In this work, phase transition temperatures of La–Fe and Ce–Fe alloys were determined using differential thermal analysis (DTA), while phase transition temperatures, microstructure, and phase compositions of La–Ce–Fe and Ce-Nd-Fe alloys were studied using DTA and scanning electron microscopy with energy dispersive spectroscopy (SEM-EDS). Based on the available experimental data reported in the literature and the experimental results determined in this work, the La–Fe and Ce–Fe systems were re-assessed thermodynamically using the CALPHAD (CALcuation of PHAse Diagrams) method, and then the Ce–La–Fe and Ce-Nd-Fe systems were calculated by combining the re-assessed La–Fe and Ce–Fe systems with the previously assessed Nd–Fe, Ce–La, and Ce–Nd systems. The calculated phase diagrams and thermodynamic properties of the La–Fe and Ce–Fe systems are consistent with the experimental data. The calculated isothermal sections and vertical sections in the Ce–La–Fe and Ce-Nd-Fe systems are in good agreement with the experimental results. The solidification behaviors of Ce–La–Fe and Ce-Nd-Fe as-cast alloys were analyzed through the experimental examination and thermodynamic calculation with Scheil-Gulliver non-equilibrium model. The simulated results agree well with the experimental results. It indicates that the reasonable thermodynamic parameters of the Ce–La–Fe and Ce-Nd-Fe systems were obtained finally, which would be fundamental to developing a thermodynamic database of the multi-component Nd-RE-Fe-B alloy systems and then to designing novel Nd-Fe-B permanent magnets with light rare-earth metals La and Ce.

Phase relations of the Mg-Gd-Er system at the Mg-rich corner were investigated experimentally through alloy sampling approach. Isothermal sections at 673 K and 773 K were determined according to electron probe microanalysis (EPMA) and X-ray diffraction (XRD) results. No ternary compounds were detected at the investigation temperatures. MgEr and MgGd can form a continuous solid solution. Five three-phased fields were measured and deduced in both isothermal sections at 673 K and 773 K.

Abstract not available

Bi–Sb–Se is an important thermoelectric material system. However, a phase diagram of the entire compositional range is not available in the literature. This study determines the Bi–Sb–Se liquidus projection and its phase equilibria isothermal section at 400 °C. Ternary Bi–Sb–Se alloys are prepared. Their primary solidification phases and invariant reactions are determined. There are seven primary solidification phases, (Se), Bi2Se3, Sb2Se3, (Bi,Sb), (Bi2)m(Bi2Se3)n, Bi3Sb5Se2 and Bi3Sb12Se5. Both Bi3Sb5Se2 and Bi3Sb12Se5 are newly found ternary compounds. In the Bi–Sb–Se isothermal section at 400 °C, there are eight three-phase regions. Besides these two ternary compounds, the other single phases are, Sb2Se3, Bi2Se3, (Bi2)m(Bi2Se3)n, Liquid (Bi,Sb), Liquid (Se), and (Bi,Sb) phases. It has been found the solubilities of Sb in the (Bi2)m(Bi2Se3)n and Bi2Se3 compounds are significant.

Thermodynamic reassessment of the ternary Fe–Nb–V system was carried out after a critical review of the available experimental and ab initio data. The primary solidified phases of Fe–Nb–V alloys were investigated using scanning electron microscopy and X-ray diffraction analysis. The C14 Laves phase exhibits a wide primary solidification field in the Fe-rich region, which is in contradiction with the experimental observation in the literature. Ab initio calculations were performed to obtain the enthalpies of formation of three intermetallic compounds (i.e., C14 Laves, μ, and σ), which were used to estimate the thermodynamic parameters. The end-member parameters for the three phases were improved in comparison with the previous description. The presently obtained thermodynamic description of the ternary system can reproduce the wide homogeneity range of the C14 Laves phase and the phase equilibria between the intermetallic compounds and bcc phase.

Over decades, Mg–Li alloys have been widely used in aerospace industries owing to their low density (<1.65 g/cm3), medium strength (UTS: 130–200 MPa; YS: 100–170 MPa), and exceptional ductility (elongation: 5–30%). However, their stiffness is so poor (Young's Modulus: 45–47 GPa) that cannot meet many engineering design requirements such as space exploration and Lunar/Mars landing. Therefore, increasing modulus without degrading the strength and ductility of Mg–Li alloy has been a tough problem to be solved for many years. In this study, we have successfully made a significant breakthrough in improving the performance of Mg–Li alloys by inventing a new composition and a new processing route using CALPHAD for ultra-light Mg–Li alloys (density∼1.57 g/cm3), achieving high-strength (UTS: 335 MPa and YS: 290 MPa) and high-modulus (62.5 GPa). The origin of modulus improvement has been discovered by using a combination of SEM, TEM, XCT, nanoindentation, and neutron scattering experiments. Thermodynamically, it was found the high strength and modulus are attributed to the enhanced Mg–Mg bonding in the matrix and the increased elastic interaction forces from the lattice mismatch between the solute atoms and the solvent Mg. Meanwhile, the solution strengthening by lithium and precipitation hardening is discovered by inhibiting dislocation motion. Interestingly, age softening in Al–Li has been found to be a result of phase transformation from high-modulus particles into low ones using TEM, SANS, and nanoindentation tests.

The binary systems Fe–Mn, Fe–Ti and Mn–Ti play remarkable roles in the development of numerous structural and functional materials. As their properties are closely related to the thermodynamic behavior of the phases in the corresponding alloys, the calculation of phase equilibria is crucial for a sustainable and straightforward alloy development. In that course, thermodynamic descriptions within the framework of 3rd generation CalPhaD databases of the binary systems Fe–Mn, Fe–Ti and Mn–Ti were developed. To ensure physically meaningful estimations of the individual thermodynamic properties, advanced sublattice models for several phases, such as A12 (α-Mn), A13 (β-Mn) and C14 (hexagonal Laves phase) were applied. The modelling was supported by heat-capacity measurements for the C14–Mn2Ti phase, to provide a thermodynamically profound basis for a subsequent description of the phase relations. Through the combination of the advanced models and reliable thermodynamic data for each system, thermodynamic descriptions could be derived, which facilitate reliable calculations from 0 K to far beyond the melting point of the individual alloys.

In the CALPHAD modeling, the Compound Energy Formalism (CEF) is widely used because it is a general formalism which allows to select the most appropriate model for different kind of phases (gas phase, solid and liquid solutions, interstitial solutions, intermetallic compounds, solid oxide phases, ionic melts, phases that show chemical order-disorder transformations …). It consists in dividing the phase into sublattices and the models differ by the nature of the constituents (atoms, ions, vacancies, etc.) present in the different considered sublattices where mixing may or may not occur. For many intermetallic compounds which present a crystal structure containing many crystallographic sites on which disorder occurs, the number of parameters to be evaluated which varies exponentially with the number of sublattices becomes a problem in multicomponent systems. This yields the necessity to simplify the description and reduce the number of sublattices by combining different sites in single sublattices and this leads to that different simplifications can be used to model a phase in different systems. It is quite known that the CEF approach does not allow continuous compatibility between the different simplifications which constitutes a problem for the development of multicomponent databases.In the present paper, we present a New Approach to the Compound Energy Formalism (NACEF) which is derived mathematically from the CEF and therefore retains its full potential while providing several important improvements. As example, considering the particular case of the complex σ phase, the present work shows that NACEF makes it possible to make compatible a model based on the full description (five-sublattice model) with simplifications which can be made of it. On the other hand, it is also shown that NACEF form combined to a simple Muggianu type extrapolation allows to obtain reliable estimates of the energies of ordered configurations in multicomponent systems using those for the binary configurations obtained from DFT calculations.

The Ti–Si–C and Ti–Si–N systems were thermodynamically reassessed by using the CALculation of PHAse Diagram (CALPHAD) approach. A more suitable Gibbs energies expression of Ti3SiC2 was obtained to fit better with heat capacity data in the Ti–Si–C system. The thermodynamic parameters of the Ti–Si–N system were adjusted based on the revised Ti–Si system. A self-consistent thermodynamic database of the quaternary Ti–Si–C–N system was established. The calculated thermodynamic data and phase diagrams agree well with the experimental data. The CVD (Chemical Vapor Deposition) process for TiSiCN coatings was simulated using the newly evaluated thermodynamic parameters of the Ti–Si–C–N system. A good agreement between the predicted coating composition and the experimental ones was achieved, verifying the reliability of the thermodynamic database obtained in the present work.

Thermodynamics is a science concerning the state of a system, whether it is stable, metastable, or unstable, when interacting with its surroundings. The combined law of thermodynamics derived by Gibbs about 150 years ago laid the foundation of thermodynamics. In Gibbs combined law, the entropy production due to internal processes was not included, and the 2nd law was thus practically removed from the Gibbs combined law, so it is only applicable to systems under equilibrium, thus commonly termed as equilibrium or Gibbs thermodynamics. Gibbs further derived the classical statistical thermodynamics in terms of the probability of configurations in a system in the later 1800's and early 1900's. With the quantum mechanics (QM) developed in 1920's, the QM-based statistical thermodynamics was established and connected to classical statistical thermodynamics at the classical limit as shown by Landau in the 1940's. In 1960's the development of density functional theory (DFT) by Kohn and co-workers enabled the QM prediction of properties of the ground state of a system. On the other hand, the entropy production due to internal processes in non-equilibrium systems was studied separately by Onsager in 1930's and Prigogine and co-workers in the 1950's. In 1960's to 1970's the digitization of thermodynamics was developed by Kaufman in the framework of the CALculation of PHAse Diagrams (CALPHAD) modeling of individual phases with internal degrees of freedom. CALPHAD modeling of thermodynamics and atomic transport properties has enabled computational design of complex materials in the last 50 years. Our recently termed zentropy theory integrates DFT and statistical mechanics through the replacement of the internal energy of each individual configuration by its DFT-predicted free energy. The zentropy theory is capable of accurately predicting the free energy of individual phases, transition temperatures and properties of magnetic and ferroelectric materials with free energies of individual configurations solely from DFT-based calculations and without fitting parameters, and is being tested for other phenomena including superconductivity, quantum criticality, and black holes. Those predictions include the singularity at critical points with divergence of physical properties, negative thermal expansion, and the strongly correlated physics. Those individual configurations may thus be considered as the genomic building blocks of individual phases in the spirit of the materials genome®. This has the potential to shift the paradigm of CALPHAD modeling from being heavily dependent on experimental inputs to becoming fully predictive with inputs solely from DFT-based calculations and machine learning models built on those calculations and existing experimental data through newly developed and future open-source tools. Furthermore, through the combined law of thermodynamics including the internal entropy production, it is shown that the kinetic coefficient matrix of independent internal processes is diagonal with respect to the conjugate potentials in the combined law, and the cross phenomena that the phenomenological Onsager flux and reciprocal relationships are due to the dependence of the conjugate potential of a molar quantity on nonconjugate molar quantities and other potentials, which can be predicted by the zentropy theory and CALPHAD modeling.

The Ni-Ti-Cr ternary is a critical system for shape memory alloys. In this work, a thermodynamic reassessment of the Ni-Ti-Cr system was performed using the CALPHAD (CALculation of PHAse Diagrams) method. For the first time the martensitic transformation product phase B19′ was successfully introduced into the description of the Ni-Ti-Cr system. Firstly, the thermodynamic descriptions of the binary Ni-Ti and Ti-Cr systems were updated by focusing on the heat capacities of the Ni3Ti and NiTi2 phases and the homogeneity range of the Laves phases, respectively. The determinations of the end-member parameters of the Laves phases were supported by ab initio calculations. The present model parameters can well reproduce the phase equilibria and thermochemical properties of the Ni-Ti and Ti-Cr systems. Then, the Ni-Ti-Cr system was assessed by combining the updated binaries with available experimental data. The B19′ phase was described using a two-sublattice model, i.e., (Cr,Ni,Va)0.5(Cr,Ni,Ti)0.5, based on the atomic occupancy reported in the literature. The B2/B19′ martensitic transformation starting temperatures (MS) of the Ni-Ti and Ni-Ti-Cr alloys were accurately predicted with the assumption that the energy barrier for the martensitic transformation was 150 J/mol. The reliable prediction of MS is expected to promote the development of shape memory alloys.

The thermodynamic description of the Yb-Sb binary system is developed by means of the CALculations of PHAse Diagrams (CALPHAD) method by combining experimental data in the literature and predictions from first-principles calculations based on density functional theory (DFT) in the literature and the present work. Two pseudopotentials of Yb are compared in the present DFT-based calculations with 14 and 13 f-electrons frozen in the core, i.e., 5p66s2 and 5p66s25 d1 electrons as valence electrons, termed Yb_2 and Yb_3, respectively. It is shown that the phonon spectrum of the YbSb phase calculated using the Yb_3 pseudopotential does not have imaginary phonon modes and is subsequently used to predict its temperature dependent thermodynamic properties by the DFT-based quasiharmonic phonon calculations. The present thermodynamic database includes the Yb16Sb11 phase in addition to five intermetallic phases that were considered in previous modeling studies, i.e., YbSb2, YbSb, Yb11Sb10, Yb4Sb3, and Yb5Sb3. The high temperature orthorhombic structure of the Yb5Sb3 phase is not considered in the present work as it was stabilized by hydrogen. The associate solution model is used to describe the short-range ordering behavior in the liquid phase. The calculations from the present thermodynamic model show good agreement with thermochemical and phase equilibrium data from both the present work and the literature.

Magnesium-zinc-antimony (Mg–Zn–Sb) system is a promising cast Mg alloy system. Herein, the thermodynamic assessment of Mg–Zn–Sb ternary system was carried out using the CALculation of PHAse Diagram (CALPHAD) approach coupled with the experimental data in this work. The transformation temperatures and the isothermal section of the Mg-rich corner at 300 °C was investigated. Afterwards, a set of self-consistent parameters of Mg–Zn–Sb system were achieved combined with the thermodynamic descriptions of Mg–Sb and Mg–Zn systems as well as the updated description of Zn–Sb system in the present work. Meanwhile, the ternary Al–Zn–Sb system was re-assessed based on the available literature data. A set of reliable parameters describing the ternary Al–Zn–Sb system were obtained, and the present calculated results can reproduce most of the phase equilibria data. The thermodynamic description of Mg–Al–Zn–Sb systems was then developed using the extrapolation method from the descriptions of the constituent sub-ternary systems.

The Nb–Ni system is remodeled with uncertainty quantification (UQ) using software tools of PyCalphad and ESPEI (the Extensible, Self-optimizing Phase Equilibria Infrastructure) with the presently implemented capability of modeling site fraction based on Wyckoff positions. The five- and three-sublattice models are used to model the topologically close pack (TCP) μ-Nb7Ni6 and δ-NbNi3 phases according to their Wyckoff positions. The inputs for CALPHAD-based thermodynamic modeling include the thermochemical data as a function of temperature predicted by first-principles and phonon calculations based on density functional theory (DFT), ab initio molecular dynamics (AIMD) simulations, together with phase equilibrium and site fraction data in the literature. In addition to phase diagram and thermodynamic properties, the CALPHAD-based predictions of site fractions of Nb in μ-Nb7Ni6 agree well with experimental data. Furthermore, the UQ estimation using the Markov Chain Monte Carlo (MCMC) method as implemented in ESPEI is applied to study the uncertainty of site fraction in μ-Nb7Ni6 and enthalpy of mixing (ΔHmix) in liquid.

Steels of different classes (austenitic, martensitic, pearlitic, etc.) have different applications and characteristic areas of properties. In the present work two methods are used to predict steel class, based on the composition and heat treatment parameters: the physically-based Calphad method and data-driven machine learning method. They are applied to the same dataset, collected from open sources (mostly steels for high-temperature applications). Classification accuracy of 93.6% is achieved by machine learning model, trained on the concentration of three elements (C, Cr, Ni) and heat treatment parameters (heating temperatures). Calphad method gives 76% accuracy, based on the temperature and cooling rate. The reasons for misclassification by both methods are discussed, and it is shown that the part of them caused by ambiguity/inaccuracy in the data or limitations of the models used. For the rest of cases reasonable classification accuracy is demonstrated. We suggest that the reason of the supremacy of machine learning classifier is the small variation in the data used, which indeed does not change the steel class: the properties of steel should be insensitive to the details of the manufacturing process.

Diffusion behaviors in bcc Ti–V–Fe ternary alloys have been investigated at 1273 K and 1473 K by the diffusion couple technique. The composition-distance profiles have been retrieved from Electron Probe MicroAnalysis (EPMA) and analytically represented by the ERror Function EXpansion (ERFEX), and then the ternary inter-diffusion and impurity diffusion coefficients have been extracted by Whittle–Green and generalized Hall methods, respectively. The extracted inter-diffusion coefficients of D˜VVTi and D˜FeFeTi range from 0.89 × 10−13 m2/s to 1.86 × 10−13 m2/s and from 1.33 × 10−12 m2/s to 1.76 × 10−12 m2/s, respectively, at 1273K, and from 6.41 × 10−13 m2/s to 15.98 × 10−13 m2/s and from 82.46 × 10−13 m2/s to 147.08 × 10−13 m2/s at 1473K. D˜VVTi exhibits the similar compositional variation at both temperatures, which decrease with decreasing Fe and minimize at the Ti–V binary. D˜FeFeTi has the maximum value at Ti–corner and eventually decreases with the increasing V and Fe at 1273K, but has the maximum value at Fe-rich Ti–Fe binary and decreases with the increasing V. The atomic mobility parameters of Ti–V, Ti–Fe and V–Fe binary systems have been revised to associate with the lasted ternary interaction parameters in thermodynamics. The ternary atomic mobility parameters for diffusion have been assessed for the first time by reproducing the extracted diffusion coefficients. The composition-distance profiles and the diffusion paths have been successfully reproduced by using the optimized atomic mobility parameters showing a consistency with the experimental data and indicating the accuracy of this work.

We develop new computational and experimental methods to determine materials properties at high temperature, such as melting temperature, heat of fusion, heat capacity, and lattice constant. From density functional theory, we construct the small-size coexistence method and the SLUSCHI package to compute the properties accurately, as well as to fully automate the computation process. From experiment, we build experimental approaches, including ultra-high-temperature Drop-n-Catch (DnC) calorimetry and synchrotron X-ray diffraction on solid laser-heated aerodynamically levitated samples. Employing deep learning techniques, we build an ensemble graph-neural-networks model that predicts materials properties in milliseconds. The simultaneous development of computational and experimental approaches allows us to integrate these methods and the data generated by them.

Mg-Sr alloys are promising to fabricate orthopedic implants. The alloying of rare earth elements such as Gd may improve the comprehensive mechanical properties of Mg-Sr alloys. The information on the phase diagram and the microstructure development are required to design chemical composition and microstructure of Gd alloyed Mg-Sr alloys. The phase equilibria and the microstructure development in Mg-rich Mg-Gd-Sr alloys (Gd, Sr < 30 at. %) are experimentally investigated via phase identification, chemical analysis, and microstructure observation with respect to the annealed ternary alloys. The onset temperatures of liquid formation are measured by differential scanning calorimetry. A thermodynamic database of the Mg-rich Mg–Gd–Sr ternary system is developed for the first time via CALPHAD (CALculation of PHAse Diagram) approach assisted by First-Principles calculations. The thermodynamic calculations with the developed database enable a well reproduction of the experimental findings and the physical-metallurgical understanding of the microstructure formation in solidification and annealing.

Bi–Sb–Se–Te is one of the most important material systems for thermoelectric applications. Phase diagrams of its constituent binary and ternary systems are reviewed and assessed. The Bi–Sb–Se–Te isothermal section tetrahedron at 400 °C and liquidus projection tetrahedron were proposed. Ternary compounds are only found in the Bi–Sb–Se system. There are eight three-phase regions at 400 °C and seven primary solidification phases in the Bi–Sb–Se system, including (Bi,Sb), (Bi2)m(Bi2Se3)n, Bi2Se3, Se, Sb2Se3, Bi3Sb5Se2, Bi3Sb12Se15. In the Bi–Sb–Te system, there are four three-phase regions at 400 °C. The (Bi,Sb)2Te3 and (Bi,Sb) are continuous solid solutions. There are six primary solidification phases, including (Bi,Sb)2Te3, (Te), γ, δ, (Bi,Sb), and (Bi2)m(Bi2Te3)n. In the Bi–Se–Te system, there is one three-phase region. (Bi2)m(Bi2Se3)n and (Bi2)m(Bi2Te3)n form a continuous solid solution phase at 400 °C. There are four primary solidification phases, including Bi, (Bi2)m(Bi2(Se,Te)3)n, Bi2(Se,Te)3 and (Se,Te). In the Sb–Se–Te system, there are five three-phase regions and six primary solidification phases, including (Sb), δ-(Sb2Te), γ-(SbTe), Sb2Te3, and (Se,Te). A wide range of (Bi,Sb)2(Se,Te)3 single-phase region was observed in the Bi–Sb–Se–Te quaternary system.

The increasing concentrations of metallic Al or Al2O3 in secondary copper resources like electronic wastes motivate research into the slag chemistry of high-Al2O3 iron silicate slags for optimizing the industrial smelting process. In this study, the effect of Al2O3 in slag on the phase equilibria of the FeOx-SiO2-Al2O3 slag system was experimentally investigated at 1200 °C and pO2 of 10−8.6 atm. The high-temperature experiments were undertaken in silica and spinel crucibles in a controlled CO–CO2 gas atmosphere, followed by rapid quenching and Electron Probe Microanalysis. The equilibrium compositions of liquid slags in the tridymite primary phase field, spinel primary phase field, and the three-phase invariant point were determined. The 1200 °C isothermal section was constructed for the FeOx-SiO2-Al2O3 system, and the results showed that Al2O3 can dissolve in the liquid slags to a maximum concentration of 17 wt%. The present experimental results were compared with the predictions by MTDATA and FactSage. It was found that the present experimentally determined liquid domain agreed well with the calculations by FactSage except the invariant point. However, the present isotherm in the spinel primary phase field of spinel displayed lower Al2O3 concentrations. The present results help regulating fluxing strategies of FeOx-SiO2-Al2O3 slags and optimizing smelting operations of recycling high-alumina concentration copper resources.