Thermodynamic assessment of the cesium–oxygen system by coupling density functional theory and CALPHAD approaches

The thermodynamic properties of cesium oxides were calculated by combining ab initio calculations at 0 K and a quasi-harmonic statistical thermodynamic model to determine the temperature dependency of the thermodynamic properties. In a second approach, the CALPHAD method was used to derive a model describing the Gibbs energy for all the cesium oxide compounds and the liquid phase of the cesium–oxygen system. For this approach, available experimental data in the literature was reviewed and it was concluded that only experimental thermodynamic data for Cs₂O are reliable. All these data together with the thermodynamic data calculated by combining ab initio and the statistical model were used to assess the Gibbs energy of all the phases of the cesium–oxygen system. A consistent thermodynamic model was obtained. The variation of the relative stability of the different oxides is discussed using structural and bond data for the oxides investigated by ab initio calculations. This work suggests that the melting point for Cs₂O₂ reported in the literature (863 K) is probably overestimated and should be re-measured.     

Interface between quantum-mechanical-based approaches,experiments, and CALPHAD methodology

The increased application of quantum-mechanical-based methodologies to the study of alloy stability has required a re-assessment of the field. The focus is mainly on inorganic materials in the solid state. In a first part, after a brief overview of the so-called ab initio methods with their approximations, constraints, and limitations, recommendations are made for a good usage of first-principles codes with a set of qualifiers. Examples are given to illustrate the power and the limitations of ab initio codes. However, despite the “success” of these methodologies, thermodynamics of complex multi-component alloys, as used in engineering applications, requires a more versatile approach presently afforded within CALPHAD. Hence, in a second part, the links that presently exist between ab initio methodologies, experiments, and the CALPHAD approach are examined with illustrations. Finally, the issues of dynamical instability and of the role of lattice vibrations that still constitute the subject of ample discussions within the CALPHAD community are revisited in the light of our current knowledge with a set of recommendations.     

Re-modeling of Laves phases in the Cr–Nb and Cr–Ta systems using first-principles results

Total energies of Laves phases Cr₂X, CrX₂, CrCr₂ and XX₂ (X=Nb,Ta) in all three structural forms C14, C15 and C36 have been calculated ab initio by pseudopotential VASP code with a complete relaxation of structural parameters. The calculated values were used in a two-sublattice model for re-modeling of Gibbs energies of Laves phases and subsequently for calculation of phase diagrams of Cr–Nb and Cr–Ta systems by CALPHAD method. It turns out that application of ab initio calculated values of total energy of hypothetical “end-members” in a two-sublattice model substantially simplifies the modeling and lowers the number of necessary parameters. Comparison of phase diagrams obtained by a model using first-principles results with previous empirical approach as well as relative stability of Cr₂X polytypes is presented.     

Modeling of Ni–Cr–Mo based alloys: Part I—phase stability

The CALPHAD approach is applied to the study of the stability of several candidate Ni–Cr–Mo based alloys for use in waste disposal canisters for the Yucca Mountain Project (YMP). The stability of occurring phases in the as-fabricated materials is investigated since temporal evolution of precipitates at service temperatures can significantly alter their corrosion resistance and mechanical properties. To account for the stability of the Ni₂Cr phase and the hypothetical Ni₂Mo and Ni₂W phases, available thermodynamic data and ab initio electronic structure-based results were employed. A comparative study of property diagrams for the nominal composition of several YMP candidate alloys is presented. These results form the basis for experimental and theoretical studies of kinetics of phase transformation and aging to be presented in Part II.     

A critical test of ab initio and CALPHAD methods: The structural energy difference between bcc and hcp molybdenum

Ab initio calculations of the enthalpy of formation of bcc, fcc, and hcp Ru–Mo alloys have been performed for random, ordered, and partially ordered structures. The lattice stability of the bcc and hcp forms of Mo is isolated in order to compare the hcp–bcc difference calculated by ab initio and CALPHAD methods with experimental measurements of the enthalpy of formation of Ru–Mo alloys. The significance of this comparison in calculating the Mo–Ru phase diagram is illustrated. The results of these considerations suggest a rational method for coupling ab initio and CALPHAD techniques might be utilization of the ab initio methods for calculation of the isostructural energies of formation for binary bcc, hcp, and fcc solutions while retaining the CALPHAD lattice stabilities in the calculation of phase diagrams.     

Thermodynamic optimization of Si-Zr-N system using Calphad approach coupled with ab initio methods

Gibbs energy model parameters for Si-Zr-N system were obtained using Calphad approach coupled with ab initio calculations. The enthalpies of formation of $\alpha$ and $\beta$Si₃N₄ in Si-N system and Zr₅Si₃N end-member in Si-Zr-N system were calculated using density functional theory (DFT). The finite temperature thermodynamic properties were calculated using quasiharmonic approximation (QHA). The computed heat capacities were fitted to appropriate expressions valid down to 0 K. The ab initio thermochemical data obtained in the present work and the experimental thermochemical and constitutional data from literature were used for the thermodynamic optimization of Si-N and Si-Zr-N systems. The calculated phase equilibria and thermodynamic properties are in good agreement with the input data.     

Modelling the thermodynamic data for hcp Zn and Cu–Zn alloys– an ab initio and calphad approach

The phase diagrams of systems between zinc and elements such as Cu, Ag and Au show two distinct hcp phases on the Zn side of the system. Because of this, it is difficult to model the thermodynamic properties of these phases within a single dataset. As a result it is common to assess the data for these systems with two hexagonal phases, a phase HCP_A3 with a near ideal c/a ratio and the terminal solid solution of Zn with an anomalously high value for this ratio designated as HCP_ZN. We have examined the effect of additions of Cu on the enthalpy of mixing and lattice parameters of HCP_ZN in order to verify, using ab initio calculations, the origin of the above mentioned thermodynamic model for the alloy. The analysis of the calculations allows us to suggest a possible alternative to the state-of-the-art two hcp phases approach akin to the magnetic model used with success within the CALPHAD modelling.     

Thermodynamic assessment of the Ni–Co–Cr system and Diffusion Study of its fcc phase

In the present work, the liquidus and solidus for a series of Ni_xCo_1-2xCr_x alloys were measured by means of differential scanning calorimetry, and the first-principles calculations were performed to obtain total energies for all solid solutions and end-members of the intermediate phases in the Ni–Co–Cr ternary system. Various types of data from the present work and the literature were used in the assessments of the Ni–Co–Cr ternary system and sub-binary systems by the CALPHAD method, and were well reproduced by the present thermodynamic database. In addition, diffusion couples of fcc Co–Cr and Ni–Co–Cr alloys were assembled and annealed at different temperatures to extract interdiffusion coefficients. Experimental diffusion data from the present work and the literature, in conjunction with thermodynamic parameters, were adopted to assess the atomic mobilities of the fcc phase in the Ni–Co–Cr system. The calculated and experimental diffusion coefficients reach a satisfactory agreement. The diffusional kinetic database developed was further validated by appropriate predictions of composition profiles and diffusion paths.     

Thermodynamic analysis of the Zr–Be system using thermochemical properties based on ab initio calculations

A thermodynamic analysis of the Zr–Be system has been carried out by combining ab initio energetic calculations with the CALPHAD approach. The energy of formation of the binary compound phases and some bcc-based ordered phases was calculated using the Full Potential Linearized Augmented Plane Wave method. The CrB-type ZrBe phase, which has been reported as a metastable phase, was found to be stable in the ground state, while the ZrBe phase with a CsCl-type B2 structure was found to be metastable. The Gibbs free energy of formation of the bcc phase was obtained by applying the cluster expansion and the cluster variation methods. To describe the B2 ordering state, the Gibbs energy of the bcc phase was represented using the two-sublattice model with the formula (Zr,Be)_0.5(Zr,Be)_0.5. Although the thermodynamic parameters for the CrB-type ZrBe phase did not satisfy both the experimental data and the ab initio calculations, the calculated phase diagram reproduced the experimental results. In addition, the glass-forming ability of this binary alloy was evaluated by incorporating the thermodynamic quantities from the phase diagram calculation into the Davies–Uhlmann kinetic approach. The evaluated glass-forming compositional range was narrower than the experimental results.