Substitutional Solution Model Research Articles

Phase diagrams of permanent magnet alloys: Binary rare earth alloy systems

Thermodynamic optimization of the binary rare earth alloy (Ce–La, Ce–Pr, Ce–Nd and La–Nd) systems was performed in this work through the CALPHAD method based on the critical evaluation of all available phase diagram and thermodynamic data reported in the literature. During the thermodynamic modeling, the solution phases including liquid, bcc, fcc and dhcp, were treated as the substitutional solution model. Thermodynamic parameters of the stable phases in the Ce–La, Ce–Pr, Ce–Nd and La–Nd binary systems are obtained finally and would be used directly to develop the thermodynamic database of the multi-component Nd–Fe–B-based alloys, which is indispensable for designing alloy compositions and processes of Nd–Fe–B permanent magnets with highly abundant rare earth metals.

Thermodynamic description and solidified microstructure of the Co-Ge system

The Co–Ge binary system was reassessed by CALPHAD method in view of new phase diagram data and drawbacks of the previous modeling. A three-sublattice model (Co,Va)1(Co)1(Ge)1 was used to describe the B82-type βCo5Ge3-phase based on its crystal structure. In order to describe the transformation between ordered L12-type phase (Co3Ge) and the disordered fcc_A1 phase (αCo), one single Gibbs energy function was used for both ordered and disordered phases. In almost all the previous thermodynamic calculations, the ordered phase with a negligible homogeneity range, such as Co3Ge, is treated as a stoichiometric compound. Such a treatment is not physically sound. For (αCo) and (εCo), the magnetic contribution to Gibbs energy is taken into account. Both substitutional solution model and associated model were applied to describe the liquid phase, and thus two sets of self-consistent thermodynamic parameters for this system were obtained. It was found that the associated model can account for the experimental data more satisfactorily than the substitutional solution model, especially for the partial enthalpy of mixing data for the liquid phase. Five representative as-cast Co-Ge alloys were prepared to compare the solidified microstructure with that predicted according to thermodynamic calculations. According to the calculated Scheil solidification curves of two key alloys, the solidified microstructure for the alloys was analyzed. Also, the calculated amounts of the solidified phases in five as-cast alloys are compared with the experimental data resulting from automatic image analysis of the BSE images, showing a good agreement between the calculation and experiment, in particular for the case in which the associated model is used to describe the properties of the liquid phase. It is demonstrated that the combined use of the thermodynamic calculation and decisive experiment is an efficient strategy to obtain the desired microstructure.

CALPHAD-type thermodynamic modeling of the Ti-W-B and Ti-W-Si refractory systems

Based on the thermodynamic descriptions of binary sub-systems as well as the experimental phase equilibria data available in the literature, the Ti-W-B and Ti-W-Si ternary systems have been assessed by the CALPHAD (CALculation of PHAse Diagram) method. The solution phases, i.e. liquid and bcc (βTi, W), are described by the substitutional solution model. The binary phases TiB2, TiB, W2B, αWB, W2B5, W2B9, Ti5Si3 and W5Si3 with the solubilities of the third element are modeled using the sublattice models. Two ternary compounds τ with composition of (Ti1-xWx)B and Ti3W2Si10 are described by two sublattice model (Ti,W)1B1 and (Ti,W)1Si2, respectively. A set of self-consistent thermodynamic parameters is finally obtained for each of the two ternary systems. Some representative isothermal and vertical sections as well as liquidus surface projections are calculated. Comparisons between the calculated and measured phase diagrams show that the most of experimental data are satisfactorily reproduced by the present thermodynamic descriptions.

Experimental investigation and thermodynamic assessment of the RbCl[sbnd]CsCl[sbnd]ZnCl2 system and its subsystems

The RbClZnCl2 and CsClZnCl2 binary systems with different proportions are firstly prepared to determine the phase equilibrium relation using differential scanning calorimetry (DSC) and X-ray diffraction (XRD) methods. The phase diagrams of RbClZnCl2, CsClZnCl2 and RbClCsCl are thus critically optimized by thermodynamic modeling based on the available experimental values. The associate solution model (ASM) is used to describe the liquid phases of RbClZnCl2 and CsClZnCl2 binary system, and the substitutional solution model (SSM) is applied for the RbClCsCl binary system. All Gibbs energies of intermediate compounds are treated with the Neumann-Kopp rule. In light of this, a set of self-consistent thermodynamic database is obtained and applied to predict the thermodynamic properties of the RbClCsClZnCl2 ternary system by merging the Gibbs energies in its subsystems via the Kohler-Toop interpolation method. These results could provide effective data supporting for the selection and optimization of multi-component molten salts for the concentrating solar power (CSP) plants.

Experimental investigation and thermodynamic modeling of the Ni–Ti–V system

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.

CALPHAD-type thermodynamic description of phase equilibria in the Ti-W-M (M = Zr, Mo, Nb) ternary systems

Based on the thermodynamic descriptions of constitutive binary systems as well as the experimental phase equilibria data available in the literature, the Ti-W-M (M = Zr, Mo, Nb) ternary systems have been thermodynamically evaluated by the CALPHAD (CALculation of PHAse Diagram) approach. The solution phases, i.e. liquid and bcc, are described by the substitutional solution model. The binary phase W2Zr with the solubilities of Ti is modeled using the sublattice model. A set of self-consistent thermodynamic parameters is finally obtained for each of these ternary systems. Some representative isothermal and vertical sections are calculated. Comparisons between the calculated and measured phase diagrams show that the most of experimental data are satisfactorily reproduced by the present thermodynamic descriptions.

Thermodynamic assessment of the SrBr2-MBr (M: alkali metal) binary systems

The phase diagrams of the SrBr2-MBr (M: alkali metal) binary systems based on phase equilibria and thermochemical data were evaluated and optimized using the CALPHAD method. The substitutional solution model was used to describe the Gibbs energies of all liquid phases. All intermediate compounds were handled to be stoichiometric of which the Gibbs energies comply with the Neumann-Kopp rule. Self-consistent datasets with all model parameters were developed, and thermodynamic properties of liquid phase (mixing enthalpy and activity) were also calculated for the whole range of compositions. Meanwhile, the empirical method based on ionic parameter function was successfully used to validate the mixing enthalpies calculated. Results manifest that the calculated results were in excellent agreement with the experimental values reported in the present work. By means of thermodynamic calculation, this database of strontium bromide system was used to predict the SrBr2-based multicomponent system with low melting points, which will design and prepare composite materials for low temperature thermal energy storage (TES).

Thermodynamic modelling of the Hf–Pt system

Abstract By means of the CALPHAD (CAlcultion of PHAse Diagram) technique, the Hf–Pt system was critically assessed. Based on the experimental data, the four solution phases (liquid, fcc, bcc and hcp) were described with the substitutional solution model. The intermetallic compounds Hf3Pt4 and αHfPt were treated as the formula (Hf,Pt) m (Hf,Pt) n by a two-sublattice model. Based on the solid solution range, the intermetallic compounds HfPt4 and Hf2Pt were treated as the formula (Hf,Pt)1(Pt)3 and (Hf)2(Hf,Pt)1, respectively. The intermetallic compound Hf2Pt3 was treated as a stoichiometric compound. The formulas (Hf,Pt)0.5(Hf,Pt)0.5 · (Va)3 and (Hf,Pt)0.25(Hf,Pt)0.75(Va)0.5 were applied to describe the compounds βHfPt with CsCl-type structure (B2) and HfPt3 with Ni3Ti-type structure (D024) to cope with the order-disorder transition from bcc-A2 to bcc-B2 and hcp-A3 to hcp-D024. A set of self-consistent thermodynamic parameters of the Hf–Pt system was obtained.

Phase equilibria and thermodynamic evaluation of BaO-TiO2-YO1.5 system

A series of ceramic samples were prepared and characterized at 1400 °C in the BaO-TiO2-YO1.5 system. The obtained experimental results were adopted to the present thermodynamic evaluation to derive a set of thermodynamic database for the BaO-TiO2-YO1.5 system. The database was constructed by the CALPHAD method where the binary parameters from BaO-TiO2 and TiO2-YO1.5 systems were re-optimized, those from the BaO-YO1.5 system were simulated by our previous assessments. Thermodynamics descriptions of all liquid and terminal solid solution phases were treated by the substitutional solution model, the H-BaTiO3 (BaTiO3, hexagonal structure) and C-BaTiO3 (BaTiO3, cubic structure) were described by the compound energy formalism model, and other binary intermediate phases were treated as the stoichiometric compounds. Finally, some thermodynamic diagrams were calculated and compared with the experimental results. Good agreement between present calculations and the experimental results demonstrates that the present thermodynamic database is self-consistent and credible.

Thermodynamic descriptions of the Ag-X (X = S, As, Lu) systems

Based on the critical evaluation of experimental data available in the literature, for the first time the Ag-X (X = S, As, Lu) binary systems were assessed thermodynamically using the CALPHAD (CALculation of PHAse Diagram) method. The solution phases including liquid, (Ag), (αS), (βS), (αAs), (εAs) and (Lu) were described by the substitutional solution model, and their excess Gibbs energies were described with the Redlich-Kister polynomial equation. In view of the possible existence of associate species Ag2S and the difficulty to describe the liquid phase by the substitutional solution model, an associate model was used to describe the liquid phase in the Ag-S system. The intermetallic compounds, i.e. αAg2S, AgLu, Ag2Lu and Ag4Lu, were all modeled as stoichiometric compounds. And the intermetallic compounds βAg2S and γAg2S were described by two-sublattice model (Ag, S)2S1 in view of the solubilities. Self-consistent thermodynamic parameters to describe the Gibbs energies of phases in the Ag-S, Ag-As and Ag-Lu systems were obtained. The calculated results were in good agreement with the reported phase equilibria and thermodynamic properties.

Thermodynamic description of phase equilibria in the C–Mo–W–N quaternary system

The Mo–W–N and C–Mo–W ternary systems have been critically evaluated by means of the CALPHAD approach, wherein the C–Mo–W system was reassessed to ensure the model consistency. The Gibbs energies of individual phases in the ternary systems are described with corresponding models, such as substitutional solution model and sublattice model. A set of self-consistent thermodynamic parameters is obtained. Comprehensive comparisons between the calculated and reported phase diagram data show that the reliable information is satisfactorily accounted for by the present modeling. Based on the present work together with the previously reported assessments of C–Mo–N and C–W–N ternary sub-systems, a thermodynamic description of the C–Mo–W–N quaternary system is obtained and used to calculate the phase equilibria and thermodynamic stability of the hexagonal structured (Mo,W)(C,N) carbonitride phase.

Thermodynamic optimization of the Ni–Ta system supported by the key experiments

The Ni–Ta system was experimentally determined and thermodynamically optimized in the present work. In experimental work, 12 as-cast and 12 annealed alloys were measured using scanning electron microscopy (SEM) with energy dispersive X-ray spectroscopy (EDX) and powder X-ray diffraction (XRD) techniques, and 5 crucial annealed alloys were determined using differential scanning calorimetry (DSC) method. It was found that Ni8Ta did not exist in both as-cast and annealed samples, and then two invariant reactions liq. → fcc(Ni) + Ni3Ta and liq. → Ni2Ta + μ-NiTa were modified. In the thermodynamic optimization, the solution phases liquid, fcc(Ni) and bcc(Ta) were treated with a substitutional solution model, three compounds Ni3Ta, Ni2Ta and NiTa2 were described using a two-sublattice model, and the compound μ-NiTa was modeled as (Ni,Ta)1Ta4(Ni,Ta)2(Ni,Ta)6 using a four-sublattice model. A set of self-consistent thermodynamic parameters of the Ni–Ta system was obtained.

Experimental Investigation and Thermodynamic Modeling of the NaCl-NaNO3-Na2SO4 Ternary System

Molten salts as heat transfer and storage materials have been used to nuclear energy and concentrated solar power(CSP) applications. In this work, the system of molten salt mixture based on thermodynamic principles was designed as thermal energy storage(TES) materials. The substitutional solution model can be employed to describe the Gibbs energies of all liquid phase. Thermodynamic model parameters for the NaCl-NaNO3-Na2SO4 subsystems were conducted by thermodynamic evaluation and optimization based on experimental phase-equilibria data. Thus, a set of self-consistent thermodynamic database was eventually obtained to reliably calculate the whole phase diagram and thermodynamic properties for the NaCl-NaNO3-Na2SO4 ternary system. The results manifest that the eutectic point of theternary system located at T=280 °C and xNaCl=8.4%, $${x_{NaN{O_3}}}$$ =86.3% and $${x_{N{a_2}S{O_4}}}$$ =5.3%. Moreover, the results predicted were verified experimentally using differential scanning calorimetry(DSC) and the agreement between the measured value[T=(287±2) °C] and predicted value(T=280 °C) was satisfactory. Thus, the thermodynamic calculation method will be used to design and develop novel molten salt mixture as thermal energy storage materials.

Thermodynamic Evaluation of NaF-MF n (M=Be, U, Th) Systems for Molten Salt Reactor

The phase diagrams of NaF-BeF2, NaF-ThF4 and NaF-UF4 systems were assessed based on thermodynamic principles, and diverse thermodynamic models were adapted to different systems. Associate solution model (ASM) was used to describe the Gibbs energies of liquid phase of the NaF-BeF2 system, whereas other systems(NaF-ThF4 and NaF-UF4) were treated with the substitutional solution model(SSM) and intermediate compounds were described as stoichiometric compounds according to the Neumann-Kopp rule. All the thermodynamic model parameters were optimized by the least squares procedure until good coincidence was achieved between the calculated results and the experimental data. The derived thermodynamic parameters will be merged into the NaF-BeF2-ThF4-UF4 database to develop the thorium molten salt reactor(TMSR) project.

Thermodynamic assessment of the C–Zr–Nb ternary system

The whole C–Zr–Nb ternary system was assessed by means of CALPHAD method for the first time. All of the experimental phase diagram data available from the literature were critically reviewed and assessed using thermodynamic models for the Gibbs energies of individual phases. The solution phases including the liquid, fcc, hcp and bcc were described by the substitutional solution model. There is no ternary compound in this system. A set of self-consistent thermodynamic parameters for the C–Zr–Nb system is obtained. Comparisons between the calculated and measured diagrams show that the present thermodynamic description can account for the experimental information satisfactorily. The liquidus projection and reaction scheme for the C–Zr–Nb system were generated by using the present thermodynamic parameters. The effect of microstructure on the mechanical properties of C–Zr–Nb alloys homogenized at 1800 °C reported in the literature was discussed from a phase diagram point of view.

Thermodynamic Assessment of the Mn-H and Mg-Mn-H Systems

Mn-H and Mg-Mn-H are two important sub-systems in the Mg-Mn-Ni-H system, which is considered as a very promising hydrogen storage system. In this work, the available phase diagram and thermodynamic data of the Mn-H and Mg-Mn-H systems were critically reviewed and the thermodynamic assessments of the two systems were carried out using the CALPHAD method. In the thermodynamic modeling, the liquid phase was described by a substitutional solution model, the solid solution phases and binary e hydride were described by the two-sublattice model, and the two ternary hydrides (Mg3MnH7 and Mg3MnH6) were treated as stoichiometric phases. Using the obtained thermodynamic parameters for the Mn-H and Mg-Mn-H systems, the phase diagrams of the two systems at different pressure were calculated. Comparisons between the calculated and experimental results showed that the reliable experimental data can be reasonably described by the present thermodynamic description. Some predictions of the two systems were also presented according to the obtained thermodynamic parameters.

Thermodynamic re-assessment of the Re–X (X=Al, Co, Cr, Ta) binary systems

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.

Thermodynamic optimizations of the Nd-Sn and Sn-Tb systems

The thermodynamic optimizations of the Nd-Sn and Sn-Tb binary systems were carried out by means of the Calculation of Phase Diagram (CALPHAD) method on the basis of the available experimental data including the thermodynamic properties and phase equilibria. The Gibbs free energies of the liquid, bcc, bct, dhcp and hcp phases were described by the substitutional solution model with the Redlich-Kister equation, while all of the intermetallic compounds (Nd5Sn3, Nd5Sn4, Nd11Sn10, NdSn, Nd3Sn5, NdSn2, Nd3Sn7, Nd2Sn5, NdSn3, Sn3Tb, βSn7Tb3, αSn7Tb3, Sn2Tb, Sn5Tb4, SnTb4, Sn10Tb11, Sn4Tb5 and Sn3Tb5) were described by the sublattice model. A set of self-consistent thermodynamic parameters of each phase in the Nd-Sn and Sn-Tb binary systems has been obtained, and the calculated results are in good agreement with the available experimental data.

Thermodynamic assessment of the Ge-Yb binary system

ABSTRACTThe Ge-Yb binary system was thermodynamically assessed by CALPHAD (CALculation of PHAse Diagram) approach based on the available experimental data of thermochemical properties and phase equilibria. The liquid phase and the terminal phases, Fcc_A1 (Yb), Bcc_A2 (Yb) and Diamond_A4 (Ge), are described by a substitutional solution model, and their excess Gibbs free energies are formulated with the Redlich-Kister expression. The intermetallic compounds, Yb3Ge8, Yb3Ge5, Yb11Ge10, Yb5Ge4, Yb5Ge3 and Yb2Ge, are treated as strict stoichiometric compounds. A set of self-consistent thermodynamic parameters for the system is obtained. The calculated phase diagrams and thermochemical properties are in good agreement with the available experimental data.

Thermodynamic modeling of the In-Sc and In-Y systems supported by first-principles calculations

Based on an assessment of the phase equilibria and thermodynamic data in the literature, the thermodynamic modeling of the In?Sc and In?Y systems was carried out by means of the calculation of phase diagram (CALPHAD) method supported by first-principles calculations. The solution phases, i.e., liquid, (In), (?Sc), (?Sc), (?Y) and (?Y), were modeled with the substitutional regular solution model. Ten intermetallic compounds, including InSc3, InSc2, In4Sc5, InSc, In2Sc, In3Sc, InY2, InY, In5Y3, and In3Y were described as stoichiometric phases, while In3Y5 was modeled with a sublattice model with respect to its homogeneity range. The enthalpies of formation of the intermetallic compounds at 0 K were computed using firstprinciple calculations and were used as input for the thermodynamic optimization. A set of self-consistent thermodynamic parameters for both the In?Sc and In?Y systems were obtained and the calculated phase diagrams are in good agreement with the experimental data.