A CALPHAD-type Young's modulus database of Ti-rich Ti–Nb–Zr–Mo system

Accurate Young's modulus is the necessity for the design of biomedical Ti alloys. A combinatorial method of the diffusion couple, nanoindentation, electron probe microanalysis (EPMA), and CALculation of PHAse Diagrams (CALPHAD) techniques has been utilized to construct the Young's modulus database of Ti alloys with various compositions in the present work. Two groups of body-centered cubic (bcc) Ti–Nb–Zr–Mo quaternary diffusion couples annealed at 1273 K for 25 h were experimentally prepared. Subsequently, the composition-dependent mechanical properties in the wide compositional range of Ti-based alloys were obtained by using EPMA and nanoindentation probes. Finally, on the basis of the measured Young's moduli in the present and previous work and the modeling parameters of Young's modulus of Ti–Nb–Zr system, the Young's modulus database of bcc Ti–Nb–Zr–Mo system was established through the CALPHAD approach. The CALPHAD-type database of bcc Ti–Nb–Zr–Mo system can provide the accurate Young's moduli of Ti alloys with wide compositions.     

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 calculation of the T0 curve and metastable phase diagrams of the Ti–M (M = Mo,V, Nb,Cr, Al) binary systems

Based on the experimental data available in the literature, the β-α′/α′′⬠ martensitic transformation and athermal ω formation of the Ti–M (M = Mo, V, Nb, Cr, Al) binary systems at low temperature are thermodynamically described. According to β-α′/α′′⬠ martensitic transformation and metastable ω phase formation temperatures, thermodynamic parameters of these systems are assessed by means of the CALPHAD (CALculation Phase Diagram) approach supported by first-principles calculations. In addition to the metastable ω phase, only solution phases, i.e. liquid, α(hcp), β(bcc) or γ(fcc) are included and their thermodynamic parameters are adopted in the literature or revised in this work. The metastable phase diagrams of the Ti–M (M = Mo, V, Nb, Cr, Al) systems with T₀(β/α) and T₀(β/ω) curves are calculated using the obtained parameters. Comparisons between the calculated results and experimental data reported in the literature show that almost all the reliable experimental information can be satisfactorily accounted for by the present modeling.     

Computational modeling of elastic constants as a function of temperature and composition in Zr–Nb alloys

We used CALPHAD-type model to describe single crystal elastic constants of bcc solution phases in Zr–Nb system. The model parameters were evaluated by utilizing least square algorithm based on available experimental and first-principles data. The composition-polycrystalline elastic properties profiles of the Zr–Nb alloys of full composition were predicted and are in agreement with experimental data. The critical temperature corresponding to the dynamical stabilization of bcc pure Zr can be estimated to 600 K and the critical composition corresponding to the dynamical stabilization of bcc Zr–Nb alloys at room temperature is about 4 at% Nb. The current calculations present an effective strategy to design biomedical alloys using a computational method.     

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.     

Assessments of molar volumes of Co-, Ni- and Ti- related bcc and fcc phases

In the present work, we modeled the molar volumes of the bcc phases in Co-X (X = Fe, Mo and Zr), Ni-X (X = Fe, Mo and V) and Ti-X (X = Fe, Mo, Nb, Ta and V) binary systems, as well as fcc phases in Co-X (X = Au, Fe, Ge, Hf, Ir, Mo, Nb, Ni, Os, Pd, Pt, Re, Rh, Ru, Sc, Sn, V, Zn and Zr), Ni-X (X = Al, Au, Cr, Cu, Fe, Ga, Ir, Mo, Nb, Os, Pd, Pt, Rh, Ru, Sb, Sn, Tc, Ti, V and Zn) and Ti-X (X = Au, Cu, Ni and Pd) binary systems, at room temperature and atmospheric pressure by using the CALPHAD method combined with first-principles calculations. The model parameters involve molar volumes of pure constituent elements in their stable, metastable or unstable bcc and fcc structures and excess molar volumes. Specifically, the molar volumes of the pure constituent elements in their stable or metastable bcc and fcc structures were directly adopted from the previous CALPHAD assessments or experimental measurements from the literature; for their unstable bcc and fcc structures, a new method of volume extrapolation was proposed, which avoids trial-and-error fitting. Once the molar volumes of pure constituent elements were fixed, the excess term was easily assessed according to the experimental measurements from the literature combined with the first-principles calculation results. The thus assessed model parameters can well reproduce most experimental data.     

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.     

Experimental investigation and thermodynamic description of the Fe–Mo–Zr system

The phase relations at 1273 K and liquidus surface projection of the Fe–Mo–Zr system were investigated by means of electron probe micro-analyzer (EPMA), scanning electron microscopy coupled with energy dispersive spectroscopy (SEM-EDS) and X-ray diffraction (XRD) methods. The composition range of C14 Laves phase was determined at 1273 K. The maximum solubility of Mo in C15–Fe₂Zr, Mo in Fe₂₃Zr₆, Fe in C15–Mo₂Zr and Zr in μ phase is about 4.8, 0.6, 17.7 and 4.6 at.% at 1273 K, respectively. The isothermal section at 1273 K of the Fe–Mo–Zr system on the whole composition ranges was constructed using 30 annealed alloys. In the liquidus surface projection, the primary solidification phase regions of bcc(Fe), C15–Fe₂Zr, C14, μ, R, σ, bcc(Zr), C15–Mo₂Zr and bcc(Mo) were experimentally confirmed using 31 as-cast alloys. Based on the experimental data in literature and the present work, the Fe–Mo–Zr system was optimized using CALPHAD method, and a set of self-consistent reliable thermodynamic parameters was obtained.     

Thermodynamic description and phase selection for the Mo–Ti–Zr biomedical alloys

Thermodynamic information of the Mo–Ti–Zr ternary system is extremely useful to provide guidance for biomedical alloy development. In the present work, the experimental phase diagram data available from the literature were critically reviewed, and a thermodynamic modeling of the Mo–Ti–Zr system was performed using the CALPHAD (CALculation of PHAse Diagram) approach. The solution phases including liquid, bcc_A2 (β) and hcp_A3 (α) were modelled by the substitutional solution model, and the laves_C15 phase was modelled using a two sublattice model. A set of self-consistent thermodynamic parameters was developed. Comprehensive comparisons between the calculated and measured phase diagrams demonstrate that the experimental information is satisfactorily accounted for by the present thermodynamic modeling. The discrepancies between the calculated and measured phase equilibria have been well explained in this work. With regard to the β phase, the miscibility gap and related phase relations are well described by the present calculation. The liquidus projection and Scheil solidification simulation were generated using the present thermodynamic parameters. The presently calculated phase diagrams of the Mo–Ti–Zr alloys can be used to guide the development of Mo–Ti–Zr biomedical alloys. Based on the present calculations, two guidelines were formulated to avoid the formation of laves phase in these frequently studied Mo–Ti–Zr biomedical alloys.     

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.     

Thermodynamic modeling of selected ternary systems containing Y and CALPHAD simulation of CoNiCrAlY metallic coatings

Metallic coatings can improve the high temperature resistance of superalloys serving in the gas turbines. In general they are Al–Co–Cr–Ni alloys with small Y additions to improve oxide scale adherence.In order to complete the construction of a thermodynamic database for coatings, thermodynamic assessments of four ternary systems have been performed by means of the CALPHAD method, namely Al–Co–Y, Al–Ni–Y, Al–Cr–Y and Co–Ni–Y. All of the experimental phase diagrams and thermodynamic data available in the literature were critically reviewed. The liquid, fcc, bcc and hcp phases were modeled as substitutional solutions. The order-disorder model has been adopted to describe the A1/L1₂ and A2/B2 phase relations. A series of ternary compounds have been modeled during the present work according to the crystal structure or composition. As a result a satisfactory agreement was obtained between our calculations and the experimental data used in the assessment.Finally, interaction parameters calculated in this work have been merged in the thermodynamic database for the simulation of Al–Co–Cr–Ni–Y alloys. This has been validated by comparing our calculations with experimental data regarding selected Ni-based and Co-based alloy coatings.     

Molar volumes of Al,Li, Mg and Si

The molar volumes of the fcc, bcc, hcp and liquid phases of Al, Li, Mg and Si as well as diamond Si have been evaluated as functions of temperature based on experimental data from the literature. The molar volume of each element in each structure is described by a single polynomial expression as a function of temperature. These polynomials can be used above around 150 K. The molar volumes of the liquids were described by a linear temperature dependence. The molar volumes of nonstable structures were evaluated with the help of lattice parameter measurements of the corresponding solid solutions. A large majority of the solid solutions studied showed negative excess volumes. The molar volumes of the relatively closely packed fcc, bcc and hcp structures were always found to be very close to each other, and a reasonably good approximation would be to set them as equal.