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The phase equilibria at 1473, 1373 and 1273 K in the Fe–Ta–Ti system were investigated by means of Scanning Electron Microscopy (SEM) with Energy Dispersive Spectrometer (EDS) and X-ray Diffraction (XRD). The experimental results indicated that Fe2Ta and Fe2Ti formed the continuous solid solution phase Fe2(Ta,Ti). No ternary compound was found in this system. The temperature of invariant reaction liquid + Fe7Ta6 → FeTi + bcc(Ta,Ti) was measured as 1431 K by Differential Thermal Analysis (DTA). Based on the experimental results in the present work, the Fe–Ta–Ti system was optimized using CALPHAD technique. A set of self-consistent thermodynamic parameters for the Fe–Ta–Ti system was obtained, and the experimental isothermal sections at 1473, 1373 and 1273 K were reproduced well.

The liquidus surface projection and isothermal section at 1273 K of the Ni–Ti–V system were established using X-ray diffraction (XRD), scanning electron microscopy (SEM) coupled with energy dispersion spectroscopy (EDS), electron probe micro-analyzer (EPMA) and differential thermal analysis (DTA) techniques. Six primary solidification regions and four invariant reactions were deduced in the liquidus surface projection, and six three-phase regions were derived in the isothermal section at 1273 K. No ternary compound was observed. According to the experimental results in the present work and literatures, the Ni–Ti–V system was modeled by means of the CALPHAD (CALculation of PHAse Diagram) method. Two-sublattice model (Ni,Ti)10(Ni,Ti)20 for binary σ phase was used, and the thermodynamic parameters of the σ and NiV3 phases in the Ni–V system was reassessed. Solution phases (liquid, fcc, bcc and hcp) were modeled with the substitutional solution model in the Ni–Ti–V system. The compounds, Ni3Ti, NiTi2, Ni3V and σ, were treated as (Ni,Ti,V)m(Ni,Ti,V)n, and B2 were treated as (Ni,Ti,V)0.5(Ni,Ti,V) 0.5Va3. A set of self-consistent thermodynamic parameters of individual phases was obtained.

The isothermal section at 1273 K of the Co–Ni–Ti system was determined using equilibrated alloys by means of scanning electron microscopy (SEM) with energy dispersive X-ray spectroscopy (EDX) and X-ray diffraction (XRD). The experimental results indicated that eight three-phase regions and a ternary compound τ with Co3V-structure existed. Meanwhile, this system was thermodynamically modeled using the CALculation of PHAse Diagram (CALPHAD) method. The isothermal section at 1273 K measured in this work as well as the four isothermal sections at 1373, 1273, 1223 and 1173 K and the liquidus surface projection from the literatures were taken into account in the present optimization work. A set of self-consistent thermodynamic parameters of the Co–Ni–Ti system was first derived.

On the basis of all the presently determined experimental data, as well as the formerly reported isothermal sections of the Ce-Co-Sb ternary system, it was thermodynamically assessed by means of CALPHAD technique. For Ce-Co and Co-Sb binary systems, thermodynamic parameters previously obtained were considered and a re-assessment work of the Ce-Sb system is presented. In particular, the Kaptay exponential model, rather than the Redlich-Kister polynomials, is used to describe the interactions between Ce and Sb in the liquid phase of the Ce-Sb system, mainly to avoid the abnormal miscibility gap of liquid phase in the Ce-Co-Sb system at the Co rich region, and also to guarantee the consistent phase equilibria between the experimental isothermal sections and the calculated ones of the Ce-Co-Sb ternary system. The ternary compound CeCouSb2 is modeled as Ce2/9(Co%,Va)1/3Sb4/9 in order to describe the fluctuation of Co atoms in this compound. The ternary compound CeCoSb3 is regarded as a stoichiometric phase. The previously optimized thermodynamic parameters of other phases were adopted in the present work. A set of self-consistent thermodynamic parameters were finally obtained and fit well with the experimental results from various literatures, which will contribute to the further development of the CoSb3-based thermoelectric materials, especially from the perspective of the couping of electron-phonon interactions.

Rhenium was one of important alloying elements in the Ni-based superalloys. Based on the molar Gibbs energy of the pure Re updated in SGTE Pure 5 database, the Re–X(X=Al, Co, Cr, Ta) systems were re-optimized by means of CALPHAD (CALculation of PHAse Diagrams) technique. In the present work, the phases liquid, fcc, bcc and hcp were described using a substitutional solution model. The phases AlRe, Al3Re, Al6Re, Al12Re, AlRe2 and Al11Re4 in the Re–Al system were described as stochiometric compound. The Al4Re_H and Al4Re_L instead of Al4Re were evaluated in the present work. The phases σ in the Re–Cr and Re–Ta systems and χ in the Re–Ta system were modeled as (X, Re)10(X, Re)20 (X=Cr or Ta) and Re24(Re, Ta)10(Re, Ta)24, respectively. A set of self-consistent thermodynamic parameters of the Re–X systems were obtained and the optimized results were in good agreement with the experimental data.

The Ce-Co-Sb ternary system is experimentally investigated, including the isothermal section at 823 K and the supplementary measurements at 673 K, as well as the liquidus projection of this system. For the isothermal section at 823 K, the phase equilibria in the whole composition range are determined using the equilibrated alloys prepared by the combination of arc melting method and high frequency melting method under the protection of an argon atmosphere, in case of the volatility of Sb and the oxidation of Ce. To confirm the phase relations at 673 K from previous literature report, some phase equilibria are re-determined on the basis of supplementary experiments. Two ternary compounds, (CeCouSb2) and CeCoSb3 are observed and the isothermal sections of the Ce-Co-Sb system at 823 and 673 K are finally constructed. The solidification paths of the Ce-Co-Sb as-cast alloys are analyzed to ascertain the primarily solidified phases from liquid. Both the liquidus projection and the isothermal sections of the Ce-Co-Sb system are determined based on the microstructures, crystal structures and phase compositions determined by scanning electron microscopy, X-ray diffraction and electron probe micro-analyzer, respectively. Some phase transition temperatures of the Ce-Co rich region are determined by means of differential scanning calorimeter.

The phase equilibria and thermodynamic properties of the Al–Ge–Ni system are useful for understanding the diffusion process during the transient liquid phase (TLP) bonding. In this work, the thermodynamic description of the Al–Ge–Ni system over the whole composition and temperature ranges was performed by means of the CALPHAD (CALculation of PHAse Diagrams) method. The enthalpies of mixing of the liquid phase, three isothermal sections at 973, 823, and 673 K and nine vertical sections at 10, 20, 35, 55, 60, 70, 75, and 80 at% Ni and at a constant Al:Ni ratio of 1:3 were taken into account in the present optimization work. A set of self-consistent thermodynamic parameters of the Al–Ge–Ni system was first obtained. The liquidus projection and reaction scheme were constructed according to the thermodynamic parameters obtained in this work. The phase equilibria and thermodynamic properties calculated by the present thermodynamic description show satisfactory agreement with the available experimental information.

The Fe–Zr and Al–Fe–Zr systems were critically assessed by means of the CALPHAD technique. The solution phases, liquid, face-centered cubic, body-centered cubic and hexagonal close-packed, were described by the substitutional solution model. The compounds with homogeneity ranges, hex.- Fe2Zr, Fe2Zr, FeZr2 and FeZr3 in the Fe–Zr system, were described by the two-sublattice model in formulas such as hex.- Fe2(Fe,Zr), (Fe,Zr)2(Fe,Zr), (Fe,Zr)Zr2 and (Fe,Zr)(Fe,Zr)3 respectively. The compounds AlmZrn except Al2Zr in the Al–Zr system were treated as line compounds (Al,Fe)mZrn in the Al–Fe–Zr system. The compounds FeZr2 and FeZr3 in the Fe–Zr system were treated as (Al,Fe,Zr)Zr2 and (Al,Fe,Zr)(Fe,Zr)3 in the Al–Fe–Zr system, respectively. All compounds in the Al–Fe system and hex.- Fe2Zr in the Fe–Zr system have no solubilities of the third components Zr or Al, respectively, in the Al–Fe–Zr system. The ternary compounds λ 1 with C14 structure and λ 2 with C15 structure in the Al–Fe–Zr system were treated as λ 1 - (Al,Fe,Zr)2(Fe,Zr) with Al2Zr in the Al–Zr system and λ 2 - (Al,Fe,Zr)2(Fe,Zr) with Fe2Zr in the Fe–Zr system, respectively. And the ternary compounds τ 1 , τ 2 and τ 3 in the Al–Fe–Zr system were treated as (Al,Fe)12Zr, Fe(Al,Zr)2Zr6 and Fe7Al67Zr26, respectively. A set of self-consistent thermodynamic parameters of the Al–Fe–Zr system was obtained.

This paper focuses on the theoretical descriptions of the Gibbs energies concerning the unary systems of Antimony and Germanium. The Gibbs energy expressions of the vapor phase and the condensed phases are included in order to study the influences of temperature and pressure on the phase equilibrium status. The vapor phase is treated as a real solution of the constituent species by considering the fugacity coefficients of gas species in order to assure the reasonability of the P-T (Pressure-Temperature) diagrams at high pressure. The condensed phases are described in the light of the pressure dependent Murnaghan equation which is presented as a pressure dependent contribution Gpres in the present work. The pressure correction factors for the condensed phases include the molar volume, the parameters of the variations of the thermal expansivity and the compressibility with temperature, and the variation of the bulk modulus with pressure. The related parameters of the vapor phase and all the condensed phases are thermodynamically optimized for Sb and Ge unary systems, on the basis of experimental measurements in literature. The calculated results of the P-T relations and the P-V (Pressure-Volume) curves agree well with the experimental data. The calculated slope of liquid-solid coexistence curve at the low pressure of the P-T phase diagram of Sb unary system follows Clausius-Clapeyron equation. The calculated heat capacity curves of condensed phases of Sb and Ge are analyzed and discussed to assure the reasonability of the thermodynamic models described in the paper. The present thermodynamic descriptions of both unary systems are of great benefit to the preparation process of the CoSb3-based thermoelectric materials, especially for the prevention of the volatilization of Sb containing vapor species and the determination of the sintering temperature under certain pressure.

The isothermal sections at 1073 and 1273 K, and liquidus surface projection of the Ti–V–Zr system were investigated using X-ray powder diffraction (XRD), scanning electron microscopy (SEM) with energy dispersive spectrometer (EDS) and differential thermal analysis (DTA). Combined with experimental results in the present work and literature, the Ti–V–Zr system was optimized by the means of the CALPHAD (CALculation of PHAse Diagram) method. The solution phases, liquid, hcp (Ti, Zr) and bcc (Ti, V, Zr), were modeled with the substitutional solution model, and the compound V 2 Zr was treated as (Ti, V, Zr) 2 (Ti, V, Zr) based on its thermodynamic model in the V–Zr system and the experimental solid solubility of Ti in the Ti–V–Zr system. A set of self-consistent thermodynamic parameters of the Ti–V–Zr system was obtained.

The experimentally observed aging precipitation sequence of the Mg–Y alloys from literatures is α(Mg)SS(supersaturated)→ β''(c-bco like GP zones)→ β′(c-bco, metastable coherent Mg7Y)→ β(bcc, stable incoherent Mg24Y5). Compared with the equilibrium phase diagram, Mg7Y is not a stable precipitate, but from the view point of the crystal structure, Mg7Y is an ordering product of HCP_A3(Mg). The Mg–Y binary system has been thermodynamically re-assessed by means of CALPHAD technique in order to study its aging precipitation sequence in the present work. Three solution phases, HCP_A3, BCC_A2 and Liquid, are modeled with the Redlich-Kister polynomial. The intermetallic compounds Mg2Y and Mg24Y5 have been described by the following formulas, (Mg, Y)0.6667(Mg, Y)0.3333 and Mg0.8276(Mg, Y)0.1379 Y0.0345, respectively. Especially for order-disorder transitions, the following two-sublattice models (Mg, Y)0.5(Mg, Y)0.5 and (Mg, Y)0.875(Mg, Y)0.125 have been used to describe the stable MgY/BCC_A2 and the metastable Mg7Y/HCP_A3, respectively. Based on the satisfied optimization of equilibrium phase diagram, the order-disorder transition between the metastable intermediate Mg7Y and the stable phase HCP_A3 is assessed to ensure that Mg7Y has a big enough driving force but not a lowest Gibbs free energy change to precipitate preferentially from the supersaturated α(Mg). Meanwhile, in order to match the aging precipitation sequence of the Mg–Y alloys, the effective nucleation driving forces for the precipitated phases are calculated by subtracting the interfacial energy and the elastic strain energy from the thermodynamic driving force. A set of self-consistent thermodynamic parameters has been obtained to agree satisfactorily with the reported experimental data from literatures, including the phase diagram, the thermochemical properties, and the aging precipitation sequence of the typical Mg–Y alloys with x(Y) = 0.0232 at 473 K.

The V–Zr system was investigated by experiments and thermodynamic modeling. In experiments, 10 crucial alloys were selected, and the microstructure, crystal structure and phase transformation temperatures of the alloys were obtained using scanning electron microscopy (SEM) with energy dispersive spectrometer (EDS), X-ray powder diffraction (XRD) and differential thermal analysis (DTA). A thermodynamic modeling of the V–Zr system was then performed by considering the reviewed literature data and incorporating the present experimental results. The solution phases, liquid, hcp (Zr) and bcc(V, Zr), were modeled with the substitutional solution model, and the compound V 2 Zr was treated as (V, Zr) 2 (V, Zr) by means of two-sublattice model. A set of self-consistent thermodynamic parameters of the V–Zr system was obtained.

The Ga–Pt–Sb system was modeled using the CALPHAD (CALculation of PHAse Diagram) technique. Solution phases, liquid, fcc(Pt) and rhom(Sb), of the Ga–Pt–Sb ternary system were modeled with the substitutional solution model. The compounds, Ga 6 Pt, Ga 7 Pt 3 , Ga 2 Pt, Ga 3 Pt 2 , GaPt, Ga 3 Pt 5 , and GaPt 2 of the Ga–Pt system, and PtSb 2 , PtSb, Pt 3Prod. Type: FTP Sb 2 , and Pt 7 Sb of the Pt–Sb system, were treated as line compounds Pt m (Ga,Sb) n in the Ga–Pt–Sb system. The compounds Ga 3 Pt and Pt 5 Sb were treated as (Ga%,Pt,Sb) 3 (Ga,Pt%) and (Pt%,Sb) 5 (Ga,Pt,Sb%) in the Ga–Pt–Sb system on the basis of thermodynamic models of the respective binary system. A set of self-consistent thermodynamic parameters was obtained. Using these thermodynamic parameters, the experimental isothermal section phase relation at 773 K, the vertical sections at 25 at% Pt, the GaPt–GaSb, Ga 2 Pt–GaSb, Ga 7 Pt 3 –GaSb, and GaPt–PtSb 2 sections and the partial invariant reactions in the Ga–Pt–Sb system were well reproduced.

The Yb–Bi and the Yb–Te systems are thermodynamically assessed by CALPHAD approach based on the available experimental data including thermochemical properties and phase equilibria. All of the intermetallic phases in these two systems are treated as stoichiometric compounds due to their unnoticeable homogeneity ranges, and their formation enthalpies at 0 K are calculated by the first-principles method to supplement the thermochemical property data for the present optimization. The values of the first-principles calculation are confirmed by the lattice constants and the equilibrium modulus of the compounds available from experiments. Given the asymmetric shapes of the liquidus around the compounds Yb4Bi3/YbTe in the Yb–Bi/Te phase diagrams, both the associate model and the sub-regular solution model are used to describe the Gibbs free energy of the liquid phase for the two systems. The comparison between the optimization results and the experimental data for the Yb–Bi/Te systems shows that a much better agreement is obtained by means of the associate model than the sub-regular solution model.

The microstructure and phase constituents at 1023 and 1123 K of the annealed Ag–Cu–Y alloys were studied by using scanning electron microscopy (SEM) with energy dispersive X-ray spectroscopy (EDX) and X-ray diffraction (XRD). A new phase τ2 at 1023 K was found at ~33.3 at.% Y between Ag2Y and Cu2Y. Another ternary compound τ1(Ag3Cu12Y5) was confirmed to be stable at 1023 and 1123 K, and the liquidus temperatures of two annealed alloys with single phase τ1 were determined by thermal analysis techniques. Based on the obtained solubilities of compounds and phase equilibria data, the complete isothermal section at 1023 K and partial isothermal section with Y

According to the available experimental phase equilibrium relations, thermochemical properties and the experimentally observed ageing precipitation sequence from literatures, the thermodynamic re-modeling and re-assessment of the Mg–Gd binary system have been carried out by means of CALPHAD (CALculation of PHAse Diagram) technique. The metastable phases, including GP zones (D019-type) and Mg7Gd intermediate, are described by the models (Gd,Mg%)0.75(Gd%,Mg)0.25 and (Gd,Mg%)0.875(Gd%,Mg)0.125, respectively, in order to deal with the order-disorder transitions between the ordered phases GP zones (D019-type)/Mg7Gd intermediate and the disordered solid solution HCP_A3(Gd,Mg). With the considerations of the influences of elastic strain energies and interface energies on the thermodynamic driving forces, the effective driving forces for the precipitates to nucleate are calculated to match the ageing precipitation sequence, α(Mg)SSSS(supersaturated solid solution)→ GP zones (D019-type)→ β′(c-bco, Mg7Gd)→ β1(D03, Mg3Gd)→ β(fcc, Mg5Gd). A set of the self-consistent thermodynamic parameters has been obtained. The present calculations of the phase diagram, the thermochemical properties, and the precipitation driving forces of the typical Mg–Gd alloys with x(Gd) = 0.01819 and 0.0318 at 498 K, are not only well consistent with the reported experimental data but also in a satisfactory agreement with the precipitation sequence in the early stage of ageing observed by high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM).

The CaO–CaF 2 , Al 2 O 3 –CaF 2 , and CaO–CaF 2 –Al 2 O 3 systems are modeled using the CALculation of PHAse Diagram (CALPHAD) approach. The liquid phase is described by a two-sublattice ionic model (Al +3 , Ca +2 ) p (F − , O −2 , AlO 3/2 ) q . The compounds Al 2 O 3 , CaF 2 , and CaO are treated as stoichiometric phases. The Gibbs energy of formation of (CaO) 11 (Al 2 O 3 ) 7 (CaF 2 ) is predicted by means of first-principles calculations. Phase diagrams of the CaO–CaF 2 and Al 2 O 3 –CaF 2 quasibinaries as well as the isothermal and vertical sections and the liquid surface projection of the CaO–CaF 2 –Al 2 O 3 system are calculated, showing favorable accord with available experiments in the literature.

On the basis of the experimental data of the phase equilibria and the thermochemical properties, a critical evaluation for the Ni–Sc binary system has been carried out using the CALPHAD (Calculation of Phase Diagrams) method. The associated model is used for the liquid phase containing the constituent species Ni, Sc and ScNi. The terminal solid solutions Fcc_A1 (Ni), Hcp_A3 (Sc), and Bcc_A2 (Sc) are described by the solution model with the Redlich–Kister polynomial. The intermetallic compounds, ScNi5, Sc2Ni7, ScNi and Sc2Ni, are treated as strict stoichiometric compounds. The compound with a homogeneity range, (ScNi2), is modeled using two sublattices as (Sc,Ni)0.333(Sc,Ni)0.667. A set of self-consistent thermodynamic parameters for the Ni–Sc binary system is obtained. The calculated results agree well with the available experimental data from literatures.

The Nd-B binary system was re-assessed thermodynamically based on the available experimental information from literatures using the CALPHAD method. The solution phases including liquid, fcc, bcc and dhcp, were modeled as substitutional solutions and their excess Gibbs energies were expressed by the Redlich-Kister polynomial. Combined with the previous assessments of the Fe-Nd and the Fe-B binary systems, the thermodynamic description of the Nd-Fe-B ternary system was performed on the basis of the reported experimental information. The solubility of third element in the binary intermetallic compounds was not taken into account due to the lack of experimental data. A set of self-consistent thermodynamic parameters formulating the Gibbs energies of various phases in the Nd-Fe-B ternary system was obtained in the present work. The liquidus projection, several isothermal sections and vertical sections of this ternary system were calculated, which are in reasonable agreement with the reported experimental data.

The experimental determinations of the isothermal section at 823 K and the supplementary measurements of the liquidus projection of the Ce-Co-Fe ternary system were presented in the present study. In the Ce-Co-Fe ternary system, in consideration of the temperature dependent solubilities of the linear phases such as Ce2(Co,Fe)17 and Ce(Co,Fe)2, as well as the specific locations of the univariant lines between each two primary solidification surfaces, it is necessary to study more than one isothermal section and some particular as-cast alloys to construct the phase equilibria in the temperature-composition space of the Ce-Co-Fe system. The samples for determining the liquidus projection were prepared by arc-melting method under high purity argon atmosphere in a water-cooled copper hearth, and then those for measuring the isothermal section at 823 K were isothermally treated and quenched in ice water. The microstructures and the phase compositions of samples were measured by means of scanning electron microscopy (SEM), X-ray diffraction (XRD) and electron probe micro-analyzer (EPMA). Some primary solidification regions and univariant lines of the Ce-Co-Fe ternary system were complementally determined, and the reasonability of the liquidus projection reported in previous literature was further confirmed. The phase equilibrium relations at 823 K were determined, including two-phase and three-phase equilibria. No ternary compounds were discovered in the present study. Based on the experimental results of both the previous literature reports (the reported liquidus projection and isothermal sections at 723 and 1173 K) and the present experimental study, the Ce-Co-Fe ternary system was thermodynamically assessed using the CALPHAD method. The isothermal sections, the vertical sections and the liquidus projection were calculated using the present optimized thermodynamic parameters, and a reasonable agreement between the calculated results and the experimental data was obtained.

Based on the experimental measurements of the liquidus projection and the isothermal sections, the Nb–Si–W ternary system is thermodynamically assessed using the CALPHAD (CALculation of PHAse Diagram) approach. The parameters for the thermodynamic descriptions of all the phases in the system are optimized, following the reported descriptions of the three constituent binaries the Nb–Si, the Nb–W and the Si–W systems. Some of the parameters of the Si–W system are modified according to its related binary experimental data. The comparisons between the calculated liquidus projection and isothermal sections and the experimental ones show that the present modeling can finely describe the experimental results of the Nb–Si–W ternary system. The solidification analyses can explain the microstructure evolutions for typical ternary alloys.

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 Gd 5 Pb 3 , αGd 5 Pb 4 , βGd 5 Pb 4 , Gd 11 Pb 10 , Gd 6 Pb 7 , GdPb 2 , GdPb 3 were treated as stoichiometric compounds. Two sets of self-consistent thermodynamic parameters of the Gd–Pb system were obtained.

The Gd–Bi and the Ho–Bi systems were thermodynamically assessed by CALPHAD (CALculaton of PHAse Diagram) approach based on the available experimental data including thermochemical properties and phase equilibria. The formation enthalpies (at 0 K) of the compounds of the Gd–Bi and the Ho–Bi systems were calculated by the first-principles method and used in the present thermodynamic optimization. The excess Gibbs energies were formulated with the Redlich–Kister polynomial. The intermetallics, Gd5Bi3, Gd4Bi3, GdBi, GdBi2, Ho5Bi3 and HoBi, were treated as stoichiometric compounds. A good agreement between the calculated results and the experimental data was obtained.

A complete thermodynamic description of the Bi–Mg–Sn ternary system was obtained by means of the CALPHAD (CALculation of PHAse Diagram) technique. The solution phases, Rhomb, Bct_A5 and Hcp_A3, are modeled with Redlich–Kister equation. The associate model has been used to describe the liquid phase with the constituents of the species Bi, Bi2Mg3, Mg, Mg2Sn and Sn. The non-stoichiometric compounds Mg2Sn and α-Bi2Mg3 as well as its high temperature variant β-Bi2Mg3 were described by the sublattice models (Bi,Mg)2(Bi,Sn)1, (Bi,Sn,Va)2Mg3 and (Bi,Sn)1(Bi,Sn,Va)3Mg6, respectively. The Bi–Mg–Sn ternary system has been modeled thermodynamically for the first time and the calculated phase equilibrium relations and thermochemical properties are consistent with the experimental results available in literatures.

The criterion for coherent equilibria is investigated. The expressions of driving force considering elastic energy for further formation of infinitesimal amount of one phase from the other in the two-phase coherent system are obtained. Zero driving force is shown to be a useful thermodynamic criterion for coherent equilibria, which not only has a clear physical meaning, but also is really convenient in coherent equilibria calculations. The coherent equilibrium equations are derived to directly demonstrate that the equilibrium compositions of phases in coherent equilibria depend on the overall alloy composition. In particular, the influence of the deviation of lattice parameter from Vegard’s law on the coherent spinodal, the field lines and the equilibrium compositions of coherent miscibility gap is examined in a regular binary solution. The calculated results show that even small deviation of lattice parameter from Vegard’s law can make the equilibrium compositions strongly dependent on the overall alloy composition. Therefore, the deviation of lattice parameter from Vegard’s law should be carefully considered in the calculation of coherent phase diagrams of the realistic systems.

The thermodynamic reassessment of the Al–Mo–Si system was performed using the CALPHAD technique. The solution phases (liquid, bcc, fcc and diamond) were modeled as a substitutional solution. The compounds AlMo3 in the Al–Mo system and Mo3Si in the Mo–Si system had the same A15 crystal structure, and were treated as one phase and described by a two-sublattice model (Al,Mo,Si)(Al,Mo)3. The compound Mo5Si3 with D8m crystal structure was treated as the formula Mo0.5(Mo,Si)0.125(Al,Mo,Si)0.375 in the Al–Mo–Si system. Other compounds, Al63Mo37, Al8Mo3, Al3Mo, Al4Mo, Al17Mo4, Al22Mo5, Al12Mo and Al5Mo in the Al–Mo system, MoSi2 in the Mo–Si system, and ternary compounds C40 and C54 were treated line compounds (Al,Si)mMon in the Al–Mo–Si system. Based on the published experimental isothermal sections and the liquidus surface projection, the Al–Mo–Si system was re-optimized, and a set of self-consistent thermodynamic parameters was obtained.

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.

The Er–Fe, Er–Sb and Er–Fe–Sb systems were optimized by means of the CALPHAD (CALculation of PHAse Diagram) technique. The solution phases, liquid, bcc, fcc, hcp and rhom, were described by the substitutional solution model. The binary compounds, Er 2 Fe 17 , Er 6 Fe 23 , ErFe 3 , ErFe 2 , Er 5 Sb 3 , αErSb, βErSb, ErSb 2 and ternary compound Er 6 FeSb 2 were treated as stoichiometric compounds. The thermodynamic description of the Fe–Sb system was taken from the literature. A self-consistent thermodynamic description of the Er–Fe–Sb system was obtained.

The Al–Li–Zn system was critically assessed using the CALPHAD technique. The solution phases (liquid, bcc, fcc and hcp) were described by the substitutional solution model. The compounds Al2Li3 and Al4Li9 in the Al–Li system had homogeneity ranges of Zn and were treated as (Al,Zn)2Li3 and (Al,Zn)4Li9 in the Al–Li–Zn system, respectively. The compounds αLi2Zn3, βLi2Zn3, αLi2Zn5, βLi2Zn5 and αLiZn4 in the Li–Zn system had no solubility of the third component Al in the Al–Li–Zn system. A two-sublattice model (Al,Li,Zn)0.2(Al,Li,Zn)0.8 was applied to describe the compound βLiZn4 in the Al–Li–Zn system in order to cope with the order–disorder transition between hexagonal close-packed solution (hcp-A3) and βLiZn4 with the Mg-type structure. The ternary compound τ 2 with a NaTl-type structure (B32) had the same structure with the compounds AlLi in the binary Al–Li system and LiZn in the binary Li–Zn system. In the present work, the three compounds AlLi, LiZn and τ 2 were treated as one phase by a two-sublattice model (Al,Li,Zn)0.5(Al,Li,Zn)0.5 in order to cope with the order–disorder transition between B32(AlLi, LiZn and τ 2 ) and body-centered cubic solid solution (bcc-A2). The ternary intermetallic compounds τ 1 and τ 3 in the Al–Li–Zn system were treated as the formula Li(Al,Zn)2 and (AlLi,Zn)Zn3, respectively. A set of self-consistent thermodynamic parameters describing the Gibbs energy of each individual phase as a function of composition and temperature in the Al–Li–Zn system was obtained.

With the consideration of the existence of the metastable miscibility gaps, the thermodynamic parameters of the Mg-based solid solution have been assessed for the Mg–Zn and Mg–Nd systems. The new semi-empirical equation proposed by Kaptay was used to describe the interaction parameters between components to prevent the appearance of the artificial miscibility gap at high temperature. The obtained thermodynamic descriptions can reproduce the metastable miscibility gaps and the spinodal decomposition curves in the Mg–Zn and Mg–Nd systems. The driving force calculation can explain well the precipitation of the second hcp_A3 phase (Guinier–Preston zones) in the primary Mg-based hcp_A3 matrix, as well as the decomposition sequence following a quench from a single-phase supersaturated solid solution to a two-phase regime. The miscibility gap of the Mg-based phase in the Mg–Zn–Nd ternary system was predicted on the basis of the descriptions of the Mg–Zn and Mg–Nd binary systems without consideration of the three-component interactions.

Based on the experimental isothermal sections at 800, 1000, 1700 and 1800 °C and the liquidus projections from both the literature and the present measurements, the ternary Nb–Si–Mo system is thermodynamically assessed using the Calculation of Phase Diagram method (CALPHAD). The constituent Mo–Si binary system is reassessed according to its related binary experimental data. The sublattice model of the phase Nb5Si3 in the constituent Nb–Si binary system is modified as (Nb)0.5(Nb,Si)0.125(Nb,Si)0.375, in order to give the solid phase β ( Nb , Mo ) 5 Si 3 in the Nb–Si–Mo ternary system a better description. The parameters of the thermodynamic descriptions of the Nb–Si–Mo ternary system are optimized following the reported Nb–Mo, the modified Nb–Si and the reassessed Mo–Si binary systems. The comparison between the calculated results and the experimental investigations shows that the present modeling can finely describe the experimental information of the Nb–Si–Mo ternary system.

The Co–Ni–Sb ternary system is critically assessed using the CALPHAD technique on the basis of three optimized binaries, the Co–Ni, the Co–Sb and the Ni–Sb systems. The NiAs-type structure phase γ-(Co,Ni)Sb with a wide non-stoichiometric range is described by three sublattices, (Sb) 1/3 (Co,Ni,V a) 1/3 (Co,Ni,V a) 1/3 , following the extension of the sublattices of the phases γ(CoSb) and γ(NiSb) in the related binary systems. The sublattice model is also used to describe the terminal ternary solid solution phases: three sublattices (Sb) 1/4 (Co,Ni,V a) 1/2 (Ni,V a) 1/4 for β(Co,Ni) 3 Sb and two sublattices (Co,Ni) 3/4 (Ni,Sb) 1/4 and (Co,Ni) 5/7 (Ni,Sb) 2/7 for δ(Co,Ni) 3 Sb and θ(Co,Ni) 5 Sb 2 respectively. The intermetallic compounds ζ(Co,Ni)Sb 2 and η(Co,Ni)Sb 3 are treated as linear stoichiometric compounds. The liquid, the α Fcc_A1 and the e Hcp_A3 phases are assumed to be substitutional solutions with Redlich–Kister polynomials for the expressions of the excess Gibbs energies. The Sb Rhombo_A7 phase is considered to be the pure Sb due to the very small and negligible solubilities of Co and Ni in it. A set of self-consistent thermodynamic parameters of the Co–Ni–Sb system is obtained according to eleven experimental isothermal sections and one isoplethal section. Most of the experimental data can be satisfactorily reproduced by the present thermodynamic modeling. The liquidus projection and the reaction scheme are also presented.

The Ni–Sb system was critically assessed by means of the CALculation of PHAse Diagram (CALPHAD) technique. The solution phases, Liq and (αNi), were modelled as the substitutional solutions with the Redlich–Kister equation. The intermediate phases, (γNiSb) and (βNi3Sb), with homogeneity ranges were described respectively using three-sublattices (Sb)1/3(Ni%,V a)1/3(V a%,Ni)1/3 and (Sb)1/4(Ni%,V a)1/2(Ni%,V a)1/4 based on their structure features. Corresponding to the phase (βNi3Sb), the two low-temperature phases of (δNi3Sb) and (θNi5Sb2) with narrow homogeneity ranges were modelled as two-sublattice, (Ni)3/4(Sb,Ni)1/4 and (Ni)5/7(Sb,Ni)2/7. The intermetallic compound ζNiSb2 with no homogeneity ranges was treated as stoichiometric compound. The phase e Sb was considered as pure Sb for the solubility of Ni in e Sb is very low. A set of self-consistent thermodynamic parameters of the Ni–Sb system was obtained. The optimized phase diagram and thermodynamic properties were presented and compared with experimental data.

The Co–Sb system has been assessed thermodynamically based on the experimental data of both the phase equilibria and the thermochemical properties. The reasonable models were constructed for all the phases of the system. The liquid and the terminal solid solution phases, (αCo) and (eCo), were described as the substitutional solution model while the (Sb) phase was treated as the pure element Sb for lack of the solubility data of Co in (Sb). The (βCoSb) phase was modelled by three-sublattices, Sb: Co, Va: Va, Co, since its structure is of the NiAs-type (B81) with the Co vacancies on the Sb-rich side and the interstitial Co atoms on the Co-rich side. The compounds γCoSb2 and δCoSb3 were expressed as the stoichiometric compounds. Much attention has been paid to the consistence between the experimental measurements and the theoretical calculations. Using the present thermodynamic data, the calculation results can reproduce the experimental data well.

Being the structural cause of hardening, Guinier–Preston (GP) zones in many alloys still attract much interest. The expression of energy for GP zones in the Al–Cu alloy is established by combining the essential Gibbs energy with the interfacial energy and the strain energy. Based on the equilibrium between GP zones and the surrounding matrix, a quantitative analysis on the sizes, concentrations, aging temperatures and their relationships can be predicted. The size and the concentration of GP zones calculated with defined composition and aging temperature accord with the experimental results well.

The thermodynamic modeling and optimization of the Mg–Pr, Pr–Y binary systems and Mg–Pr–Y ternary system were critically carried out by means of the CALPHAD (CALculation of PHAse Diagram) technique. The solution phases (liquid, body-centered cubic, hexagonal close-packed and double hexagonal close-packed) were modeled with the Redlich–Kister equation. The Compound Energy Model has been used to describe the thermodynamic functions of the intermetallic compounds in these systems. The compounds Mg2Y,Mg12Pr,Mg41Pr5,Mg3Pr,Mg2Pr,Mg5M and Mg24Y 5 in the Mg–Pr–Y system were treated as the formulae (Mg,Y )2(Mg,Pr,Y ), Mg12(Pr,Y ),Mg41(Pr,Y )5,Mg3(Mg,Pr,Y ),Mg2(Pr,Y ),Mg5(Pr,Y ) and Mg24(Mg,Pr,Y )4Y, respectively. A two-sublattice model (Mg,Pr,Y )0.5(Mg,Pr,Y )0.5 was applied to describe the compound MgM formed by MgPr and MgY in order to cope with the order–disorder transition between body-centered cubic solution (A2) and MgM with CsCl-type structure (B2) in the Mg–Pr–Y system. A set of self-consistent thermodynamic descriptions of the Mg–Pr–Y system was obtained. The projection of the liquidus surfaces for the Mg–Pr–Y system and the complete reaction scheme for the Mg–Pr–Y system were predicted.

The eutectic temperature between the NiZr and NiZr2 compounds was experimentally determined using Differential Scanning Calorimetry (DSC) to evaluate definitely the discrepancy among different sources. Based on the data of the literature reports and the present experiment results, the Ni–Zr binary system was re-assessed thermodynamically by means of CALPHAD (CALculation of PHAse Diagrams) technique. The substitutional solution model was used to describe the liquid phase and the three terminal solid solutions, fcc-A1 (Ni), bcc-A2 ( β Zr ) and hcp-A3 ( α Zr ), and their excess Gibbs energies were formulated with the Redlich–Kister polynomial. The sublattice-compound energy model was employed to describe the intermetallic compound (Ni5Zr) with a homogeneity range. The other compounds Ni7Zr2, Ni3Zr, Ni21Zr8, Ni10Zr7, Ni11Zr9, NiZr and NiZr2 were treated as stoichiometric phases. The parameters of all the Gibbs energy expressions were optimized. A set of self-consistent thermodynamic parameters has been obtained. The calculated phase diagram and thermodynamic properties agree well with the available experimental data.

Thermodynamic modeling and optimization of the Gd–Mg, Mg–Y, Gd–Y binary systems and the Gd–Mg–Y ternary system have been critically carried out by means of the CALPHAD (CALculation of PHAse Diagrams) technique. The solution phases (liquid, body-centered cubic, and hexagonal close-packed) are modeled with the Redlich–Kister equation. The Compound Energy Model has been used to describe the thermodynamic functions of the intermetallic compounds in these systems. The compounds Mg2Y, Mg5Gd, Mg3Gd, Mg2Gd and Mg24Y5 in the Gd–Mg–Y system have been treated as the formulae (Mg, Y)2(Gd, Mg, Y), Mg5(Gd, Y), Mg3(Gd, Y), Mg2(Gd, Y) and Mg24(Gd, Mg, Y)4Y, respectively. A model (Gd, Mg, Y)0.5(Gd, Mg, Y)0.5 has been applied to describe the compound MgM formed by MgGd and MgY in order to cope with the order–disorder transition between the body-centered cubic solution (A2) and MgM with CsCl-type structure (B2) in the Gd–Mg–Y system. A set of self-consistent thermodynamic descriptions of the Gd–Mg–Y system have been obtained. The projection of the liquidus surfaces and the entire reaction schemes for the Gd–Mg–Y system have been constructed.

A mathematical analysis of the constraints on the thermodynamic parameters of the compound energy formalism for an ( A , B ) m ( A , B ) n ↔ ( A , B ) m + n type order–disorder transformation is studied. Both the ordered phase and the disordered phase can be described using the two-sublattice model after the formula ( A y A ′ B y B ′ ) m ( A y a ″ B y B ″ ) n , with y i ′ = y i ″ ( i = A , B ) for the disordered phase and y i ′ ≠ y i ″ for the ordered phase. The constraints between the parameters for the ordered state were derived. The thermodynamic characteristics of the transformation are illustrated using a mathematical method and the validity of the model and the constraints is tested.

Based on all the available experimental phase equilibrium and thermochemical data, the Cu–Zr system has been thermodynamically reassessed by means of the CALPHAD (CALculation of PHAse Diagrams) technique. The substitutional solution model was used to describe the liquid and three terminal solid solutions, fcc-A1(Cu), bcc - A 2 ( β Zr ) and hcp - A 3 ( α Zr ) , and their excess Gibbs energies were formulated with the Redlich–Kister polynomial. Ten intermediate compounds, Cu9Zr2, Cu51Zr14, Cu8Zr3, Cu2Zr, Cu24Zr13, Cu10Zr7, CuZr, Cu5Zr8, CuZr2-h and CuZr2-l, were contained in the system, and were treated as stoichiometric phases. For the phase CuZr2, there are the high and the low temperature modifications. The parameters of all the Gibbs energy expressions were optimized. A set of self-consistent thermodynamic parameters has been obtained. The calculated phase diagram and thermodynamic properties agree well with the available experimental data.

In this paper, the effect of NH 3 decomposition rate on the composition conditions necessary to grow a single phase GaN film has been studied thermodynamically with trimethyl gallium (Ga(CH 3 ) 3 ) and ammonia (NH 3 ) as the source gases and hydrogen (H 2 ) as the carrier gas for the MOVPE process. The results indicated that the Gas+Nitride two phase region essential for the growth of GaN epitaxy layer becomes narrower with increase of the NH 3 decomposition rate. This explained the high value of V/III ratio during the practical growth process. This analysis also implied that low NH 3 decomposition rate might possibly facilitate the MOVPE growth of GaN epitaxy layer.