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 assessments of the U–Nb–Mo and U–Nb–Cr ternary systems

The thermodynamic assessments of the U–Nb–Mo and U–Nb–Cr systems have been performed by using the CALPHAD (Calculation of Phase Diagrams) method on the basis of critical evaluation of phase diagram data reported in literature. The reported individual solution phases, i.e. liquid, (αU), (βU), γ, δ and two intermetallic compounds, i.e. MoU₂ and NbCr₂, have been modeled. The modeling covers the whole composition range and a wide temperature range. By utilizing the available thermodynamic parameters of the sub-binary systems, the U–Nb–Mo and U–Nb–Cr systems have been thermodynamically assessed and a series of self-consistent parameters have been obtained for the first time, which can reproduce most of the phase diagram and thermodynamic data to provide guidance for the design of nuclear fuels.     

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 assessment of the Sr–In and Sr–Bi systems supported by first-principles calculations

Based on the available experimental phase equilibria and thermodynamic data, the Sr–In and Sr–Bi systems have been assessed by means of the CALPHAD technique. The solution phases (Liquid, (αSr), (βSr), (In) and (Bi)) were modeled with the Redlich–Kister polynomial. All the intermetallic compounds (SrIn₅, SrIn₃, Sr₂In₅, SrIn₂, Sr₂In₃, SrIn, Sr₅In₃, Sr₃In, SrBi₃, Sr₁₁Bi₁₀, αSr₅Bi₃, βSr₅Bi₃ and Sr₂Bi) were treated as stoichiometric compounds. The enthalpies of formation at 0 K for SrIn₅, SrIn₃, SrIn₂, SrIn, Sr₅In₃, Sr₃In, SrBi₃, αSr₅Bi₃, βSr₅Bi₃ and Sr₂Bi were computed by first-principles calculations in order to assist the thermodynamic modeling. A set of self-consistent thermodynamic parameters for each of the two systems has been obtained, and the present calculations can satisfactorily reproduce the available experimental data.     

Thermodynamic reassessment of the Mo–Hf and Mo–Zr systems supported by first-principles calculations

Based on the experimental phase equilibria and thermodynamic data available in the literature and enthalpies of formation computed from first-principles calculations, the thermodynamic reassessment of the Mo–Hf and Mo–Zr systems was carried out by means of the CALPHAD (CALculation of PHAse Diagram) method. The enthalpies of formation for stable and metastable Laves (C15, C36, C14) phases and enthalpy of mixing for the β(bcc) solid solution phase in dilute solution were predicted via first-principles calculations to supply the necessary thermodynamic data for the modeling in order to obtain the thermodynamic parameters with physical sound meaning. The relative stability of Laves C14, C15 and C36 in the systems was discussed. The solution phases, i.e. liquid, β(bcc) and α(hcp) were described by the substitutional solution model, and all the Laves phases in the systems were described using two sublattice model. A set of self-consistent thermodynamic parameters were obtained for these binary systems, which agrees well with the experimental data in the literature. Based on these results, the trend of the site occupancy fraction of Laves phase changing with temperature was predicted. The isothermal section and the liquidus projection of Mo-Hf-Zr system were also predicted by combing with the Zr–Hf system reported in the literature.     

Experimental investigations and thermodynamic modelling of the Cr–Nb–Sn–Zr system

This work reports the Calphad modelling of the Cr–Nb–Sn–Zr quaternary system. In a previous paper, the thermodynamic modelling of the Cr–Nb–Sn system was presented. Since no experimental data were available for the Cr–Sn–Zr ternary system, new experimental data are provided, within this study, on the isothermal section at 900 °C. A ternary C14 phase has been identified on the Sn-poor side of the phase diagram. In addition to these experimental data, Density Functional Theory (DFT) calculations are carried out in order to determine formation enthalpies of the stable and metastable compounds. At last, the Special Quasirandom Structures (SQS) method is jointly used with DFT calculations in order to estimate the mixing enthalpies of the A2 and A3 binary solid solutions. Finally, these experimental and calculated data in addition to those from the literature, are used as input data for the Calphad modelling of the Cr–Zr, Nb–Zr and Sn–Zr binary systems and the Cr–Nb–Zr, Cr–Sn–Zr and Nb–Sn–Zr ternary systems. A complete database for the Cr–Nb–Sn–Zr quaternary system is provided.     

Effect of alloying on stability of grain boundary in γ phase of the U–Mo and U–Nb systems

U–Mo and U–Nb alloys are both extensively used in nuclear industry. γ phase in U–Mo or U–Nb alloy is a solid solution, being metastable in low temperature region. In this work, the effect of alloying on stability of grain boundary in meta-stable γ phase in U–Mo and U–Nb alloys are investigate through first-principles calculations. At first, crystal structure and elastic constants of Mo, Nb and γ-U metals are calculated and the obtain results show the mechanical unstable nature of γ phase at 0 K, no matter with GGA or GGA + U treatment, which agrees with most of the theoretical results in the literature. Furthermore, from the calculated symmetric tilt grain boundary (STGB) formation energies of Σ3[110]/(111) and Σ5[001]/(310) in Mo, Nb, and γ-U, it is found that due to the mechanical unstable character of the γ-U phase, negative GB formation energy is predicted at 0 K for Σ5[001]/(310) if the STGB model is relaxed with all degrees of freedom. Therefore, by using special quasirandom structure (SQS) method, Σ5[001]/(310) and Σ3[110]/(111) STGBs with different solute concentrations in U-rich side in U–Mo and U–Nb systems are further investigated. It is found that, when alloying with Mo or Nb, unlike Σ3[110]/(111), although the fixed-atom constraint is applied, the GB formation energy of Σ5[001]/(310) STGB is becoming negative when the solute concentration is in U-rich side. Only when the concentration of Mo or Nb is larger than 27 at.% or 30 at.%, respectively, or sufficient small, the GB formation energy is becoming positive, suggesting a cooperative effects of solute concentration, unstable character, and grain size on GB structures in γ phase. The predicted different stability of alloyed GB structures at 0 K suggest that although γ phase is metastable at low temperature, its metastability can be controlled through alloying with different solutes, or with different GBs. And grain refinement should be relatively easy in U-rich part than U-poor part of the U–Mo and U–Nb systems.     

The phase equilibria of the Cu–Cr–Ni and Cu–Cr–Ag systems: Experimental investigation and thermodynamic modeling

The phase equilibria of the Cu–Cr–Ni and Cu–Cr–Ag systems were investigated by a combination of key experiments and thermodynamic modeling. Eleven and fourteen ternary alloys were prepared to determine the isothermal sections of the Cu–Cr–Ni system at 800 and 1000 °C, and the Cu–Cr–Ag system at 500, 600, 650 and 700 °C, respectively, by means of X-ray diffraction (XRD) and scanning electron microscopy with energy dispersive X-ray spectroscopy (SEM/EDX). The three- and two-phase regions were determined. The solubility of the third elements in the phases of the binary systems was measured. No ternary compound was found in these two ternary systems. Based on the experimental equilibria data from the literature and the present work, a thermodynamic modeling of the Cu–Cr–Ni and Cu–Cr–Ag systems was performed by the CALPHAD (CALculation of PHAse Diagrams) method. The substitutional solution model was used to describe the solution phases. A set of self-consistent thermodynamic parameters of the Cu–Cr–Ni and Cu–Cr–Ag systems was obtained. Most of the reliable experimental data can be well reproduced by the present thermodynamic modeling.     

Asymmetric mixing behavior and stability of the predicted phases in the W–Cu system

The W–Cu system is strongly immiscible both in the solid and liquid phases. By combining cluster expansion method with first-principles calculations, present calculations clearly show that there is asymmetric mixing behavior for both FCC and HCP lattices in the W–Cu system. This asymmetric behavior affects the cohesion of the immiscible W–Cu system and is confirmed by Cu/W multilayers ion-bean mixing experiment and Compton scattering observation in the literature. Moreover, the predicted energetically metastable phases of L1₂ CuW₃ and D0₁₉ CuW₃ are confirmed to be mechanically and vibrationally stable at zero temperature. The further calculated Helmholtz free energy of formation as a function of temperature from lattice dynamics shows that the L1₂ phase is more stable than D0₁₉ phase when temperature is lower than 350 K.     

Fault detection in dynamic systems using the Kullback–Leibler divergence

This paper proposes detecting incipient fault conditions in complex dynamic systems using the Kullback–Leibler or KL divergence. Subspace identification is used to identify dynamic models and the KL divergence examines changes in probability density functions between a reference set and online data. Gaussian distributed process variables produce a simple form of the KL divergence. Non-Gaussian distributed process variables require the use of a density-ratio estimation to compute the KL divergence. Applications to recorded data from a gearbox and two distillation processes confirm the increased sensitivity of the proposed approach to detect incipient faults compared to the dynamic monitoring approach based on principal component analysis and the statistical local approach.     

Modeling of the molar volume of the solution phases in the Al–Cu–Mg system

The temperature and pressure dependences of molar volume of the solution phases in the Al–Cu–Mg system have been modeled by using the CALPHAD approach. Molar volumes of pure elements were critical assessed according to the experimental data from the literature. Subsequently, volumetric properties of the Al–Cu, Al–Mg and Cu–Mg binary systems were assessed and compared with the corresponding experimental data. The results are compatible with the assessed data.     

Study of diffusion mobilities of Nb and Zr in bcc Nb–Zr alloys

On the basis of the available thermodynamic parameters, the atomic mobilities of Nb and Zr in bcc Nb–Zr alloys are critically assessed as functions of temperatures and compositions by the CALPHAD method, where self-diffusion coefficients, impurity diffusion coefficients, tracer diffusion coefficients, interdiffusion coefficients and concentration curves are simultaneously optimized. Comparisons between the calculated and experimentally measured diffusion coefficients are made, where good agreement is evident. In addition, the obtained mobility parameters can reproduce a reasonable concentration profile for the Nb/Zr diffusion couple annealed at 1868 K for 5400 s.