Experimental investigation and thermodynamic calculation of binary Mg-Mn phase equilibria

The monotectic reaction L″=L′+δMn in the binary system Mg-Mn was measured at about 1200 °C using differential thermal analysis. This was possible by sealing the samples in arc welded Ta capsules, thus solving the problem of high vapor pressure of Mg at such high temperature and also suppressing any reaction of Mg with the environment. Problems associated with the reaction of Mn with the Ta crucible are discussed in detail. In addition, a thermodynamic assessment and calculation of the entire Mg-Mn phase diagram was performed, incorporating published experimental data on the Mn solubility in the Mg-rich corner.     

Particle size effects on the coherent phase equilibria of binary nanoparticles

The phase stability of isolated, radially symmetric nanoparticles of a binary system that exhibits a miscibility gap was analyzed by constructing coherent phase diagrams which account for both the surface stress (([^(T)]_s )(\hat T_s ))and the second-order compositional dependence of the lattice parameter (η _cc ). Although the elastic stress field in a two-phase coherent particle with a concentric core-shell structure is heterogeneous and nonhydrostatic at equilibrium, the appropriate free energy extremized for equilibrium could be expressed as a function solely of the temperature (ϕ), composition (c), and effective pressure (P), which are homogeneous in each phase at equilibrium. The construction of coherent phase diagrams in the three-dimensional ϕ-c-P space showed that the miscibility gap can be either extended or reduced by decreasing the particle radius, depending on the sign of[^(T)]_s h_cc\hat T_s \eta _{cc} , and that the tie-lines lie in thec-P plane.     

Equilibrium and diffusion in coherent multilayers

A thermodynamic treatment is presented of isothermal phase equilibria and diffusion in coherent planar multilayers. For two-component systems, coherent two-phase equilibrium compositions can often be generated using the familiar procedure in which a line is constructed doubly tangential to the coherent free energy functional(s). Only in special cases, however, are the phase equilibria so generated independent of the average composition (for unsupported multilayers) or the substrate effective composition (for multilayers coherently attached to a substrate). The common tangent construction is extended to ternary solution systems, and conditions for which the coherent ternary equilibria are independent of the average or substrate compositions defined.

The formal thermodynamic aspects of diffusion in binary and ternary solids are reviewed, and the expected effects of coherency strain on the diffusional homogenization process outlined. The diffusion formalism can often be extended to account for coherency strains simply by substituting the coherent free energy density for its incoherent counterpart. In ternary and higher order systems, it is always possible to choose an initial condition that is free of strain; it is not in general possible to maintain this strain-free condition during the course of diffusional homogenization. The general effect of strain energy on ternary diffusion is to rotate the diffusion eigenvectors such that the strain is reduced. Some sample calculations are presented for the system Cu---Au---Ag.     

Experimental investigation and thermodynamic calculation of the phase equilibria in the Co-Nb-Ta ternary system

The isothermal sections of the Co-Nb-Ta ternary system at 900 °C, 1000 °C, 1100 °C, 1200 °C, 1300 °C have been experimentally determined by electron probe microanalysis (EPMA) and X-ray diffraction (XRD) techniques on the equilibrated alloys. On the basis of the experimental data investigated in the present work, the phase equilibria in the Co-Nb-Ta system has been thermodynamically assessed by using CALPHAD (CALculation of PHAse Diagrams) method, and a consistent set of the thermodynamic parameters leading to reasonable agreement between the calculated results and experimental data was obtained.     

Experimental determination and thermodynamic calculation of phase equilibria in the Fe−Mn−Al system

The phase equilibria among the face-centered cubic (fcc), body-centered cubic (bcc), and βMn phases at 800, 900, 1000, 1100, and 1200 °C were examined by electron probe microanalysis (EPMA), and the A2/B2 and B2/D0₃ ordering temperatures were also determined using the diffusion couple method and differential scanning calorimetry (DSC). The critical temperatures for the A2/B2 and B2/D0₃ ordering were found to increase with increasing Mn content. Thermodynamic assessment of the Fe−Mn−Al system was also undertaken with use of experimental data for the phase equilibria and order-disorder transition temperatures using the CALPHAD (Calculation of Phase Diagrams) method. The Gibbs energies of the liquid, αMn, βMn, fcc, and ε phases were described by the subregular solution model and that of the bcc phase was represented by the two-sublattice model. The thermodynamic parameters for describing the phase equilibria and the ordering of the bcc phase were optimized with good agreement between the calculated and experimental results. This paper was presented at the International Symposium on User Aspects of Phase Diagrams, Materials Solutions Conference and Exposition, Columbus, Ohio, 18–20 October, 2004.     

Experimental investigation and thermodynamic calculation of the phase equilibria in the Co–Mo–W system

The isothermal sections in the Co–Mo–W ternary system at 600 °C, 800 °C, 1000 °C, 1100 °C, 1200 °C and 1300 °C have been determined using the Electron probe micro-analyzer (EPMA) and X-ray diffraction (XRD) techniques. A thermodynamic assessment of the Co–Mo–W ternary system was carried out by the CALPHAD (Calculation of Phase Diagrams) method. The Gibbs free energies of the liquid and solution phases were described by the subregular solution models, and those of intermetallic compounds were described by the sublattice model. A consistent set of thermodynamic parameters has been derived for describing the Gibbs free energies of each solution phase and intermetallic compound in the Co–Mo–W ternary system. The calculated phase diagrams and thermodynamic properties in the Co–Mo–W ternary system are in good agreement with experimental data.     

Thermodynamics of phase equilibria

Equations have been proposed and derived for calculation of thermodynamic properties of multicomponent alloys from various types of data. Effects of phase boundaries on activity measurements are discussed, and the use of activity data on condensed phase diagram determination is presented. The activity and phase boundary determinations are discussed, and advantages and shortcomings of selected experimental techniques are noted. A brief review of computer calculation of phase diagrams is presented. This research is part of the effort in material properties at the U.S. Bureau of Mines and at the Polish Academy of Sciences.     

A first-principles approach to modeling alloy phase equilibria

This paper presents a brief overview of recent developments in the application of first-principles calculations to the study of bulk and interfacial thermodynamic properties and phase equilibria in alloys. Among the applications discussed are calculations of: bulk thermodynamic properties, phase boundaries, interfacial free energies, and precipitate morphologies. The article concludes by highlighting some further recent developments that are likely to lead to increasing applications of these modeling techniques. For additional information, contact M. Asta, Northwestern University, Department of Materials Science and Engineering, 2225 North Campus Drive, Evanston, Illinois 60208-3108; telephone (847) 491-5940; fax (847) 491-7820; e-mail m-asta@northwestern.edu.     

High-temperature phase equilibria in Ti-Al-Mo system

The phase equilibria in a Ti-Al-Mo system have been investigated at 1473 and 1573 K experimentally and theoretically. Phase equilibria were experimentally investigated by in situ x-ray diffraction and thermal analysis as well as by quenching and diffusion couples. From the results, isothermal sections of this system at these two temperatures were constructed and assessed using Thermo-Calc. A part of the ternary phase diagram of Ti-Al-Mo system has revealed the following features: (1) low solution limits of Mo (a few at.% Mo) in both α (disordered hcp structure) and γ (TiAl: the L1₀ structure) phases, (2) the narrow three-phase region of the α+β (bcc Ti terminal solution) +γ phases, and (3) the role of Mo as a strong “β stabilizer.”     

High-temperature phase equilibria in Ti-Al-Mo system

The phase equilibria in a Ti-Al-Mo system have been investigated at 1473 and 1573 K experimentally and theoretically. Phase equilibria were experimentally investigated by in situ x-ray diffraction and thermal analysis as well as by quenching and diffusion couples. From the results, isothermal sections of this system at these two temperatures were constructed and assessed using Thermo-Calc. A part of the ternary phase diagram of Ti-Al-Mo system has revealed the following features: (1) low solution limits of Mo (a few at.% Mo) in both α (disordered hcp structure) and γ (TiAl: the L1₀ structure) phases, (2) the narrow three-phase region of the α+β (bcc Ti terminal solution) +γ phases, and (3) the role of Mo as a strong “β stabilizer.”     

A theoretical study of the phase equilibria in the Cu---Cr---Zr system

Small additions of chromium to Cu---Zr alloys or zirconium to Cu---Cr alloys are widely used to improve the mechanical properties of the alloys at high temperatures. There are conflicting opinions in the literature on the question of the existence of the pseudobinary system Cu---Cr₂Zr. From calculation of the Cu---Cr---Zr phase diagram using the CALPHAD method, the possibility of the existence of this pseudobinary has been ruled out. A number of isothermal and vertical sections have been calculated together with the liquidus surface. The reaction scheme for the whole system is also presented.     

Measurement of phase equilibria in Ti-Ni-Sn system

Phase relations of the Ti-Ni-Sn ternary system were investigated via alloy sampling assisted with X-ray diffractometry (XRD) and electron probe micro-analysis (EPMA). A new binary phase with composition of TiSn₄ (molar fraction, %) was detected at 508 K. In addition, a supplementary phase (Ti_1–x–yNi_xSn_y)Ni₃ (τ, AuCu₃-type) was observed at 873 and 973 K. According to the characterised microscopic structure in various annealed alloys, four ternary phases were detected in Ti-Ni-Sn ternary system: TiNiSn, TiNi₂Sn, Ti₂Ni₂Sn and (Ti_1–x–yNi_xSn_y)Ni₃. Additionally, isothermal sections of Ti-Ni-Sn ternary system at 508, 873 and 973 K were constructed. By comparing three isothermal sections, a peri-eutectic reaction, L+TiNi₂Sn→Ni₃Sn₄+TiNiSn, was deduced, which occurs at a temperature between 873 and 973 K. Furthermore, the solubility of Sn in TiNi and Ni in Ti₅Sn₃ was detected.     

Prediction of ordered superstructure phase equilibria

The approach to phase diagram calculations has changed drastically within the last few years. Previously, mean-field models (regular solution, Bragg-Williams, concentration waves) were used almost exclusively. These models rely on two very poor approximations: supe5rposition of ’point’ probabilities for pair probabilities and an ideal solution model for the configurational entropy. Today, effective cluster interactions can be calculated from first-principles electronic structure methods: the superposition approximation is avoided, and cluster formulations for the entropy are available. As will be shown by recently computed examples, such cluster methods predict first-priciples phase diagrams that are often in excellent agreement with those determined empirically. The pater was presented at the International Phase Diagram Prediction Symposium sponsored by the ASM/MSD Thermodynamics and Phase Equilibria Committee at Materials Week, October 21-23,1991, in Cincinnati, OH. The symposium was organized by John Morral, University on Connecticut, and Philip Nash, Illinois Institute of Technology.     

Solid-state phase equilibria in the Titanium-Aluminum-Nitrogen system

The 1273K isothermal section of the Ti-Al-N phase diagram was studied using modern methods of physical and chemical analyses. The data obtained by various techniques are in good agreement and in harmony with the results of thermodynamic calculations. It has been reliably established that AIN can coexist with TiN_1−x , Ti₂AIN, and TiAl₃; the ternary nitride Ti₂AIN can be in equilibrium with TiAl₃, AIN, TiAl₂, TiAl, TiN_1−x , and Ti₃AIN; the solid solution based on α(Ti) coexists with Ti₃Al, Ti₃AIN, TiN_1−x , and Ti₂N. Literature data on phase equilibria in the Ti-Al-N system were analyzed, and a 1273 K isothermal section of the phase diagram has been suggested.     

Prediction of ordered superstructure phase equilibria

The approach to phase diagram calculations has changed drastically within the last few years. Previously, mean-field models (regular solution, Bragg-Williams, concentration waves) were used almost exclusively. These models rely on two very poor approximations: supe5rposition of ’point’ probabilities for pair probabilities and an ideal solution model for the configurational entropy. Today, effective cluster interactions can be calculated from first-principles electronic structure methods: the superposition approximation is avoided, and cluster formulations for the entropy are available. As will be shown by recently computed examples, such cluster methods predict first-priciples phase diagrams that are often in excellent agreement with those determined empirically. The pater was presented at the International Phase Diagram Prediction Symposium sponsored by the ASM/MSD Thermodynamics and Phase Equilibria Committee at Materials Week, October 21-23,1991, in Cincinnati, OH. The symposium was organized by John Morral, University on Connecticut, and Philip Nash, Illinois Institute of Technology.     

Coherent phase equilibria in a bending film

The theory of coherent phase diagrams is expanded to phase transformations in films. The misfit between the phases dictates their layer arrangement with an interface parallel to film surfaces. Then, the transformational self-strain leads to the film bending. The phase equilibrium analysis shows that the polymorphic transformation in a single component system proceeds with the thermodynamic hysteresis. However, the reversible change of the two-phase equilibrium with temperature is possible, if the phase fractions are maintained within a range (1/2, 2/3). The character of polymorphic phase transformations in a binary system depends on the concentration of a second component. The quasi-single component transformation takes place at small concentrations. The reversible coherent phase transformation should be observed at the concentration larger than the critical one. The irreversible transformations take place at intermediate concentrations. The irreversible transformations begin with the formation of a finite fraction of the second phase. Then, the two-phase system evaluates with temperature to the final two-phase state that transfers discontinuously to a single-phase state. The concentration diagram does not coincide with the phase diagram. The concentrations in the transforming phases change non-monotonously with temperature.