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.     

Thermodynamic modeling of Laves phases in the Cr–Hf and Cr–Ti systems: Reassessment using first-principles results

The Cr–Hf and Cr–Ti belong to interesting systems exhibiting the existence of all polytypes of Laves phases, i.e. lower-temperature cubic C15 and higher-temperature hexagonal C14 and C36, although in the Cr–Hf phase diagram only C14 and C15 phases occur. Comparison of total energies of these structures calculated from first principles with the total energy of the ideal mixture of elemental constituents reveals the relative stability of Laves phases in these systems. The effect of magnetic order in the Laves phases is also briefly discussed.     

Thermodynamic modeling of Cr–Nb and Zr–Cr with extension to the ternary Zr–Nb–Cr system

The total energies of Laves phases in the Cr–Nb and Zr–Cr systems have been calculated by the pseudo-potential VASP code with a full relaxation of all structural parameters. The special quasirandom structures (SQSs) have been constructed and their total energies have been calculated by the VASP code to predict the enthalpies of mixing for bcc and hcp solid solution phases. The phonon calculations for the C14 and C15 Laves phases have been performed to analyze the phase stability at elevated temperatures. The experimental study on the Zr–Cr system has been carried out at different temperatures to determine the phase boundaries. Based on these results, thermodynamic models of Cr–Nb and Zr–Cr with extension to the ternary Zr–Nb–Cr systems have been developed in this work by using the CALPHAD approach.     

Thermodynamic modelling of Ti-Zr-N system

Thermodynamic modelling of Ti-Zr-N system is performed using Calphad method coupled with ab initio calculations. The energies of formation of stable and metastable end-members of sublattice formulations of solid phases in Zr-N system and enthalpy of mixing of the mixed nitride (Ti, Zr)N (δ) are calculated using ab initio method. Phonon calculations are used to compute the Gibbs energies of stoichiometric ZrN and the mixed nitride δ. With the aid of experimental thermochemical and constitutional data from the literature along with the results of ab initio calculations, thermodynamic optimization is carried out to obtain the Gibbs energy model parameters.     

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.     

Gibbs energy modeling of Fe–Ta system by Calphad method assisted by experiments and ab initio calculations

In this work we report the Gibbs energy model parameters of equilibrium phases of Fe–Ta system obtained by the Calphad approach. In order to assist the Gibbs energy modeling new constitutional and enthalpy increment data were obtained by experiments. Further, the energy of formation of intermediate phases and the energy of mixing of bcc solid solution were calculated by ab initio method. These results were combined with selected experimental data from the literature in order to optimize the model parameters of the Gibbs energy functions. Calculated phase diagram and the thermochemical properties are in good agreement with the experimental results.     

Structural stability of intermetallic phases in the Ga-Ti system

The total energies of intermetallic compounds in the Ga-Ti system are calculated employing electronic density functional theory (DFT) using pseudopotentials constructed by the projector augmented waves (PAW) method in the generalized gradient (GGA) approximation for the exchange and correlation energy. The calculations are performed for the experimentally observed compounds at their ideal stoichiometry as well as for structures which are stable in systems of early transition metals or rare earth elements with p elements of columns IIIB, IVB, and VB. The calculated formation enthalpy of the hexagonal B8₂-GaTi₂ compound is in excellent agreement with the value obtained by direct reaction calorimetry. The composition dependence of the enthalpies of formation is slightly asymmetric, the values of the enthalpies of formation being slightly more negative in the Ga-rich side. For the stable intermetallic compounds, the calculated zero-temperature lattice parameters agree well with those obtained experimentally at ambient temperature. Furthermore, for stable phases with unit-cell-internal degree(s) of freedom, the results of ab initio calculations show good agreement when compared with data obtained by structural analysis of X-ray diffraction.     

Combined ab initio/CALPHAD modeling of fcc-based phase-equilibria in the Ir–Nb system

A preliminary thermodynamic assessment of the Ir–Nb system, one of the key binary systems of the Ir-based refractory superalloys, has been performed by combining ab initio calculations and the CALPHAD (CALculation of PHAse Diagrams) technique. The ground-state formation enthalpies have been calculated by the full-potential linearized augmented plane wave method. The free energies at finite temperatures have been estimated using the cluster variation method, where the effective cluster interaction energies have been extracted from the formation enthalpies by the cluster expansion method. The liquid and A1 phases are modeled as substitutional solutions. The L1₀ and L1₂ phases are described using the four-sublattice model with the formula (Ir,Nb)_1/4(Ir,Nb)_1/4(Ir,Nb)_1/4(Ir,Nb)_1/4, while other solid phases are not considered in the present assessment. The obtained parameter set reproduces well the characteristic features of the experimental phase diagram and thermodynamic quantities.