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[Arkapol Saengdeejing](https://orcid.org/0000-0001-8739-3262), [Ryoji Sahara](https://orcid.org/0000-0003-0788-2985), [Hiori Kino](https://orcid.org/0000-0002-8912-686X), Yoshiyuki Kawazoe, Kazuyuki Higashino, [Toyohiro Chikyow](https://orcid.org/0000-0003-3860-4806)

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[The comparison between the Pt-Rh thermodynamic database from machine learning interatomic potential and first-principles calculations](https://mdr.nims.go.jp/datasets/44d29fdb-8672-48a1-abe1-9bc0f42d68ee)

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npj | computationalmaterials ArticlePublished in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Scienceshttps://doi.org/10.1038/s41524-026-02212-9The comparison between the Pt-Rhthermodynamic database frommachinelearning interatomic potential and first-principles calculationsCheck for updatesArkapol Saengdeejing1 , Ryoji Sahara1, Hiori Kino1,2, Yoshiyuki Kawazoe3,4, Kazuyuki Higashino5 &Toyohiro Chikyow1Platinum–rhodium alloys are one of the prominent alloys used in high-temperature and high-corrosionenvironments. Pt and Rh maintain a single solid-solution phase up to high temperatures. To developand design a part for use in the field, many techniques, tools, and software in combination, such as thephase-fieldmethod formicrostructure prediction and the finite elementmethod for stress analysis, arerequired. These methods require additional data, such as thermodynamic stability. Developing athermodynamic database is costly and time-consuming. To accelerate the construction of athermodynamic database, advanced computational methods are usually incorporated into theprocess to facilitate data acquisition required for thermodynamic assessment. With the advancementof machine learning techniques, many tools for computational materials science have been availableat a fraction of the computational resources required when performing similar calculations usingtraditional techniques. In this study, we employed machine learning interatomic potential (MLIP) tocalculate the thermodynamic properties required for the CALPHAD-type thermodynamic assessmentof the Pt-Rh binary system. First-principles calculations were performed to compare thethermodynamic database constructed from the MLIP and first-principles calculation data. The phasediagram calculated from MLIP achieved an accuracy similar to that of the phase diagram calculatedusing more first-principles calculations.Phase diagram is considered on of the most important tools for materialsscience community. Traditional phase diagram is typically constructeddirectly from experiment result. The modern approach of phase diagramconstruction is mostly based on the CALculation of PHAse Diagram(CALPHAD) methodology1, which is based on the construction of thethermodynamic databases. Several experimental and theoretical investiga-tions are required to construct thermodynamic databases for alloy researchand development. This is a costly and time-consuming process, particularlyin termsof experimental procedures. First-principles calculationshelp reducethe number of experiments required but still require considerable compu-tational resources. With advancements in machine learning techniques,several models of tailor-made or universal interatomic potentials have beenproposed2–5, which can achieve an accuracy similar to that of first-principlescalculationswith several orders ofmagnitude fewer computational resources.Using machine learning interatomic potential (MLIP), the construction of apreliminary thermodynamicdatabase entirelywithout experimental data canhelp accelerate the thermodynamic database development process.Only a few alloy systems are suitable for high-temperature and high-corrosion environments. Platinum–rhodium (Pt-Rh) alloys are amongthe most prominent alloys used in these environments. Owing to thenature of Pt and Rh, they maintain a single solid-solution phase acrossthe composition range up to 2000 K6. Several experimental and theore-tical investigations have been performed on Pt-Rh systems 7–17. Becausemost Pt applications involve catalytic reactions, many studies havefocused on its surface properties. Very limited thermodynamic data existfor the Pt-Rh systems. The solid-liquid equilibria have been investigatedby Müller18 and Acken19. Jacob et al.11 performed the solid state galvaniccell experiment to determine the activity of the Pt-Rh alloys between 9001National Institute for Materials Science, Tsukuba, Japan. 2The Institute of Statistical Mathematics, Tokyo, Japan. 3NICHe Tohoku University, Sendai, Japan.4Department of Physics, SRM University-AP, Amaravati, India. 5NETS Co. Ltd., Tsurugashima, Japan. e-mail: saengdeejing.arkapol@nims.go.jpnpj Computational Materials |          (2026) 12:362 11234567890():,;1234567890():,;https://doi.org/10.1038/s41524-026-02212-9http://crossmark.crossref.org/dialog/?doi=10.1038/s41524-026-02212-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41524-026-02212-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41524-026-02212-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41524-026-02212-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41524-026-02212-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41524-026-02212-9&domain=pdfmailto:saengdeejing.arkapol@nims.go.jpwww.nature.com/npjcompumatsand 1300 K. The current phase diagram of the Pt-Rh binary system6 isbased on a combination of experiments by Múller and Acker18,19 and aMonte Carlo simulation by Lu et al.9,10.In this work, we attempt to construct the thermodynamic database forthe Pt-Rh binary system via the CALPHAD methodology using either thedata from MLIP or the first-principles calculations.ResultsGround state stability calculationsFor the Pt-Rh binary system, two intermediate phases (PtRh-I41/amd andPtRh3-I4/mmm)have been reported in the literature9,10,14,15,20. Based on the0 K thermodynamic stability investigations by Lu et al.10, Yuge et al.14, andCurtarolo et al.20 using cluster expansion method (CEM)21 on the fcclattice, both PtRh-I41/amd and PtRh3-I4/mmm have been reported.Figure 1 shows the unit cell of PtRh-I41/amd and PtRh3-I4/mmm. TwoCEMs were performed using the energies calculated from first-principlescalculations based on density functional theory (DFT) and MLIP. TheCEM results are shown in Figure 2. The CEM results from DFT weregenerally similar to those reported by Curtarolo et al. 20. However, somedifferences were observed between the DFT and MLIP results. There arefour thermodynamically most stable phases (PtRh, Pt2Rh5, PtRh3, andPtRh5) along the convex hull in the DFT results, whereas there areadditional stable phases (Pt5Rh, Pt3Rh, Pt2Rh, and PtRh2) present in theMLIP that do not appear in theDFT results. In theDFT calculations,mostof the additional structures were positioned slightly above the convex hull.Only Pt3Rh,which is the anti-site structure of PtRh3-I4/mmm,was shownto be significantlymore stable than the DFT results. In the literature20 andDFT results, Pt3Rh-I4/mmm is a metastable phase, where the stability isslightly above the convex hull.Although there are some differences between the DFT and MLIPresults, the PtRh-I41/amd and PtRh3-I4/mmm phases are the com-monly observed in both calculation results. Thus, PtRh-I41/amd andPtRh3-I4/mmm will be considered for this study. In addition to pre-viously mentioned phases, the Pt3Rh-I4/mmm structure, which is theanti-site of the PtRh3-I4/mmm, was also considered. Table 1 listsinformation related to some stable and metastable fcc-ordered struc-tures from the CEM results. Figure 3 shows the calculated formationenthalpies, according to the Equation (16), of Pt3Rh, PtRh, PtRh3, andsome higher-order structures (R�3 m and P4/mmm) from the DFT andMLIPcalculations.Thehigh-symmetryordered structures are less stable thanthose of the results predicted from the CEM. In addition, all the enthalpies offormation calculated from theMLIP seems tobemorenegative than theDFTresults. It seem to be systematically different betweenMLIP and DFT. FromFigure 2, it can be seen that all the energies calculated from MLIP aresystematically lower than DFT results. The differences between MLIP andDFT values are somewhere between 0.7 and 1.5 kJ/mol-atom. Typically, theexperimental uncertainties from themeasurement can sometime varies from± 5% to ± 50% owing to the different techniques and sample preparations22.The differences betweenMLIP andDFT fall on the higher end comparing tothe experiment. However, the convex hull shapes were similar between thetwo results. Compared with other alloy systems such as Al-Ni23 and Cu-Hf24binary, the enthalpy of formation for all the structures in the Pt-Rh system isconsidered small, comparing to the Al-Ni system, which is between −60 to−40 kJ/mol-atom, or between−18 to−7 kJ/mol-atom in the Cu-Hf system.This implies that these structures are unstable at elevated temperatures.Solid solubility phase calculationsTo incorporate the solid solubility of theorderedcompounds, two-sublatticemodels, which can represent the continuous changing of the concentrationof element in each sublattice, were adopted for all three structures((Pt,Rh)3(Pt,Rh) and (Pt,Rh)(Pt,Rh)). By introducing two-sublatticemodel25 based on the compound energy formalism (CEF)26, the config-urational entropy from the mixing between different elements in eachsublattice can be considered as the long range ordering contribution to thephase. For the Pt3Rh and PtRh3 phases, only single two-sublattice model isrequired to represent both compounds. The 0 K and the finite-temperatureenergies were calculated for all end-members related to all three phases.Since the stability of all ordered compounds in the Pt-Rh system areFig. 1 | Ordered Pt–Rh crystal structures. aUnit cell of the ordered PtRh-I41/amdphase.bUnit cell of the ordered PtRh3-I4/mmmphase. Blue and red atoms representPt and Rh, respectively.-0.025-0.020-0.015-0.010-0.0050.0000.005 0  0.2  0.4  0.6  0.8  1Formation energy, eV/atomMole fraction of RhPredicted structuresCalculated structuresConvex hull(a)PtRhPtRh3PtRh5Pt2Rh5-0.040-0.035-0.030-0.025-0.020-0.015-0.010-0.0050.0000.005 0  0.2  0.4  0.6  0.8  1Formation energy, eV/atomMole fraction of RhPredicted structuresCalculated structuresConvex hullPt5RhPt3RhPt2RhPtRh2PtRh5PtRhPtRh3(b)Fig. 2 | Cluster expansion results for the fcc Pt-Rh binary system calculated usingDFT andMLIP. aDFT results. bMLIP results. The solid line represents the convexhull connecting the lowest-energy structures across the composition range. The solidsymbols represent the formation energies of the structures used for the clusterexpansion calculations, whereas the open symbols represent the predicted formationenergies of the sampled structures obtained from the cluster expansion model.https://doi.org/10.1038/s41524-026-02212-9 Articlenpj Computational Materials |          (2026) 12:362 2www.nature.com/npjcompumatspresented at the low temperature, only the harmonic-vibrational con-tribution is considered. The anharmonic-vibrational contribution, whichtypically important where the stability of the structure is extended to a veryhigh temperature, is not considered. Figure 4 shows the calculated finite-temperature vibrational free energy of PtRh-I41/amd usingMLIP andDFT.The differences between the MLIP and DFT results are very small at lowtemperatures but increase with temperature. At 1000 K, the differences areapproximately 5.5meV/atom (~ 550 J/mol-atom). Since PtRh-I41/amd isonly stable below300 K.The vibrational contribution should be comparablefrom bothMLIP andDFT calculations. Figure 5 shows the calculated Gibbsfree energy for the formation of PtRh-I41/amd. The calculated Gibbs freeenergy of formation of PtRh-I41/amd is positive for both MLIP and DFT.This indicates that PtRh-I41/amd phase is thermodynamically unstableabove room temperature. The result follows a linear relationship accordingto the Neumann–Kopp approximation in Equation (11). The slopes of theDFT and MLIP were slightly different. The formation entropy was esti-mated from the negative slope of the plot. The formation entropy dictatesthe stability of the structure in relation with their reference phases. Thepositive entropy of formation leads to the higher stability of the structure(the Gibbs free energy become more negative), conversely for the negativeentropy of formation. The enthalpy and entropy of formation required forthermodynamic modeling were derived and are listed in Table 2. Theentropy of formation for all of the structures is negative. It indicates thedecrease in the structural stability as the temperature increases.Mixing interaction calculationsTo evaluate the interaction parameters for both sublattice models, six dif-ferent SQSs are constructed to represent the random mixing of the Pt andRh elements in each sublattice. Figure 6 shows the 64-atom unit cell of theTable 1 | Crystal symmetry and lattice parameters of all compounds investigated in this workPhases Pearson SG # atom Lattice parameter Ref.a b c α β γPt3Rh-P4/mmm cP4 Pm�3 m 4 3.9340(3.9318) 3.9340(3.9318) 3.9340(3.9318) 90(90) 90(90) 90(90) 52Pt3Rh-I4/mmm tI8 I4/mmm 8 3.9367(3.9331) 3.9367(3.9331) 7.8606(7.8587) 90(90) 90(90) 90(90) This work (CEM)PtRh-m�3 m tP4 P4/mmm 4 2.7119(2.7559) 2.7119(2.7559) 3.3423(3.8926) 90(90) 90(90) 90(90) This work (CEM)PtRh-R�3m N/A R�3 m 6 2.7697(2.7620) 2.7697(2.7620) 13.4075(13.4421) 90(90) 90(90) 120(120) 52PtRh-I41/amd tI8 I41/amd 8 3.8994(3.8919) 3.8994(3.8919) 7.8111(7.8051) 90(90) 90(90) 90(90) 9PtRh3-P4/mmm tI8 Pm�3 m 8 3.8672(3.8591) 3.8672(3.8591) 3.8672(3.8591) 90(90) 90(90) 90(90) This work (CEM)PtRh3-I4/mmm cP4 I4/mmm 4 3.8673(3.8569) 3.8673(3.8569) 7.7377(7.7260) 90(90) 90(90) 90(90) 52The values in parentheses are from DFT calculations−0.5−0.4−0.3−0.2−0.10.00.1  0 200 400 600 800 1000−10−8−6−4−2 0MatlantisDFTVibrational free energy, eV/atomEML−EDFT, meV/atomTemperature, KPtRh-I41/amdFig. 4 | Calculated vibrational free energy of the PtRh-I41/amd phase obtainedusingMLIP and DFT as a function of temperature. Red and blue curves representtheMLIP andDFT results, respectively. The dashed line shows the energy differencebetween the MLIP and DFT results.−0.040−0.030−0.020−0.0100.0000.0100.0 0.2 0.4 0.6 0.8 1.0Formation enthalpy, eV/atomMole fraction of RhPtRh-m3mPt3Rh-I4/mmmPt3Rh-P4/mmmPtRh3-I4/mmmPtRh-I41/amdPtRh3-P4/mmmDFTMatlantisFig. 3 | Comparison of the calculated 0K enthalpies of formation of Pt3Rh, PtRh,and PtRh3 obtained using MLIP and DFT. Blue and red symbols represent theMLIP and DFT results, respectively. The solid line represent the convex hull con-necting the lowest-energy structures across the composition range. the symbolsindicate the formation energies of the individual structures. The symbols are theformation energy of each structure.024681012 0 200 400 600 800 1000MatlantisDFTGibbs free energy of formation,              meV/mol−atomTemperature, KFig. 5 | Calculated Gibbs free energy of formation of the PtRh-I41/amd phaseobtained from MLIP and DFT. The red and blue curves represent the MLIP andDFT results, respectively.https://doi.org/10.1038/s41524-026-02212-9 Articlenpj Computational Materials |          (2026) 12:362 3www.nature.com/npjcompumatsPt(Pt,Rh)-I41/amd SQS. The calculated correlation functions of the pair-and triplet-cluster show that from the smallest to the 15th pair-cluster arematched with the theoretical correlation function of random configuration.The triplet-cluster correlation functions are also matched from smallest upto the 17th cluster. Table 3 lists the calculated Pt(Pt,Rh)-I41/amd correlationfunctions for different types and sizes of clusters compared with the theo-retical random values. Table 4 lists all the SQSs generated for this study.Owing to limited computational resources, the finite-temperature ther-modynamic propertieswere not evaluated usingDFT.Only the 0 K energiesof the SQSs were calculated. However, both the 0 K and finite-temperaturefree energies of all SQSs were calculated using MLIP. The results listed inTable 5 were used to evaluate the interaction parameters of the thermo-dynamic model. Regular solution models (only the temperature-independent 0th-order interaction parameters) were considered for bothPtRh-I41/amd and PtRh3-I4/mmm. The evaluated interaction parametersare listed in Table 6.Fordisorderedphases,Table 4 lists the SQSsused for the calculationsofmixing energy. Figure 7 shows the mixing enthalpy of the fcc and hcpdisordered states at different Rh concentrations calculated using DFT. Themixing enthalpies of the fcc-disordered samples were not symmetrical,whereas those of the hcp-disordered samples were small and remainedmostly positive. The inset shows a comparison of the bcc-disordered results.The bcc-disordered enthalpies ofmixingwere significantlymore stable thanthe fcc- and hcp-disordered enthalpies. This characteristic is commonlyobserved in the system that sizable displacements occurred27. For the bcclattice that contained amoderately difference in the atomic size between thetwo elements, it is likely that this situationmight appear. The SQS tended tocollapse into amore stable ordered structure.An example andverificationofsimilar behaviorwere explained anddiscussed in our previous publication28.Thus, these bcc-disordered results were excluded from the evaluation. Thehcp-disordered enthalpies of mixing were relatively small. Thus, the hcp-disordered model was treated as an ideal solution model (the interactionTable 2 | Calculated enthalpies of formation (ΔFH), entropies of formation (ΔFS), and k-point meshes for all intermediatecompounds and end-members obtained from MLIPStructures Compositions Formation energy ΓΔFH, eV/atom ΔFS, × 10−6 eV/atom-K k-point meshPt3Rh-P4/mmm Pt3Rh −0.0197(−0.0090) −5.7999(−3.9969) 13 × 13 x 13Pt3Rh-I4/mmm Pt3Rh −0.0235(−0.0200) −6.6686/−6.6228 (−4.3144/−3.6410) 15 × 15 x 7PtRh-R�3 m PtRh −0.0055(0.0001) −6.7451(−1.8621) 17 × 17 x 5PtRh-I41/amd PtRh −0.0376(−0.0236) −10.9029/−10.8742 (−6.0105/−5.9512) 13 × 13 x 7PtRh3-P4/mmm PtRh3 −0.0270(−0.0156) −5.6501(−6.1509) 13 × 13 x 13PtRh3-I4/mmm PtRh3 −0.0280(−0.0200) −6.6463/−6.6524 (−6.1059/−6.1377) 15 × 15 x 7The values in parentheses are from DFT calculations. The underline numbers are the entropy of formation calculated from CALPHAD.Fig. 6 | Illustration of PtRh-I41/amd SQS unit cell. Blue and red atoms represent Ptand Rh, respectively.Table 3 | Calculated pair and triplet correlation functions ofPt(Pt0.5Rh0.5)-I41/amd SQS compared with those of thecorresponding perfectly random configurationCluster type Cluster size, Correlation functionÅ Calculated RandomPair 2.7558 0.0000 0.00003.8919 0.0000 0.00004.7688 0.0000 0.00004.7753 0.0000 0.00005.5040 0.0000 0.00006.1554 0.0000 0.00006.1688 0.0000 0.00007.2826 0.0000 0.00007.2868 0.0000 0.00007.2939 0.0000 0.00007.7839 0.0000 0.00007.8051 0.0000 0.00008.2573 0.0000 0.00008.2673 0.0000 0.00008.7026 0.0000 0.00008.7216 -0.5000 0.0000Triplet 3.8919 0.0000 0.00004.7688 0.0000 0.00004.7688 0.0000 0.00004.7753 0.0000 0.00004.7753 0.0000 0.00004.7753 0.0000 0.00004.7753 0.0000 0.00004.7753 0.0000 0.00005.5040 0.0000 0.00005.5040 0.0000 0.00005.5040 0.0000 0.00006.1554 0.0000 0.00006.1554 0.0000 0.00006.1554 0.0000 0.00006.1688 0.0000 0.00006.1688 0.0000 0.00006.1688 0.0000 0.0000https://doi.org/10.1038/s41524-026-02212-9 Articlenpj Computational Materials |          (2026) 12:362 4www.nature.com/npjcompumatsparameters were equal to zero). The symbols in Fig. 8 show the calculated 0K fcc-disorderedmixing energy comparedwith bothDFT andMLIP. The 0K mixing energy of fcc-disordered calculated from the MLIP appearedsymmetrical, which slightly differed from the DFT results, where the 0 Kmixing enthalpy was slightly asymmetric. There are some differencesbetween the mixing energies calculated using MLIP and DFT. At equia-tomic concentrations, the MLIP value is approximately 700 J/mol-atomlower than the DFT value. The solid and dashed lines represent the mixingenergies calculated from the CALPHADmodel fitted to theMLIP andDFTdata, respectively. There were some differences between theMLIP andDFTresults; however, as the temperature increased, the mixing energy behaviorbecame similar to that ofDFT.Thedifference in the temperature-dependentterms of the interaction parameters was approximately 3 J/mol-atom-K.The extracted disordered interaction parameters are listed in Table 6. Boththe MLIP and the DFT temperature-dependent parameters are positive.This indicates that the fcc-disordered excess energy term become smaller asincreasing temperature. However, the fcc-disordered phase is most stabi-lized owing to the larger negative contribution from the ideal mixing term(Eq. (7)). In the case of the MLIP phase diagram, the fcc-disordered idealmixing energy is almost double in value of the excess energy at 500 K and 50at%Rh. Owing to the lack of data on the liquid phase, only an ideal mixingwas considered.Phase diagram calculationsFigure 9 shows the calculated phase diagram of the Pt-Rh binary systemusing only DFT data. Typically, the Gibbs free energy descriptions ofCALPHAD are valid up to room temperature (298.15 K). The T−1 poly-nomial term used to describe the Gibbs free energy in CALPHAD results inthe Gibbs free energy reaching infinity near 0 K. For the past decade, theCALPHADcommunity have beendeveloped the 3rd generationmodel anddescriptions for the thermodynamic database by adopting the physicalrepresentation to fix the problem for the low-temperature region of thethermodynamic description29. However, with the current generation data-bases, the free energy down to 50 K should be accurate. Table 6 lists all theparameters of the Pt-Rh thermodynamic database. The two ordered phases(PtRh-I41/amd and PtRh3-I4/mmm) were stable, starting at very lowtemperatures and destabilizing around −62 ∘C and −117 ∘C, respectively.The solubility range for both ordered structures was narrower than thatreported in the literature15. There are a fewpossibilities that could contributeto thenarrowingof the solubilityof theorderedphases.Thefirst is the lackoftemperature dependence of the interaction parameters. This can result innarrow solubility ranges for both ordered structures. The other possibility isunaccounted-for short-range ordering. The ordered–disordered transitiontemperature between PtRh-I41/amd and fcc-disordered is−63 ∘C, whereasthe result from Monte Carlo simulations from Pohl et al.15 is −35 ∘C. Thedifference is less than 30 ∘C between the literature and this work. Thisindicates that the energy descriptions of the ordered and disordered phasesare similar to those reported in the literature. All the invariant reactions arelisted in Table 7. Figure 10 shows the calculated phase diagram from−200 ∘C to 100 ∘C, comparing the results obtained using MLIP and DFT.The solid line is the phase diagram calculated from the MLIP, whereas thedashed line is the phase diagram calculated from the DFT result. The threeordered phases were stable below 0 ∘C (Pt3Rh, PtRh, and PtRh3). All threephases exhibited solubility ranges. However, Pt3Rh exhibits a narrowsolubility range (less than 0.03 at% Rh). Comparing to the solubility rangefrom Lu et al.9, it is clearly seen that our model underestimates the solid-solubility in bothPtRh andPtRh3. The exclusionof the short-rangeorderingcontribution might be the reason for the largely underestimation of thesolid-solubility range for both phases. Incorporating the order-disordermodel30 might improve the solid-solubility range in both phases. Only oneeutectoid reaction, fcc↔PtRh+PtRh3 at 69.3 at%Rh and−124 ∘C, appearsin the DFT phase diagram. The order-disorder transition temperature forboth PtRh and PtRh3 were -62.6 ∘C and −117.1 ∘C, respectively. Twoeutectoid reactions appear in the MLIP phase diagram. The first is fcc ↔Pt3Rh+PtRh at 25.8 at%Rh and−128 ∘C.The other is fcc↔PtRh+PtRh3at 65.3 at%Rh and −62 ∘C. The order-disorder transition temperatures forPt3Rh, PtRh, and PtRh3 are −127.46 ∘C, −15.32 ∘C, and −38.43 ∘C,respectively. Table 7 lists all the temperatures and compositions of theinvariant reactions in the Pt-Rh binary system. Figure 11 shows the calcu-lated phase diagram, including the liquid phase. The two-phase regionbetween the fcc phase and liquid is very narrow. This indicates the similaritybetween the Gibbs free energy between liquid and fcc phases. The shapes ofthe liquidus and soliduswere also very close to the ideal solutionmodel. Thisis because of the very small fcc-disordered interaction parameters. Owing toTable 5 |Mixing enthalpies calculated fromSQSs for the selected interaction parameters of the sublatticemodels. The values inparentheses are from DFT calculationsMixing structures SQS composition Enthalpy of mixing, Entropy of mixingmeV/atom meV/atom-KPt(Pt0.5Rh0.5) Pt48Rh16 −0.3093(2.3269) −3.5471 × 10−6Rh(Pt0.5Rh0.5) Pt16Rh48 −3.1216(-3.6378) −3.0634 × 10−6Pt(Pt0.5Rh0.5)3 Pt40Rh24 −5.9900(−2.4355) −7.1602 × 10−6Rh(Pt0.5Rh0.5)3 Pt24Rh40 −11.8169(−10.1693) −5.7584 × 10−6(Pt0.5Rh0.5)Pt3 Pt56Rh8 1.0571(1.3506) −1.8027 × 10−6(Pt0.5Rh0.5)Rh3 Pt8Rh56 1.1278(-0.8052) −1.0496 × 10−6Table 4 | Mixing sites, compositions, and size of all SQSsinvestigated in this workStructures Mixing site SQS composition No. atoms Ref.fcc (A0.875,B0.125) A56B8 64 23(A0.75,B0.25) A12B4 16 53(A0.625,B0.375) A40B24 64 23(A0.5,B0.5) A8B8 16 53bcc (A0.875,B0.125) A56B8 64 23(A0.75,B0.25) A12B4 16 54(A0.625,B0.375) A40B24 64 23(A0.5,B0.5) A8B8 16 54hcp (A0.875,B0.125) A56B8 64 23(A0.75,B0.25) A12B4 64 55(A0.625,B0.375) A40B24 64 23(A0.5,B0.5) A8B8 64 55PtRh-I41/amd (Pt,Rh)Rh Pt16Rh48 64 This workPt(Pt,Rh) Pt48Rh16 64 This workPtRh3-I4/mmm Pt(Pt,Rh)3 Pt40Rh24 64 This workRh(Pt,Rh)3 Pt24Rh40 64 This work(Pt,Rh)Pt3 Pt56Rh8 64 This work(Pt,Rh)Rh3 Pt56Rh8 64 This workhttps://doi.org/10.1038/s41524-026-02212-9 Articlenpj Computational Materials |          (2026) 12:362 5www.nature.com/npjcompumatsthe positive temperature-dependent term in the fcc interaction parameters,it is possible that fcc phase might destabilized or formed amiscibility gap athigh temperature. We perform the phase diagram calculation without theliquid phase and found that the fcc-disordered phase transforms to bcc-disordered at the temperature above 4000 K. This temperature is well abovethe melting temperature of the typical Pt-Rh binary system. This kind ofbehavior is often observed in many thermodynamic database assessments.Another system that has the similar characteristic as the Pt-Rh is theAu-Cu system30. Both systems contain stable fcc-ordered phases (L10 andL12) at low temperature. The stability of fcc-ordered phases in the Au-Cu ishigher, reaching around 430 ∘C for L10. The liquid–solid phase boundaryalso differs owing to negative liquid interactions.Both the MLIP and DFT phase diagrams showed similar results andagreed with those of previously published studies15. This proves that, usingeither MLIP or DFT data, a thermodynamic database can be accuratelyestablished. Similar to the implementation of the DFT data in the extendedCALPHAD model, MLIP data can be further implemented in the similarmanner. In this work, only a simple solid-solution model is implemented.There aremore sophisticatedmodels suchas the order-disordermodel31, theeffective bond energy formalism for a multi-lattice system32, or the para-equilibrium in multicomponent system33, which typically need the manydata from computationally demanding DFT calculations. With MLIP, theimplementation of further complex CALPHAD modeling can beaccelerated.DiscussionsIn this study, we constructed and compared a thermodynamic database ofthe Pt-Rh binary systemusing only data obtained from eitherMLIP orDFTcalculations. The MLIP-calculated thermodynamic properties were com-parable to the results of theDFT calculations. The phase diagram calculatedusing theMLIP results was similar to that calculated using the DFT data. Anoticeable difference is the stability of the Pt3Rh phase in the MLIP phaseTable 6 | CALPHAD parameters for the Pt-Rh thermodynamicdatabasePhases Parameters ValuesMLIP DFTfcc 0Gfccpta0Gfccpt0GfccRha0GfccRh0LfccPt;Rh−8725 + 4.53T −5000 + 1.3T0LfccPt;Rh+1.25T +3000bcc 0GbccPta0GbccPt0GbccRha0GbccRhhcp 0GhcpPta0GhcpPt0GhcpRha0GhcpRhLiquid 0GliqPta0GliqPt0GliqRha0GliqRh0LliqPt;Rh −8725 + 4.53T −5000 + 1.3T0LliqPt;Rh +1.25T +3000PtRh-I41/amd0GPtRhPt22a0Gfccpt + 2000 + 2T 2a0Gfccpt + 2000 + 2T0GPtRhPtRha0Gfccpt +a0GfccRh−7252 + 2.10Ta0Gfccpt +a0GfccRh-5000 + 1.16T0GPtRhRhPta0Gfccpt +a0GfccRh−7252 + 2.10Ta0Gfccpt +a0GfccRh-5000 + 1.16T0GPtRhRh22a0GfccRh + 2000 + 2T 2a0GfccRh + 2000 + 2T0LPtRhðPt;RhÞ:Pt −125 + 1.37T 10000LPtRhðPt;RhÞ:Rh −1200 + 1.18T −1400PtRh3-I4/mmm0GPtRh3Pt44a0Gfccpt + 4000 + 4T 4a0Gfccpt + 4000 + 4T0GPtRh3PtRh3a0Gfccpt +3a0GfccRh−10807 + 2.57Ta0Gfccpt +3a0GfccRh-8000 + 2.38T0GPtRh3Rh3Pt3a0Gfccpt +a0GfccRh−9075 + 2.57T3a0Gfccpt +a0GfccRh-4500 + 1.550GPtRh3Rh44a0GfccRh + 4000 + 4T 4a0GfccRh + 4000 + 4T0LPtRh3ðPt;RhÞ:Pt +400 +0.70T 5000LPtRh3ðPt;RhÞ:Rh+435 + 0.41T −3000LPtRh3Pt:ðPt;RhÞ −2300 + 2.76T −10000LPtRh3Rh:ðPt;RhÞ−4560 +2.22T −4000a SGTE pure elements database42−0.020−0.015−0.010−0.0050.0000.0050.0100.0150.0200.0 0.2 0.4 0.6 0.8 1.0Formation enthalpy, eV/atomMole fraction of Rhfcchcp−0.35−0.30−0.25−0.20−0.15−0.10−0.050.000.0 0.2 0.4 0.6 0.8 1.0fccbcchcpFig. 7 | Calculated enthalpies of mixing from DFT for fcc and hcp SQS obtainedfrom DFT as a function of composition, with the bcc SQS results shown in theinset for comparision. Upward triangles, downward triangles, and diamondsrepresent the fcc, bcc, and hcp structures, respectively.−2500.0−2000.0−1500.0−1000.0−500.00.0500.00.00 0.25 0.50 0.75 1.00Mixing energy, J/mol−atomMole fraction of Rh0K2000KDFTMatlantisFig. 8 | Calculated mixing energy of fcc-disordered phase obtained from SQSsusing MLIP and DFT. Closed and opened symbols represent the MLIP and DFTresults, respectively. Solid and dashed lines represent the mixing energies calculatedfrom the CALPHAD models parameterized using the MLIP and DFT data,respectively. The line color varies from blue (0 K) to red (2000 K), indicating thecalculation temperature.https://doi.org/10.1038/s41524-026-02212-9 Articlenpj Computational Materials |          (2026) 12:362 6www.nature.com/npjcompumatsdiagram, which is not stable in the DFT phase diagram or in previouslypublished literature10. However, the DFT results showed that the stability ofthePt3Rhphasewas between the stable andmetastable phases on the convexhull. The underestimated solubility range for both PtRh-I41/amd andPtRh3-I4/mmm was owing to neglecting the contribution of short-rangeordering in the sublattice model. The ordered–disordered transition tem-peratures for both PtRh and PtRh3 are similar to those reported in theliterature. The liquid phase remained an ideal solution model and wascomparable to that reported in the literature.MethodsThermodynamic databaseThe thermodynamic database was created using the CALPHAD metho-dology. The core concept of CALPHAD is the systematic parameterizationof the Gibbs free energy of all the phases present in the system. The molarGibbs free energy of each stoichiometric phase in the system was para-meterized according to the following polynomial:Gφm ¼ aþ bT þ cTlnT þ dT2 þ eT3 þ fT�1 þ . . . ð1Þwhere a, b, c, d, e, i, and f are coefficients that must be evaluated from theexperimental and calculation results. Polynomial terms can be expanded oromitted to improve thefit based on evaluated data points. The polynomial isgenerally applied throughout the CALPHAD description not limited to theGibbs free energy but also for the interaction parameter presented in thesystem. FrommolarGibbs free energy such as entropy (S), enthalpy (H) andheat capacity (Cp) can be systematically derived:S ¼ � dGdT¼ �b� c 1þ lnTð Þ � 2dT � 3eT2 þ fT�2 þ . . . ð2ÞH ¼ Gþ TS ¼ a� cT � dT2 � 2eT3 þ 2fT�1 þ . . . ð3ÞCp ¼dHdT¼ �c� 2dT � 6eT2 � 2fT�2 þ . . . ð4ÞThe molar Gibbs free energy for solid-solution phase can be described asfollows:Gφm¼0GφmþidealGφmixþexGφmix ð5ÞThe molar Gibbs free energy of φ, the solid solution phase (Gφm), is parti-tioned into different contributions. 0Gφm represents a linear combination ofthe Gibbs free energies of the end-members. Pt-Rh binary system, 0Gφm canbe written as:0Gφm ¼ x0PtGφPt þ x0RhGφRh ð6Þwhere xPt and xRh denotes themole fraction of the Pt and Rh in theφ phase,respectively. 0GφPt and0GφRh are the Gibbs free energy of the end-member,which in this case are the Gibbs free energy of pure Pt and Rh in φ phase,respectively. idealGφmix is the ideal mixing energy obtained from the randomdistribution of atoms in each phase according to the Bragg–Williamapproximation34.The idealmixingenergywas calculatedusing the followingformula:idealGφmix ¼ RT xPt lnxPt þ xRhlnxRh� � ð7ÞUsing only two terms in Equation (5), which represent themolar Gibbs freeenergy, is called the ideal solution model. exGφmix is the excess energy andrepresents the deviation from the ideal solution model. The Redlich–Kisterpolynomial35 is typically used to represent the excess terms.exGφmix ¼ xPtxRhXNn¼0nLφPt;Rh xPt � xRh� �n ð8Þwhere nLφPt;Rh is thenth-orderedbinary interactionparameterbetweenPt andRh in φ phase, which are typically evaluated using first principles and/orexperimental data36. The excess term corresponds to the deviation from theideal mixing. If the reference state is the same phase as φ, excess term isdirectly corresponding to the enthalpy of mixing in φ phase at 0 K. For themulti-sublattice phase, the ideal mixing energy (idealGφmix) and the excessenergy (exGφmix) are slightlymodified to accommodate themixing indifferentfccLiquidPtRh PtRh3Fig. 9 | Calculated phase diagram of the Pt-Rh binary system based on the ther-modynamic database developed using DFT data.Table 7 | Calculated temperatures and compositions of the invariant reactions in the Pt-Rh binary system obtained fromMLIPReactions Temperature, ∘C Compositions, mole fraction of Rhx1 x2 x3fcc ↔ Pt3Rh + PtRh −127.58 2.576 × 10−1 2.506 × 10−1 4.935 × 10−1fcc ↔ PtRh + PtRh3 −62.17 (−124.01) 6.533(6.929) × 10−1 5.144(5.094) × 10−1 7.419(7.449) × 10−1fcc ↔ Pt3Rh −127.46 2.505 × 10−1 2.505 × 10−1 N/Afcc ↔ PtRh −15.32 (−62.57) 5.018(4.981) × 10−1 5.018(4.981) × 10−1 N/Afcc ↔ PtRh3 −38.43 (−117.12) 7.480(7.479) × 10−1 7.480(7.479) × 10−1 N/AThe values in parentheses are from DFT calculations.https://doi.org/10.1038/s41524-026-02212-9 Articlenpj Computational Materials |          (2026) 12:362 7www.nature.com/npjcompumatssublattices37. The ideal mixing energy can be expressed as follow:idealGφmix ¼ iRT yIPt lnyIPt þ yIRhlnyIRh� �þ jRT yIIPt lnyIIPt þ yIIRhlnyIIRh� �ð9Þwhere yIx and yIIx are the site fraction of x element in the 1st and 2ndsublattice, respectively. The i and j are the constituent on 1st and 2ndsublattice, respectively. The excess energy term is expanded to the following:ex Gφmix ¼ yIPtyIRh yIIPt0 LPt;Rh:Pt þ yIIRh0 LPt;Rh:Rh� �þyIIPtyIIRh yIPt0 LPt:Pt;Rh þ yIRh0 LRh:Pt;Rh� � ð10Þwhere nLPt:Pt,Rh is the nth-order interaction parameter which has the mixingbetween Pt and Rh elements in the 2nd sublattice. The “:" represent theseparation between sublattices in each phase and “," represent the mixingbetween elements in the sublattice.Compound energy formalism (CEF)26 is used for describing the Gibbsfree energy of the end-member of the sublattice model. Instead of directcalculation of the Gibbs free energy for each end-member, theNeumann–Kopp rule38 is used for the approximation for the Gibbs freeenergy:0GAaBbm ¼ x0AGAm þ x0BGBm þ ΔFH � TΔFS ð11ÞΔFG ¼ ΔFH � TΔFS ð12Þwhere 0GAaBbm is the molar Gibbs free energy of the AaBb structure, and0GAmand 0GBm are the molar Gibbs free energies of pure A and B, respectively.ΔFG, ΔFH, and ΔFS are the Gibbs free energy, enthalpy, and entropy offormation, respectively. The entropy of formation was calculated from thenegative slope of the Gibbs free energy of the formation plot for eachstructure with respect to the standard element reference energy.For a thermodynamic assessment of the interaction parameters, phasediagram, and related thermodynamic properties, the Thermo-CalcSoftware39 was used.Thermodynamic propertiesIn solid-state physics, the Helmholtz free energy can be partitioned as afunction of volume and temperature of a single structure (F(V, T)) into thefollowing contributions:F V;Tð Þ ¼ E0K Vð Þ þ Fvib V ;Tð Þ þ . . . ð13Þwhere E0K(V) and Fvib(V, T) are the ground-state energy and latticevibrational contributions, respectively. Additional contribution terms, suchas thermal electron excitation and spin-ordering magnetic contributions,can also be incorporated. Typically, the CALPHAD-type thermodynamicassessment is preferred using the Gibbs free energy. Using the Legendretransformation, the Gibbs free energy can be expressed asG P;Tð Þ ¼ F V;Tð Þ � ∂F∂V� �TV ð14ÞThe lattice vibrational contribution is calculated from the followingequation40:Fvib V ;Tð Þ ¼ kBTZ 10ln 2 sinhℏω2kBT� �g ωð Þdω ð15Þwhere ℏ, ω, and g ωð Þ are the reduced Planck constant, phonon frequency,and phonondensity of states (PDOS), respectively. The PDOSwas obtainedby solving the dynamic matrix of the force constants41. The enthalpy offormation of each endmember was calculated using the following formula:ΔFH ¼ EAaBb0 �XixiEi0 ð16ÞwhereEAaBb0 is the 0Kenergyof theAaBb end-memberwitha andb atoms,xiis the mole fraction of element i, and Ei0 is the 0 K energy of the i element inits stable structure at room temperature. The crystal structures of thereference energy (Ei) are selected following the same criteria using in theCALPHAD methodology, which is called Standard Element Reference(SER)42. If the reference structures are the same as the end-member (AaBb),it can be called enthalpy of mixing. The mixing Gibbs free energy is alsoderived from the similar manner. The binary interaction parameters fromEq. (8) are evaluated from the calculated mixing energy.The energy and associated thermodynamic properties are calculatedusing the Matlantis platform43 which is based on a machine learninginteratomic potential (MLIP) that is trained on reference data obtainedfrom quantum-mechanical calculations such as DFT for energies, forces,PtRh PtRh3Pt3RhfccMatlantisDFTFig. 10 | Comparison of the calculated Pt-Rh binary phase diagrams from −200∘C to 100 ∘C obtained using MLIP and DFT. Solid and dashed lines represent theMLIP and DFT results, respectively.fccPtRh PtRh3Pt3RhLiquidMole fraction RhFig. 11 | Calculated phase diagram of the Pt-Rh binary system based on the ther-modynamic database developed using MLIP data.https://doi.org/10.1038/s41524-026-02212-9 Articlenpj Computational Materials |          (2026) 12:362 8www.nature.com/npjcompumatsand stresses when necessary. It allows the prediction of energies andforces at substantially reduced computational cost while maintainingnear-DFT accuracy. The MLIP used in Matlantis is called PreFerredPotential (PFP), which is a neural network potential (NNP) designed as a“universal" MLIP applicable to multi-element, multi-system materials.PFP is based on the message passing neural network (MPNN) andtrained on a dataset that covering broad combinations across 96 elementsand including diverse structure types (molecules, crystals, slabs, clusters,disordered structures, adsorption structures, etc.)44. Its performance isvalidated via energy/force benchmarks on an out-of-training test set andproperty-level evaluations including liquid densities (25 organic com-pounds) and phase-stability metrics based on energy above hull,demonstrating broad applicability across material classes44. The DFTcalculations were performed using the Vienna Ab initio SimulationPackage (VASP)45 on the generalized gradient approximation (GGA) andthe Perdew–Burke–Ernzerhof (PBE)46 exchange-correlation functional.The DFT results are used for comparing with the data calculated fromMLIP. The PAW potential sets comprised of a 5s15d9 configuration ofvalence electrons for Pt and 4s14d8 for Rh. All calculations were per-formed with a spin-polarized enabler and all atomic positions were fullyrelaxed. Radial distribution function is performed on the relaxed struc-tures to ensure that the relaxed structures remain similar to the parentstructure. A cutoff energy of 500 eV and the k-point mesh of Γ-centeredMonkhorst–Pack47 was used for the integration of the Brillouin zoneduring both the structural relaxations and the final static calculationsduring the total energy calculations to provide the convergence of elec-tronic energies to a least 0.1 meV per formula unit for all calculations.The 1st order Methfessel–Paxton smearing method48 was used with a0.2 eV temperature broadening parameter for electronic occupation. Toobtain a highly accurate energy and density of states (DOS) during thefinal static calculation, the Blöchl correction linear tetrahedron method49was used.The energy of a disordered configuration of the solid solution phasecanbe calculatedusing theCEM21which allowus to explore the ground stateof the randomlymixing in a certain lattice by expanding the internal energyof a system at specific k configuration, Ek, as the following equationEk ¼XαmαJασα ð17Þwhere mα is the multiplicities for α cluster, Jα is the effective clusterinteraction (ECI) for α cluster and σα is the correlation function of the αcluster. For a set of ordered structures, the internal energies are calculatedusing first-principles calculations, and a set of ECIs can be extracted bymatrix inversion. Furthermore, these ECIs can be used to estimate theenergies of completely disordered of the system as a function ofcomposition from the above expansion. However, a large numbers ofordered structures are needed to obtain an accurate prediction of theenergy for cluster expansion method. Thus, a special quasi-randomstructure (SQS)50 which is the structure within a certain number of atomsthat has the correlation function (σα) closely matched the correlationfunction of random configuration is used. The Monte Carlo-based code(mcsqs) from the Alloy-theoretical automated toolkit (ATAT)51 was usedfor SQS generation. The 0 K mixing energies of the fcc, bcc, and hcplattices were calculated for the SQSs at 12.5, 25, 37.5, 50, 62.5, 75, and87.5 at% that used for the evaluation of the interaction parameters fromEq. (8). Additional SQSs were generated to represent the mixing energiesof the different elements in the sublattices of the solid-solutioncompounds. Table 3 shows an example of the calculated correlationfunctions for the (Pt,Rh)Rh SQS. Up to 12th pairs (pair cluster size of8.7026 Å and below) of (Pt,Rh)Rh SQS pair correlation functions werealigned with the pair correlation functions of the theoretical randomconfiguration, and all 11th three-body clusters (triplet with cluster size of6.1688 Å and below) were matched. In this study, the calculatedcorrelation functions up to at least the 12th smallest pair- and 11thsmallest triplet-cluster matched the theoretical random correlationfunctions of the same composition. The details of all the generatedSQSs are listed in Table 4.In this work, short-range ordering contribution and state-of-the-artapproaches for liquid free energy calculations are not considered owing tothe limited computational resources. 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B 76, 144204 (2007).AcknowledgementsThis work was supported by the Grants-in-Aid for Scientific Research(KAKENHI; Grant Number 24K01149), the Innovative Science and Tech-nology Initiative for Security Project of the Acquisition, Technology &Logistics Agency (ATLA; Grant Number JPJ004596), which supported theDFT calculations, and the MEXT Program: Data Creation and Utilization-Type Material Research and Development Project (Grant NumberJPMXP1122715503), which supported the purchase of Matlantis. Theauthors gratefully acknowledge the Numerical Materials Simulator super-computing resources provided by the Research Network and Facility Ser-vices Division (RNFS) of the National Institute for Materials Science(NIMS), Japan.Author contributionsA.S. performed DFT calculations, thermodynamic assessment, and wrotethe main manuscript. H.K. performed all the MLIP calculations. R.S., Y.K.,K.H., and T.C. provided useful discussions, reviewed, and revised themanuscript.Competing interestsThe authors declare no competing interests.Additional informationCorrespondence and requests for materials should be addressed toArkapol Saengdeejing.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’snoteSpringerNature remainsneutralwith regard to jurisdictionalclaims in published maps and institutional affiliations.https://doi.org/10.1038/s41524-026-02212-9 Articlenpj Computational Materials |          (2026) 12:362 10https://matlantis.com/en/product/validation/https://matlantis.com/en/product/validation/https://matlantis.com/en/product/validation/http://www.nature.com/reprintswww.nature.com/npjcompumatsOpenAccessThisarticle is licensedunder aCreativeCommonsAttribution-NonCommercial-NoDerivatives 4.0 International License, which permits anynon-commercial use, sharing,distributionand reproduction inanymediumorformat, as long as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indicate if youmodified the licensedmaterial. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2026https://doi.org/10.1038/s41524-026-02212-9 Articlenpj Computational Materials |          (2026) 12:362 11http://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/npjcompumats The comparison between the Pt-Rh thermodynamic database from machine learning interatomic potential and first-principles calculations Results Ground state stability calculations Solid solubility phase calculations Mixing interaction calculations Phase diagram calculations Discussions Methods Thermodynamic database Thermodynamic properties Data availability References Acknowledgements Author contributions Competing interests Additional information