Compound Energy Formalism Research Articles

Extracting metal oxide redox thermodynamics from TGA measurements requires moving beyond the linearized van ‘t Hoff approach

Thermodynamic modeling of metal oxide reduction is crucial for optimizing chemical processes and materials in systems dependent on off-stoichiometric reduction/re-oxidation cycling. Two prevalent methods for extracting reduction thermodynamics from thermogravimetric data are linearized van ‘t Hoff (VH) analysis and the compound energy formalism (CEF). This work evaluates the accuracy of these methods by constructing invertible ground truth thermodynamic models, generating hypothetical thermogravimetric data, and determining the reduction thermodynamic using both VH and CEF methods. Our findings reveal that the VH method produces absolute errors 3–5 times higher than the CEF in kJ/mol O or J/mol O K for enthalpy and entropy of reduction, respectively. In contrast, the CrossFit CEF (CF-CEF) method yields errors often less than 10 kJ/mol O or J/mol O K. Moreover, the CF-CEF method provides models based on mole fraction, temperature, and extent of reduction, while a typical VH analysis provides thermodynamics of only the specific compositions measured. Although simple to implement, the VH method suffers from significant, non-systematic errors due to entropy/enthalpy compensation and defect modeling. Consequently, we recommend the more complex but robust, CF-CEF method for extracting redox thermodynamics from thermogravimetric measurements.

Critical Evaluation and Thermodynamic Optimization of the Cr–P and Cr–Fe–P Systems

Existing thermodynamic descriptions of the whole Cr–Fe–P system are insufficiently accurate for understanding the thermodynamic behavior of the Cr–Fe–P materials during the manufacturing process. To construct a more precise and consistent thermodynamic database of the Cr–Fe–P system, thermodynamic modeling of the Cr–P and Cr–Fe–P systems was conducted using the CALculation of PHAse Diagrams (CALPHAD) approach based on critical evaluation of the experimental data. The modified quasichemical model and compound energy formalism were employed to describe the liquid and solid solutions, respectively. The Gibbs energies of stoichiometric compounds Cr3P(s), Cr2P(s), CrP(s), and CrP2(s) were carefully determined based on reliable experimental data. The ternary (Cr,Fe)3P, (Cr,Fe)2P, and (Cr,Fe)P phosphides were modeled as solid solutions considering mutual substitution between Cr and Fe atoms. In addition, the phase equilibria of BCC_A2 and FCC_A1 solutions and the liquid phase of the ternary Cr–Fe–P system were also optimized for more accurate descriptions of existing phase equilibria and thermodynamic properties data. As an application of the present database, the experimentally unexplored thermodynamic properties and phase diagrams of the Cr–Fe–P system are predicted.

Experimental investigations and thermodynamic modelling of the ternary system Pb-Mo-O

A combined experimental and modelling study into the Pb-Mo-O system has been conducted in view of the safety analysis for lead-cooled nuclear systems. The thermal expansion and low-temperature heat capacity of the ternary compounds PbMoO4 and Pb2MoO5 have been determined experimentally, as well as the melting enthalpy of PbMoO4. Moreover, XANES measurements have confirmed the hexavalent oxidation state of Mo in PbMoO4 and Pb2MoO5.A thermodynamic model of the ternary system including the ternary phases PbMoO4, Pb2MoO5 and Pb5MoO8 has also been developed in this work based on the CALPHAD methodology. For the first time, an ionic two-sublattice model is used for the liquid phase, while the compound energy formalism is used for the solid phases.

Thermodynamic modeling of Bi2Te3 in the defect energy formalism

Bi2Te3 is a promising thermoelectric material that is often touted as one of the best-performing low-temperature thermoelectric materials. As a result, it has been widely used commercially, both for clean energy generation and in cooling devices. Like many other thermoelectric materials, defects play a key role in the performance of Bi2Te3. As a result, numerous studies have attempted to experimentally and computationally map out the dominant defects in the phase, these include efforts to determine the dominant defect, estimate defect energies, and predict their concentration. The computer coupling of phase diagrams and thermochemistry (CALPHAD) is one of many tools under the auspices of the materials genome initiative (MGI) that enables the rapid design of new functional materials with improved properties. The Defect energy formalism (DEF) with a charged sublattice, an offshoot of the Compound energy formalism (CEF), provides a way to directly include first-principle defect energy calculations into CALPHAD descriptions of solid phases. The introduction of the charge sublattice enables the estimation of the free carrier concentrations in the phase. Here we apply the DEF to the Bi2Te3 system, emphasizing the robustness of the DEF in describing meaningful endmembers and the elimination of fitting parameters. Unlike previous assessments using the Wagner–Schottky defect model, we include the description of the charged defects in our assessment. The DEF with a charged sublattice provides a good prediction of the non-stoichiometry of the phase when compared with experimental data and also predicts a thermodynamic defect concentration at low temperature that is physically reasonable.

Coupled Experimental Study and Thermodynamic Modeling of the Fe–Si–N System

Some key experiments are carried out to determine the solubility of N in Fe–Si liquid solutions at various compositions, temperatures, and N2 pressures using the sampling method. The Fe–Si–N system is thermodynamically modeled using the CALculation of PHAse Diagrams (CALPHAD) approach for the first time based on critical evaluation of all available experimental data. In the present thermodynamic modeling, the liquid and solid solutions of the Fe–Si–N system are described using the modified quasichemical model and compound energy formalism, respectively. In particular, the Gibbs energies of liquid, FCC_A1, and BCC_A2 solutions are carefully determined to resolve the discrepancies among existing experimental data. The solubility product of β‐Si3N4(s) in Si steels and ferrosilicon alloys, activity coefficients, and interaction parameters of N in liquid and solid Fe–Si–N solutions are determined over a wide temperature range. As an application of the database, the N solubility in Si steels and phase constitution of nitrided ferrosilicon alloys in a wide composition, temperature, and N2 pressure ranges are predicted.

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In the present work, phase equilibria in the Li2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document}O–Al2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document}O3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_3$$\\end{document} system were experimentally studied and calorimetric measurements were performed. Based on obtained results and data from literature, thermodynamic parameters of the system were assessed. The solid solution phases were modeled using Compound Energy Formalism (CEF) and liquid phase was described by two-sublattice partially ionic liquid model. The experimental investigations for selected compositions of isothermally heat-treated samples were performed using x-ray Diffraction (XRD) and Scanning Electron Microscopy (SEM). Differential Thermal Analysis (DTA) and Differential Scanning Calorimetry (DSC) were used to measure the temperature of the reactions as well as the heat capacities, respectively. After DTA, the microstructure was analyzed using SEM. Temperature of peritectic melting of h-LiAl5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_5$$\\end{document}O8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_8$$\\end{document} was determined to be 2222 K and temperature of eutectic reaction Liq ↔\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\leftrightarrow$$\\end{document}γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma$$\\end{document}-LiAlO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document} + h-LiAl5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_5$$\\end{document}O8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_8$$\\end{document} to be 1965 K. Heat capacity of LiAlO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document} and LiAl5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_5$$\\end{document}O8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_8$$\\end{document} was measured in the temperature range of 100-1300 K. The degree of inversion of spinel phase (Al+3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{+3}$$\\end{document}, Li+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{+1}$$\\end{document})1T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_1^T$$\\end{document}:(Al+3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{+3}$$\\end{document}, Li+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{+1}$$\\end{document}, Va)2O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2^O$$\\end{document}:O4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_4$$\\end{document} was modelled assuming Al+3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{+3}$$\\end{document} and Li+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{+1}$$\\end{document} can occupy tetrahedral (T) and octahedral (O) cationic sublattices while its composition extension in Al2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document}O3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_3$$\\end{document} enriched region was described by introducing vacancies in octahedral sites. Thermodynamic description derived in the present study reproduces the degree of inversion close to that of the high-temperature spinel phase, which means that Al+3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{+3}$$\\end{document} ions preferentially occupy the tetrahedral sites. The calculated phase diagram satisfactorily agrees with the experimental results. Available experimental thermodynamic data are also reproduced within uncertainty limits.

Experimental investigation and thermodynamic assessment of the Na2O–Al2O3–CaO system

Thermodynamic investigation of the Na2O–Al2O3–CaO system plays an important role in the applications of HACs and refractory. However, the phase relationships in the Al2O3-rich corner of the isothermal section at 1473 K for this system are uncertain. Key experimental investigation was performed and an improved method was developed to prepare the samples. Four three-phase regions of NC3A8+CA2+β-Al2O3, CA2+CA6+β-Al2O3, CA6+β-Al2O3+Al2O3 and NC3A8+NA6+β-Al2O3 were confirmed by X-ray diffraction. The Na2O–Al2O3 system was reassessed by CALPHAD method and the Na2O–Al2O3–CaO system was optimized by using the ionic two-sublattice liquid model (Na+1,Al+3,Ca+2)P(AlO2−1,O−2)Q to describe the liquid phase and a sublattice compound energy formalism model (Na+1)2(Ca+2,Na+1)1(Al+3)4(O−2)8(Va,Na+1)1 to describe the ternary compound NCA2. A set of self-consistent thermodynamic parameters was finally obtained, satisfactorily reproducing the experimental data and predicting the liquidus projection and primary phase regions. The present thermodynamic investigation of the Na2O–Al2O3–CaO system contributes to compositional design in HACs and refractory.

Phase equilibria of advanced technology uranium silicide-based nuclear fuel

The phases in uranium-silicide binary system were evaluated in regards to their stabilities, phase boundaries, crystal structures, and phase transitions. The results from this study were used in combination with a well assessed literature to optimize the U-Si phase diagram using the CALPHAD method. A thermodynamic database was developed, which could be used to guide nuclear fuel fabrication, could be incorporated into other nuclear fuel thermodynamic databases, or could be used to generate data required by fuel performance codes to model fuel behavior in normal or off-normal reactor operations. The U3Si2 and U3Si5 phases were modeled using the Compound Energy Formalism model with 3 sublattices to account for the variation in composition. The crystal structure used for the USi phase was the tetragonal with an I4/mmm space. Above 450°C, the U3Si5 phase was modeled. The composition of the USi2 phase was adjusted to USi1.84. The calculated invariant reactions and the enthalpy of formation for the stoichiometric phases were in agreement with experimental data.

Thermodynamic study of the Ca-Cu-Zn ternary system with key experimental measurements

Zinc-based alloys, particularly those alloyed with elements like Ca, Cu, and others, have emerged as promising candidates for biodegradable implants. Therefore, the purpose of this study was to investigate the phase equilibrium relationship of the Ca-Cu-Zn ternary system in the Cu-Zn rich region for Zn-based biodegradable materials development. In this work, 21 and 19 samples were prepared and annealed at the temperatures of 350 and 400 °C, respectively. And Then, these samples were analyzed using analytical techniques including scanning electron microscopy (SEM) coupled with energy dispersion spectroscopy (EDS), electron probe microanalysis (EPMA), and powder X-ray diffraction analysis (XRD). The present obtained results indicate the existence of five regions exhibiting three-phase equilibrium and sixteen regions exhibiting two-phase equilibrium within these two isothermal sections. Additionally, differential scanning calorimetry (DSC) was utilized to determine the phase transition temperatures of alloys with constant values of 12 at% Cu and 40 at% Ca, respectively. Moreover, the CALPHAD method was employed to optimize the thermodynamic parameters of the phases of Ca-Cu-Zn ternary system. The model of the liquid phase was performed using the modified quasi-chemical model (MQM). The compound energy formalism (CEF) model was used to represent the Gibbs free energy of the solid phases. Finally, the experimental measured results could be reproduced well using the present obtained thermodynamic parameters.

Thermodynamic Evaluation and Optimization of the As–Cd, As–Zn and As–Cd–Zn Systems

In this work, a critical evaluation of all available phase diagrams and thermodynamic data for the As–Cd, As–Zn and As–Cd–Zn systems has been performed and thermodynamic assessments over the whole composition ranges are presented using the CALPHAD method. To predict thermodynamic properties and phase equilibria for these systems, the Modified Quasichemical Model for short range ordering was used for the liquid phase and the Compound Energy Formalism was used for the solid solutions. The optimized binary systems are in good agreement with existing experimental data. Within the ternary system predicted phase equilibria, the {text{ZnAs}}_{2}{-}{text{CdAs}}_{2} and {{text{Zn}}_{3}text{As}}_{2}{-}{{text{Cd}}_{3}text{As}}_{2} sections are in good agreement with the experimental data. Also, the eutectic temperature is accurately optimized for {text{ZnAs}}_{2}{-}{{text{Cd}}_{3}text{As}}_{2} and {{text{Zn}}_{3}text{As}}_{2}{-}{text{CdAs}}_{2}. However, the calculated liquidus of these two joins are less satisfactory compared to the experimental data. This is most likely due to the polymerization behavior of arsenic and its multivalence, which is not considered by the model used in this work.

Thermodynamic evaluation and optimization of the As–Co, As–Fe and As–Fe–S systems

A critical evaluation of all available phase diagram and thermodynamic data for the As–Co and As–Fe binary systems as well as the As–Fe–S ternary system has been performed and thermodynamic assessments over the whole composition ranges are presented using the CALPHAD method. To predict thermodynamic properties and phase equilibria for these systems, the Modified Quasichemical Model (MQM) for short range ordering was used for the liquid phases. The Compound Energy Formalism (CEF) was used for the solid solutions. Since Co and Fe are ferromagnetic, magnetic contributions were added to describe the Gibbs energy of cobalt and iron rich solid solutions. Important uncertainties remain for the liquidus of As-rich regions in the binary subsystems.

Critical evaluation and thermodynamic modeling of the Al–Li binary system

Al–Li alloys have extensive applications in aircraft and spacecraft due to their high mechanical performance and lightness in weight. In this work, the thermodynamic property and phase equilibria data of the Al–Li system available in the literature were critically reviewed and evaluated, thermodynamic assessments of the entire Al–Li system were then carried out based on the reliable experimental thermochemistry and phase equilibria data. The liquid phase was described using both the modified quasi-chemical model in pair approximation and the regular solution model. The compound energy formalism and the regular solution model were used to describe the Gibbs energies of the solid phases. The split four sublattice compound energy formalism was used to model the Gibbs free energies of the ordered AlLi (B32) and Al3Li (L12) superlattice structures. The reliable thermodynamic property and phase equilibria data can generally be represented by the currently obtained sets of self-consistent thermodynamic descriptions.

Critical Evaluation and Thermodynamic Re-Optimization of the Si-P and Si-Fe-P Systems.

Thermodynamic modeling of the Si-P and Si-Fe-P systems was performed using the CALculation of PHAse Diagram (CALPHAD) method based on critical evaluation of available experimental data in the literature. The liquid and solid solutions were described using the Modified Quasichemical Model accounting for the short-range ordering and Compound Energy Formalism considering the crystallographic structure, respectively. In the present study, the phase boundaries for the liquidus and solid Si phases of the Si-P system were reoptimized. Furthermore, the Gibbs energies of the liquid solution, (Fe)3(P,Si)1, (Fe)2(P,Si)1, and (Fe)1(P,Si)1 solid solutions and FeSi4P4 compound were carefully determined to resolve the discrepancies in previously assessed vertical sections, isothermal sections of phase diagrams, and liquid surface projection of the Si-Fe-P system. These thermodynamic data are of great necessity for a sound description of the entire Si-Fe-P system. The optimized model parameters from the present study can be used to predict any unexplored phase diagrams and thermodynamic properties within the Si-Fe-P alloys.

Experimental investigation and thermodynamic assessment of the Zn–Cu–Sr ternary system

Zn–Cu–Sr alloys play a crucial role in the development of biodegradable implant materials based on zinc. The current study aimed to investigate the phase equilibria of the Zn–Cu–Sr ternary system in the Cu–Zn-rich region, through experimental analysis. For this purpose, fifteen and fourteen samples were respectively prepared and equilibrated at 350 and 400 °C, to determine the isothermal sections. The equilibrated alloys were then subjected to various analytical techniques such as scanning electron microscopy (SEM) equipped with energy dispersive spectrometry analysis (EDS), electron probe microanalysis (EPMA), and powder X-ray diffraction analysis (XRD). The analysis revealed the presence of five three-phase equilibria and ten two-phase equilibria in the two isothermal sections. Differential scanning calorimetry (DSC) was used to investigate the phase transformation temperature with constant values of 8 at. % Sr and 30 at. % Cu. The obtained experimental results were used to perform a thermodynamic assessment of the Zn–Cu–Sr system especial in Zn-rich region using the calculation of phase diagrams (CALPHAD) method. The modified quasi-chemical model (MQM) was used to model the liquid solution, while the compound energy formalism (CEF) was used to represent Gibbs free energies of the solid phases. The present obtained thermodynamic parameters were found to accurately reproduce the experimentally measured phase relationships in the Zn–Cu–Sr ternary system.

Thermodynamic description of molten salt systems: KCl-LiCl-NaCl and KCl-LiCl-NdCl3

The mixture of KCl and LiCl has been used as electrolyte in the electrorefining process to recover uranium from used nuclear fuels due to the low melting point. However, lanthanides and sodium in the reactor waste continuously dissolve into it and thus alter its thermodynamic properties. To understand the thermodynamic behavior evolution of the electrolyte with the accumulation of impurities, thermodynamic modeling for KCl-LiCl-NaCl and KCl-LiCl-NdCl3 and four constituent binary systems in the entire composition space was performed using the CALPHAD (CALculation of PHAse Diagrams) approach. The ionic liquid was described by the two-sublattice model, where neutral species were introduced to consider short-range ordering (SRO) within the melt, whereas the solid solution was modeled based on the Compound Energy Formalism. Literature data on phase equilibria and thermochemical properties were critically evaluated and used during the optimization of thermodynamic parameters for KCl-LiCl-NaCl and KCl-LiCl-NdCl3 and their subsystems. The calculated phase diagrams and mixing enthalpies are in good agreement with the experimental data. The thermodynamic modeling for the KCl-LiCl-NdCl3 system was carried out for the first time. To fill the gap in experimental measurement, enthalpy of mixing for the KCl-LiCl-NdCl3 melt was estimated using the surrounded-ion model. These data then served as critical inputs for thermodynamic optimization. The present study can provide insights into thermodynamic property evolution of the electrolyte and solubility limit of various impurities during the electrorefining process.

Thermodynamic Evaluation and Optimization of the Ag-As-S system

A critical evaluation of all available phase diagram and thermodynamic data has been performed for Ag-As, Ag-S, As-S and Ag-As-S systems, including a detailed review of the gaseous species involved in the As-S system. Thermodynamic assessments over the whole composition range for these four systems are presented using the CALPHAD method. To predict thermodynamic properties and phase equilibria, the Modified Quasichemical Model for short range ordering was used for the liquid phases, and the Compound Energy Formalism was used for the solid solutions. For the As-S binary system, natural logarithm terms for the temperature dependence of the excess Gibbs energy of the liquid solution have been used. This led to a significant improvement compared with the previous assessment of this system. The optimization of Ag-As-S systems is in good agreement with the existing experimental data.

Stabilization of Equiatomic Solutions Due to High-Entropy Effect

The stability of solid-solution phases in FCC and BCC lattices was examined in multi-component alloys based on the CALPHAD technique using the compound energy formalism and regular solution model. From the thermodynamic calculations, it was found in ternary systems that the single solid-solution phase became stable around the equiatomic composition where the configurational entropy was the largest value. The transition temperature from the disordered phase to ordered phase(s) or miscibility gap(s) decreased with the increasing number of elements in the system. The order-disorder transition temperature on the FCC lattice was affected by the end member of the ordered phases existing at the equiatomic composition, whereas it was not significant for the order-disorder transition in the BCC lattice. The single solid-solution phase region at equiatomic compositions was affected by variations in the interaction parameters. In multi-component systems, the variations were averaged with increasing the number of elements in the system. This suggests that high-entropy alloys can afford a variety of elements. This study shows that the disordered state can be formed in multicomponent systems around the equiatomic composition and suggests clearly that due to the high-entropy effect, the solution phases are stabilized.

Thermodynamic modeling of the Mg–Nd–Sr ternary system with key experimental investigation

The phase equilibria in the Mg-rich region of the Mg–Nd–Sr ternary system at 300 and 350 °C were established using equilibrated-sample method. Powder X-ray diffraction (XRD) technique and scanning electron microscopy (SEM) equipped with energy-dispersive spectroscopy (EDS) were used for phase composition determination. Four three-phase equilibria and four two-phase equilibria have been experimentally determined at both isothermal sections of 300 and 350 °C. The phase equilibria relationships in the Mg-rich side were studied. The major invariant reaction temperatures of vertical sections with 80 at. % Mg and 10 at. % Sr were determined with differential scanning calorimetry (DSC) test. Moreover, thermodynamic modeling of Mg–Nd–Sr ternary system has been carried out by CALPHAD method based on the present key experimental results. The liquid solution was described using the modified quasi-chemical model in the pair approximation (MQMPA). The compound energy formalism (CEF) was used for the solid phases. The present obtained thermodynamic database of Mg–Nd–Sr ternary system will provide an important support for the Mg-based biodegradable implant development.

Investigating the Al/Si mixed site occupancy in the β-AlFeSi phase.

This work investigates the mixed site occupancy of aluminium and silicon atoms in the β-AlFeSi phase. For this purpose, the six mixed Al/Si sites of the β-AlFeSi structure were considered independent and alternatively substituted by Al or Si, thus generating 64 ordered structures or end-members. The enthalpy of formation of each end-member was calculated by DFT. These calculations allowed us to derive the enthalpy of mixing of the solid solution at 0 K, over a wide range of chemical compositions, from the Al-Fe binary system to the Si-Fe binary system. In addition, the heat capacities of the solid solution were determined using a Debye model based on the calculation of the elastic constants and the equation of state of each end-member. These heat capacity values were used along with the enthalpy of formation we calculated to determine the Gibbs free energies of all the end-members of the β-AlFeSi structure. Finally, the configurational entropy of mixing from the Compound Energy Formalism (CEF) for the configurational entropy of mixing was subsequently used to calculate the occupation fractions of the Si sites on the Al sites of the β-AlFeSi structure, at 300 K and 938 K, the latter being the thermal decomposition temperature of this compound. These original site occupancy data were used to quantify the chemical ordering of the solid solution and to compare different sublattice (SL) models. We thus highlight that the SL model of the β-AlFeSi solution most commonly accepted in the literature generates considerable errors in its thermodynamic description, contrary to the model proposed in this paper, which is both simple and particularly accurate, consisting in merging the sites Al(1)-Al(6), the sites Al(2)-Al(3) as well as the sites Al(4)-Al(5).

Overcoming significant challenges in extracting off-stoichiometric thermodynamics using the compound energy formalism through complementary use of experimental and first principles data: A case study of Ba1-xSrxFeO3-δ

The compound energy formulism (CEF) is a powerful framework to describe the thermodynamics of metal oxides as a function of off-stoichiometry, temperature, and composition. The thermodynamic properties are crucial materials design attributes in metal oxide-based oxygen-exchange chemical processes. Despite the richness of information an accurate CEF model provides, a method to determine a unique and accurate fit for oxygen-exchange materials remains elusive. This contribution details a method for fitting the CEF model that overcomes the current fitting challenges through three innovations: 1) the combination of density functional theory calculations with experimental data decorrelates excess terms and delineates the enthalpic/entropic contributions to the Gibbs free energy; 2) a systematic determination of the important CEF model terms, removing thermodynamic predetermining human intervention; 3) a self-consistent solution of the starting oxygen off-stoichiometry (δ0) of thermogravimetric measurements. Thus, our method enables the reliable extraction of off-stoichiometric metal oxide thermodynamic properties and facilitates rapid materials compositional screening, and reliable process design of systems dependent on off-stoichiometric redox-active metal oxides. We apply this method to a BaxSr1-xFeO3-δ test case. We find by systematically examining the performance of the CEF model fit with and without each innovation that all three innovations are necessary for an accurate fit. We determined that reduction enthalpy is higher and more sensitive to off stoichiometry when the Sr fraction is large (139.5 and 185.3 kJ/mol O2 for SrFeO3 at δ = 0 and δ = 0.5, respectively vs. nearly constant 83 kJ/mol O2 for BaFeO3). However, the reduction entropy is mostly insensitive to Sr fraction, but highly dependent on δ suggesting larger contributions in the non-configurational entropy parameters.