Thermodynamic assessments of the Bi-Lu and Lu-Sb systems

The phase diagrams and thermodynamic properties in the Bi-Lu and Lu-Sb binary systems have been assessed by using the CALPHAD (Calculation of Phase Diagrams) method on the basis of experimental data including the thermodynamic properties and phase equilibria. The Gibbs free energies of the liquid, hexagonal close-packed (hcp) and rhombohedral phases were described by the substitutional solution model, and the intermetallic compounds (Bi₂Lu, BiLu, Bi₃Lu₅, Lu₃Sb, Lu₅Sb₃, αLuSb, βLuSb and LuSb₂ phases) were treated as stoichiometric compounds. The thermodynamic parameters of the Bi-Lu and Lu-Sb binary systems were obtained, and agreement between the calculated results and experimental data was obtained for each binary system.     

Thermodynamic description of the Cr–Ge system

The Cr–Ge binary system was thermodynamically optimized using the CALPHAD method. The liquid phase was described by means of an associate solution model. The BCC terminal solid solution was described by the substitutional solution model. The two-sublattice model was used to describe the non-stoichiometric compounds Cr₃Ge, αCr₅Ge₃ and βCr₅Ge₃. The Cr₁₁Ge₈, CrGe and Cr₁₁Ge₁₉ phases were modeled as stoichiometric compounds. A set of thermodynamic parameters for the Cr–Ge system was obtained via thermodynamic optimization using assessed experimental data. The calculated phase diagram and thermodynamic properties agree well with most of the experimental data.     

Thermodynamic re-assessment of the Fe-Dy and Fe-Tb binary systems

In this work, based on the critical evaluation of previous optimizations and available experimental data in the published literature, the Fe-Dy and Fe-Tb binary systems were re-assessed thermodynamically using the CALPHAD method. The solution phases including liquid, fcc-Fe, bcc-Fe, bcc-Dy, bcc-Tb, hcp-Dy and hcp-Tb, were described by the substitutional solution model and their excess Gibbs energies were expressed with the Redlich-Kister polynomial. Due to their narrow homogeneity ranges, the intermetallic compounds, Fe₁₇Dy₂, Fe₂₃Dy₆, Fe₃Dy, Fe₂Dy, Fe₁₇Tb₂, Fe₂₃Tb₆, Fe₃Tb and Fe₂Tb, were modeled as stoichiometric compounds. Self-consistent thermodynamic parameters to describe the Gibbs energies of various phases in the Fe-Dy and Fe-Tb binary systems were obtained finally. The calculated results are in good agreement with the reported phase equilibria and thermodynamic properties.     

Thermodynamic description of the Dy–Ni system

The Dy–Ni binary system has been thermodynamically assessed by means of the computer program Thermo-Calc. The Redlich–Kister polynomial was used to describe the solution phase, liquid (L). Ten compounds, Dy₃Ni, Dy₃Ni₂, DyNi, DyNi₂, DyNi₃, Dy₂Ni₇, DyNi₄, Dy₄Ni₁₇, DyNi₅ and Dy₂Ni₇, were treated as stoichiometric phases. The parameters of the Gibbs energy expressions were optimized according to all the available experimental information of both the equilibrium data and the thermodynamic results. A set of self-consistent thermodynamic parameters of the Dy–Ni system has been obtained. The calculations agree well with the respective experimental data.     

Thermodynamic reassessment of the Au–Dy system supported by first-principles calculations

Based on the available experimental phase equilibria and thermodynamic data and enthalpies of formation computed via first-principles calculations, thermodynamic reassessment of the Au–Dy system was carried out by means of the CALPHAD method. The enthalpies of formation at 0 K for AuDy₂, αAuDy, βAuDy, Au₂Dy, Au₃Dy, Au₅₁Dy₁₄ and Au₆Dy were computed via first-principles calculations to supply the necessary thermodynamic data for the modeling in order to obtain the thermodynamic parameters with sound physical meaning. The solution phases, i.e. liquid, (Au), (αDy) and (βDy), were described by the substitutional solution model, and all the intermetallic compounds in the Au–Dy system were treated as stoichiometric phases. A set of self-consistent thermodynamic parameters for the Au–Dy system was finally obtained. The calculated phase diagram and thermodynamic properties agree reasonably with the literature experimental data and the present first-principles calculations.     

Thermodynamic assessment of the Eu–Pb and Lu–Pb systems

The thermodynamic assessment of the Eu–Pb and Lu–Pb binary systems has been carried out by using the Calculation of Phase Diagrams (CALPHAD) method based on the available literature data including the phase equilibria and thermodynamic properties. The Gibbs free energies of the liquid, bcc, fcc and hcp phases were described by the subregular solution model with Redlich–Kister formula, and those of the intermetallic compounds (Eu₂Pb, Eu₅Pb₃, βEuPb, αEuPb, EuPb₃, Lu₅Pb₃, βLu₅Pb₄, αLu₅Pb₄, Lu₆Pb₅, and LuPb₂) in the binary systems were described by the two-sublattice model. A consistent set of the thermodynamic parameters leading to a reasonable agreement between the calculated results and literature data was obtained.     

Thermodynamic optimization of the Gd–Pb system using random solution and associate models

The Gd–Pb system was critically modeled by means of the CALPHAD technique on the basis of experimental data in the literature. Given the asymmetric shape of the liquidus in the Gd–Pb phase diagram, the associate model for the liquid phase was tested and compared with the substitutional solution model. The results of the optimization show that a better agreement with the available experimental data is obtained by means of the associate model than the substitutional solution model. The solution phases (liquid, bcc, fcc, hcp) were treated with the Redlich–Kister equation. The intermetallic compounds Gd₅Pb₃, αGd₅Pb₄, βGd₅Pb₄, Gd₁₁Pb₁₀, Gd₆Pb₇, GdPb₂, GdPb₃ were treated as stoichiometric compounds. Two sets of self-consistent thermodynamic parameters of the Gd–Pb system were obtained.     

Thermodynamic assessment of the Ce-Pt system

The thermodynamic assessment of the Ce-Pt binary system has been carried out by using the CALPHAD (Calculation of Phase Diagrams) method based on the available literature data including the phase equilibria and thermodynamic properties. The Gibbs free energies of the liquid, bcc_A2 and fcc_A1 phases were described by the subregular solution model with the Redlich-Kister formula, and those of the intermetallic compounds (Ce₇Pt₃, Ce₃Pt₂, Ce₅Pt₄, CePt, Ce₃Pt₄, CePt_x and CePt₅) in the Ce-Pt binary system were described by the sublattice model, and two intermetallic compounds (Ce₅Pt₃,CePt₂) in the Ce-Pt binary system were described by the two-sublattice model. A consistent set of thermodynamic parameters leading to a reasonable agreement between the calculated results and literature data was obtained.     

Experimental study and thermodynamic calculation of the Mn-Dy and Mn-Ho binary systems

In this work, twenty-four Mn-Dy and Mn-Ho alloys were investigated experimentally using the differential thermal analysis (DTA), scanning electron microscopy (SEM) and electronic probe microanalysis (EPMA) to study phase equilibria of the Mn-Dy and Mn-Ho binary systems. The temperatures of some invariant reactions and liquidus in the Mn-Dy and Mn-Ho binary systems were determined. Based on the experimental results obtained in the present work and the critical evaluation of the experimental data reported in the literature, the Mn-Dy and Mn-Ho binary systems were calculated using the CALPHAD method. In the thermodynamic calculation, the solution phases including liquid, α(Mn), β(Mn), γ(Mn), δ(Mn), α(Dy), β(Dy), α(Ho) and β(Ho), are treated as the substitutional solution model with the Redlich-Kister formula. The intermetallic compounds, Mn₂Dy, Mn₁₂Dy, Mn₂₃Dy₆, Mn₂Ho, Mn₁₂Ho and Mn₂₃Ho₆, are modeled as the stoichiometric compounds. Self-consistent thermodynamic parameters were obtained finally to describe the Gibbs energies of various stable phases in the Mn-Dy and Mn-Ho binary systems. The calculated results reproduce well the phase equilibria and thermodynamic properties.     

Thermodynamic re-assessment of the Fe-Tm and Fe-Ho binary systems

Thermodynamic re-assessment of the Fe-Tm and Fe-Ho binary systems were carried out with the help of the CALPHAD method based on the previous optimizations and the critical review of the available experimental information in the published literature. The substitutional solution model was used to describe liquid phase and fcc-Fe, bcc-Fe, hcp-Tm, bcc-Ho and hcp-Ho solid solution phases and their excess terms of Gibbs energies were expressed with the Redlich-Kister polynomial. The intermetallic compounds, Fe₁₇Tm₂, Fe₂₃Tm₆, Fe₃Tm, Fe₂Tm, Fe₁₇Ho₂, Fe₂₃Ho₆, Fe₃Ho and Fe₂Ho, were treated as stoichiometric compounds considering the experimental heat capacity of two intermetallic compounds (Fe₂Tm and Fe₂Ho) in the low temperature range. Self-consistent thermodynamic parameters to describe the Gibbs energies of various phases in the Fe-Tm and Fe-Ho binary systems were obtained finally. The calculated results in this work are in good agreement with available phase equilibira data and thermodynamic data.     

A thermodynamic reassessment of the La–Sn system

Thermodynamic modelling of the La–Sn binary system was carried out with the help of the CALPHAD (CALculation of PHAse Diagram) method. The liquid phase has been described with the association solution model with a ‘ La₁Sn₁’ associated complex. The intermetallic compounds were treated as stoichiometric phases. The calculated phase diagram and the thermodynamic properties of the system are in satisfactory agreement with the majority of the experimental data.     

Critical evaluation and thermodynamic optimization of Mg–Ga system and effect of low pressure on phase equilibria

A complete literature review and critical evaluation of the Mg–Ga binary system are presented. All stable phases known in this system were thermodynamically modeled in the framework of CALPHAD. Liquid phase was modeled using the Modified Quasichemical Model in the pair approximation which takes into account short-range ordering. α-(Mg) solid solution of hcp structure was modeled as a substitutional solution. All intermetallic phases Mg₅Ga₂, Mg₂Ga, MgGa, MgGa₂, Mg₂Ga₅, as well as Ga-rich phase, were treated as stoichiometric compounds. Gas phase was assumed to behave as an ideal solution. Thermodynamic optimization was carried out by evaluating enthalpic and entropic contributions to the Gibbs free energy of all phases independently. It showed that the optimized model parameters along with the model equations could reproduce the available and reliable experimental data in the Mg–Ga system. Representation of partial excess Gibbs free energy of Mg in Ga-rich liquid was improved compared to the previous thermodynamic modeling. Low pressure phase equilibria in this system were analyzed using the developed thermodynamic model, and it was compared with a recent observation of an Mg nanopillar fabricated by Ga⁺ ion beam in a focused ion beam (FIB), followed by in-situ heating in a transmission electron microscope (TEM). Reported surface melting of Mg nanopillar in the TEM column pressure (~10⁻¹⁰ bar) is attributed to the fact that decreasing pressure enhanced sublimation of Mg from the Mg nanopillar significantly, leaving Ga-rich liquid.     

Thermodynamic assessment of the Te-X (X = As,Si, Co) systems

Thermodynamic assessment of the Te-X (X = As, Si, Co) binary systems has been carried out using the CALculation of Phase Diagrams (CALPHAD) method. The thermodynamic parameters were evaluated based on the available experimental data in the literature. Thermodynamic models were constructed for all the phases in the three systems. The substitutional solution model was employed to describe the thermodynamic properties of solution phases including liquid, (Te), (As), (Si), and (Co). Due to the existence of associate species Si₂Te₃ and the difficulty of describing the liquid phase by the substitutional solution model, the associate model was adopted to describe the liquid phase in the Te-Si system. The intermetallic compounds in the Te-As and Te-Si systems were treated as stoichiometric phases and the compounds βCoTe and γCoTe₂ in the Te-Co system were modeled by using sublattice model due to the homogeneity range. A set of self-consistent thermodynamic parameters for the Te-X (X = As, Si, Co) systems was obtained. Comparisons between the calculated results and experimental data available in the literature show that almost all the reliable experimental information can be satisfactorily accounted for by the present modeling.     

Thermodynamic assessment of Sn–Cu–Ce system

Phase relations of Sn–Cu–Ce system are important in understanding metallurgical role of Ce in Sn–Cu based lead-free solder alloys. Thermodynamic assessment of Sn–Cu–Ce ternary system has been done based on experimental data about phase equilibria and thermodynamic properties by using the CALPHAD approach combined with first-principle calculations of formation enthalpy of key compounds. The solution phases (liquid, Fcc_A1, Bcc_A2 and Bct_A5) were treated as substitutional, of which the excess Gibbs energies were modeled by the Redlich–Kister polynomial. Considering its crystal structure and solid solubility range, intermetallic compound Ce₁₁Sn₁₀ was described with a three-sublattice model (Ce)_0.429(Sn)_0.429(Ce,Cu,Sn)_0.142. Other binary and ternary intermetallic compounds were described as stoichiometric phases because of their limited homogeneity ranges. During optimization, Ce–Sn binary system was first assessed; then phase relations in Sn–Cu–Ce ternary system were modeled by combining with the optimized Ce–Cu and Cu–Sn binary systems in literatures. A set of thermodynamic parameters for all known phases were obtained, which can reproduce most experimental data. The Scheil model was used to simulate the process of non-equilibrium solidification for a series of alloys.     

Thermodynamic description of the Mg–Eu binary system

Thermodynamic assessment of the Mg–Eu binary system has been carried out by combining first-principles calculations and Miedema’s theory with CALPHAD method. Firstly, the mixing enthalpy of the liquid alloys was calculated by using Miedema’s theory, and formation enthalpies of the intermetallic compounds were calculated by using the projector augmented-wave (PAW) method within the generalized gradient approximation (GGA). Subsequently, the liquid phase was described employing a simple substitutional model, of which the excess Gibbs energy was formulated with a Redlich-Kister expression. And the solubility of Eu in HCP_(Mg) and Mg in BCC_(Eu) were neglected. While the intermetallic compounds Mg₁₇Eu₂, Mg₅Eu, Mg₄Eu, Mg₂Eu and MgEu, were treated as stoichiometric compounds. Consequently, a set of self-consistent thermodynamic parameters for all stable phases in the Mg–Eu binary system were obtained, which can reproduce most of the thermodynamic and phase boundary data.     

A thermodynamic modeling of the Pd-Ti system

The thermodynamic properties of the Pd-Ti system were optimized using the CALPHAD (CALculation of PHAse Diagram) technique. The solution phases, liquid, bcc, fcc and hcp, were described by the substitutional-solution model. Both compounds Pd₂Ti and PdTi₂ with tetragonal MoSi₂-type structure were treated as one phase with the formula (Pd,Ti)₂(Pd,Ti) by a two-sublattice model. The intermetallic compounds Pd₃Ti, Pd₃Ti₂, and PdTi₃ were treated as stoichiometric compounds. The intermetallic compound αPdTi, which had a homogeneity range, was treated as the formula (Pd,Ti)(Pd,Ti) by a two-sublattice model. A two-sublattice model (Pd,Ti)_0.5(Pd,Ti)_0.5 was applied to describe the compound βPdTi in order to cope with the order-disorder transition between βPdTi with CsCl-type structure (B2) and body-centered cubic solution (A2) in the Pd-Ti system. A set of self-consistent thermodynamic parameters was obtained.     

Thermodynamic description of the Pb–Yb binary system

Thermodynamic modelling of the Pb–Yb binary system was carried out with the help of the CALPHAD method. The liquid phase has been described with the association solution model with ‘ Pb₁Y b₂’ as an associated complex. The solution phases BCC_A2 and FCC_A1 were modelled with the sublattice formalism. The αPbYb_LT and βPbYb_HT Pb sub-stoichiometric intermetallic compounds, which have a homogeneity range, were treated with the formula (Pb,Y b)_0.5(Y b)_0.5 by a two-sublattice model with Pb and Yb on the first sublattice and Yb on the second one. Pb₃Y b, Pb₃Y b₅ and PbY b₂ have been treated as stoichiometric compounds. The calculations based on the thermodynamic modelling are in good agreement with the phase diagram data and experimental thermodynamic values.