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.

Through the judicious use of “virtual elements” that have zero atomic mass but that are included in the materials balances, it is possible to apply a variety of constraints to chemical equilibrium calculations without the necessity of writing dedicated software for each individual application. Several examples are presented, including the suppression of decomposition of metastable molecules and ions or redox reactions in aqueous solutions, the suppression of internal equilibria in molten salts and ceramics, the calculation of the surface tension of solutions, following the course of reactions with time, paraequilibrium calculations and limiting the extent of a reaction.

The materials genome initiative was announced by US President Obama 2011. Its actual meaning was initially rather unclear and it seemed as it could contain everything. Nevertheless, in their Scripta paper 2014 Kaufman and Ågren argued that the materials genome, analogously with biological genomes, should be defined as a set of information (databases) allowing prediction of materials structure as well as its response to processing and usage conditions. It thus seemed natural that CALPHAD thermodynamics and kinetics should play an essential role.In the present paper the role of CALPHAD as a way of processing different kind of materials data will be emphasized. The extension from thermochemistry to other properties, industrial applications and extrapolation far from equilibrium will be reviewed.

Light steel profiles are steels with medium C, Mn and Si contents. In these steels internal cleanness is important for mechanical properties and surface quality, both during hot rolling and during the coating of the final product. Production costs of these steels are defined by the total heat time, from charging the Electric Arc Furnace (EAF) to starting continuous casting. Ladle furnace (LF) processing has a large influence on heat time and costs related to deoxidation, meeting composition requirements with a high yield of oxidizing alloying additions and proper rinsing to promote homogenization and cleanness. Thermodynamics and kinetics control how slag and steel oxidation levels evolve, as well as the evolution of the non-metallic inclusion population since at this stage primary oxides are the main inclusions. A model based on solid thermodynamic and kinetics fundamentals would be extremely useful for the producer of these steels, so that the evolution of the heat in the ladle furnace could be properly understood and, afterward, improve. In this work, we present the development of an “Effective Equilibrium Reaction Zone” (EERZ) for the LF processing of these steels, using a well-established thermodynamic database for metal and slags and a module of commercial computational thermodynamic software. We aim at demonstrating that these easily accessible and useable tools can be very effective to help in industrial process development. Thus, based on three selected industrial heats we adjust the kinetic parameters to describe the processes occurring for tapping from the EAF to releasing the heat for continuous casting. Volumetric mass transfer coefficients, heat transfer coefficients and other less critical variables are adjusted based on these heats and the model is then tested in seven further heats of the same steel grade. Results indicate that it is possible to achieve a good fit of the model to the industrial process and this helps identifying critical process variables to optimize the process schedule of these steels. Furthermore, we identify possible current limitations of the modeling strategy as suggested in the preliminary steps for the development of a EERZ model that may be coupled for on-line use, the support LF operators.

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.

Calculation of Phase Diagrams (Calphad) is a method of using thermodynamic models obtained from experimental data to perform thermodynamic calculations. The next step to advancing this methodology is to account for the inconsistency or lack of data when assessing thermodynamic systems. A generalized method of propagating uncertainty through Calphad calculations by local expansion is proposed such that any type of Gibbs free energy model or number of components can be used. This method is faster than Monte Carlo approaches as only a single equilibrium calculation is needed for uncertainty propagation and also improves upon previous approaches to uncertainty propagation by its generalization to any thermodynamic system. As a case study, the Mg–Si system was assessed using a Bayesian approach and various thermodynamic calculations were performed comparing uncertainty quantification by Monte Carlo and the method in this work. Sensitivities (derivatives with respect to model parameters) were calculated on the Fe–Cr–Ni system and compared with sensitivities determined by finite different method.

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.

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 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.

In this paper, the thermodynamic descriptions of the binary Al–V and ternary Al–Cr–V systems were updated by using the CALculation of PHAse Diagram (CALPHAD) approach. For the Al–V sub-binary system, in order to unify the models of Al8V5 phase and βAl8Cr5 phase with the same structure, Al8V5 phase was described by a two-sublattice model of (Al,V)8(Al,V)5. The calculated phase equilibria and thermodynamic properties in the binary Al–V system showed better agreement with more experimental information than those in the previous assessment. As for the Al–Cr–V ternary system, the newly updated thermodynamic descriptions of the Al–V system as well as those of the Al–Cr and Cr–V binary systems from the literature were adopted. A set of reliable and self-consistent thermodynamic parameters for the Al–Cr–V ternary system were obtained. Good agreement between the calculated results and the experimental data was achieved.

The thermodynamic properties of the intermediate compound KCaCl(NO3)2 are important for the thermochemical description of the KCl–CaCl2–KNO3–Ca(NO3)2 reciprocal system, for its utilization as a phase change material or heat transfer fluid in thermal energy storage. The formation of KCaCl(NO3)2 as intermediate compound in this reciprocal system was confirmed by X-Ray diffraction analysis. Aside from the main phase of KCaCl(NO3)2, other crystalline phases were also identified. Kinetic effects were observed during the study of this compound. For example, the enthalpy of fusion of the 50KCl–50Ca(NO3)2 mixture depends on the cooling rate of the previous cycle. The samples, which crystallized with 10 K/min, show higher enthalpy of fusion (22.4 kJ/mol) compared to samples cooled with 0.5 K/min - 16.3 kJ/mol. This effect is related to the different ratio of crystallized phases (KCaCl(NO3)2, KCaCl3, Ca(NO3)2 and various Potassium Calcium nitrates) in this mixture. The heat capacity and enthalpy increment of the KCaCl(NO3)2 compound in solid and liquid phases are determined with two different DSC instruments. The selected value of the fusion enthalpy for KCaCl(NO3)2 compound is 22.4 ± 1.2 kJ/mol, which was obtained after fast (10 K/min) cooling. The comparison of the experimental results with calculated values based on Neumann-Kopp rule confirms the reliability of the obtained data.

The Ni–Ti-Hf ternary system has been assessed by CALPHAD method combined with first-principles calculations. Based on experimental results in the literature, the whole system is evaluated completely. The liquid, γ-Ni, β-(Hf, Ti), α-(Hf, Ti) are modeled using the substitution solution model. Ordered and disordered bcc phases use single Gibbs energy description, while the other intermetallic compounds use corresponding sublattice model to describe. The formation enthalpies of the hypothetical end members of each phase in Ni–Ti-Hf ternary system have been calculated using first principles and incorporated into the modeling. According to the experimental results, the sublattice model of the newly discovered ternary phase τ is determined as (Ni)11(Hf, Ti)14 and the phase is added to the thermodynamic optimization. By comparing the experimental data with the calculated results, the thermodynamic description of this work is in good agreement with the reasonable experimental results. The calculated Ni–Ti-Hf ternary system is summarized by means of isothermal sections, vertical sections, liquidus projection and reaction scheme.

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.

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.

Ni–Ti–V is one key sub-system of the Ni-based superalloys. However, there is a lack of the investigation on diffusion kinetics for Fcc Ni–Ti–V alloys. In order to remedy this situation, we combined the diffusion couple experiment and calculation by CALTPP (CALculation of ThemoPhysical Properties) software to investigate diffusion kinetics for Fcc Ni–Ti–V alloys. Totally, 12 ternary Fcc Ni–Ti–V diffusion couples were prepared and annealed at 1273, 1373 and 1473 K, for which the composition profiles were measured by EPMA (Electron Probe MicroAnalysis). Based on the measured composition profiles and available thermodynamic descriptions, the interdiffusivities and atomic mobility parameters for Fcc Ni–Ti–V system were then calculated by the numerical inverse method integrated with CALTPP software. The simulated composition profiles by CALTPP software agree well with the measured ones. Moreover, the interdiffusivities obtained by CALTPP software were confirmed to be reliable through the good consistence with those calculated by M − K (Matano-Kirkaldy) method. Thus, the present numerical calculation by CALTPP software was verified to be reasonable. Finally, based on the presently obtained mobility parameters and available thermodynamic descriptions for Fcc Ni–Ti–V alloys, the three-dimensional surfaces of the interdiffusivities, activation energies and frequency-factors against Ti and V composition were predicted. The present work provides reliable kinetic parameters for Fcc Ni–Ti–V alloys, which can be included in the diffusion database of Ni-based superalloys for material design.

Application of the Neumann-Kopp rule to aluminum containing compounds produces a kink in the specific heat caused by the description of pure aluminum in the SGTE Unary database. Two ways to get rid of this problem are investigated. In a first step, we tried to redefine the description of aluminum above its melting point using a reverse Neumann-Kopp approach. After a systematic review of the experimental Cp data for all the aluminum based compounds, we could evaluate the accuracy of the Neumann-Kopp rule. We could find a nearly systematic overestimation of the Cp of the order of 15%. This makes the use of the reverse Neumann-Kopp approach inapplicable. In a second step, based on this analysis of the available data, we propose another approach consisting in defining the Cp of a compound by a composition average of the Cp of the pure elements, not at the same temperature as in the Neumann-Kopp rule, but rather at a temperature normalized to the melting point of each pure element and the compound. Not only are the results in much better agreement with the experimental data than the conventional Neumann-Kopp rule, but also, there is no longer a need to define the Cp of aluminum above its melting point.

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.

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.

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.