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.     

Thermodynamic evaluation of the phase equilibria and glass-forming ability of the Ti−Be system

The glass-forming ability of Ti−Be alloys is of great interest. Experimental and theoretical evaluations of the glass-forming ability of this binary alloy show that the formation of a metastable TiBe phase with a CsCl-type B2 structure controls the glass-forming ability in this system. However, there is no information on the thermochemical properties of metastable TiBe for the quantitative evaluation of the glass-forming ability using Davies-Uhlmann kinetic formulations. We have carried out a thermodynamic analysis using experimental phase diagram data and the energy of formation of the stoichiometric compounds from ab initio calculations. Furthermore, the Gibbs energy of formation for the body-centered cubic (bcc) phase was evaluated over the entire composition range by applying the cluster expansion method (CEM) to the total energy of some bcc-based ordered structures obtained from ab initio calculations. For the bcc phase, the two-sublattice formalism, (Ti, Be)_0.5(Ti,Be)_0.5, was adopted to describe the A2/B2 transformation. A good agreement between the calculated values and experimental phase equilibria was obtained. Evaluation of the glass-forming ability was also attempted utilizing the thermodynamic quantities obtained from the phase diagram assessment. The calculated glass-forming ability agrees well with the 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 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.     

TerQuat: Approximate prediction of ternary and quaternary phase diagrams comprising stoichiometric phases

The TerQuat method is an easy-to-use tool for a first prediction of ternary and quaternary phase diagrams. A number of simplifications (only stoichiometric compounds; Δ_fG° can be approximated by the Miedema model, ΔH_Miedema, data) allows the user to get an impression of a ternary or quaternary system with only a minimum of knowledge about the system, namely a list of existing compounds. Stable tie lines are calculated by comparison of the Gibbs energy of reaction of all possible reactions between four phases in the ternary. In the quaternary system, all the competing two- and three-phase equilibria are compared with each other, and a list of stable equilibria is generated. In addition, the intersections of stable two- and three-phase equilibria with a user specified plane in the tetrahedron can be calculated and plotted. This enables a simple generation of a space model for the visualization of the quaternary system. 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.     

Experimental and Thermodynamic Analysis on Phase Equilibria of Ni-rich Region in Ni-Al-Ti System

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Critical assessment and thermodynamic calculation of the ternary system C-Hf-Zr (Carbon-Zirconium-Hafnium)

Based on an assessment of the available experimental thermochemical and phase diagram information available, the phase equilibria of the C-Hf-Zr system were calculated. The G of the individual phases was described with thermodynamic models. The liquid phase was described as a substitutional solution using the Redlich-Kister formalism for excess G. Graphite was treated as a stoichiometric phase. The solid solutions of carbon in α(Hf,Zr) and β(Hf,Zr), as well as the non-stoichiometric phase (Hf,Zr)C_1−x, were represented as interstitial solid solutions using the compound energy model with two sublattices. The parameters in the models were determined by computerized optimization using selected experimental data. A detailed comparison was made between calculation and experimental data.     

Experimental investigation and thermodynamic description of the In–Sb–Sn ternary system

Phase equilibria in the In–Sb–Sn ternary system have been studied experimentally and calculated by the CALPHAD method. Solubility of indium in the SbSn phase was experimentally established. Our own SEM–EDX results were used together with the literature data for thermodynamic modeling of the Gibbs energy of the SbSn intermediate phase. Optimized phase diagrams of the isothermal sections at 100 °C and 300 °C were compared with the experimental results from this work and literature. Three calculated vertical sections were compared with the DTA results from this work and with available thermal analysis data reported in the literature. Some calculated thermodynamic functions are compared with experimental values reported in the literature. Reasonable agreement between calculations and experimental data was observed in all cases.     

A thermodynamic analysis of the Al-Re system

Phase equilibria and thermodynamic data of the Al-Re system are critically reviewed. In addition to the three solution phases, liquid, fcc Al, and hcp Re, there exist six intermetallic compounds in this binary. The thermodynamic properties of the system are analyzed using thermodynamic models for the Gibbs energy of individual phases of the system. A regular solution model is used for the three substitutional solution phases, and the intermetallic phases are treated as stoichiometric compounds. The model parameters are optimized from a limited amount of experimental data. The calculated phase diagram and thermodynamic values are in accord with the available experimental values.     

Experimental Investigation and Thermodynamic Calculation of Phase Equilibria in the Mg-Pb-Sn Ternary System

The phase equilibria of the Mg-Pb-Sn ternary system were investigated using a combined method of electron probe microanalyzer and x-ray diffraction. Three isothermal sections of the Mg-Pb-Sn ternary system at 200, 300 and 400 °C were experimentally established. The phase equilibria of Mg-Pb-Sn ternary system were thermodynamically assessed by using CALPHAD (Calculation of Phase Diagrams) method on the basis of the presently determined experimental data. A consistent set of thermodynamic parameters has been derived for describing the Gibbs free energies of each solution phase and intermetallic compound in the Mg-Pb-Sn ternary system. The calculated phase diagrams and thermodynamic properties in the Mg-Pb-Sn ternary system are in good agreement with experimental data.     

Thermodynamic assessments of the Cu–Th and Mo–Th systems

The thermodynamic assessments of the Cu–Th and Mo–Th binary systems were carried out by using Calculation of Phase Diagrams (CALPHAD) method on the basis of the experimental data including the thermodynamic properties and phase equilibria. The Gibbs free energies of the liquid, bcc, and fcc phases are described by the subregular solution model with the Redlich–Kister equation and those of the four intermetallic compounds Cu₆Th, Cu_3.6Th, Cu₂Th and CuTh₂ in the Cu–Th binary system were described by the sublattice model. A set of self-consistent thermodynamic parameters are obtained, and the calculated phase diagrams and thermodynamic properties are presented and compared with the experimental data from literatures. The calculated thermodynamic properties as well as phase diagrams are in good agreement with the experimental data.     

Coherent phase diagrams

Phase equilibria in coherent solids differ significantly from equilibria in fluids or incoherent solids. Gibbs' phase rule usually does not apply, the field lines on a phase diagram do not coincide with the equilibrium phase compositions, phase compositions in a two-phase field depend on the bulk alloy composition and, for a given set of imposed thermodynamic variables, several equilibrium states may exist. These conditions place significant constraints on the use of coherent phase diagrams. This paper discusses some aspects of phase equilibria that are unique to coherent systems and how phase information may be conveyed when phase stability diagrams are used in lieu of coherent phase diagrams. The utility of phase stability diagrams lies in their ability to display graphically the equilibrium states of a two-phase coherent system as a function of alloy composition (in much the same way that Gibbs energy vs composition curves show equilibrium states for a fluid system).     

Thermodynamic assessments of the phase diagrams of the hafnium-vanadium and vanadium-zirconium systems

Thermodynamic assessments have been made for the hafnium-vanadium (Hf-V) and vanadium-zirconium (V-Zr) systems using the Calphad-Thermocalc approach. The Gibbs energies of the liquid, body-centered cubic, and close-packed hexagonal phases were described by a substitution solution model with a Redlich-Kister formalism to express the excess Gibbs energy. The C15-Laves phase was treated first as stoichiometric and then with a composition range. A consistent set of optimized thermodynamic parameters was obtained, and calculated phase equilibria were compared with the experimental data. The enthalpy of formation of the C15-Laves phase was calculated equal to approximately −3 and −5 kJ/mol, respectively, in the Hf-V and V-Zr systems, which is in good agreement with predicted values.     

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.     

Thermodynamic study of phase equilibria in strained III–V alloy semiconductors

Phase equilibria in GaSb-GaAs and InAs-GaAs thin films grown on GaAs and InP substrates were calculated, taking into consideration the elastic contribution caused by the lattice mismatch between the film and substrate. The basic concept used in describing the elastic free energy is that strain due to a lattice mismatch accumulates in a thin film and interfacial misfit dislocations form when the strain energy exceeds a certain energy barrier. Our calculations showed that the single-phase region of a zinc-blend-type compound forms in the GaSb-GaAs/InP and InAs-GaAs/InP systems. This result is in fairly good agreement with the experimental examination of epitaxially grown thin film. The compositional latching phenomenon frequently observed in alloy semiconductor systems can be interpreted in terms of the phase separation generated by the accumulated strain in the thin films.     

Thermodynamic study of phase equilibria in strained III–V alloy semiconductors

Phase equilibria in GaSb-GaAs and InAs-GaAs thin films grown on GaAs and InP substrates were calculated, taking into consideration the elastic contribution caused by the lattice mismatch between the film and substrate. The basic concept used in describing the elastic free energy is that strain due to a lattice mismatch accumulates in a thin film and interfacial misfit dislocations form when the strain energy exceeds a certain energy barrier. Our calculations showed that the single-phase region of a zinc-blend-type compound forms in the GaSb-GaAs/InP and InAs-GaAs/InP systems. This result is in fairly good agreement with the experimental examination of epitaxially grown thin film. The compositional latching phenomenon frequently observed in alloy semiconductor systems can be interpreted in terms of the phase separation generated by the accumulated strain in the thin films.     

Phase equilibria in the Al–Si–V system: The vanadium rich part

The V-rich part of the Al–Si–V phase diagram was determined by a combination of optical microscopy, powder X-ray diffraction (XRD), differential thermal analysis (DTA) and electron probe microanalysis (EPMA). Phase equilibria were investigated at two isothermal sections at 850 and 1300 °C. High temperature DTA was performed to identify the ternary invariant reactions yielding a ternary reaction scheme and the vertical section at 50 at.% V. As cast samples were investigated in order to gain additional information about primary crystallization fields. A liquidus surface projection was constructed for the entire ternary system by combining our experimental data with those from literature.