Phase equilibria and microstructure development in Mg-rich Mg-Gd-Sr alloys: Experiments and CALPHAD assessment

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.     

Construction of modified embedded atom method potentials for the study of the bulk phase behaviour in binary Pt–Rh, Pt–Pd, Pd–Rh and ternary Pt–Pd–Rh alloys

A first attempt is made to simulate the solid part of the phase diagram of the ternary Pt–Pd–Rh system. To this end, Monte Carlo (MC) simulations are combined with the Modified Embedded Atom Method (MEAM) and optimised parameters entirely based on Density Functional Theory (DFT) data. This MEAM potential is first validated by calculating the heat of mixing or the demixing phase boundary for the binary subsystems Pt–Rh, Pt–Pd and Pd–Rh. For the disordered alloy systems Pt–Rh and Pt–Pd, the MC/MEAM simulation results show a slightly exothermic heat of mixing, thereby contradicting any demixing behaviour, in agreement with other theoretical results. For the Pd–Rh system the experimentally observed demixing region is very well reproduced by the MC/MEAM simulations. The extrapolation of the MEAM potentials to ternary systems is next validated by comparing DFT calculations for the energy of formation of ordered Pt–Pd–Rh compounds with the corresponding MEAM energies. Finally, the validated potential is used for the calculation of the ternary phase diagram at 600 K.     

Experimental investigation and thermodynamic modeling of the Mo–Co–B ternary system

Among the ternary borides, the Mo–Co–B system is of great interest because of its excellent hardness, toughness, and stability performance. Eight samples with 60 at.% Co were designed to investigate the isothermal section of Mo–Co–B system at 1073 K in the Co-rich portion. Scanning electron microscopy, energy-dispersion spectroscopy, electron probe microanalysis, differential scanning calorimetry, and X-ray diffraction were used to investigate the phase equilibria of the samples. The formation enthalpies of the ternary borides were obtained by first-principles calculations to serve as key information for thermodynamic assessment. By coupling the reviewed experimental data from the literature, the presently determined phase equilibria, and the calculated formation enthalpies of the compounds, the thermodynamic parameters for the Mo–Co–B ternary system were optimized and used to calculate the isothermal sections, vertical section, and liquidus projection of the system. Comprehensive comparisons showed that the calculated results are in reasonable agreement with the reported phase diagram and thermodynamic data.     

Phase diagrams and elastic properties of the Fe-Cr-Al alloys: A first-principles based study

The phase diagrams and elastic properties of the Fe-Cr-Al alloys in full-temperature and all-compositional ranges are calculated. By combining first-principles calculations and cluster variation method, binary and ternary phase diagrams are obtained. A new ternary ordered phase B32 which is different from ternary extension of binary phases appears in the ternary section around temperature of 600 K. The binary FeAl phases show an extremely high solubility for Cr, while the binary CrAl phase solid solution has a low solubility for Fe. By combining first-principles calculations and cluster expansion method, the bulk modulus, shear modulus and Poisson's ratio are calculated. The shear modulus and Poisson's ratio show a strong ordering dependency, while the ordering dependency in bulk modulus is weak. Disordered Fe-Cr alloys with a little Al solvent shows ductile property, the Al-rich corner has brittle property.     

Experimental study and thermodynamic assessment of the Zn-Fe-Ni system

The Zn-rich corner of the Zn-Fe-Ni system was experimentally determined. It was found that the T phase, which was believed previously to be an extension of the binary Γ₁ phase in the Zn-Fe system, is a true ternary compound. The composition range of this phase has been determined in this study. Based on the results of the assessments of the Zn-Fe and Zn-Ni systems carried out by the authors and using information on the Fe-Ni system available in the open literature, a preliminary assessment of the Zn-Fe-Ni system was carried out. A two-sublattice model (Fe,Ni,Zn)Zn₅ was used to describe the ternary T phase. The calculated phase boundaries in the Zn-rich corner are in good agreement with experimental results. However, reproduction of the complex homogeneity range of the ternary T phase has proven challenging.     

Thermodynamic modeling of Fe–Ti–Bi system assisted with key experiments

Phase equilibria of Fe–Ti–Bi ternary system have been studied in this work. Firstly, by using alloy sampling, the isothermal section of Fe–Ti–Bi ternary system at 773 K was determined, where the existence of a ternary phase Bi₂FeTi₄ was confirmed. Meanwhile, formation enthalpies of the intermediate phases BiTi₂, Bi₉Ti₈ and Bi₂FeTi₄, were obtained with first-principles calculations. Based on experimental data of phase equilibria and thermodynamic properties in literatures along with the calculated formation enthalpies in this work, thermodynamic modeling of Ti–Bi binary system and Fe–Ti–Bi ternary system were carried out with the CALPHAD approach. A set of self-consistent thermodynamic parameters to describe the Gibbs energy for various phases in Fe–Ti–Bi ternary system was finally obtained, with which solidification processes of two typical Fe–Ti–Bi alloys could be reasonably explained.     

Thermodynamic analysis of the Fe–Ti–P ternary system by incorporating first-principles calculations into the CALPHAD approach

A thermodynamic analysis of the Fe–Ti–P ternary system has been performed by combining first-principles calculations with the CALPHAD approach. Because of the lack of experimental information available, the enthalpies of formation of the Fe–P and Ti–P based binary phosphides were evaluated using the Full Potential Linearized Augmented Plane Wave method, and the estimated values were introduced into a CALPHAD-type thermodynamic analysis. Applying this procedure, the phase diagrams of the Fe–P and Ti–P binary systems, whose contents are uncertain so far, were calculated with a high degree of probability. The thermodynamic properties of orthorhombic anti- PbCl₂-type FeTiP were obtained following the same procedure. The calculated phase diagrams were in good accordance with previous experimental results. The ternary phosphide, FeTiP, was in equilibrium with most of the phases in the ternary system, and was dominant in the liquidus surface projection.     

Experimental investigation and thermodynamic modeling of the Al–Cu–Si system

16 ternary alloys located over the entire composition range of the Al–Cu–Si system are investigated by means of XRD, SEM/EDX and DTA. The phase equilibria associated with the kappa phase of the Cu–Si system are determined in detail and the isothermal sections at 600 and 500 ^°C are experimentally constructed. No ternary phase is observed at 600 or 500 ^°C. A thorough thermodynamic modeling for this system is then conducted based on the critically reassessed literature data and the present experimental results.     

A thermodynamic analysis of the Mo-V and Mo-V-C system

A new thermodynamic evaluation of the binary Mo-V system and the ternary Mo-V-C system using thermodynamic models for the Gibbs energy of individual phases is presented. The CALPHAD method has been used, with predictions of unknown thermodynamic quantities, to optimize a set of thermodynamic parameters taking related experimental information into consideration. The predictions are based on regularities in bonding properties and vibrational entropy of 3d-transition metal carbides. The results are summarized in tables of thermodynamic parameters, calculated binary phase diagrams and isothermal sections of the ternary phase diagram compared with experimental information. Finally the influence of ternary interaction parameters, especially in the fcc phase, on calculations of equilibria in multicomponent systems is discussed.     

Ortho-equilibrium and para-equilibrium phase diagrams for interstitial / substitutional iron alloys

A mathematical model for the evaluation of compositionally constrained thermodynamic equilibrium has recently been implemented into the computer program MatCalc. This model is applied to the calculation of para-equilibrium phase diagrams for some ternary model iron alloy systems Fe---X---C, with X = Mn, Ni, Cr and Mo. The results are compared to the corresponding full equilibrium (ortho-equilibrium) phase diagrams and the impact of each element on the austenite / ferrite / carbide transformation in steels is analyzed. The para-equilibrium phase diagrams are considerably more simple than the potentially complex ortho-equilibrium phase diagrams, showing cementite formation as the only stable carbide under para-equilibrium conditions. The driving forces for precipitation of cementite and the complex chromium carbides in the Fe---Cr---C system are evaluated as a function of the precipitate composition. The evaluation of the driving forces under para-equilibrium conditions predicts carbide precipitation behavior that agrees with experimental findings.     

Thermodynamic evaluation of the phase equilibria and glass-forming ability of the Fe–Si–B system

A thermodynamic study has been carried out on the Fe–Si–B ternary system, which is important in the development of transformer core materials and Ni-based filler metals. A regular solution approximation based on the sublattice model was adopted to describe the Gibbs energy for the individual phases in the binary and ternary systems. Thermodynamic parameters for each phase were evaluated by combining the experimental results from differential scanning calorimetry with literature data. The evaluated parameters enabled us to obtain reproducible calculations of the isothermal and vertical section diagrams. Furthermore, the glass-forming ability of this ternary alloy was evaluated by introducing thermodynamic quantities obtained from the phase diagram calculations into Davies–Uhlmann kinetic formulations. In this evaluation, the time–temperature-transformation (TTT) curves were obtained, which are a measure of the time required to transform to the minimum detectable mass of crystal as a function of temperature. The critical cooling rates calculated on the basis of the TTT curves enabled us to evaluate the glass-forming ability of this ternary alloy. The results show good agreement with the experimental data in the compositional amorphization range.     

Modified embedded-atom interatomic potential for Fe-Ni,Cr-Ni and Fe-Cr-Ni systems

A semi-empirical interatomic potential formalism, the second-nearest-neighbor modified embedded-atom method (2NN MEAM), has been applied to obtaining interatomic potentials for the Fe-Ni, Cr-Ni and Fe-Cr-Ni systems using previously developed MEAM potentials of Fe and Ni and a newly revised potential of Cr. The potential parameters were determined by fitting the experimental data on the enthalpy of formation or mixing, lattice parameter and elastic constant. The present potentials generally reproduced the fundamental physical properties of the Fe-Ni and Cr-Ni alloys. The enthalpy of formation or mixing of the disordered phase at finite temperature and the enthalpy of mixing of the liquid phase are reasonable in agreements with experiment data and CALPHAD calculations. The potentials can be combined with already-developed MEAM potentials to describe Fe-Cr-Ni-based multicomponent alloys. Moreover, the average diffusivities in the unary, some binary and ternary alloys were simulated based on present potential. Good agreement is obtained in comparison with experimental data.     

The phase equilibria in the Mg–Ni–Ca system

The three binary systems Mg–Ni, Ca–Ni and Mg–Ca have been re-optimized. A self-consistent thermodynamic database of the Mg–Ni–Ca system is constructed by combining the optimized parameters of these three constituent binaries. Lattice stability values are not added to the pure elements Mg-hcp, Ni-fcc, Ca-fcc and Ca-bcc to construct this database. The Redlich–Kister polynomial model is used to describe the liquid and the terminal solid solution phases, and the sublattice model is used to describe the non-stoichiometric phase, in this system. The constructed database is used to calculate the three binary and the ternary systems. The calculated binary phase diagrams along with their thermodynamic properties such as Gibbs energy, enthalpy, entropy and activities are found to be in good agreement with experimental data from the literature. This is the first attempt to construct the ternary phase diagram of the Mg–Ni–Ca system. The established database for this system predicted three ternary eutectic, five ternary quasi-peritectic, two ternary peritectic and two saddle points.     

Thermodynamic description of the Cu-Ge-Pb system: Experiment and modeling

Decades of scientific work dedicated to the investigation of phase diagrams gave significant benefit to industry and science. After all those years of phase diagram investigation still there is missing information about phase diagram of some ternary systems. One of those systems is Cu-Ge-Pb. It is known importance of Cu-based alloys and Ge-based alloys in electro industry. Since such combination is not tested before this work will provide information about phase diagram of ternary Cu-Ge-Pb system. In this work ternary Cu-Ge-Pb system has been tested experimentally and analytically by using Calphad model. Two isothermal sections at 600 and 400 °C and three vertical sections are experimentally tested and results were compared with calculated corresponding phase diagrams. None of the ternary compound and large solubility of third element in binary compound is not confirmed. Liquidus projection, invariant reaction and scheme of invariant reaction are presented. Scheil and Lever simulation of solid phases for Cu₈₀Ge₁₀Pb₁₀ alloy were calculated.     

Machine learning aided high-throughput first-principles calculations to predict the formation enthalpy of σ phase

The σ phase is a topologically close-packed phase and can significantly influence the performance and properties of materials. Accurate prediction the formation enthalpy of the σ phase is crucial for the development of high-performance materials. First-principles calculations based on density functional theory (DFT) have been employed to study the formation enthalpy of the σ phase, but this approach requires a amount of computational resources and time. In this study, we propose a machine learning (ML) method to predict the formation enthalpy of the σ phase. This method employs a first-principles dataset containing 1342 configurations of the binary σ phases for model training and testing. Among the algorithms used, the Multi-Layer Perceptron algorithm demonstrated the highest predictive accuracy, with the mean absolute error (MAE) of 22.881 meV/atom, which is comparable to the existing ML prediction model based on first-principles calculations. The trained model was then utilized to predict the formation enthalpy of the 1177 untrained ternary configurations, achieving a significant reduction in computational time of over 59% compared to traditional first-principles calculations. Furthermore, the model was validated for lattice parameters prediction, achieving the MAE of 0.073 Å and 0.048 Å for the a and c, respectively. A Graphical User Interface (GUI) was developed. Finally, we predicted the formation enthalpy of all the possible ternary configurations, which is comparable to the MAE of DFT-calculations itself.