ 6uiijk 3ujijk =2 3ukijk =2
N
I
II
III
GΦ
m ¼ ∑ yi ∑ yj ∑ yk 1Gi:j:k
i
j
k
ref
ref
ref
N
1GΦ
i:j:k ¼ p1Gi þ q1Gj þ r1Gk þ ΔGi:j:k
id
I
I
II
II
III
III
GΦ
m ¼ RT∑ðpyi ln yi þ qyi ln yi þ ryi ln yi Þ
xs
n Φ
n Φ
I I
I
I n
II II
II
II n
GΦ
m ¼ ∑ ∑ yi yl ∑ ∑ Li;l:j:k ðyi yl Þ þ ∑ ∑ ∑ yi yl ∑ Lj:i;l:k ðyi yl Þ
i
i l4i
j n¼0
j
i l4i
n¼0
n Φ
III
III
III n
þ ∑ ∑ ∑ yIII
i yl ∑ Lk:j:i;l ðyi yl Þ
j
n¼0
i l4i
ϕ-Al12Mo
φ-Al4Mo
η-Al5Mo
τ-Al63Mo37
θ-Al8Mo3
A15–AlMo3
D011-Al3Ni
ε-Al3Ni5
D1a-Ni4Mo
ρ-Ni2Mo
ζ-Ni8Mo
(Al)12(Mo)1
(Al)4(Mo)1
(Al)5(Mo)1
(Al)63(Mo)37
(Al)8(Mo)3
(Al)0.75(Mo)0.25
(Al)0.75(Ni)0.25
(Al)0.375(Ni)0.625
(Ni)0.8(Mo)0.2
(Ni)0.67(Mo)0.33
(Ni)0.89(Mo)0.11
ref
ref
Φ
GΦ
m ¼ ΔGi:j þ p1Gi þ q1Gj
X
(Al)0.75(Mo)0.11(Ni)0.14
0 f cc
X
GXm ¼ 0:750 GfAlcc þ 0:110 Gbcc
Mo þ 0:14 GNi þ a þ b T
Φ
ref
¼ aΦ þ b T þ p1Gref
i þ q1Gj
(p and q represent subscript values in formulation, respectively.)
X
the Gibbs free energy contribution due to ordering with
xi ¼ 3yIi =4 þ yIIi =4 as described by the two-sublattice model in
Table 4 where yIi and yIIi are sit-fraction in first and second sublattices,
respectively. It has three terms as shown in Eq. (2) and in Table 4 and
contains implicitly a contribution to the disordered state also. When
yIi ¼ yIIi ¼ xi , Gord
m ði : jÞ represents the extraneous excess energy contribution to the disorder state. Thus ΔGord
m ði; jÞ is described as the
Gord
m ði
Gord
m ði
temperature and composition ranges, Gγm' should always have an
extremum when yIi ¼ yIIi ¼ xi . When the disordered phase is stable,
this extremum is a minimum. Hence, the following equation [8,47]
has to be fulfilled:
!
∂Gγm'
γ'
ðdGm Þxi ¼ ∑ ∑
dyðsÞ
ð7Þ
i ¼0
yðsÞ
i s
i
x
i
: jÞxi ¼ yI ¼ yII in Eq. (6b) and
k ord
k ord
From this equation, the parameters 0 Gord
i:j , Li;j:l , and Li:j;l can be
k ord
k ord
must be zero when the phase is disordered. 0 Gord
i:j , Li;j:l , and Li:j;l
expressed with the parameters u1ij , u4ij , uiijk , ujijk , and ukijk (i, j, or
listed in Table 4 are the parameters for Gord
m ði : jÞ.
As described in reports [8,20,47,49], in order to favor the
stability of the ordered phase to disorder transit in certain
k¼ Al, Mo, or Ni) as shown in Table 4. The detail descriptions of the
difference between
: jÞyI ;yII and
i
i
i
i
parameter derivation for the ordered γʹ-Ni3Al phase can be found
in Refs. [20,49].
S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133
129
Table 5
Symbols and related references used in Figs. 2–4.
Figure
Figs. 3 and 5(b)
Boundary
Single phase
Symbol
Ref.
▽γ
●bcc
△ γʹ
◇δ
þγ
*
γʹ
[26]
Symbol
Two-phase
Ref.
Fig. 5(d) and (f)
Fig. 5(e) and (g)
Symbol
Ref.
Symbol
Ref.
◆ γ þγʹ
[27]
▼γ þδ þ bcc
□γ þγʹ þbcc
▲γ þγʹ þbcc
[26]
△fccþ bcc þliq
[37]
γ þγʹ þbcc
◆ bcc þ B2 þγʹ
▲ γ þ bcc þδ
B2þ bcc þ l12
fccþ l12 þbcc
[15]
✖ γ þγʹþ δ
* bcc þ δ þγʹ
◆ bcc þ B2 þ γʹ
D0a þ γʹ þδ
γ þ γʹþ D0a
✡ B2 þAlMo3 þN
[15]
⊖ γ þ γʹþ δ
*
bcc þ δþ γʹ
✶liqþ Al3Moþ N
liq þ Al3Ni2 þX
N þAl3Ni2 þ X
◆γ þγʹ þ D0a
✖ γ þγʹþ δ
[31]
▲ bcc þliq
○ bcc þliq
■ fcc þ liq
◆ bccþ liq
◇ fccþ liq
▼ Nþ liq
▽γ
△ γʹ
□ B2
δ
þ bcc
γ
✧γʹ
◯γ
◇γʹ
B2
▽γ
△ γʹ
□ B2
δ
γ
✧γʹ
B2
δ
○γ
◇γʹ
B2
▽γ
△ γʹ
○D0a
□ B2
γ
✧γʹ
δ
D0a
Y γʹ
✚γ
[25]
[27]
Fig. 4
Fig. 5(a) and (c)
Three-phase
[15]
✖N
●γ
N
[22]
[15]
[23,24]
[25]
[34]
[38]
[22]
■ γ þγʹ
δ þγ
▼ bcc þγʹ
*
bcc þ B2
[15]
■ γ þγʹ
▲ γʹþ δ
δþ γ
▼ bcc þγʹ
✰γʹþ D0a
∅δ þD0a
⊖γ þ D0a
[15]
◇ δþ γ
✰δ þ γʹ
∅ bcc þ γʹ
▲Al3Niþ X
@
▼ γ þ γʹ
[31]
[29]
[29]
[22]
[35]
[15]
[11]
[24]
[35]
[22]
● γʹ
N
■X
[28]
[22]
[32]
[36]
[18]
[36]
[32]
[35]
@γ þ γʹ two-phase field at 1048 K.
The N-NiAl8Mo3 phase has the D022 structure with the space
group I4/mmm [22,24]. Their atoms distribute in three-sublattices.
According to the first-principles data in Table 2a and experimental
data [22,24] discussed in Section 2, the N-NiAl8Mo3 phase is
described with the three-sublattice model, (Al,Ni)2(Al,Ni)1(Mo)1, with
the Gibbs free energy given in Table 4, where yI and yII are the site
fractions in the first and second sublattices, respectively. The Gibbs
free energy of formation ΔGN
i:j:Mo of the end-member i:j:Mo is
N
xs N
expressed as aN
þ
b
T.
G
m is the excess Gibbs free energy
i:j:Mo
i:j:Mo
1 N
with the interaction parameters 0 LN
Al:Al;Ni:Mo and LAli:Al;Ni:Mo being
N
assumed as a constant. Both of them as well as aN
i:j:Mo and bi:j:Mo are
evaluated with the experimental and first-principles data.
The thermodynamic models of B2-NiAl and D0a-Ni3Mo phases
were described in detail in Refs. [8,21], respectively. The experimental data [11,15,35] revealed a small solubility of Mo in B2-NiAl
and Al in D0a-Ni3Mo. The non-stoichiometric B2-NiAl and D0aNi3Mo phases were modified here with the model shown in
Table 4. Here we only consider the unstable end-members with
B2
the Gibbs free energy of formationsΔGB2
Al:Mo and ΔGMo:Ni for B2D0a
NiAl and ΔGNi:Al for D0a-Ni3Mo.
The X phase was treated as stochiometric compounds with the
X
Gibbs free energy functions as shown in Table 4, where aX and b
are the model parameters to be evaluated.
4. Determination of the thermodynamic model parameters
The parameters described in the preceding section for the
ternary Ni–Al–Mo system were determined in the order of γ-fcc
and γʹ-Ni3Al phases, B2-NiAl, D0a-Ni3Mo, liquid, N and X phases
using the available experimental data in Table 5 and first-principles
data in Table 2a. The evaluated parameters as well as those of the
binary Ni–Al [8,20], Al–Mo [9] and Ni–Mo [21] systems are listed in
Tables 6 and 7. In the present section, the methodology used for
the determination of the parameters is discussed.
In both binary Ni–Al and Ni–Mo systems, Al and Mo dissolve in
γ-fcc(Ni). The isothermal section at 1533 K is calculated only using
the parameters of the three binary systems in Tables 6 and 7 and
shown in Fig. 3, where the calculated γ and γʹ two-phase region
deviates far from the experimental data [25–27] indicating that the
130
S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133
Table 6
Gibbs free energy of formation for end-member reference states (per mole of formula unit).
Phase
Parameters
Value, J/mol of atoms
D011-Al3Ni
11
ΔGD0
Al:Ni
48,483.73þ 12.299T
13
ΔGD5
Al:Al:Ni
39,465.978þ 7.895T
D513-Al3Ni2a
ε-Al3Ni5
B2-NiAl
γʹ-Ni3Al
13
ΔGD5
Al:Ni:Ni
427,191.9þ 79.217T
13
ΔGD5
Al:Al:Va
30,000 3T
13
ΔGD5
Al:Ni:Va
357,725.92 þ 68.322T
ΔGεAl:Ni
55,507.7594 þ7.265T
0
a
ΔGB2
Al:Al
B2 a
ΔGB2
Al:Ni , ΔGNi:Al
152,397.3 þ 26.406T
a
ΔGB2
Al:Va
1000 T
a
ΔGB2
Ni:Ni
0
a
ΔGB2
Ni:Va
162,397.3 27.406T
ΔGB2
Mo:Va
150,000
ΔGB2
Mo:Mo
150,000
B2
ΔGB2
Al:Mo , ΔGMo:Al
5876
B2
ΔGB2
Mo:Ni , ΔGNi:Mo
27,844
u1AlNi
u4AlNi
u1MoNi
ϕ-Al12Mo
φ-Al4Mo
η-Al5Mo
τ-Al63Mo37
θ-Al8Mo3
ψ-AlMo
A15–AlMo3
δ-NiMo
ρ-Ni2Mo
D0a-Ni3Mo
D1a-Ni4Mo
ζ-Ni8Mo
N
X
a
[9]
This work
13,415.515 þ 2.082T
[8]
341 3.515T
This work
u4MoNi
3621.1 þ3.411T
u1AlMo
0
u4AlMo
0
uNi
AlMoNi
20,375 12.033T
ΔGφAl:Mo
ΔGϕAl:Mo
ΔGηAl:Mo
ΔGτAl:Mo
ΔGθAl:Mo
ΔGψAl:Al
ΔGψMo:Al
ΔGψMo:Mo
ΔGA15
Al:Mo
ΔGδNi:Mo:Mo
ΔGδNi:Ni:Mo
ΔGρNi:Mo
139,100 þ 26.975T
[9]
137,570 þ29.69T
139,104þ 30.156T
2,268,100 þ167.2T
412,500 þ105.05T
20,166.8 9.626aT
36,850þ 1.0T
0
89,000 þ20T 0.003T2
169,981 þ1154.981T 155.484T ln (T)
[21]
154,106þ 2855.001T 94.923T ln (T)
9421 þ 49.551T 6.231T ln (T)
a
ΔGD0
Mo:Mo
42,650
a
ΔGD0
Ni:Mo
10,131.9þ 58.132T 7.366T ln (T)
a
ΔGD0
Mo:Ni
17,060
a
ΔGD0
Ni:Ni
2840
a
ΔGD0
Ni:Al
37,910 þ5.651T
This work
a
ΔGD1
Ni:Mo
ΔGςNi:Mo
ΔGN
Ni:Ni:Mo
ΔGN
Al:Ni:Mo
ΔGN
Al:Al:Mo
ΔGXAl:Mo:Ni
9021 þ 55.004T 7.080T ln (T)
[21]
6115 þ33.258T 4.085T ln (T)
29,600þ 32.028T
[21]
This work
119,040
108,760 þ22.128T
39,464.2 þ6.521T
0
0
0
bcc
Reference state is Al-bcc and Ni-bcc, i.e. Gref
¼ Gbcc
Al or GNi .
i
[21], the thermodynamic properties of the γ and γʹ phases in the
ternary Ni–Al–Mo system are described with the ternary interacuNi
AlMoNi .
[8]
7088.7363 3.687T
binary interaction parameters for γ-fcc from both Ni–Al [8,20] and
Ni–Mo [21] systems predict a ternary γ-fcc-field that is too expansive; hence positive ternary interaction parameters here had to be
considered to reduce it. Similar behavior was also found in the
ternary Ni–Al–Cr system modeled by Dupin et al. [20], in which the
positive ternary interaction parameter for γ-fcc(Ni) was employed.
In addition to the parameters in the Al–Ni [8,20] and Ni–Mo
tion parameters
Ref.
γ
γ
0 γ
LAl;Mo;Ni , 1 LAl;Mo;Ni , 2 LAl;Mo;Ni ,
u1Mo;Ni , u4Mo;Ni and
These parameters are evaluated with the experimental
data [11,15–18,22,26–35] and listed in Tables 6 and 7. The
experimental data [11,15,22,26,35] indicate little solubility of Mo
in B2-NiAl. The model parameters ΔGB2
Mo:i (i ¼ Al,Ni), which are
considered as a constant in Table 4, were fixed with the firstprinciples data as listed in Tables 6 and 7.
Based on the thermodynamic description in Section 3, the
D0a
a
model parameters aD0
and bNi:Al
in Table 4 are used for the
Ni:Al
a
was determined with
compound D0a-Ni3Mo. The parameter aD0
Ni:Al
D0
a
was evaluated with the phase
the first-principles data, while bNi:Al
equilibrium data [15], i.e. the four-phase δ þ γ-γʹ þD0a reaction at
11637 5 K. The evaluated parameters are listed in Table 6.
According to the experimental phase equilibrium data [22–24]
and first-principles data in Fig. 2, the composition range of the
S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133
131
Table 7
Excess Gibbs free energy interaction parameters.
Phase
Parameters
Value, J/mol
Ref.
Liquid
0 Liq
LAl;Ni
1 Liq
LA;Ni
2 Liq
LA;Ni
3 Liq
LA;Ni
4 Liq
LA;Ni
0 liq
LAl;Mo
1 liq
LAl;Mo
0 liq
LMo;Ni
1 liq
LMo;Ni
2 liq
LMo;Ni
0 liq
LAl;Mo;Ni
1 liq
LAl;Mo;Ni
2 liq
LAl;Mo;Ni
0 bcc
LMo;Ni
1 bcc
LMo;Ni
0 bcc
LAl;Mo
0 γ
LAl;Mo
0 γ
LAl;Ni
1 γ
LAl;Ni
2 γ
LAl;Ni
3 γ
LAl;Ni
0 γ
LTðAl;NiÞ
1 γ
LTðAl;NiÞ
0 γ
LMo;Ni
1 γ
LMo;Ni
2 γ
LMo;Ni
0 γ
LAl;Mo;Ni
1 γ
LAl;Mo;Ni
2 γ
LAl;Mo;Ni
0 D513
LAl:Al;Ni:Ni
0 D513
LAl:Al:Ni;V a
0 D513
LAl:Ni:Ni;Va
0 D513
LAl:Al;Ni:V a
0 B2
LAl;Ni:Ni
0 B2
LAl:Ni;Va
0 B2
LNi:Ni;Va
0 B2
LAl;Ni:Va
0 ψ
LAl;Mo:Al
0 ψ
LAl:Al;Mo
0 ψ
LMo:Al;Mo
1 ψ
LMo:Al;Mo
0 ψ
LAl;Mo:Mo
1 ψ
LAl;Mo:Mo
0 δ
LNi:Mo;Ni:Mo
1 δ
LNi:Mo;Ni:Mo
0 D0a
LMo;Ni:Mo
0 D0a
LNi:Mo;Ni
0 N
LAl:Al;Ni:Mo
1 N
LAl:Al;Ni:Mo
207,109þ 41.315T
[8]
bcc
γ-Fcc
D513-Al3Ni2
B2-NiAl
ψ-AlMo
δ-NiMo
D0a-Ni3Mo
N
10,186þ 5.871T
81,205 31.957T
4365 2.516T
22,101.64 þ 13.163T
100,000 þ 35T
[9]
15,000 þ 6.3T
39,597þ15.935T
[21]
7373þ 4.102T
12,123þ 5.551T
50,748
This work
Fig. 3. Isothermal section of the Ni–Al–Mo phase diagram calculated at 1533 K
using the parameters of the three binary systems in Tables 6 and 7 in comparison
with experimental data with the symbols listed in Table 5.
70,748
115,748
27,691
[21]
18,792
75,000þ 25T
[9]
92,220 þ 20T
162,407.75 þ16.213T
[20]
73,417.798 34.914T
33,471.014 9.837T
30,758.01 þ10.253T
1112
[8]
1745
8916 þ3.591T
[21]
5469 0.249T
1549 2.741T
41,546 50.349T
This work
91,047
193,074 12.948T
193,484.18 þ 131.79T
[8]
22,001.7þ 7.033T
22,001.7þ 7.033T
Fig. 4. Liquidus projection of the Ni–Al–Mo system calculated in the composition
triangle using the parameters in Tables 6 and 7 indicating the primary phases and comparing with experimental data as the symbols in Table 5 and dash-line by Lu et al. [11].
193,484.18 þ 131.79T
52,440.88 þ 11.3012T
64,024.38þ 26.4949T
64,024.38þ 26.494T
52,440.88 þ 11.301T
5000
[9]
5000
25,000
10,000
25,000
10,000
829,211þ 825.923T
[21]
417,368 þ 326.504T
three binary systems and ternary parameters, 0 Lliq
, 1 Lliq
Al;Mo;Ni
Al;Mo;Ni
6710
and 2 Lliq
. Henry [25] studied the two alloys of Ni65.7Al17.6Mo16.7
Al;Mo;Ni
1198–0.401T
140,870
0 N
function of the N phase in which ΔGN
LAl:Al;Ni:Mo and
Al:Ni:Mo ,
assumed to be constant. The parameters,
and aN
Al:Al:Mo , were fixed by the first-principles
N
N
data. The parameters, bNi:Ni:Mo and bAl:Al:Mo , were evaluated with
the phase equilibrium data [22,24]. The interaction parameters,
0 N
LAl:Al;Ni:Mo and 1 LN
Al:Al;Ni:Mo , were evaluated with the first-principles
data in Fig. 2. For the X phase, due to the lack of experimental data,
the parameter aX in Table 4 had to be assumed by calculating the
enthalpy of mixing of the compounds N, D011-Al3Ni and D513X
Al3Ni2. The parameter b was evaluated with the phase equilibrium data [36].
The liquid phase was described with the parameters of the
1 N
LAl:Al;Ni:Mo were
N
aN
Ni:Ni:Mo , aAl:Ni:Mo
This work
60,570
stable N phase is estimated between Ni12.5Al62.5Mo25 and
Ni3Al72Mo25 at high temperatures. Fig. 2 shows the composition
with the minimum enthalpy of formation for the N phase being at
NiAl8Mo3. Therefore, the interaction parameters, 0 LN
Al:Al;Ni:Mo and
1 N
LAl:Al;Ni:Mo , and the Gibbs free energies of formation, ΔGN
Ni:Ni:Mo ,
ΔGNAl:Ni:Mo and ΔGNAl:Al:Mo were considered for the Gibbs free energy
and Ni65.6Al14.4Mo20 using the optical microscopy and observed
that the primary phase of the two alloys is the Mo-bcc phase
during the solidification. Svetlov et al. [34] studied the
liquid þsolids (solid¼ fcc or bcc) two-phase field at the Ni-rich
corner using DTA and optical microscopy. Yoshizawa et al. [37]
studied the eutectic reaction (liquid-bcc þ γ) using SEM and TEM.
, 1 Lliq
and 2 Lliq
, were evaluated
The parameters, 0 Lliq
Al;Mo;Ni
Al;Mo;Ni
Al;Mo;Ni
with the primary solidification phase data [25,34,37]. The para1 bcc
2 bcc
meters 0 Lbcc
Al;Mo;Ni , LAl;Mo;Ni and LAl;Mo;Ni were considered to be zero.
132
S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133
5. Phase equilibrium results
The Ni–Al–Mo phase diagrams computed from the Gibbs free energy functions
of individual phases described above are shown in Figs. 4 and 5. Relevant
Fig. 5. The isothermal section calculated at different temperatures in comparison
with experimental data as the symbols in Table 5 and the calculated results in
dash-line by Lu et al. [11]. (a) At 1573K, (b) at 1533K, (c1) overall composition at
1473K, (c2) Ni–rich corner at 1473K, (d1) overall composition at 1373K, (d2) Ni–
corner at 1373K and at 1273K, (e) at 1273K, (f) at 1153K and (g) at 1073K.
experimental data in Table 5 as well as the phase diagram calculated using the
parameters by Lu et al. [11] are also shown in Fig. 1(a), Figs. 4 and 5 for comparison.
The phase diagrams we propose here include several dramatic differences from
previously suggested phase diagrams as described briefly here.
Our model describes the ternary compound N and X phases using a threesublattice model and as a stoichiometric compound, respectively, which were
ignored by Lu et al. [11]. The first-principles calculated enthalpies of formation of
the N phase in Fig. 2 were used to evaluate the thermodynamic parameters of the N
phase. In addition, our first-principles data were used to determine the Gibbs
energy of the end-members of the B2-NiAl and D0a-Ni3Mo phases. This differs
greatly from the modeling by Lu et al. [11]. The other major contributor of this work
is that the recent descriptions of the binary Ni–Al [20] and Ni–Mo [21] in Fig. 1
were adopted for better description of γ and γʹ in the Ni–Al–Mo system.
The current model liquidus projection of the Ni–Al–Mo system is plotted in
Fig. 4, showing a good agreement with experimental data [22,25,34,37,38]. In
comparison with our results, the liquidus projection calculated using parameters by
Lu et al. [11] shows two significant differences in Fig. 4. In our calculation, the N
phase is a primary phase on Al rich side while the calculated primary phase by Lu
et al. [11] is the θ-Al8Mo3 phase as marked with “Al8Mo3” in gray in Fig. 4. Second, a
curve as marked “A-line” in Fig. 4 is calculated using the parameters by Lu et al.
[11]. This curve starts from the binary Ni–Mo system as shown in Fig. 1
(a) indicating that the B2 phase can be the primary phases in the liquidus
projection.
Fig. 6. γ-fcc and γʹ-Ni3Al single-phase domains plotted in 3D as computed using the
present model where symbol ϑ (ϑ ¼ liq, bcc, B2, and δ) indicates the phase
equilibrium with γ-fcc or γʹ-Ni3Al.
S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133
The calculated isothermal sections at 1573, 1533 and 1473 K along with the
experimental data in Table 5 are shown in Fig. 5(a–c), in which the calculated equilibrium phases are in good agreement with the experimental data
[15,22–27,29,35]. The calculated isothermal sections at 1473 and 1373 K plotted
in Fig. 5 (c and d) show the invariant reaction γ þbcc-γʹ þ δ, which occurs at
1400 K. According the experimental data, the invariant reaction γ þbcc-γʹþ δ
temperature is between 1356 and 1420 K by Miracle et al. [26], between 1300 and
1400 K by Hong et al. [30] and at 1403 K by Wakashima et al. [31]. This means that
our calculated invariant reaction γ þ bcc-γʹþ δ is in good agreement with the
experimental data [26,30,31]. Furthermore, our calculated results show an agreement with the experimental data [15] of the γ þ γʹ two-phase region in Fig. 5(c),
while the calculated results by Lu et al. [11] are in the single fcc phase region.
Fig. 5(d) and (e) shows the isothermal section calculated at 1373 and 1273 K
with experimental data [11,15,18,22,24,28,31,32,35,36]. There are two significant
differences between Fig. 5(d) and (e). First, the X phase appears in Fig. 5(e). Our
calculated results indicate that the X phase forms at 1282 K, which is in reasonable
agreement with the experimental data 1288 K measured by Grushko et al. [36].
Second, the invariant reaction, liqþ θ-Nþ φ, occurs at 1355 K. It should be noted
that the data by Grushko et al. [36] plotted in Fig. 5(e) indicate the liqþ Al3Mo þN
three phases in equilibrium at 1273 K. This is inconsistent with our calculated
liq þ φ-Al4Mo þN three-phase equilibrium. According to the thermodynamic
description for the binary Al–Mo system [9] adopted in this work with the
associated phase diagram in Fig. 1(d), the liquid, Al4Mo, Al8Mo3, AlMo3 and bcc
phases are stable at 1273 K, while the Al3Mo phase is not considered as a stable
phase. The stability of the Al3Mo phase needs to be confirmed in future work.
Fahrmann et al. [16,17] investigated the Ni79.2Al6.4Mo14.4 and Ni77.97Al12.9Mo9.13
alloys. In the Ni79.2Al6.4Mo14.4 alloy aged at 1258 K, minor amounts of the δ-NiMo
phase was observed [16,17] which is consistent with our calculated result i.e. the
mole fraction of the δ-NiMo phase in Ni79.2Al6.4Mo14.4 alloy is 0.0057 which is very
little. The volume fraction of the γʹ phase for the Ni77.97Al12.9Mo9.13 alloy calculated
at 1258 K in this work is 0.61, which is in good agreement with the experimental
data 0.60 7 0.06 [16,17], while that calculated using parameters by Lu et al. [11] is
0.54. It is clear that our model yields better agreement with experimental data
[16,17].
Fig. 5(f) and (g) shows the isothermal sections calculated at 1153 and 1073 K,
respectively, with the experimental data [15,22,24,32,35,36]. Comparing with Fig. 5
(e), the calculated four phase δ þ γ-γʹþ D0a reaction occurs at 1167 K, which is in a
good agreement with the experimental phase transformation data 1163 75 K [15].
In Fig. 5(f), the experimental data [15], which indicated the γ þ γʹ two-phase region,
are in agreement with our calculated results, but outside the calculated γ þ γʹ twophase region using the parameters by Lu et al. [11]. A similar problem is also shown
in Fig. 5(g), where our calculated results shown better agreement with the
experimental γ þ γʹ two-phase field data [18] than those by Lu et al. [11]. To
understand the stabilities of the γ and γʹ phases, we plotted the stable γ and γʹ
phase field in the related composition triangle over the temperature range 1200–
1800 K in Fig. 6, which is useful for Ni-based superalloy design by showing the
stable γ, γʹ and γ þ γʹ phase domains with related composition and temperature
dimensions.
6. Summary
By combining CALPHAD approach with the first-principles
calculations, a thermodynamic model was developed for the
ternary Ni–Al–Mo system. The present work exhibits several key
differences relative to previously reported models: (i) firstprinciples calculations were used to determine some thermodynamic parameters, (ii) both ternary compounds N and X, and the
first-principles calculations confirm that the N-compound is only
marginally stable; (iii) calculated phase diagrams exhibit improved
agreement with the experimental data.
Acknowledgment
This work was supported by the Pennsylvania State University
by the NSF Grants (DMR-9983532, DMR-0122638, and DMR0205232). First-principles calculations were carried out on the
LION clusters at the Pennsylvania State University supported in
part by the NSF Grants (DMR-9983532, DMR-0122638, and DMR0205232).
133
Appendix A. Supporting information
Supplementary data associated with this article can be found in
the online version at http://dx.doi.org/10.1016/j.calphad.2014.03.002.
References
[1] S. Miyazaki, Y. Murata, M. Morinaga, Tetsu To Hagane—J. Iron Steel Inst. Jpn. 80
(1994) 166.
[2] S. Miyazaki, Y. Kusunoki, Y. Murata, M. Morinaga, Tetsu To Hagane–J. Iron Steel
Inst. Jpn. 81 (1995) 1168.
[3] R. Hashizume, A. Yoshinari, T. Kiyono, Y. Murata, M. Morinaga, Development of
Ni-based single crystalsuperalloys for power-generation gas turbines, in:
K.A. Green, H. Harada, T.M. Pollock, T.E. Howson, R.C. Reed (Eds.), Superalloys
2004, Minerals, Metals & Materials Soc, Champion, PA, 2004, p. 53.
[4] R. Yamamoto, Y. Kadoya, H. Kawai, R. Magoshi, S. Ueta, T. Noda, S. Isobe, J. Iron
Steel Inst. Jpn. 90 (2004) 37.
[5] J.E. Gould, F.J. Ritzert, W.S. Loewenthal, Preliminary investigations of joining
technologies for attaching refractory metals to Ni-based superalloys, in:
M.S. ElGenk (Ed.), Space Technology and Applications International Forum
(STAIF 2006), Amer Inst Physics, Albuquerque, NM, 2006, p. 757.
[6] J. Lapin, J. Marecek, M. Kursa, Kov. Mater. Met. Mater. 44 (2006) 1.
[7] L. Kaufman, H. Nesor, Metall. Trans. 5 (1974) 1623.
[8] I. Ansara, N. Dupin, H.L. Lukas, B. Sundman, J. Alloy. Compd. 247 (1997) 20.
[9] N. Saunders, J. Phase Equilib. 18 (1997) 370.
[10] K. Frisk, CALPHAD 14 (1990) 311.
[11] X. Lu, Y. Cui, Z. Jin, Metall. Mater. Trans. A 30A (1999) 1785.
[12] R.E.W. Casselton, W. Hume-Rothery, J. Less-Common Met. 7 (1964) 212.
[13] G. Grube, H. Schlecht, Z. Elektrochem. 44 (1938) 413.
[14] C.P. Heijwegen, G.D. Rieck, Z. Metallkd. 64 (1973) 450.
[15] S.B. Maslenkov, N.N. Burova, V.A. Rodimkina, Russ. Metal (1988) 179.
[16] M. Fahrmann, W. Hermann, E. Fahrmann, A. Boegli, T.M. Pollock, H.G. Sockel,
Mater. Sci. Eng. A 260 (1999) 212.
[17] M. Fahrmann, E. Fahrmann, T.M. Pollock, W.C. Johnson, Metall. Mater. Trans.
A 28 (1997) 1943.
[18] M. Fahrmann, P. Fratzl, O. Paris, E. Fahrmann, W.C. Johnson, Acta Metall. Mater.
43 (1995) 1007.
[19] S.H. Zhou, Y. Wang, C. Jiang, J.Z. Zhu, R.A. MacKay, L.Q. Chen, Z.K. Liu, in: K.A.
Green, H. Harada, T.M. Pollock, T.E. Howson, R.C. Reed (Eds.), Superalloys 2004,
Champion, PA, TMS (The minerals, Metals & Materials Society 2004), 2004, p.
969.
[20] N. Dupin, I. Ansara, B. Sundman, CALPHAD 25 (2001) 279.
[21] S.H. Zhou, Y. Wang, C. Jiang, J.Z. Zhu, L.-Q. Chen, Z.-K. Liu, Mater. Sci. Eng. A 397
(2005) 288.
[22] V.Y. Markiv, V.V. Burnashova, L.I. Pryakhina, K.P. Myasnikova, Izv. Akad. Nauk
SSSR Metall 5 (1969) 180.
[23] K. Schubert, A. Raman, W. Rossteutscher, Naturwissenschaften 51 (1964) 506.
[24] A. Raman, K. Schubert, Z. Metallkd 56 (1965) 99.
[25] M.F. Henry, Scripta Metall 10 (1976) 955.
[26] D.B. Miracle, K.A. Lark, V. Srinivasan, H.A. Lipsitt, Metall. Trans. A 15A (1984)
481.
[27] S. Chakravorty, D.R.F. West, Met. Sci 18 (1984) 207.
[28] M.P. Arbuzov, I.A. Zelenkov, Fiz. Metal. Metalloved. 15 (1963) 725.
[29] P. Nash, S. Fielding, D.R.F. West, Met. Sci 17 (1983) 192.
[30] Y.M. Hong, Y. Mishima, T. Suzuki, Mater. Res. Soc. Symp. Proc 133 (1989) 429.
[31] K. Wakashima, K. Higuchi, T. Suzuki, S. Umekawa, Acta Metall. 31 (1983) 1937.
[32] S. Chakravorty, D.R.F. West, J. Mater. Sci. 19 (1984) 3574.
[33] S. Chakrovorty, D.R.F. West, Mater. Sci. Technol. 1 (1985) 61.
[34] I.L. Svetlov, A.L. Udovskii, E.V. Monastyrskaya, I.V. Oldakovskii, M.P. Nazarova,
Russ. Metal (1987) 186.
[35] C.C. Jia, K. Ishida, T. Nishizawa, Metall. Mater. Trans. A 25 (1994) 473.
[36] B. Grushko, S. Mi, J.G. Highfield, J. Alloy. Compd. 334 (2002) 187.
[37] H. Yoshizawa, K. Wakashima, S. Umekawa, T. Suzuki, Scr. Metall. 15 (1981)
1091.
[38] H. Sprenger, H. Richter, J.J. Nickl, J. Mater. Sci. 11 (1976) 2075.
[39] Y. Jin, M.C. Chaturvedi, Y.F. Han, Y.G. Zhang, Mater. Sci. Eng. A 225 (1997) 78.
[40] G. Kresse, T. Demuth and F. Mittendorfer. ⟨http://cms.mpi.univie.ac.at/vasp/⟩
vol. 2003, 2003 .
[41] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892.
[42] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh,
C. Fiolhais, Phys. Rev. B 46 (1992) 6671.
[43] A. van de Walle, M. Asta, G. Ceder, CALPHAD 26 (2002) 539.
[44] A.T. Dinsdale, CALPHAD 15 (1991) 317.
[45] O. Redlich, A.T. Kister, Ind. Eng. Chem. 40 (1948) 345.
[46] M. Hillert, M. Jarl, CALPHAD 2 (1978) 227.
[47] I. Ansara, B. Sundman, P. Willemin, Acta Metall. 36 (1988) 977.
[48] B. Sundman, I. Ansara, M. Hillert, G. Inden, H.L. Lukas, K.C.H. Kumar, Z. Metallk.
92 (2001) 526.
[49] C.R. Li, J.C. Hu, F.M. Wang, Z.M. Du, W.J. Zhang, CALPHAD 30 (2006) 387.