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.     

Configurational Entropies of Mixing in Solid Alloys

Entropy becomes an increasingly important contributor to the Gibbs energy at high temperatures with both non-configurational and configurational contributions to be considered. Some examples of where configurational entropies alone are important in determining the domain of phase stability of a solution phase are given. In phenomenological calculations, the modeling of configurational entropy should allow for short range order and be readily applicable to multicomponent systems. The use of Fowler-Yang-Li transforms is important in this regard by providing the opportunity for changing the functional variables in cluster calculations of the Gibbs energy from cluster probabilities or correlation functions to the considerably fewer point probabilities, just as in the Bragg-Williams approximation. This article was presented at the Multi-Component Alloy Thermodynamics Symposium sponsored by The Alloy Phase Committee of the joint EMPMD/SMD of The Minerals, Metals, and Materials Society (TMS), held in San Antonio, Texas, March 12-16, 2006, to honor the 2006 William Hume-Rothery Award recipient, Professor W. Alan Oates of the University of Salford, UK. The symposium was organized by Y. Austin Chang of the University of Wisconsin, Madison, WI. Patrice Turchi of the Lawrence Livermore National Laboratory, Livermore, CA, and Rainer Schmid-Fetzer of the Technische Universitat Clausthal, Clauthal-Zellerfeld, Germany.     

The influence of solution-model complexity on phase diagram prediction

In the derivation of phase diagrams using solid- and liquid-solution thermodynamic equilibria, thermodynamic models are used to extrapolate experimental data to a broader range of compositions and temperatures. Such models have historically increased in complexity from the regular-solution formalism, with regard to both temperature and composition dependence of parameters. The regular, quasi-regular, subsubregular, and quasi-subsubregular models have been applied to existing thermodynamic data for four“simple” metallic systems of differing types (Cu-Ni, Pb-Ag, Sn-Zn, and Ge-Mg) to illustrate the effect of more accurate thermodynamic modeling on the resulting predicted phase diagrams. Comparison with the actual phase diagrams demonstrates the relative importance of“nonregularity” in the solution with regard to both composition and temperature in the accuracy of phase diagram prediction. This paper 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 of Connecticut, and Philip Nash, Illinois Institute of Technology.     

Characterization of Nucleation Behavior in Temperature-Induced BCC-to-HCP Phase Transformation for High Entropy Alloy

Phase transformation is one of the essential topics in the studies on high entropy alloys (HEAs).However,characterization of the nucleation behavior in the phase transformation for HEAs is still challenging through experimental methods.In the present work,HfNbTaTiZr HEA was chosen as the representative material,and molecular dynamics/Monte Carlo (MD/MC) simulations were performed to investigate the nucleation behavior in temperature-induced BCC-to-HCP transformation for this HEA system.The results indicate that Nb-Ta,Ti-Zr,Hf-Zr and Hf-Ti atom pairs are preferred in the BCC solid solution of HfNbTaTiZr HEA and Hf-Ti-Zr-rich atom cluster with chemical short range order acts as the nucleation site for HCP phase.The nucleation process follows the non-classical two-step nucleation model:BCC-like structure with severe lattice distortion forms first and then HCP structure nucleates from the BCC-like structure.Moreover,at low temperature,the BCC-to-HCP nucleation hardly occurs,and the BCC solid solution is stabilized.The present work provides more atomic details of the nucleation behavior in temperature-induced BCC-to-HCP phase transformation for HEA,and can help in deep understanding of the phase stability for HEAs.     

Phase equilibria of the cu-in system II: Thermodynamic assessment and calculation of phase diagram

The phases in the Cu-In binary were modelled thermodynamically using the Redlich-Kister expression for the Gibbs energies of the solution phases, the Wagner-Schottky model for those of the η (η)’)-Cu₂ln phase (taking η and η)’ to be a single phase), and assuming line compound behavior for the other intermetallic phases. The model parameters were obtained using primarily the thermodynamic data, as well as the phase equilibrium data. The thermodynamic values for the various phases calculated from the models are in reasonable agreement with the experimentally determined thermodynamic data that are available in the literature. The entropies of melting for the intermetallic phases obtained from the models are in accord with the values calculated from the empirical formulas suggested by Kubaschewski. The calculated phase diagram is also in reasonable agreement with the experimentally determined diagram, with the calculated temperatures for all the invariant equilibria within 1°C of the experimental values. The discrepancies between the calculated and experimental phase boundaries at the invariant temperatures are less than 1 at.% except those involving βCu₄Inn and γCu₇ln₃. These two phases were taken to be line compounds in the present study, although experimentally they exist over appreciable ranges of homogeneity. Current address: Dept. of Chemical Engineering, National Tsing Hua University, Taiwan.     

Computed multicomponent phase diagrams for hardenability (H) and HSLA steels with application to the prediction of microstructure and mechanical properties

Approximate methods for the calculation of the phase equilibria of multicomponent dilute systems designed for minimal claims on computer time and memory are reviewed. These results are required as the constraints on semiempirical kinetic models for the prediction of the TTT and CCT diagrams, which are used in turn for the prediction of microstructure and mechanical properties. Comparisons of predictions and observations based on the work of Kirkaldy and coworkers in hardenability and HSLA steels are presented. In the theoretical part, expressions for the chemical potentials of the elements in substitutional and substitutional- interstitial solid solutions are developed via the partition function method assuming random positioning of the atoms. This approach leads for substitutional solid solutions to the BalePelton “unified formalism” in thermodynamically consistent second- order approximation. It is shown that the quadratic term contributed by the solvent in substitutional solutions does not appear in the expression of the activity coefficient for the interstitial elements in the substitutional-interstitial solid solution as reported in 1962 by Kirkaldy and Purdy. This paper 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, Ohio. The symposium was organized by John Morral, University of Connecticut, and Philip Nash, Illinois Institute of Technology.     

First-principles calculations of phase equilibria and transformation dynamics of Fe-based alloys

Theoretical procedures of first-principles calculations of phase stability and phase equilibria are summarized. The present scheme is shown to be able to reproduce the transition temperatures with surprisingly high accuracy for Fe−Pd and Fe−Pt systems. The main emphasis of the present report is placed on the extension of the first-principles calculation to transition dynamics calculations. This is performed by combining the cluster variation method with the phase-field method via a coarse graining operation. The time evolution process of antiphase boundaries associated with L1₀ ordering for Fe−Pd system is demonstrated. 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.     

Stability and thermodynamic properties of supersaturated solid solution and amorphous phase formed by ball milling in the Zr- Al system

Zr ₁₀₀-_xA1_x (x ≤ 40) metastable alloys were synthesized by high- energy ball milling of elemental Zr and Al powders: supersaturated substitutional cph solid solution for x ≤ 15 and an amorphous phase for x ≥ 17.5. We performed a calorimetric study of the thermodynamics and kinetics of the metastable- to- equilibrium transformations of these phases. Their stability range (temperature/composition), as well as the apparent activation energies associated with the transformations, were determined. The transformation enthalpies were measured and used to determine the enthalpy of formation for these metastable phases. For both as- milled and relaxed amorphous phases, the measured enthalpy of crystallization is compared with those estimated for an undercooled liquid. Different amounts of retained entropy at the glass transition temperature were used to estimate the enthalpy loss upon undercooling due to the excess specific heat. This paper was presented at the Thermodynamics and Phase Equilibria of Metastable Phases Symposium at the Spring TMS Meeting, March 1-4,1992, in San Diego. The symposium was organized by Philip Nash, Illinois Institute of Technology, and Ricardo Schwarz, Los Alamos National Laboratory.     

Thermodynamically Improbable Phase Diagrams

Phase diagrams showing very unlikely boundaries, while not explicitly violating thermodynamic principles or phase rules, are discussed. Phase rule violations in proposed phase diagrams often become apparent when phase boundaries are extrapolated into metastable regions. In addition to phase rule violations, this article considers difficulties regarding an abrupt change of slope of a phase boundary, asymmetric or unusually pointed liquidus boundaries, location of miscibility gaps, and gas/liquid equilibria. Another frequent source of phase diagram errors concerns the initial slopes of liquidus and solidus boundaries in the very dilute regions near the pure elements. Useful and consistent prediction can be made from the application of the van’t Hoff equation for the dilute regions.     

A figure of merit for predicted phase diagrams

A figure of merit is proposed that can be used to evaluate the accuracy of a predicted diagram. The figure of merit is the“probability that a phase will be predicted correctly” in a randomly selected alloy. The figure of merit depends on both the phase and the boundaries of the phase diagram being considered. It can be determined in two ways, either by the use of ZPF (zero phase fraction) lines or from the results of phase analyses on selected alloys. This paper 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 of Connecticut, and Philip Nash, Illinois Institute of Technology.     

A figure of merit for predicted phase diagrams

A figure of merit is proposed that can be used to evaluate the accuracy of a predicted diagram. The figure of merit is the“probability that a phase will be predicted correctly” in a randomly selected alloy. The figure of merit depends on both the phase and the boundaries of the phase diagram being considered. It can be determined in two ways, either by the use of ZPF (zero phase fraction) lines or from the results of phase analyses on selected alloys. This paper 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 of Connecticut, and Philip Nash, Illinois Institute of Technology.     

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.”     

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.     

Factors affecting the validity of both calculated and experimental phase diagrams

The validity of phase diagram information is affected by many factors. With regard to experimental determination of phase information, any experimental approach must have adequate resolution to provide accuracy and adequate precision to define the specific equilibria that are in question. Such things as possible side reactions, strain effects, purity, homogeneity, atomic ordering, magnetic and/or electric dipole interactions, kinetic inhibition to equilibria, and environmental effects can individually or in combination lead to erroneous interpretations. Calculation of phase equilibria is also subject to various sources of uncertainty. To do a complete calculation from first principles is indeed a difficult undertaking because the energy difference between one phase form and another is a tiny fraction of the total cohesive energy. Thus, in practice, calculated phase equilibria are generally dependent upon some experimental input data in combination with approximations. The calculated results are thus weighted by the quality of input. Computer people describe this as, “Garbage in = Garbage out” The choice of mathematical and physical models may also affect the results. This paper was presented at the International Phase Diagram Prediction Symposium sponsored by the ASM/MSD Thermodynamics and Phase Equilibria Committee at Materials Week, October21–23,1991, in Cincinnati, Ohio. The symposium was organized by John Morral, University of Connecticut, and Philip Nash, Illinois Institute of Technology.     

Factors affecting the validity of both calculated and experimental phase diagrams

The validity of phase diagram information is affected by many factors. With regard to experimental determination of phase information, any experimental approach must have adequate resolution to provide accuracy and adequate precision to define the specific equilibria that are in question. Such things as possible side reactions, strain effects, purity, homogeneity, atomic ordering, magnetic and/or electric dipole interactions, kinetic inhibition to equilibria, and environmental effects can individually or in combination lead to erroneous interpretations. Calculation of phase equilibria is also subject to various sources of uncertainty. To do a complete calculation from first principles is indeed a difficult undertaking because the energy difference between one phase form and another is a tiny fraction of the total cohesive energy. Thus, in practice, calculated phase equilibria are generally dependent upon some experimental input data in combination with approximations. The calculated results are thus weighted by the quality of input. Computer people describe this as, “Garbage in = Garbage out” The choice of mathematical and physical models may also affect the results. This paper was presented at the International Phase Diagram Prediction Symposium sponsored by the ASM/MSD Thermodynamics and Phase Equilibria Committee at Materials Week, October21–23,1991, in Cincinnati, Ohio. The symposium was organized by John Morral, University of Connecticut, and Philip Nash, Illinois Institute of Technology.     

Phase equilibria in Fe-Cu-X (X: Co, Cr, Si, V) ternary systems

Phase equilibria on the Fe-Cu side in the Fe-Cu-X (X: Co, Cr, Si, V) system were experimentally determined over the temperature range of 1073–1273 K. Based on the present results and previous works, the thermodynamic assessments of the phase equilibria in the Fe-Cu-X system were evaluated using the Calculation of Phase Diagram (CALPHAD) method. The Gibbs energies (G) of the bcc, fcc, and liquid phases are described by the subregular solution model, and a set of thermodynamic parameters enable us to calculate various isothermal and vertical sections and the miscibility gaps of the solid and liquid phases.     

Phase equilibria in Fe-Cu-X (X: Co, Cr, Si, V) ternary systems

Phase equilibria on the Fe-Cu side in the Fe-Cu-X (X: Co, Cr, Si, V) system were experimentally determined over the temperature range of 1073–1273 K. Based on the present results and previous works, the thermodynamic assessments of the phase equilibria in the Fe-Cu-X system were evaluated using the Calculation of Phase Diagram (CALPHAD) method. The Gibbs energies (G) of the bcc, fcc, and liquid phases are described by the subregular solution model, and a set of thermodynamic parameters enable us to calculate various isothermal and vertical sections and the miscibility gaps of the solid and liquid phases.     

Activities and immiscibility in the system Cu-Rh

The thermodynamic activity of rhodium in solid Cu-Rh alloys is measured by the electromotive force method in the temperature range from 1050 to 1325 K with a solid-state cell: