# Fileset [STAMMethods4(2024)2412968.pdf](https://mdr.nims.go.jp/filesets/18d3dfe9-b0ba-499b-9129-a1bc6d965767/download) ## Creator [Arkapol Saengdeejing](https://orcid.org/0000-0001-8739-3262), [Ryoji Sahara](https://orcid.org/0000-0003-0788-2985), [Yoshiaki Toda](https://orcid.org/0000-0002-8343-2890) ## Rights [Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/) ## Other metadata [First-principles thermodynamic modeling for the Al-Nb-Ni ternary system](https://mdr.nims.go.jp/datasets/152c4ce3-451c-44c7-b4e3-e54251e1268f) ## Fulltext First-principles thermodynamic modeling for the Al-Nb-Ni ternary systemScience and Technology of Advanced Materials: MethodsISSN: (Print) (Online) Journal homepage: www.tandfonline.com/journals/tstm20First-principles thermodynamic modeling for theAl-Nb-Ni ternary systemArkapol Saengdeejing, Ryoji Sahara & Yoshiaki TodaTo cite this article: Arkapol Saengdeejing, Ryoji Sahara & Yoshiaki Toda (2024) First-principlesthermodynamic modeling for the Al-Nb-Ni ternary system, Science and Technology ofAdvanced Materials: Methods, 4:1, 2412968, DOI: 10.1080/27660400.2024.2412968To link to this article: https://doi.org/10.1080/27660400.2024.2412968© 2024 The Author(s). Published by NationalInstitute for Materials Science in partnershipwith Taylor & Francis GroupPublished online: 25 Nov 2024.Submit your article to this journal Article views: 169View related articles View Crossmark dataFull Terms & Conditions of access and use can be found athttps://www.tandfonline.com/action/journalInformation?journalCode=tstm20https://www.tandfonline.com/journals/tstm20?src=pdfhttps://www.tandfonline.com/action/showCitFormats?doi=10.1080/27660400.2024.2412968https://doi.org/10.1080/27660400.2024.2412968https://www.tandfonline.com/action/authorSubmission?journalCode=tstm20&show=instructions&src=pdfhttps://www.tandfonline.com/action/authorSubmission?journalCode=tstm20&show=instructions&src=pdfhttps://www.tandfonline.com/doi/mlt/10.1080/27660400.2024.2412968?src=pdfhttps://www.tandfonline.com/doi/mlt/10.1080/27660400.2024.2412968?src=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1080/27660400.2024.2412968&domain=pdf&date_stamp=25%20Nov%202024http://crossmark.crossref.org/dialog/?doi=10.1080/27660400.2024.2412968&domain=pdf&date_stamp=25%20Nov%202024https://www.tandfonline.com/action/journalInformation?journalCode=tstm20First-principles thermodynamic modeling for the Al-Nb-Ni ternary systemArkapol Saengdeejing , Ryoji Sahara and Yoshiaki TodaComputational Structural Materials Group, Materials Evaluation Field, Research Center for Structural Materials, National Institute for Materials Science, Tsukuba, JapanABSTRACTIn CALPHAD methodology, used for thermodynamic database construction, the first-principles calculations based on the density functional theory have increasingly become important tool to provide an input data for assessing the thermodynamic database. As the advancement in computational power, it is evident that the first-principles calculations has become integral part for determining the thermodynamic properties of the phases within the multi-component system than the time consuming and costly experimental procedures. The alloys development process can be significantly accelerated, especially in the complex multi-component systems. With first-principles data for both end-members thermodynamic descriptions and interaction parameters, the Al-Nb-Ni ternary thermodynamic database construction can be rapidly established. Without relying on any experimental data for solid-state phases, the first-principles Al- Nb-Ni thermodynamic database can exhibits most of the features comparing with the experimental phase diagram.IMPACT STATEMENTThe first-principles phase diagram can reproduced most of the features presented in published experimental phase diagram. It enable rapid construction of thermodynamic database where experiments are time consuming and costly.ARTICLE HISTORY Received 23 April 2024 Revised 29 September 2024 Accepted 1 October 2024 KEYWORDS CALPHAD; DFT; SQS; first- principles; thermodynamic database; Al-nb-ni1. IntroductionPhase diagram is an important tool for both academic and commercial development of new alloys. Modern phase diagrams calculated from open source or commercialized programs are not only a figure that illustrate phase relations with respect to the variables presented along the axis in the phase diagram but also are the result of calculations based on the thermodynamic principles derived from the thermodynamic databases. CALPHAD methodology [1], where the multi-component thermodynamic database is established, has become an essential tool for alloys development since established in 1970s. Traditionally, the assessment of thermodynamic parameters are solely conducted through the collective of experimental data [2–6]. However, it is time consuming and expensive experimental procedures are required to obtain accurate thermodynamic descriptions for all phases appearing in any specific system. With the advancement in computational techniques, coupled between first-principles calculations and experimental data are CONTACT Arkapol Saengdeejing saengdeejing.arkapol@nims.go.jp Computational Structural Materials Group, Materials Evaluation Field, Research Center for Structural Materials, National Institute for Materials Science, Tsukuba, JapanSCIENCE AND TECHNOLOGY OF ADVANCED MATERIALS: METHODS 2024, VOL. 4, NO. 1, 2412968 https://doi.org/10.1080/27660400.2024.2412968© 2024 The Author(s). Published by National Institute for Materials Science in partnership with Taylor & Francis Group This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent.http://orcid.org/0000-0001-8739-3262http://orcid.org/0000-0003-0788-2985http://orcid.org/0000-0002-8343-2890http://www.tandfonline.comhttps://crossmark.crossref.org/dialog/?doi=10.1080/27660400.2024.2412968&domain=pdf&date_stamp=2024-11-09increasingly becoming an attractive procedure to help shorten the assessment of the multi-component thermodynamic database. Enthalpy and entropy of formation of a single compound used to describe the free energy of the end-members in a complicated sublattice model are calculated by first-principles data [7,8]; the flexibility and robustness of first-principles calculations in aiding the development of the CALPHAD- type thermodynamic database is increasingly appealing. In our previous attempt, we successfully constructed the thermodynamic database of the Al-Ni-Ti ternary system by employing only first-principles calculations results [9].In this work, the Al-Nb-Ni ternary system is chosen because Nb is an important alloying element in the Ni- based superalloys. The binary database of the Al-Nb, Al-Ni, and Nb-Ni systems are combined to create the Al-Nb-Ni ternary database. Previously, the first- principles Al-Ni binary thermodynamic database has been published [8]. Thus, there is no need to reconstruct the Al-Ni binary system. There are several published thermodynamic databases for both Al-Nb [4,10–12] and Nb-Ni [13–15] binary system but not a single one that have been assessed using only first- principles results or incorporated some of the first- principles calculations. Because there are several available literature, the data can be used for the validation of the established thermodynamic database. The Al- Nb-Niternary system have been modeled using experimental data by Du et al. [16]. The Al-Nb-Ni ternary system, as well as the Al-Ni-Ti system, are major constituent atoms of practical nickel-based superalloys. Intermetallic compounds, such as Al3Nb and NbNi3, with high melting points are expected to apply to ultrahigh-temperature materials. Therefore, the phase diagram of the Al-Nb-Ni ternary system is important for engineering purposes.2. MethodologyThe CALPHAD methodology is an approach used to calculate the thermodynamic properties, including the phase diagram, of any materials ranging from pure element to multi-component systems. The general principle of CALPHAD is to parameterize the temperature, composition, and/or pressure dependent Gibbs free energy of the individual phase using the thermochemical data, such as heat capacity, mixing energy, formation enthalpy, activity, etc., of individual phases coupled with phase equilibria data between phases [17]. Typically, the molar Gibbs free energy description of φ phases (Gφm) as a function of temperature (T) can be express as: where a, b, c, d, e, and f are the fitted parameters. The function can be expanded to obtain a better fitting of the free energy description. For a pure element, at least three sets of either experimental or first-principles calculations data are required for evaluating all the parameters of the Gibbs free energy function defined in Equation 1. Entropy (S) can be derived from the following expression: Enthalpy (H) can be expressed as: Finally, the heat capacity (Cp) can be derived from the first derivative to the temperature of the enthalpy: Typically, formation enthalpy (�FH), entropy (ST), and temperature dependent heat capacity (CpðTÞ) are used to fit all the parameters of the Gibbs free energy description. For a phase with solubility, the Gibbs free energy can be described as follows: 0Gφm is the summation of the weighted end-member’s Gibbs free energy, that is in case of binary system, it is the summation between the Gibbs free energy of pure components weighted and their respected mole fraction of each component. idealGφmix is the mixing energy according to W. L. Bragg et al. [18] in each phase. exGφmix is the excess Gibbs energy that represents the deviation from an ideal solution. The typical Gibbs energy for the binary solution phase is expressed as: where 0Gφi , xφi , and R are the molar Gibbs free energy of the i element in the φ phase, mole fraction of i element within the φ phase, and molar gas constant, respectively. To ensure that all the CALPHAD thermodynamic databases are compatible with each other, the pure element Gibbs free energies are standardized. The pure element Gibbs energy descriptions can be obtained from Dinsdale et al. [19]. A Redlich-Kister polynomial [20] is used for the excess Gibbs energy: where nLφA;B is the non-ideal interactions between A and B elements, and typically defined as: Sci. Technol. Adv. Mater. Meth. 4 (2024) 2 A. SAENGDEEJING et al.where nAφ, nBφ, and nCφ are the parameters that should be evaluated [21].For the intermediate compounds with solubility, the free energy is described using compound energy formalism (CEF) [22]. The end-member free energy is formulated using Neumann-Kopp rule [23]: where 0GAiBjm , �FH, and �FS are molar Gibbs free energy of AiBj structure, formation enthalpy, and formation entropy, respectively.Thermo-Calc Software [24] is used for thermodynamic database construction and assessment.2.1. First-principles calculationsThe first-principles calculations, based on the density functional theory (DFT), are performed to obtain the data needed for the CALPHAD thermodynamic database assessment. In this work, all the calculations were conducted using the Vienna Ab initio Simulation Package (VASP) [25,26]. The electron- ion interactions are described using the projector augmented wave (PAW) method [27]. The generalized gradient approximation (GGA) of Perdew-Burke -Ernzerhof (PBE) [28] is selected to describe the exchange and correlation energy. The PAW potential sets are comprised of a 2p12s2 configurations of valance electron for Al, 4p65s14d4 for Nb and 3d10 for Ni. All the calculations are performed with spin polarized enable. For the total energy calculations, the k-point mesh of Γ-centered Monkhorst-Pack [29] is used for integration of the Brillouin zone during both structure relaxations and final static calculations. The 1st order Methfessel-Paxton smearing method [30] is used for with a 0.2 eV temperature broadening parameter for the electronic occupation. To obtain highly accurate energy and density of states (DOS) during the final static calculation, the Blöchl correction linear tetrahedron method [31] is performed. Table 1 lists k-points meshes for each structure. Each k-point mesh is selected to ensure that convergence of the electronic energies is lower than 0.1 meV. All the calculations were performed with the cutoff energy of 500 eV.The Helmholtz free energy as a function of volume (V) and temperature of a single structure (FðV;TÞ) can be partitioned into the different contributions, such as ground state energy (E0KðVÞ), thermal electron excitation (FelðV;TÞ), and vibrational contribution (FvibðV;TÞ): The formation enthalpy for each end-member is calculated using the following formula: where EAaBb0 is the 0K energy of the AaBb end-member with number of a and b atoms, xi is the mole fraction of element i, and Ei0 is the 0K energy of i element in their stable structure at 298.15 K.In CALPHAD, instead of Helmholtz energy, Gibbs energy is primarily used. Using Legendre transformation, the Gibbs free energy (GðP;TÞ) is expressed as: The contribution to the free energy from thermal electron excitation (Fel) where the energy emerge from the random occupancy of thermally excited electron in the energy level above Fermi energy is calculated using the following equation [32]: where Eel V;Tð Þ is the thermal electron excitation energy, n ε;Vð Þ is the electronic density of states (DOS) at ε energy, Sel is the bare electronic entropy, εF is Fermi energy, kB is Boltzmann constant, and f is Fermi distribution function: where μe is the electron chemical potential, which is equivalent to the Fermi energy at 0 K.Vibrational contribution (Fvib V;Tð Þ) is corresponded to the lattice vibration energy and calculated through the phonon density of states (PDOS). Supercell method, as implemented in the Alloy Theoretical Automated Toolkit (ATAT) [33], are used for the PDOS calculations. The minimum separation between perturbed atoms are set to be approximately 10–12 Å to ensure that the interference from perturbed atom are isolated from the same perturbed atom from periodic boundary condition. The 0.1 Å displacement for each perturbed atom is selected for the phonon calculations. For better accuracy in force constant matrix calculation during the phonon calculations, the 1st order Methfessel-Paxton is used. In this work, quasi-harmonic phonon approximation is not considered, and only harmonic phonon approximation is performed to reduce the computational time. The lattice vibrational contribution is calculated from the following equation [34]: Sci. Technol. Adv. Mater. Meth. 4 (2024) 3 A. SAENGDEEJING et al.where �h, ω, and g ωð Þ are the reduced Planck constant, phonon frequencies, and PDOS, respectively.A disordered configuration in the solution phase is obtained using a special quasi-random structure (SQS) [35] based on cluster expansion method (CEM) [36]. The mixing energy of the fcc, bcc, and hcp lattice is calculated from the SQSs at 12.5, 25, 37.5, 50, 62.5, 75, and 87.5 at.%. The mixing energy (�mixGÞ) is calculated by the following equation: where Gφi is the Gibbs free energy of i element in φ structure. Additional SQSs have been generated to represent the mixing energy of the different elements in the sublattice of intermediate compounds. The calculated correlation functions of Al3(Al,Nb) SQS are listed in Table 2. Up to the 9th pairs of the Al3 (Al,Nb) SQS pair correlation functions are perfectly allied with the pair correlation functions of theoretical random configuration and all 10th three-body clusters (triple) are matched. In this work, the calculated correlation functions of all SQSs, up to at least 6th pair and 4th triplet, are perfectly matched with the theoretical random correlation functions of the Table 1. Formation enthalpy, formation entropy and correspond k-point mesh for all structures used in this work and from published thermodynamic databases (in parenthesis).Formation energyStructures Compositions ΔF H, kJ/mol ΔF S, J/mol-K k-point meshAl3Nb Al4 0.00 0.00 13×13x15Al3Nb −41.52(−41.75a) −7.14Al3Ni −21.02 3.14Nb3Al −6.33 −0.27Nb4 13.97 −1.47Nb3Ni 9.20 2.85Ni3Al −39.10 0.16Ni3Nb −27.71 −19.79Ni4 0.00 0.00AlNb2 Al30 6.44 4.44 7×7x7Al10Nb20 −28.67(−26.77a) −4.88Al10Ni20 −31.64 −3.27Nb10Al20 −14.54 −1.85Nb30 8.03 −1.06Nb10Ni20 4.65 5.29Ni10Al20 −30.89 2.37Ni10Nb20 −10.51 0.59Ni30 9.59 4.59AlNb3 Al8 7.61 6.04 11×11x11Al2Nb6 −17.46(−19.59a) 4.19Al2Ni6 −25.84 0.18Nb2Al6 −6.39 −2.12Nb8 10.14 2.22Nb2Ni6 −21.46 4.33Ni2Al6 −23.14 5.04Ni2Nb6 −5.33 1.10Ni8 13.22 6.96NbNi3 Al8 3.17 0.00 11×11x11Al2Nb6 −9.93 −3.08Al2Ni6 −39.52 −2.65Nb2Al6 −32.10 −6.18Nb8 19.04 0.55Nb2Ni6 −28.47(−30.80b) −0.53Ni2Al6 −24.81 0.20Ni2Nb6 11.13 4.12Ni8 2.38 0.91Nb7Ni6 Al39 10.34 5.18 7×7x7Al18Nb21 −24.03 −3.83Nb18Al21 2.55 1.71Nb18Nb21 16.64 −0.90Ni18Al21 −45.72 0.32Ni18Nb21 −19.96(−21.96b) −0.50NbNi8 NbNi8 −12.44(−14.28b) −0.01 11×11x7AlNbNi2 AlNbNi2 −42.06(−76.19c) 4.00 13×13x13AlNbNi Al8Nb8Al8 −28.25(−11.24c) −4.68 9×7x7Al8Nb8Ni8 −41.50 −3.22Ni8Nb8Al8 −38.55 −2.88Ni8Nb8Ni8 −18.76(−25.70c) −1.00a He et al. [12]. b Chen et al. [15]. c Du et al. [16].Sci. Technol. Adv. Mater. Meth. 4 (2024) 4 A. SAENGDEEJING et al.same composition. The details of all generated SQSs are listed in Table 3.In this work, short-range ordering contribution and state-of-the-art approaches for liquid free energy calculations are not considered owing to the limited computational resources. We discussed the short- range ordering and state-of-the-art liquid calculations in our previously published work [9].3. Results and discussions3.1. Al-NiThe thermodynamic descriptions for the Al-Ni binary system is taken from Davey et al. [8]. The bcc and fcc two sublattice model will be used instead of the original four sublattice model. All other parameters remain unchanged. Figure 1 shows the Al-Ni binary phase diagram calculated using two sublattice model for both fcc and bcc phases.3.2. Al-NbAccording to the data presented in the Materials Project database [40], there are four existing compounds but only three are experimentally observed, which are AlNb2, AlNb3, and Al3Nb. To model the solubility of all three structures, the two sublattice model will be implemented for all intermediate phases. Table 4 shows details of crystal structures for all compounds. (Al,Nb)3(Al,Nb), (Al,Nb)(Al, Nb)2, and (Al,Nb)(Al,Nb)3 sublattice models are implemented to describe the solid solubility for all three phases in the binary system. To reduce the complexity of the calculations, (Al,Nb)(Al,Nb)2 sublattice model is chosen instead of more complex (Al, Nb)5Nb2(Al,Nb)8, which can correctly represent the solid solubility in the σ phase. Table 4 lists the sublattice model used for the Al-Nb binary system. Figure 2 shows the calculated Gibbs energy of formation as a function of temperature for Al3Nb, AlNb2, and AlNb3. It can be observed that the Gibbs energy of formations are fairly linear at the elevated temperature. Therefore, we can use Equation 9 to approximate the Gibbs free energy of all the end-members. Using linear regression fitting, the slope of the plot in Figure 2 is approximated as the negative of entropy of formation. Table 1 lists all the calculated formation enthalpy and entropy for all the end-member structures required to represent sublattice model of all three phases in the Al-Nb binary system.Table 2. Correlation function of Al3(Al0:5,Nb0:5) SQS compared with theoretical random configuration.Cluster typeCluster size, Correlation functionÅ Calculated RandomPair 3.8520 0.0000 0.00005.1075 0.0000 0.00005.4476 0.0000 0.00007.4674 0.0000 0.00007.7040 0.0000 0.00008.6134 0.0000 0.00008.6411 0.0000 0.00009.2433 0.0000 0.00009.4608 0.0000 0.000010.2149 −0.5000 0.000010.7291 0.0000 0.000010.8952 0.0000 0.0000Triplet 5.1075 0.0000 0.00005.4476 0.0000 0.00005.4476 0.0000 0.00007.4674 0.0000 0.00007.4674 0.0000 0.00007.4674 0.0000 0.00007.4674 0.0000 0.00007.7040 0.0000 0.00007.7040 0.0000 0.00007.7040 0.0000 0.0000Table 3. Mixing site, compositions, and size of all SQSs used in this work.Structures Mixing site SQS composition No. atoms Ref.fcc (A0:875B0:125) A56B8 64 [8](A0:75B0:25) A12B4 16 [37](A0:625B0:375) A40B24 64 [8](A0:5B0:5) A8B8 16 [37]bcc (A0:875B0:125) A56B8 64 [8](A0:75B0:25) A12B4 16 [38](A0:625B0:375) A40B24 64 [8](A0:5B0:5) A8B8 16 [38]hcp (A0:875B0:125) A56B8 64 [8](A0:75B0:25) A12B4 64 [39](A0:625B0:375) A40B24 64 [8](A0:5B0:5) A8B8 64 [39]Al3Nb A3(A0:5B0:5) A56B8 64 This workAlNb2 (A1=3B2=3)3 A90B180 270 This workAlNb3 (A0:5B0:5)B3 A8B56 64 This workNb7Ni6 A7(A0:5B0:5)6 A120B36 156 This workNbNi3 (A0:5B0:5)B3 A8B56 64 This workA(A0:5B0:5)3 A40B24 64 This work(A0:5B0:5)C3 A8B8C48 64 This workL12 (A0:5B0:5)B3 A8B56 64 This workA(A0:5B0:5)3 A40B24 64 This work(A0:5B0:5)C3 A8B8C48 64 This workAlNbNi (A0:5C0:5)BC A8B16C24 48 This workAB(A0:5C0:5) A8B16C24 48 This work(A0:5C0:5)B(A0:5C0:5) A16B16C16 48 This workSci. Technol. Adv. Mater. Meth. 4 (2024) 5 A. SAENGDEEJING et al.Figure 3 illustrates the formation enthalpy for all three structures, including their anti-site configurations. The solid line represents the convex hull where the lowest possible energy of any combinations for all the structures exists in the Al-Nb binary system. In experimental phase diagram [10], all three phases (Al3Nb, AlNb2, and AlNb3) are stable but only for data available above 500 �C. The calculated formation g’A1-NiLiquidB2D0 11D5 13Al 3Ni 5A1-AlAl 4Ni 3Figure 1. First-principles phase diagram of the Al-Ni binary system [8].Table 4. Information about crystal symmetry, lattice parameters, and CALPHAD sublattice model for all compounds appear in this work.Lattice parameterPhases Prototypes Pearson SG # atom Models a b c α β γ Ref.Al3Nb Al3Ti tI8 I4/mmm 4 (Al,Nb,Ni)3(Al,Nb,Ni) 5.1075 5.1075 5.1075 136 136 64 [41]AlNb2 (σ) (Cr0:5Fe0:5) tP30 P42/mmm 30 (Al,Nb,Ni)(Al,Nb,Ni)2 10.0028 10.0028 5.2014 90 90 90 [10]AlNb3 Cr3Si cP8 Pm�3n 8 (Al,Nb,Ni)(Al,Nb,Ni)3 5.2133 5.2133 5.2133 90 90 90 [42]NbNi3 Cu3Ti oP8 Pmmn 8 (Al,Nb,Ni)(Al,Nb,Ni)3 4.2560 4.5616 5.1240 90 90 90 [43]Nb7Ni6 (μ) Fe7W6 hR39 R�3m 39 (Al,Nb)7(Al,Nb,Ni)6 4.9436 4.9436 27.0921 90 90 120 [64]NbNI8 V4Zn5 tI18 I4/mmm 9 NbNi8 5.6739 5.6739 5.6739 96 96 143 [44]AlNbNi2 AlCu2Mn cF16 Fm�3m 4 AlNbNi2 4.2460 4.2460 4.2460 60 60 60 [45]AlNbNi MgZn2 hP12 P63/mmc 24 (Al,Ni)Nb(Al,Ni) 5.0545 8.4908 8.0346 90 90 90 [46]−0.5−0.4−0.3−0.2−0.10.0 0 200 400 600 800 1000Gibbs energy of formation, eV/atomTemperature, KAl3NbAlNb2AlNb3Figure 2. Gibbs energy of formation as a function of temperature for Al3Nb, AlNb2, and AlNb3 compounds.−0.5−0.4−0.3−0.2−0.10.00.10.20.0 0.2 0.4 0.6 0.8 1.0Enthalpy of formation, eV/atomMole fraction of NbAl3NbAlNb2AlNb3Figure 3. Formation enthalpy for binary compounds and their corresponding end-members of sublattice model, (Al,Nb)3(Al, Nb), (Al,Nb)(Al,Nb)2, and (Al,Nb)(Al,Nb)3, with connected convex hull of the Al-Nb binary system.Sci. Technol. Adv. Mater. Meth. 4 (2024) 6 A. SAENGDEEJING et al.energy of AlNb3 is above the convex hull. Experimental measurement and estimation for the enthalpy of formations at 298.15 K (�FH�) show some discrepancy for the stability of AlNb3 phase [47–52]. The recent enthalpy of formation measurement shows that AlNb3 is not formed in the convex hull [50]. The published thermodynamic database based on experimental assessment listed the enthalpy of formation of AlNb3 end-member at 298.15 K above the convex hull. Because there are no experimentally verification of AlNb3 stability at considerably low temperature (below 0 �C), we can assume that AlNb3 is unstable at 0 K. Table 1 shows the enthalpy of formation of stable compounds obtained from the published thermodynamic database [12] in parenthesis. The differences between the enthalpies of formation calculated from the DFT and the values extracted from the assessed thermodynamic database are typically within �3 kJ/mol of each other. According to the sublattice models that represent the solid solution of all three intermediate compounds in Al-Nb binary system, 5 types of binary interaction parameters, for example (Al3Nb), nLAl3NbðAl;NbÞ3Al, nLAl3NbðAl;NbÞ3Nb, nLAl3NbAl3ðAl;NbÞ, nLAl3NbNb3ðAl;NbÞ, nLAl3NbðAl;NbÞ3ðAl;NbÞ, are available for each sublattice model. Not all the interaction parameters will be significantly affected on the solubility of the solution phases. The SQSs are generated according to the selected interaction parameters. The SQS with the equiatomic configurations within the sublattice will be used for obtaining mixing energy, which will be used to evaluate the interaction parameters. Table 3 lists the generated SQSs for the Al-Nb binary system.Table 5 lists the calculated enthalpy of mixing of the SQSs for Al-Nb binary system. We encountered a convergence problem while calculating the energy of AlNb2 SQS. Thus, the mixing energy of AlNb2 SQS is currently unavailable. The interaction parameters within each sublattice model will be evaluated based on the calculated enthalpy of mixing with the assumption of regular solution (only 0th interaction parameter is used). The interaction parameter can be directly obtained from the following equation:Within the regular solution model, the temperature dependent interaction parameter will not be obtained owing to the computational expensive of the SQS for finite temperature properties.For disordered phase, the mixing energy calculated from first-principles calculations corresponded to the excess term presented in Equation 5. For simplicity, only 0th-ordered interaction parameter will be used. Both temperature independent and dependent terms in the interaction parameters are fitted through the Equation 19. Figure 4 shows calculated SQS enthalpy of mixing (0 K) across the composition range for fcc, bcc, and hcp structures. There are some anomaly with the results, especially at higher at.% Nb in fcc and hcp lattice. The mixing enthalpy seems to be more stable compared to the lower at.% Nb. The mixing enthalpy at 50 at.% Nb of bcc and hcp seem to be significantly more stable than that at 37.5 and 62.5 at.% Nb. This can occurred when relaxing the atomic position of the SQS by DFT, especially when the atomic size between two mixing elements are moderately different. The SQS, which should represent the disordered Table 5. Mixing enthalpy of SQS for interaction parameters evaluation.Mixing structures SQS composition Mixing enthalpy, eV/atomAl3(Al,Nb)-Al3Nb Al56Nb8 0.0507(Al,Nb)(Al,Nb)2 Al90Nb180 N/A(Al,Nb)Nb3-AlNb3 Al8Nb56 0.0532(Nb,Ni)Ni3-NbNi3 Nb8Ni56 0.0201Nb(Nb,Ni)3-NbNi3 Nb40Ni24 0.0441Nb7(Nb,Ni)6-Nb7Ni6 Nb120Ni36 N/A(Al,Nb)Ni3-L12 Al8Nb8Ni48 −0.0938(Al,Ni)Ni3-L12 Al8Ni56 0.0123Nb(Nb,Ni)3-L12 Nb40Ni24 −0.0999(Al,Nb)Ni3-NbNi3 Al8Nb8Ni48 0.0113(Al,Ni)NbNi-AlNbNi Al8Nb16Ni24 −0.0267AlNb(Al,Ni)-AlNbNi Al24Nb16Ni8 −0.0940(Al,Ni)Nb(Al,Ni)-AlNbNi Al16Nb16Ni16 −0.1773−0.5−0.4−0.3−0.2−0.10.00.00 0.25 0.50 0.75 1.00Mixing enthalpy, eV/atomMole fraction of NbfccbcchcpFigure 4. Mixing enthalpy of fcc-, bcc-, and hcp-disordered phases from SQS for the Al-Nb binary system.Sci. Technol. Adv. Mater. Meth. 4 (2024) 7 A. SAENGDEEJING et al.configuration, tends to collapse into ordered structure. By comparing the calculated radial distribution function (RDF) between collapsed SQS and initial disordered structure, it usually shows largely deviation from the initial disordered structure. Further discussion about this problem is presented in our previous work [9]. If any calculated SQSs show both abnormally enthalpy of mixing and the deviation from the initial structure RDF, we excluded all the anomaly results from the interaction parameters evaluations. Finite temperature phonon calculations were performed on SQSs to obtain the vibrational contribution to the free energy. The total free energy can be obtained from combining 0K, electron excitation, and vibrational contributions as presented in Equation 10. Figure 5 shows finite temperature mixing energy of fcc SQS at 0, 500, 1000, 1500 and 2000 K. Only the finite temperature at 25, 50 and 75 at.% Nb were obtained owing to the higher computational resource required at other concentration. Only the data from 12.5 to 50 at.% Nb were excluded owing to the anomaly mention earlier. With these data points, temperature dependent interaction parameters were evaluated according to Equation 7. The solid lines in Figure 5 show the calculated excess term at 0, 500, 1000, 1500 and 2000 K from the interaction parameters listed in Table 6.Figure 6 shows the Al-Nb binary phase diagram calculated from the first-principles data. The liquid interaction parameters are all equal to zero, corresponds to an ideal solution model. The AlNb2 only stabilize up to 730 �C. As aforementioned, AlNb3 is unstable at 0 K. The stability of AlNb3 begins from 240 �C. Without the liquid interaction parameters, AlNb3 stability is extended to high temperature. Even with the ideal solution for the liquid phase, the calculated Al-Nb phase diagram exhibits all the distinguish features similar to the calculated phase diagram from He et al. [12] as shown in Figure 7. At the temperature above 3500 �C, bcc-disordered phase become stable again. It it typical to see this kind of behavior occurs in assessed thermodynamic database. Owing to the limited number of terms used in the interaction parameters, result in the incorrect extrapolation in the high temperature region. Thus, we adjust the liquid interaction parameters based on the data from Witusiewicz et al. [11]. The Al-Nb liquid interaction parameters are listed in Table 6. Figure 8 illustrates the Al-Nb binary phase diagram calculated with the interaction parameters of the liquid phase listed in Table 6. All the features that comparable with the calculated phase diagram from He et al. [12] are presented in the calculated phase diagram. In published phase diagram, the decomposition of both AlNb2 and AlNb3 are from peritectic reaction; however, in our calculated phase diagram, AlNb2 in not decompose into liquid phase owing to the lower stability of AlNb2 phase. AlNb3 is melting congruently instead of peritectic reaction. It is possibly from the lower liquid stability at high temperature from the liquid interaction parameters. By adjusting the liquid interaction parameters, it is possible to achieve the peritectic reaction for the AlNb3 decomposition. At high temperature, the bcc- disordered phase boundary is deviated from He et al. [12] phase diagram. It is likely owing to the simplification of the bcc-disordered interaction parameters. Because only 0th-ordered parameter is used, where He el al. [12] used both 0th- and 1st-ordered interaction parameters. To accurately represent the bcc- disordered phase boundary at high temperature, more interaction parameters and accurate DFT calculations might needed.3.3. Nb-NiAccording to Materials Project database [40], there are only two stable intermediate compounds (Nb7Ni6 and NbNi3). However, based on experimental data [53– 56], there is one more stable compound (NbNi8). According to the literature, NbNi8 is stable from low temperature and decomposed via peritectoid reaction into NbNi3 and fcc-disordered at 515 �C. There is no report of solid solubility in NbNi8. Thus, the stoichiometric description will be used for NbNi8 phase. Nb7Ni6 and NbNi3 phases appear to have solid solubility. To accommodate the solid solubility in NbNi3, (Nb,Ni)(Nb,Ni)3 sublattice will be employed. Nb7Ni6 is typically called μ phase. To correctly model the solubility in Nb7Ni6 phase, a four sublattice model ((Nb,Ni)Nb4(Nb,Ni)2(Nb,Ni)6) is required. However, −50−40−30−20−1000.00 0.25 0.50 0.75 1.00Mixing energy, kJ/mol−atomMole fraction of Nb0K2000KFigure 5. Finite temperature mixing energy of the Al-Nb fcc- disordered SQS (point) with CALPHAD excess energy from fitted interaction parameters using the mixing energy (line).Sci. Technol. Adv. Mater. Meth. 4 (2024) 8 A. SAENGDEEJING et al.to reduce to complexity of the sublattice model and shorten the calculation times, Nb7(Nb,Ni)6 sublattice model is selected to represent the solubility between Nb and Ni for Nb7Ni6 phase. Only Ni site can be occupied with Nb atom but Nb site cannot be occupied by Ni atom. Table 4 lists the sublattice model used for the Nb-Ni binary system. Chen et al. [15] also selected Nb7(Nb,Ni)6 sublattice model to represent the solubility in the Nb7Ni6 phase. The calculated DFT results for all the end-member structures in the Nb-Ni binary system are listed in Table 1.Figure 9 plots all the calculated formation enthalpy at different Ni concentration for all two structures, including their anti-site configurations. There are only two structures that formed the convex hull (Nb7Ni6 and NbNi3). The enthalpy of formation for Table 6. CALPHAD parameters for the Al-Nb-Ni thermodynamic database (part 1).Phases Gibbs/Interactions Parametersfcc 0GfccAla0GfccAl0GfccNba0GfccNb0GfccNia0GfccNi0LfccAl;Nb− 130000 + 18T0LfccAl;Ni− 145582 + 8.17T [8]1LfccAl;Ni+59552 - 4.7T [8]2LfccAl;Ni+65528 - 2.29T [8]0LfccNb;Ni− 75000bcc 0GbccAla0GbccAl0GbccNba0GbccNb0GbccNia0GbccNi0LbccAl;Nb− 75000 + 18T0LbccAl;Ni− 93693 + 6T [8]1LbccAl;Ni+82380 - 7T [8]2LbccAl;Ni+87090 - 4.95T [8]0LbccNb;Ni− 12250 + 3.5Thcp 0GhcpAla0GhcpAl0GhcpNba0GhcpNb0GhcpNia0GhcpNi0LhcpAl;Nb− 140000 + 23TLiquid 0GliqAla0GliqAl0GliqNba0GliqNb0GliqNia0GliqNi0LliqAl;Nb− 110000 + 20T1LliqAl;Nb+85000LliqAl;Ni− 207109 + 41.32T [8]1LliqAl;Ni− 10186 + 5.87T [8]2LliqAl;Ni+81205 - 31.96T [8]3LliqAl;Ni+4365 - 2.52T [8]4LliqAl;Ni− 22102 + 13.16T [8]0LliqNb;Ni− 80000 - 6.3T1LliqNb;Ni+100000 - 19T3LliqNb;Ni+10000Al3Nb 0GAl3 NbAl44 a0GfccAl +1000 + T0GAl3 NbAl3 Nb3 a0GfccAl + a0GbccNb − 166076 + 28.56T0GAl3 NbNb3 Ala0GfccAl +3 a0GbccNb − 25301 + 1.07T0GAl3 NbNb44 a0GbccNb +55883 + 5.9T0LAl3 NbAl:Al;Nb+200000LAlNb3Al;Nb:Nb+20000AlNb2 0GAlNb2Al33 a0GfccAl +19326 - 13.33T0GAlNb2AlNb2a0GfccAl +2 a0GbccNb − 86001 + 14.63T0GAlNb2AlNi2a0GfccAl +2 a0GfccNi − 94927 + 9.82T0GAlNb2NbAl22 a0GfccAl + a0GbccNb − 43614 + 5.56T0GAlNb2Nb33 a0GbccNb +24080 + 3.19T0GAlNb2NbNi2a0GbccNb +2 a0GfccNi +13964 - 15.88T0GAlNb2NiAl22 a0GfccAl + a0GfccNi − 92670 - 7.10T0GAlNb2NiNb22 a0GbccNb + a0GfccNi − 31522 - 1.76T0GAlNb2Ni33 a0GfccNi +28756 - 13.77Ta SGTE pure elements database [19].Sci. Technol. Adv. Mater. Meth. 4 (2024) 9 A. SAENGDEEJING et al.NbNi8 is slightly above the convex hull. According to literature [53–55], the formation of NbNi8 requires a long annealing time and numerous excess vacancies from either rapid quenching or charged-particle irradiation. Therefore, NbNi8 might be stabilized by vacancy mechanism. Because the DFT calculation of NbNi8 does not contain any vacancy, the NbNi8 is not sufficiently stable to form the convex hull in the Nb-Ni binary system. Table 1 shows the enthalpy of formation of compounds obtained from the published thermodynamic database [15] in parenthesis. The two stable intermediate compounds are Nb7Ni6 and NbNi3, which agree with published experimental phase diagram [57] and assessed thermodynamic database [13]. Other end-member energies are all above the convex hull. To evaluate the interaction parameters between end-members, the correspond SQSs are generated. Table 5 lists the calculated mixing energy from (Nb0:5Ni0:5)Ni3 and Nb(Nb0:5Ni0:5)3 SQS. The Nb7(Nb0:5Ni0:5)6 SQS is generated but the convergent problem occurs during the DFT calculations. Thus, the mixing energy of Nb7Ni6 SQS is unavailable. LiquidAlNb 2AlNb3bccfccAl 3NbFigure 6. Phase diagram of the Al-Nb binary system with ideal liquid model.LiquidAlNb 2AlNb 3bccfccAl 3NbFigure 7. Phase diagram of the Al-Nb binary system with liquid interaction parameters from Witusiewicz et al. [11].Sci. Technol. Adv. Mater. Meth. 4 (2024) 10 A. SAENGDEEJING et al.Similar to the Al-Nb binary system, only the temperature independent interaction parameters are evaluated and listed in Table 7.Figure 10 shows the calculated enthalpy of mixing across the composition range for fcc-, bcc-, and hcp- disordered structures. For fcc results, it seems that the data at 75 and 87.5 at.% Nb are outlier because they are more stable than other data points. For hcp results, similar to fcc, the data at 50 at.% Nb are more stable than the nearest data points. Applying the same analysis for SQS in the Al-Nb section, we will exclude the outlier data points from the further evaluations. For bcc results, the data at 12.5, 25, 37.5, and 50 at.% Nb are considered outlier. Because the ground state structure of Nb is bcc, the data point at higher at.% Nb can be considered more reliable. The data at 12.5, 25, 37.5, and 50 at.% Nb will be excluded. Figure 11 shows finite temperature mixing energy of fcc SQS at 0, 500, 1000, 1500 and 2000 K. Assuming regular mixing behavior, the bcc-disordered interaction parameter was obtained using only the 75 at.% Nb datasets. The interaction parameters of fcc- and hcp-disordered phases can be obtained from the same procedure and are listed in Table 6.By employing the ideal solution for the liquid phase, the Nb-Ni phase diagram with the first- principles calculations data is plotted in Figure 12. Both Nb7Ni6 and NbNi3 phases are stable while NbNi8 is unstable. NbNi3 stability extends to a high temperature before decompose into hcp-disordered and liquid phase through the peritectic reaction. The hcp-disordered phase is presented at high temperature but is not presented in any published phase diagrams [13–15,57–59]. Figure 13 shows the calculated phase diagram from published thermodynamic database from Chen et al. [15] that have the NbNi8 phase stable up to 515 �C. The solubility of Ni-rich fcc- disordered phase seems to be narrower compared to published phase diagram. This might be the consequence from the vibrational contribution to the free energy of the reference phases. In case of fcc Nb, imaginary phonon frequencies have appear. This results in error when performing the LiquidAlNb2AlNb 3bccfccAl 3NbbccFigure 8. CALPHAD assessed phase diagram of the Al-Nb binary system from He et al. [12].−0.5−0.4−0.3−0.2−0.10.00.10.20.0 0.2 0.4 0.6 0.8 1.0Enthalpy of formation, eV/atomMole fraction of NiNb7Ni6NbNi3NbNi8Figure 9. Formation enthalpy for binary compounds and their corresponding end-members of sublattice model, (Nb,Ni)(Nb, Ni)3 and Nb7(Nb,Ni)6, with connected convex hull of the Nb-Ni binary system.Sci. Technol. Adv. Mater. Meth. 4 (2024) 11 A. SAENGDEEJING et al.integration for vibrational energy owing to the ignoring of the imaginary frequencies. This results in the inaccuracy on the finite temperature part of the interaction parameter of the disordered phases. The parameters for the Nb-Ni binary system are listed in Table 7. Using the liquid interaction from Matsumoto et al. [60], the Nb-Ni phase diagram is shown in Figure 14. With liquid interaction parameters, the stability of high temperature hcp- disordered phase is suppressed. The stability of both Nb7Ni6 and NbNi3 phases are similar to that appearing in the published phase diagram [15]. The solubility of Nb7Ni6 is slightly smaller. This probably resulted from the lack of interaction Table 7. CALPHAD parameters for the Al-Nb-Ni thermodynamic database (part 2).Phases Gibbs/Interactions ParametersAlNb3 0GAlNb3Al44 a0GfccAl +30453 - 24.17T0GAlNb3AlNb3a0GfccAl +3 a0GbccNb − 69820 - 16.78T0GAlNb3AlNi3a0GfccAl +3 a0GfccNi − 103361 - 0.72T0GAlNb3NbAl33 a0GfccAl + a0GbccNb − 25564 + 8.49T0GAlNb3Nb44 a0GbccNb +40578 - 8.88T0GAlNb3NbNi3a0GbccNb +3 a0GfccNi − 85851 - 17.32T0GAlNb3NiAl33 a0GfccAl + a0GfccNi − 92574 - 20.18T0GAlNb3NiNb33 a0GbccNb + a0GfccNi − 21320 - 4.38T0GAlNb3Ni44 a0GfccNi +52889 - 27.85T0LAlNb3Al;Nb:Nb+20000Al3Ni-D011 0GAl3 NiAl12 Ni416 a0GfccAl − 620774 + 19.91T + 4.67�10� 2T2Al3Ni2-D513 0GAl0:6 Ni0:4Ala0GfccAl +17122 - 3.50T + 2.19�10� 3T20GAl0:6 Ni0:4Al0:6 Ni0:40.6 a0GfccAl +0.4 a0GfccNi − 58466 + 2.23T + 3.15�10� 3T20GAl0:6 Ni0:4Ni0:6 Al0:40.4 a0GfccAl +0.6 a0GfccAl − 28881 - 1.44T + 2.31�10� 3T20GAl0:6 Ni0:4Nia0GfccNi +31967 - 3.86T − 1.08�10� 4T20L0LAl0:6 Ni0:4Al;Ni:Al− 10388 + 1.02T + 2.08�10� 4T20LAl0:6 Ni0:4Al:Al;Ni− 16305 + 1.17T + 3.77�10� 4T20LAl0:6 Ni0:4Al;Ni:Ni− 10388 + 1.02T + 2.08�10� 4T20LAl0:6 Ni0:4Ni:Al;Ni− 16305 + 1.17T + 3.77�10� 4T2Al4Ni3 0GAl4 Ni3Al4 Ni34 a0GAlfcc +3 a0GfccNi − 418712 + 9.04T + 2.24�10� 2T2Al3Ni5 0GAl3 Ni5Al3 Ni53 a0GfccAl +5 a0GfccNi − 432983 + 8.15T + 1.49�10� 2T26*Nb7Ni6 0GNb7 Ni6Al1313 a0GfccAl +134471 - 67.40T0GNb7 Ni6Al7 Nb67 a0GfccAl +6 a0GbccNb +33138 - 22.23T0GNb7 Ni6Al7 Ni67 a0GfccAl +6 a0GfccNi − 594334 - 4.11T0GNb7 Ni6Nb7 Al66 a0GfccAl +7 a0GbccNb − 312429 + 49.76T0GNb7 Ni6Nb1313 a0GbccNb +216308 + 11.64T0GNb7 Ni6Nb7 Ni67 a0GbccNb +6 a0GfccNi − 259502 + 6.46TNbNi3 0GNbNi3Al44 a0GfccAl +126880GNbNi3AlNb3a0GfccAl +3 a0GbccNb − 39713 + 12.30T0GNbNi3AlNi3a0GfccAl +3 a0GfccNi − 158086 + 10.6T0GNbNi3NbAl33 a0GfccAl + a0GbccNb − 128407 + 24.700GNbNi3Nb44 a0GbccNb +76149 - 2.21T0GNbNi3NbNi3a0GbccNb +3 a0GfccNi − 113880 + 2.12T0GNbNi3NiAl33 a0GfccAl + a0GfccNi − 99228 - 0.82T0GNbNi3NiNb33 a0GbccNb + a0GfccNi +44509 - 16.48T0GNbNi3Ni44 a0GfccNi +9539 - 3.63T0LNbNi3Nb;Ni:Ni+8000bcc-B2 0Gbcc� B2Al22 a0GfccAl +18500 - 3.94T0Gbcc� B2AlNba0GfccAl + a0GbccNb − 6774 + 6.18T0Gbcc� B2AlNia0GfccAl + a0GfccNi − 127901 + 2.60T0Gbcc� B2NbAla0GfccAl + a0GbccNb − 6774 + 6.18T0Gbcc� B2Nb22 a0GbccNb +1000 + T0Gbcc� B2NbNia0GbccNb + a0GfccNi − 3492 - 14.01T0Gbcc� B2NiAla0GfccAl + a0GfccNi − 127901 + 2.60T0Gbcc� B2NiNba0GbccNb + a0GfccNi − 3492 - 14.01T0Gbcc� B2Ni22 a0GfccNi +18203 - 8.01T0Lbcc� B2Ni:Al;Ni− 60000 - 4T0Lbcc� B2Al;Ni:Ni− 60000 - 4TaSGTE pure elements database[19].Sci. Technol. Adv. Mater. Meth. 4 (2024) 12 A. SAENGDEEJING et al.parameters. The solid solubility of NbNi3 is much larger compared to the published phase diagram. The lack of the temperature dependent term in the interaction parameters and the error in the fcc- disordered energy might be the causes for larger solubility range. The bcc-disordered phase boundary is comparable to the published phase diagram.3.4. Al-Nb-NiThe thermodynamic database for the Al-Nb-Ni ternary system is established by combining the Al-Ni, Al- Nb, and Nb-Ni binary systems together. All the stable ternary compounds listed in Materials Project database [40] and indicated in experimental data [16,61], including intermediate ternary compounds and extended solid solution from binary, are added to database. There are only two intermediate ternary phases presented (AlNbNi2 and AlNbNi). Another ternary phase (M phase) that has been identified as high temperature phase by Du et al. [16,62–64]. It has the structure that similar to μ-Nb7Ni6 with higher Al content. Owing to the complexity of the M phase structure, it will be excluded from the modeling. Based on the experimental investigation [16,61], there are limited solubility range for AlNbNi2 ternary phase. Thus, to simplify the model, stoichiometric description will be used for AlNbNi2. AlNbNi phase has small solubility range for Nb but large solubility between Al and Ni. The (Al,Ni)Nb (Al,Ni) sublattice will be used to represent the solubility of Al and Ni elements in AlNbNi phase. Table 4 lists all the ternary sublattice models used in the Al-Nb-Ni system. Table 1 −0.5−0.4−0.3−0.2−0.10.00.00 0.25 0.50 0.75 1.00Enthalpy of formation, eV/atomMole fraction of NbfccbcchcpFigure 10. Mixing enthalpy of fcc-, bcc-, and hcp-disordered phases from SQS for the Nb-Ni binary system.−20−1000.00 0.25 0.50 0.75 1.00Mixing energy, kJ/mol−atomMole fraction of Nb0K2000KFigure 11. Finite temperature mixing energy of the Nb-Ni bcc- disordered SQS (point) with CALPHAD excess energy from fitted interaction parameters using the mixing energy (line).LiquidNbNi3Nb 7Ni 6fccbcchcpFigure 12. Phase diagram of the Nb-Ni binary system with ideal liquid model.Sci. Technol. Adv. Mater. Meth. 4 (2024) 13 A. SAENGDEEJING et al.presents formation enthalpy and entropy for all ternary compounds and the end-members of the sublattice models. Additional mixing enthalpy from SQS for ternary ((Al0:5Ni0:5)NbNi, AlNb(Al0:5Ni0:5), and (Al0:5Ni0:5)Nb(Al0:5Ni0:5)) and extended solubility phases from binary are listed in Table 5. Similar to the binary interaction parameter evaluations, the SQS mixing energy of corresponding each sublattice model were used. As aforementioned, only 0 K mixing energy calculations are performed owing to the high computational cost for the finite temperature properties of the SQS. Owing to the unavailable of the first-principles liquid data, only binary liquid interaction parameters from the binaries are included. The ternary liquid LiquidNbNi3Nb 7Ni 6fccbccFigure 13. Phase diagram of the Nb-Ni binary system with liquid interaction parameters from Matsumoto et al. [60].LiquidNbNi 3Nb 7Ni 6fccbccNbNi 8Figure 14. CALPHAD assessed phase diagram of the Nb-Ni binary system from Chen et al. [15].Sci. Technol. Adv. Mater. Meth. 4 (2024) 14 A. SAENGDEEJING et al.interaction parameters are all zero. Tables 6, 7 , and 8 list the available parameters of the Al-Nb-Ni ternary system.Figure 15 shows the isothermal section at 900 �C of the Al-Nb-Ni ternary system. Two ternary intermediate compounds (AlNbNi2 and AlNbNi) are stable at this temperature, which agrees with the published ternary phase diagram from Bhedwar et al. [61] and the assessed thermodynamic database from Du et al. [16]. The extended solubility from the binary are presented in AlNb3, AlNb2, AlNi-B2, AlNi3, Nb7Ni6, and NbNi3 phases. The solubility range of AlNbNi phase is limited (<5%) comparing to the phase diagrams from Bhedwar et al. [61] and Du et al. [16] (>50%). The absence of finite temperature interaction parameter might contribute to the narrower solubility range of the AlNbNi phase. It is proved that the Al-Nb-Ni thermodynamic database from the first-principles data can correctly reproduce the published phase diagram. Figure 16 shows the isothermal section at 1127 �C. As the temperature increases, the liquid phase field become larger and all the extended solubility from binary compounds seem to become larger Table 8. CALPHAD parameters for the Al-Nb-Ni thermodynamic database (part 3).Phases Gibbs/Interactions Parametersfcc-L12 0Gfcc� L12Al44 a0GfccAl0Gfcc� L12AlNb3a0GfccAl +3 a0GbccNb − 44216 + 6.01T0Gfcc� L12AlNi33 a0GfccAl + a0GfccNi − 167212 + 3.67T0Gfcc� L12NbAl33 a0GfccAl + a0GbccNb − 102105 + 10.21T0Gfcc� L12Nb44 a0GfccNb0Gfcc� L12NbNi3a0GbccNb +3 a0GfccNi − 51723 - 13.76T0Gfcc� L12NiAl33 a0GfccAl + a0GfccNi − 83972 - 10.80T0Gfcc� L12NiNb33 a0GbccNb + a0GfccNi +39720 + 10.21T0Gfcc� L12Ni44 a0GfccNi0Lfcc� L12Al;Nb:Ni− 500000Lfcc� L12Al;Ni:Ni+20000AlNbNi2 0GAlNbNi2AlNbNi2a0GfccAl + a0GbccNb +2 a0GfccNi − 168240 - 16TAlNbNi 0GAlNbNiAlNbAl2 a0GfccAl + a0GbccNb − 84747 + 14.40T0GAlNbNiAlNbNia0GfccAl + a0GbccNb + a0GfccNi − 124500 + 9.66T0GAlNbNiNiNbAla0GfccAl + a0GbccNb + a0GfccNi − 115641 + 8.64T0GAlNbNiNiNbNia0GbccNb 2 a0GfccNi − 56280 + 3T0LAlNbNiAl;Ni:Nb:Ni− 310000LAlNbNiAl:Nb:Al;Ni− 110000a SGTE pure elements database[19].NbNiAlNbNi3Nb7Ni6AlNi3AlNiAl3Ni2AlNb2AlNb3Al3NbAlNbNi2AlNbNiFigure 15. Isothermal section of the Al-Nb-Ni phase diagram at 900 �C.Sci. Technol. Adv. Mater. Meth. 4 (2024) 15 A. SAENGDEEJING et al.as well. Based on the literature, M phase is presented at this temperature. Owing to the exclusion of the M phase from the modeling, the calculated phase diagram cannot accurately reproduce the phase diagram in the region where M phase is presented. However, in other regions, the first-principles phase diagram is similar to the published phase diagram [16,61].4. ConclusionIn this work, we demonstrate that, by employing only the data from first-principles calculations, the thermodynamic database for the Al-Nb-Ni ternary system can be correctly and rapidly constructed. All the invariant reactions can be successfully reproduced. Although, it cannot accurately reproduce the solubility limit and transformation temperatures of every phases, it helps by rapidly establishing a preliminary thermodynamic databases. To achieve more accurate prediction of the calculated phase diagram, higher approximations for vibrational contribution, such as quasi-harmonic or even anharmonic approximations might be required. Other contributions to the free energy, such as magnetic ordering should be considered. Traditional CALPHAD thermodynamic assessment requires several experimental data, which are costly and time consuming. By integrating the DFT calculations data, which will help accelerate the thermodynamic database development, it provides a guidance on which additional data point need to further improve the accuracy of the database. By using along side with the uncertainty quantification (UQ) [65], the area of phase diagram that need more accurate experimental or DFT calculations data point can be identified. The semi-automated or automated generation of thermodynamic database for many unknown systems can be achieved. As previously mention, the sublattice models for both σ and μ phases are simplified in order to reduce the complexity of the first-principles calculations. To enable the higher compatibility with other thermodynamic databases, the modification of the sublattice models might needed. However, it is possible to directly replace the sublattice model of both phases in the future work without any re- optimization of the parameters in the thermodynamic database. The main challenge for rapid development of the first-principles thermodynamic database is the free energy description of the liquid phase. It is possible to calculate the free energy of liquid phase from DFT; however, it is rather computational expensive owing to the nature of the liquid phase that usually require anharmonic contribution because it existed at high temperature near or above melting point [66]. Currently, performing experimental measurement of the temperature and compositions upon the melting of the liquid phase are simpler and faster than using first-principles calculations. Therefore, experimental data point is critical in achieving highly accurate thermodynamic assessment for any system. First- principles calculations play an important role on reducing the number of required experimental data points and help saving time and cost for the multi- component thermodynamic database development.NbNiAlNbNi3Nb7Ni6AlNi3AlNiAlNb2AlNb3Al3NbAlNbNi2AlNbNiFigure 16. Isothermal section of the Al-Nb-Ni phase diagram at 1127 �C.Sci. Technol. Adv. Mater. Meth. 4 (2024) 16 A. SAENGDEEJING et al.AcknowledgementsThe authors gratefully acknowledge Numerical Materials Simulator supercomputing resources from the Research Network and Facility Services Division (RNFS), National Institute for Materials Science (NIMS), Japan. We would like to thank Editage (www.editage.com) for English language editing.Disclosure statementNo potential conflict of interest was reported by the author(s).FundingThis work was supported by Council for Science, Technology and Innovation(CSTI), Cross-ministerial Strategic Innovation Promotion Program (SIP), “Materials Integration for revolutionary design system of structural materials”, and the Grants-in-Aid for Scientific Research (KAKENHI) grant number [21H01607].Funding agency: Japan Science and Technology Agency (JST) and Japan Society for the Promotion of Science (JSPS).ORCIDArkapol Saengdeejing https://orcid.org/0000-0001-8739- 3262Ryoji Sahara https://orcid.org/0000-0003-0788-2985Yoshiaki Toda https://orcid.org/0000-0002-8343-2890Data availability statementThe raw data required to reproduce these findings cannot be shared at this time because they are being used on an ongoing study. 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SAENGDEEJING et al.https://doi.org/10.1107/S0365110X6700252Xhttps://doi.org/10.1515/ijmr-1968-590903https://doi.org/10.1016/j.actamat.2019.05.017https://doi.org/10.1103/PhysRevB.96.224202 Abstract Abstract 1. Introduction 2. Methodology 2.1. First-principles calculations 3. Results and discussions 3.1. Al-Ni 3.2. Al-Nb 3.3. Nb-Ni 3.4. Al-Nb-Ni 4. Conclusion Acknowledgements Disclosure statement Funding ORCID Data availability statement References