Subregular Solution Model Research Articles

Gibbs–Schreinemakers–Meijering Mechanisms for 3-Phase Miscibility Gap Formation

A review is given of work by J.W. Gibbs, F.A.H. Schreinemakers and J.L. Meijering on the stability of solutions to a continuous change in a phase. Gibbs laid the theoretical foundation with mathematical equations for multicomponent systems written in terms of the Gibbs energy. Later, Schreinemakers and Meijering predicted mechanisms by which 3-phase miscibility gaps could form in a ternary system. Their mechanisms were described in a series of phase diagram sketches, but the mechanisms have not yet been confirmed by experiment or calculation. In this work, phase diagrams were calculated using regular and subregular solution models and compared with the sketches. Several sketches contained unusual features like multiple critical points that form spontaneously at a single composition. In one mechanism a miscibility gap forms spontaneously in the metastable supersaturated phase of an existing miscibility gap. The calculated diagrams show from a thermodynamic standpoint that Schreinemakers’ and Meijering’s mechanisms can occur. However, there is no experimental evidence that they have occurred.

Thermodynamics and vaporization of ceramics based on the Y2O3-ZrO2 system studied by KEMS

The ceramics based on the Y2O3-ZrO2 system is promising for development of refractory materials including ultra-high temperature ceramics. In the present study the samples of the Y2O3-ZrO2 system were synthesized by the solid-state method. The vaporization processes and thermodynamic properties of the Y2O3-ZrO2 system were studied by the Knudsen effusion mass spectrometric method. The vapor composition, the YO, ZrO, ZrO2 and O partial pressures over the samples as well as the Y2O3 and ZrO2 activities of the Y2O3-ZrO2 system were determined at the temperature 2660 K. Thermodynamic properties (i.e. excess Gibbs energies) of the Y2O3-ZrO2 system obtained in the present study can be described in terms of the subregular solution model.

Stable structure and electronic properties of Ru1−xTixO2 rutile type solid solutions from DFT calculations

AbstractTo design solid solution is an effective strategy for functional materials. The complex electron correlation and non‐equilibrium atomic interaction are the primary factors to impact the macro‐/micro‐structure and application performances of solid solution. Hence, this article concentrates on the stable crystal structure and the corresponding electronic structure of Ru1−xTixO2 solid solution by using density functional theory calculations. Based on analyzing the atomic interaction between solute and solvent atoms, the most stable supercell models for the specific solid solubility were obtained. In Ru1−xTixO2 solid solution, solute (and solvent) atoms preferentially arrange along the [110] direction, and tend to occupy the nearest sites of identical atoms, to arrange the symmetric ordered substitution configurations. Using the subregular solution model, the mixing enthalpy can be represented by the following expression: . The calculated phase diagram is consistent with the experimental results. Thus, the calculated results in this article are reliable and can provide more meaningful information. In the cases of Ru‐rich solid solutions, the metastable phase exists in the range of x < 0.27; while in the cases of Ti‐rich solid solutions, the metastable phase exists in the range of x > 0.81. Interestingly, the electron density of Ru1−xTixO2 solid solutions at Fermi level has a stronger dependence on the Ru composition, which intrinsically influences its electronic structure and optical properties. According to the basic application principle, Ru1−xTixO2 solid solutions with lower (x < 0.27) and higher (x > 0.81) solid solubility could act as suitable candidate for the applications in field electrochemistry and photocatalysis.

Experimental thermodynamic investigation of the liquid Cu-Li-Sn system by Knudsen Effusion Mass Spectrometry

The thermodynamic activity of lithium was measured by Knudsen Effusion Mass Spectrometry (KEMS) within the Cu-Li-Sn system in the liquid region. For this purpose, eleven samples were studied between 773 and 1078 K in the Sn-rich part of the Cu-Li-Sn phase diagram. Mixing enthalpies, entropies and Gibbs energies were derived directly from the measured thermodynamic activity. The measured thermodynamic activities of Li were fitted to the Redlich-Kister-Muggianu (RKM) sub-regular solution model, by taking ternary interactions into account, using a mathematical regression procedure. Thereby, the partial and integral excess thermodynamic properties, the activities of Li, Cu and Sn and the ternary interaction L-parameters, as a function of temperature, were obtained.

Calorimetric Study of the Formation of Liquid Ag–Cu–Sn Alloys. Enthalpy of Mixing for the Boundary Binary Cu–Ag, Cu–Sn, and Ag–Sn Systems at 1150°C

Drop calorimetry is used to study the mixing of metallic Cu–Ag, Cu–Sn, and Ag–Sn liquid alloys at 1150°C. The calorimeter sensitivity is calibrated using pure components of the alloys and sapphire standards. The experimental results are described by analytical expressions in terms of quasi-chemical approximation for the subregular solution model. The obtained data are necessary to study the alloy formation in the ternary model Ag–Cu–Sn system.

Measurement and Thermodynamics of Carbon Solubilities of Molten Si-Fe, Si-Ni and Si-Fe-Cr Alloys at 2073 K

A study has been carried out to evaluate the equilibrium phase relations of molten Si–Fe, Si–Ni, and Si–Fe–Cr alloys saturated with either silicon carbide (SiC) or graphite. The measured carbon solubilities at 2073 K were 0.19-6.6 mol% for Si– (24.1–70.1) mol%Fe, 0.061–5.2 mol% for Si– (30.0–85.0) mol%Ni, and 1.1–3.9 mol% for Si– (50-x) mol%Fe – x mol%Cr (x=10.4–40.1) alloys, respectively. Quasi-chemical model, which assumes carbon atoms to be introduced into interstitial sites of Si-Fe, Si-Ni, and Si-Fe-Cr solvents and obstruct the bonding between solvent atoms, was adopted to evaluate the activity coefficient of carbon in each alloy, and its estimation reproduced fairly well the tendency of measured carbon solublities. On the other hand, estimation using the sub-regular solution model often overestimated the carbon solubilities. It was thus found that the carbon behaviors in silicon-transition metal alloys can be well described by using the quasi-chemical model.

Thermodynamic description for the NaF-KF-RbF-ZnF2 system

Thermodynamic evaluations and optimizations of the NaF-ZnF2, KF-ZnF2, RbF-ZnF2, NaF-RbF and KF-RbF systems were determined by use of the CAlculation of PHAse Diagrams (CALPHAD) approach in this work. The subregular solution model was applied to model liquid phase, while the stoichiometric compound model was used to describe the intermediate phases. All the model parameters were evaluated on the basis of the required thermochemical and phase equilibrium data from experimental measurements in literature and theoretical predictions from first-principles method. A set of self-consistent and reliable thermodynamic database for the NaF-KF-RbF-ZnF2 quaternary system was finally derived by merging the model parameters of relevant binary systems (model parameters of NaF-KF were adopted from the literature) based upon the Muggianu extrapolation model. The invariant reactions of ternary systems from NaF-KF-RbF-ZnF2 were obtained according to the database. This work will provide beneficial instructions in the relevant multi-components molten salt design in industrial fields.

A statistical thermodynamic model with strong adaptability for liquid mixtures

How to accurately predict thermodynamic data of highly irregular liquid mixtures, such as Ca-Pb-Sb and Fe-C-Cr, is always an interesting but knotty problem no matter in the field of metallurgy or chemistry. Based on the same theory foundation (Scott's two-fluid theory and Scatchard-Hildebrand theory) but on different assumptions, a model with two different forms for excess Gibbs energy of liquid mixtures is derived from statistical thermodynamics, in which volume parameter and energy parameter are separated. Under the same conditions, compared with classical local composition models and sub-regular solution model (SRSM), this model not only has a significant increase (more than 30%) in predictive capability, but also shows stronger adaptability to various highly irregular systems, especially to super negative deviation systems (SNDS) whose activity can be as low as 10−11 and to low concentration saturated solution (LCSS) with strong asymmetry.

Thermodynamic study of fcc-fct-fco multi-step structural transformation in Mn-Ni antiferromagnetic shape memory alloys

Shape memory effect in Mn-Ni antiferromagnetic alloys is closely related to the structural transformations that happen in this system, especially the fcc→fct→fco multi-step structural transformation. In order to discuss the change of Gibbs free energy of this alloy system during the multi-step transformation process, thus study the mechanism of shape memory effect from the standpoint of thermodynamics of fcc→fct→fco transformation, the unknown thermodynamic parameters of fct and fco phases in Mn-Ni alloys must be determined. In this work, based on the application of subregular solution model of the binary system and equilibrium relationships between different phases given in the Mn-Ni phase diagram, these unknown thermodynamic parameters were determined by solving overdetermined equation systems, and dependencies of Gibbs free energies of every phase in Mn-Ni antiferromagnetic alloys on temperature were obtained. The results of calculations show that the critical driving force of fcc→fct martensitic transformation is about 10 J/mol and higher than that of fct→fco transformation (about 1 J/mol), while the enthalpy, entropy and heat capacity of fcc→fct transformation are several times higher than the equivalent parameters for the fct→fco transformation. The dependences of critical chemical driving force, enthalpy, entropy and heat capacity on the alloy composition and temperature were calculated and discussed for the fcc→fct→fco multistep structural transformation. Based on these results, the orders of the two transformations in the fcc→fct→fco multi-step transformation happening in this system were compared with that of the single-step phase transformation, and were found to be all of first-order and to be weakened during the multi-step transformations.

CALPHAD modeling of metastable phases and ternary compounds in Ti-Al-N system

The development and evaluation of thermodynamic databases for the Ti-Al-N system is important for a wide range of engineering alloys. In this work, we resorted to the integration of ab initio thermodynamics, experiments and CALPHAD modeling for a comprehensive understanding of the metastable and the ternary phases in Ti-Al-N system. The supersaturated solutions of (Ti,Al)N with both rocksalt and wurtzite structures were well described by using the sub-regular solution model. Thereby, the formation, the phase separation and the meta-solubility of TiAlN coatings were successfully elaborated by the metastable pseudo-binary TiN-AlN section and the PVD (physical vapor deposition) phase diagram. Moreover, a number of isothermal and isopleth sections were also calculated and compared with available experiments and previous thermodynamic assessments. For the first time the ternary compound of τ3 was corrected as Ti4AlN3 in our proposed database.

Thermodynamic assessment of Al2O3–SiO2–Ce2O3system

The Al2 O3 –SiO2 , Ce2 O3 –Al2 O3 and Ce2 O3 –SiO2 binary systems were assessed by Redlich–Kister expression based on the sub-regular solution model. Then, the Al2 O3 –Ce2 Si2 O7 system was calculated on the basis of existing experimental data. All the interaction parameters related to the Al2 O3 –SiO2 –Ce2 O3 system were given. The calculated phase diagrams were proved to be accurate when compared with the experimental points or the similar phase diagrams. Finally, a series of phase diagrams were present about the Al2 O3 –SiO2 –Ce2 O3 system.

Solution growth of silicon carbide using unary chromium solvent

Solution growth of silicon carbide (SiC) using unary chromium (Cr) solvent was studied because the system enables a high solubility difference and a low degree of supersaturation, which would lead to rapid growth with a stabilized growth interface. The liquidus composition at SiC saturation in a quasi-binary Cr–SiC system was studied at 1823–2173K. The measured carbon (C) contents are in good agreement with the thermodynamic evaluation using the sub-regular solution model. In addition, growth experiments using a unary Cr solvent were performed by the bottom-seeded travelling solvent method. The obtained growth rates at 1803–1923K with a temperature difference of 15–70K were proportional to the solubility difference between the seed and source temperatures, indicating that the growth was controlled by the mass transfer of C in the solution. The maximum growth rate of 720µm/h at 1803K was much higher than the growth rate by Si-rich solvents, suggesting that the Cr-rich solvent is suitable for the rapid growth at a low temperature.

Thermodynamic and experimental study of corrosion behavior of vanadium-based alloy in liquid sodium-potassium coolant

A preliminary assessment of oxygen effect on vanadium solubility in Na–K melt eutectic composition has been carried out using mathematical framework of the subregular solution model and equations of coordination-cluster model. The effect of oxygen on the solubility of vanadium in the Na–K alloy can be considered as the result of short-range ordering in liquid metal solution. The negative deviations from the ideality for dilute oxygen solutions in Na–K solvent is one reason that explains the quantitative differences between Na and Na–K coolants, when we need to estimate the threshold oxygen concentration for the formation of ternary oxide NaVO2 on the surface of the solid vanadium in liquid sodium and in Na–K alloy. Isothermal capsule experiments qualitatively confirmed the results of calculations of vanadium solubility in Na0.32K0.68 alloy.

Application of a simple subregular solution model to the computation of phase boundaries and free dendritic growth in the Ag-Cu system

Adopting of a simple, but dependable analytic thermodynamic solution model in the simulation of phase transformation kinetics reduces the complexity of computation and the need for extensive thermodynamic data and hence is desired in the practical application of kinetic theories in materials processing. A simple subregular solution model with linear temperature dependency, which can calculate G curves with limited information extracted from an equilibrium phase diagram, is presented and applied to the calculation of (1) the binary Ag-Cu phase diagram with metastable phase boundaries and (2) the kinetics of free dendritic growth in supercooled Ag-Cu melts. The simple T-dependent subregular solution model can duplicate the published Ag-Cu phase diagram with the predicted metastable extensions to the same accuracy as that of calculations with highly structured models that require more computation and wider range of thermodynamic data. Its integration with a free dendritic growth model permits the calculation of correct values of the driving force at non-Henrian interfacial solute concentrations that occur in rapid solidification. The use of the simple T-dependent subregular solution model to calculate the interfacial driving force greatly improves the mathematical stability in the transition stage from mass transfer-limited growth to heat transfer-limited crystal growth.

The Calculation for Saturated Solubility of Oxygen in Mn–Si Melts Equilibrated with MnO–SiO2 Slags

Abstract The MnO–SiO2 slag in equilibrium with Mn–Si melts with a Si molar ratio between 0.1 and 0.3 at 1,700 K was a single liquid phase. However, it was liquid phase saturated with solid MnO or solid SiO2 in the case of X Si less than 0.1 or greater than 0.3, respectively. Based on a subregular solution model, the calculated saturated solubility of oxygen in Mn–Si melts was 0.0236 mass % in pure manganese at 1,700 K and increased with the increasing silicon in the range of X Si from 0.1 to 0.3. However, the saturated solubility of oxygen decreases as the silicon exceeded 0.3, and the minimum solubility was 0.0041 mass % in pure silicon at 1,700 K. The results could be used to predict the compositions of endogenous oxide inclusions and evaluate the cleanliness of manganese ferroalloy.

Thermodynamic description of the Ni–Yb and Ni–Eu binary systems

Thermodynamic descriptions of the Ni–Yb and Ni–Eu binary systems have been performed by means of the CALculation of PHAse Diagrams approach on the basis of the experimental data including the phase equilibria and thermodynamic properties. The Gibbs free energies of the liquid and the terminal solid solutions (fcc_A1, bcc_A2 and hcp_A3) are described by the subregular solution model with the Redlich–Kister formalism, and those of the intermetallic compounds αNiYb=NiYb_LT, βNiYb=NiYb_HT, Ni2Yb, Ni3Yb, Ni5Yb and Nil7Yb2 in the Ni–Yb system and NiEu, Ni2Eu, Ni5Eu and Ni17Eu2 in the Ni–Eu system were described as strict stoichiometric compounds. As a result, two self-consistent thermodynamic data sets for describing the Ni–Yb and Ni–Eu binary systems have been obtained, and the calculated results show a satisfactory agreement with the corresponding experimental data.

Thermodynamic Calculation of Bi-Au-Lu, Nd Ternary Systems

With available melting point and hardness, the Bi-based filler alloy is considered as one choice of high-temperature Pb-free solder. Phase diagram can play an important role in the design of new type of Pb-free solder.In the present work, the thermodynamic assessments of the Au-Nd and the Au-Lu binary systems have been carried out by the Calculation of Phase Diagram (CALPHAD) method based on the available experimental data. The Gibbs free energies of the solution phases were described by subregular solution models with the Redlich-Kister equation, and those of the intermetallic compounds were described by sublattice models. A set of self-consistent and reasonable thermodynamic parameters is obtained for the binary systems, which describes the Gibbs energies of the solution phases and the intermetallic compounds phases. Additionally, combined the reported Bi-Au, Bi-Lu and Bi-Nd binary systems, the thermodynamic database of the Bi-Au-Lu and the Bi-Au-Nd ternary systems have been developed, which will provide important thermodynamic information for the phase equilibria of the multicomponent Bi-based alloy systems.

Thermodynamic Database of the Phase Diagrams in the Sc-X Binary Systems

Sc-based alloys have been used as aero-structures materials, and Sc addition on wrought alloys has positive effects on weldability and welding properties. Phase diagrams play an effective role during the development of new type of alloy with good properties. Therefore, to establish the thermodynamic database of Sc-X binary systems is important and necessary.In the present work, the thermodynamic assessments of Sc-X (Ag, B, Cr, Er, Gd, Mo, Th, W, Y, Zr) binary systems was carried out by using CALPHAD (Calculation of Phase Diagrams) method based on the experimental data including thermodynamic properties and phase equilibria. The Gibbs free energies of the solution phases were described by the sub-regular solution model with the Redlich-Kister equation, and those of the intermetallic compounds were described by the sub-lattice model. A set of self-consistent thermodynamic parameters were derived for describing the Gibbs free energies of each solution phase and intermetallic compound. The calculated phase diagrams and thermodynamic properties are in good agreement with experimental data. The thermodynamic database of the Sc-X (Al, Ag, B, Ca, Co, Cr, Cu, Er, Fe, Ga, Gd, Ge, La, Li, Mg, Mn, Mo, Ni, Pb, Pd, Ru, Sn, Th, W, Y, Zn, Zr) binary systems have been developed, which can provide much important thermodynamic information for multicomponent Al-Sc based alloys.

Thermodynamic assessment of binary systems Tris(hydroxymethyl)aminomethane-Pentaglycerine (TRIS-PG) and 2-amino-2-methyl-1,3-propanediol-Pentaglycerine (AMPL-PG) phase diagrams

Solid state phase change materials are considered as potential candidates for thermal energy storage. Among the solid–solid heat storage materials, the most promising heat storage materials are organic polyalcohol globular tetrahedral molecular crystals such as (CH3)C(CH2OH)3 for PG, (NH2)C(CH2OH)3 for TRIS, and (CH3)(NH2)C(CH2OH)2 for AMPL, which store a large amount of thermal energy in their solid state high temperature phases with orientationally disorder crystal structure. In this study, we obtained the Gibbs energies of pure PG and pure TRIS derived utilizing the available experimental data including temperature-dependency of heat capacity, enthalpy, and transitions temperatures, as well as obtained optimized binary phase diagrams of TRIS-PG and AMPL-PG by CALPHAD calculations. The phase boundaries are verified by in-situ X-ray diffraction (XRD). Optimized database of these two binaries will allow exploration of ternary and higher order systems using CALPHAD methodology that will provide a wider selection of materials for practical thermal energy storage applications. Regular and sub-regular solution models are used for calculations in which the excess Gibbs energies are expressed by the Redlich–Kister–Muggianu polynomial. A set of self-consistent interaction parameters formulating the Gibbs energies of various phases in binary systems are obtained. The TRIS-PG binary phase diagram was calculated from room temperature to the liquid phase temperature, the modeled phase diagram and the experimental data from our work are in good agreement. We also used experimental data of TRIS-PG from the literature which shows a good agreement between the two data sets. Our optimized TRIS-PG phase diagrams show a wider solid-state two phase region, where there is de-mixing of phases, whereas the calculations from the literature show a very narrow region. We also present optimized AMPL-PG binary phase diagram using PARROT module in Thermal-Calc software. The calculated phase diagrams for the two systems mentioned above are presented along with the experimental data.

Thermodynamic modeling of the UO2–ThO2 phase diagram

The phase diagram in the UO2–ThO2 system has been assessed by thermodynamic modeling with existing data from the literature. The subregular solution model was used to represent the Gibbs free energies of the liquid and the solid phases. By considering the liquidus, solidus and miscibility gap data, the interaction parameters of the liquid and the solid phases were optimized through a multiple linear regression method. A consistent set of interaction parameters were derived for describing the miscibility gap as well as the liquidus/solidus. The phase diagram calculated in the present work is in good agreement with experimental data in the literature.