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Page 6 of 18 Lu et al. J Mater Inf 2022;2:11 I http://dx.doi.org/10.20517/jmi.2022.15

Gibbs energy, expressed in the Redlich-Kister polynomial.

Although the Ni 3Mo compound with the D0 structure is stoichiometric in the Ni-Mo system, the phase with

thesamestructureisnon-stoichiometricinsometernarysystems,suchasNi-Mo-Ta,modeledas(Ni,Mo,Ta) 3(Ni,Mo,Ta) 1
by Cui et al. [48] . To extend to high-order systems for a multi-component Ni-based database, the two-sublattice
model of (Ni,Mo) 3(Ni,Mo) 1 was also adopted in the present work.

Fornon-stoichiometricphaseNi 3Mo,theKopp-Neumannrulewasappliedtoestimatethe ofend-members
based on the composition average of the of the pure elements in their stable element reference (SER) at 298
K, i.e., fcc Ni, bcc Mo and hcp Re. Its Gibbs energy per mole of the formula of the end-members can be
expressed as:

, = 3 + + + (2)

:
Following the previous assessment for the Ni-Mo binary, the phase was modeled as (Ni) 24(Ni,Mo) 20(Mo) 12,
a simplified model of (Ni) 24(Ni,Mo) 20(Mo) 8(Mo) 4. The coordination numbers (CNs) of the three sublattices
are 12, 14 and 15/16, respectively. Since the atomic size of Re is medium among the three elements, Yaqoob [2]
assumed that Re only occupies the second sublattice (i.e., CN = 14), which was adopted in the present work.
Therefore, the model for the phase was extended to (Ni) 24(Ni,Mo,Re) 20(Mo) 12.

The end-members of the phase can be described by Eq. (3), in which the parameter a was fixed to the
[2]
enthalpy data from Zhou et al. (except that of the end-member Ni:Re:Mo, which was quoted from Yaqoob )
and b is the entropy term to be evaluated based on phase equilibria [32] . As the enthalpy data measured at high
temperature are available, a TlnT term may be needed as follows:

= 24 + 20 + 12 + + + ln (3)

: :
Because of the limited solubility range reported by Heijwegen and Rieck [12] , Kobayashi et al. and Zhu et al.,
the Ni 4Mo phase was treated as a stoichiometric compound [15,16] . Since for the Ni 4Mo phase is known, the
Gibbs energy per mole of formula Ni 4Mo is parameterized to fit the data with a power series in temperature
as follows:
4
= + + ln + 2 (4)

where the parameters c and d were obtained from the data and the parameters a and b were evaluated based
on phase equilibria and enthalpy data.

There exist two intermetallic phases (i.e., and ) in the binary Mo-Re system. The phase is the most
important phase in the Ni-Mo-Re system. Ideally, the phase should be modeled as a 5SL model. In con-
trast, the simplified three-sublattice model (3SL), i.e., (Ni,Mo,Re) 10(Ni,Mo,Re) 4(Ni,Mo,Re) 16 is also widely
used to reduce the end-members. Figure 1 shows the calculated Gibbs energy of the phase at 298 K along
0.5Mo0.5Re-Ni and binary Mo-Re using the 5SL and 3SL models, respectively, with the formation enthalpies
of all the end-members in Eq. (1) determined by VASP data calculated in the present work. Obviously, the
5SL model manifests two composition sets of the phase in the ternary system and theoretically predicts the
[6]
composition sets, which is consistent with the experimental finding of Yaqoob et al. .
Thesimplified3SLfailstocapturethisstructuralrelationshipusingtheVASPdata(mainlyrepresentingnearest-
neighbor interactions) only. One can use positive interaction parameters within a sublattice to produce the
composition sets. However, since there are only 23 binary end-members in the 3SL model, no suitable in-
teraction parameters can be used to adjust the composition sets according to the experimentally determined
composition range. For a detailed explanation, see Appendix I.