Limitations of a regular associated solution approximation in describing the thermodynamic behaviour of complex liquid alloys

The validity of the regular associated solution (RAS) model for describing the thermodynamic properties of complex liquid alloys has been analysed by applying it to the Cu-In system. The presence of Cu₃In as the associated species in equilibrium with the free atoms in the liquid state was assumed. Liquid copper-indium alloys exhibit a very complex thermodynamic behaviour where the deviations from ideality of activities and integral enthalpy of mixing vary from positive to strongly negative and the properties are strongly temperature dependent. It is seen that a regular solution type interaction among species is insufficient to model the thermodynamic properties of liquid Cu-In alloys and the temperature dependence of interaction energies among the species has to be taken into account. A modified associated solution model incorporating volume effects and temperature dependence of interaction energies could successfully describe the thermodynamic properties of liquid Cu-In alloys.     

A Calphad-compatible method to calculate liquid/liquid interfacial energies in immiscible metallic systems

For the calculation of liquid/liquid interfacial energies in monotectic metallic alloys only simplified models are known, which suppose the validity of the regular solution model for the liquid solution. In the present paper a Calphad-compatible method is developed to calculate the liquid/liquid interfacial energies in binary, ternary and multicomponent liquid monotectic alloys. The method can be considered as the extension of the Butler equation to the liquid/liquid interfaces. The first order interfacial phase transition has been predicted at a certain critical temperature, which is about 18% of the bulk critical temperature. Below this critical temperature the compositions of the two sides of the liquid/liquid interface are different, while above this critical temperature the compositions at the two sides of the interface are identical. The latter case is valid for the majority of liquid metallic systems above their monotectic temperatures. The method has been validated against the experimentally measured liquid/liquid interfacial energies in the Ga–Pb and Al–Bi systems. The method is found to be very sensitive to the correctness of the Calphad-assessment of the partial excess Gibbs energies of the components in the liquid solution. The excess Gibbs energy function of the liquid Al–Bi solution was reassessed using our recent exponential temperature dependence of the interaction energies.     

Model predictions for the enthalpy of formation of transition metal alloys II

We present a computer program in Algol 60 by means of which enthalpy effects can be calculated for binary alloys in which at least one transition metal is involved. Predictions can be made by means of this program for seven fixed ordered compositions in the solid phase and for the two limiting heats of solution and the heat of mixing at the equlatomic composition in the liquid phase. For solid alloys the predictions are representative for equilibrium compounds.As an example, we Include the output for combinations of iron and cobalt with all other metals. In addition, we include a complete table of heats of solution in liquid alloy systems.A discussion is given of both the accuracy of the predictions and the differences between predictions and the experimentally observed quantities that vary systematically with the position of the elements in the periodic table.     

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 assessment of the Be–Pu and Cd–Pu systems

Thermodynamic assessment of Be–Pu and Cd–Pu systems has been developed with the application of the CALPHAD (Calculation of Phase Diagrams) method, which is established on experimental data including thermodynamic properties and phase equilibria. The Gibbs free energies of the liquid, fcc, hcp, αPu, βPu, γPu, δPu, δ′Pu, and εPu phases were described by the subregular solution model with the Redlich–Kister equation, and those of the intermetallic compounds in the Be–Pu and Cd–Pu binary systems were described by the sublattice model. A set of thermodynamic parameters was derived for describing the Gibbs free energies of solution phases and intermetallic compounds in the Be–Pu and Cd–Pu binary systems. Calculated phase equilibria and thermodynamic parameters are in good agreement with experimental data.     

Thermodynamic assessments of the In-Eu and In-Yb systems

Based on the experimental data including thermodynamic properties and phase equilibria, thermodynamic assessments of the In-Eu and In-Yb binary systems have been carried out by using the CALPHAD (Calculation of Phase Diagrams) method. The Gibbs free energies of the solution phases (liquid, fcc, bcc and tetragonal_A6) were described by subregular solution model with the Redlich-Kister equation. The intermetallic compounds, except the InYb phase, were treated as stoichiometric phases, and the InYb phase was described using the sublattice model. A consistent set of thermodynamic parameters has been derived for describing the Gibbs free energy of each solution phase and intermetallic compound in the In-Eu and In-Yb systems. A good agreement between the calculated results and the experimental data in the In-Eu and In-Yb systems is obtained.     

Thermodynamics and phase diagram of the binary system Bi2O3-B2O3

We analyzed the thermodynamic data of the binary system Bi₂O₃-B₂O₃. The phase diagram contains five compounds, of which four melt congruently, and a miscibility gap, close to pure B₂O₃. The three sets of transformation data of Bi₂O₃, the enthalpy of mixing data in the liquid, and the critical point of the miscibility gap gave five equations for the excess Gibbs energy of mixing and five sets of values for the enthalpies of formation of the binary compounds. The critical point data yielded another set of transformation data for Bi₂O₃. Using the monotectic composition and the enthalpy of mixing data also yielded a set of transformation data for Bi₂O₃. The values obtained using the enthalpy of mixing data are the most reliable values.     

A critical thermodynamic assessment of the Mg–Ni, Ni–Y binary and Mg–Ni–Y ternary systems

A thorough review and critical evaluation of phase equilibria and thermodynamic data for the phases in the Mg–Ni–Y ternary system have been carried out over the entire composition range from room temperature to above the liquidus. This system is being modeled for the first time using the modified quasichemical model which considers the presence of short range ordering in the liquid. The Gibbs energies of the different phases have been modeled, and optimized model parameters that reproduce all the experimental data simultaneously within experimental error limits have been obtained. For the liquid phases, the modified quasichemical model is applied. A sublattice model within the compound-energy formalism is used to take proper account of the structures of the binary intermediate solid solutions. The Mg–Ni and Ni–Y binary systems have been re-optimized based on the experimental phase equilibrium and thermodynamic data available in the literature. The optimized thermodynamic parameters for the Mg–Y system are taken from the previous thermodynamic assessment of the Mg–Cu–Y system by the same authors. The constructed database has been used to calculate liquidus projection, isothermal and vertical sections which are compared with the available experimental information on this system. The current calculations are in a good agreement with the experimental data reported in the literature.     

Kinetic approach to the determination of the phase diagram of a solid solution

A binary mixture of 1,4-dichlorobenzene and 1,4-dibromobenzene, as a representative of a system that forms a solid solution, was measured in an adiabatic calorimeter. Several mixtures of different compositions were heated to 380 K, kept in the liquid phase at this temperature overnight, and then cooled slowly down to 250 K. The enthalpy path of the mixture measured during cooling carries the information that enables us to describe the crystallization process. In the first stage, crystallization develops rapidly and clearly deviates from equilibrium, while in the second stage it proceeds significantly slower, whereby we assume that the surface of the solid is in equilibrium with the existing liquid phase at the given temperature during cooling. Such an approach opposes the one that assumes equilibrium between the liquid and solid bulks, known as the equilibrium model. We show that by employing the kinetic model, the experimental enthalpy path of the mixture during cooling in the adiabatic calorimeter can be very successfully reproduced, while the equilibrium model fails in this aspect. Furthermore, we propose a procedure where the kinetic model is used to obtain the excess thermodynamic properties of the solid phase. These quantities enable the calculation of the liquid–solid phase diagram using the excess parameters obtained from the approach that does not assume the overall equilibrium between the phases during the crystallization process.     

Thermodynamics of liquid In-Ga-Zn alloys determined by vapor pressure method

Application of the Knudsen method for liquid In-Ga-Zn alloys in the range from 618 to 818 K provided experimental data which made it possible to characterize the thermodynamic properties of the liquid and gaseous phases of the In-Ga-Zn system. The examined alloys were selected along four cross sections for the mole fraction ratios x_In/x_Ga equal to 4/1, 3/2, 2/3, 1/4. For each of the cross sections, the value of the mole fraction of zinc in the tested alloys was approx.: 0.2, 0.4, 0.6 and 0.8. Based on the obtained experimental data, the values of the activity of zinc were calculated and then used in the optimization procedure, leading to the determination of parameters of the equations characterizing the Gibbs free energy of liquid In-Ga-Zn alloys as a function of temperature and composition. The Gibbs free energy of the ternary In-Ga-Zn liquid alloys was described with the application of the extrapolation method based on the binary systems as given by the Jacob-Fitzner-Muggianu approach.     

Phase equilibria and thermodynamics of Mn–C,Mn–Si,Si–C binary systems and Mn–Si–C ternary system by critical evaluation,combined with experiment and thermodynamic modeling

Critical evaluation and thermodynamic optimization of Mn–C, Mn–Si, Si–C binary systems and Mn–Si–C ternary system were carried out over the whole composition range from room temperature to above the liquidus temperature. In order to provide critical experimental input for the thermodynamic modeling, some key experiments were carried out in the present study. The liquid solution was modeled using the Modified Quasichemical Model (MQM) in the pair approximation in order to take into account the Short-Range Ordering (SRO) exhibited in the solution. In particular, the SRO observed in the Mn–C binary liquid was reasonably accounted for by the present thermodynamic model, while the conventional random mixing model was not able to properly describe the SRO. All solid solutions were modeled using the Compound Energy Formalism (CEF). Model parameters were optimized to best reproduce the important thermodynamic properties and phase equilibrium data in three binary systems. By taking a reasonable interpolation method for Gibbs free energy of the liquid solution in the ternary Mn–Si–C system, it was shown that the present model successfully reproduced thermodynamic and phase equilibrium data in the ternary system without any adjustable ternary parameter. The present database can be used as a part of larger thermodynamic database for the ferromanganese alloy.     

Thermodynamic modeling of the Mg–Al–Sb system

Thermodynamic modeling of the Mg–Al–Sb system is carried out for the first time in this work. Among the constituent binaries in this system, only the Al–Sb and Mg–Sb are re-optimized. Liquid phases are described by the Redlich–Kister polynomial model, whereas the high temperature modification of Mg₃Sb₂ compound in the Mg–Sb system is described by the sublattice model. The constructed database is used to calculate and predict thermodynamic properties, binary phase diagrams of Al–Sb and Mg–Sb, and liquidus projections of the ternary Mg–Al–Sb. The calculated phase diagrams and the thermodynamic properties such as enthalpy, entropy, Gibbs free energy of mixing, and activities are found to be in good agreement with the experimental data from the literature. The established Mg–Al–Sb database predicted a closed ternary liquid miscibility gap, six ternary eutectics, two ternary peritectics, four saddle points and a critical point.     

Thermodynamic optimization of the Ni–Sn binary system

Through the CALPHAD method and based on experimental data of thermodynamic properties and phase boundaries, the phase diagram of the Ni–Sn binary system has been reassessed. The liquid and fcc_A1 (terminal rich nickel solid solution) phases were described by using a simple substitutional model, the excess Gibbs energy being formulated with a Redlich–Kister expression. The other intermediate phases (Ni₃Sn_HT, Ni₃Sn₂_HT, Ni₃Sn₂_LT, Ni₃Sn₄), were described with a several sublattice model with different formula; the Gibbs energy of the reference compounds was assumed to be linear, and the binary interaction terms on the sub-lattices to be constant. Ni₃Sn_LT was treated as a stoichiometric compound. The solubility of Ni in the terminal phase bct_A5(Sn) was neglected because it is very small. Finally a set of self-consistent thermodynamic parameters for all condensed phases in the Ni–Sn binary system was obtained, which can reproduce most of the experimental data.     

Experimental investigation and thermodynamic assessment of the Fe-Pr and Fe-Nd binary systems

In this work, Fe-Pr alloys and Fe-Nd alloys were investigated experimentally by means of thermal analysis. The temperatures of the invariant reactions and liquidus in the Fe-Pr and Fe-Nd binary systems were measured. Based on the experimental results determined in this work and the critical review of the available experimental data in published literature, the Fe-Pr and Fe-Nd binary systems were re-assessed thermodynamically using the CALPHAD method. The solution phases including liquid, bcc, fcc and dhcp are modeled by the substitutional solution model and their excess Gibbs energies are expressed with the Redlich-Kister polynomial. The intermetallic compounds, Fe₁₇Pr₂, Fe₁₇Nd₂ and Fe₁₇Nd₅ are treated as intermetallic compounds. The self-consistent thermodynamic parameters obtained finally to describe the Gibbs energies of various phases in the Fe-Pr and Fe-Nd binary systems can be used to reproduce well the phase equilibria and thermodynamic data.     

A thermodynamic analysis of the PbSn system and the calculation of the PbSn phase diagram

Thermodynamic data of the liquid phase, the (Pb) phase and the (Sn) phases are analyzed in terms of the Marqules type of equations for the excess Gibbs energies. For the liquid phase, a quasi-sub-regular solution model is used. For the fcc (Pb) phase and the bct (Sn) phase, a quasi-regular model is used. The calculated phase diagram of the Pb---Sn binary using these data agrees well with the experimental diagram. The present analysis yields a set of internally consistent thermodynamic data and the lattice stabilities of Pb and Sn, i.e. the Gibbs energy differences between the fcc and bct forms of Pb and Sn.     

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