Miedema's Method Research Articles

STRUCTURAL-PHASE STATUS OF HIGH ENTRICITY NbTiHfVZr ALLOY

The paper presents the results of the high entropy alloy NbTiHfVZr stability study based on the analysis of critical Yum-Roseri indices, enthalpy of double alloys by Miedema method, entropy of mixing of NbTiHfVZr compound near the melting temperature. The method of inverse convex shells (InveseHubWeb) found temperature stability intervals, which for the compound NbTiHfVZr extend from the melting temperature to 1200K. In the case of NbTiHfZr the temperature stability interval extends up to room temperature. The obtained results indicate that the compounds NbTiHfVZr, NbTiHfZr belong to the high entropy lattices of which belong to the class of single-phase single-element solid solutions. Prediction of the lattices by the evolution code in the USPEX code made it possible to detect from the base of 300 standards, several simple cubic lattices with the spatial group P1 and different spatial distribution of atoms, which lattice energy turns out to be negative. The totality of the results obtained indicates that the found standards can be attributed to the stable phases in the corresponding temperature stability intervals of the compounds NbTiHfVZr, NbTiHfZr.

Theoretical and experimental study on phase stability of TiVZrMoW refractory high entropy alloy

ABSTRACT An equiatomic quinary TiVZrMoW refractory high entropy alloy (RHEA) is selected for studying phase evolution and stability. The prediction of phases that may form on synthesis is made by (i) Semi-empirical /empirical methods (based on extended Hume-Rothery rules), (ii) CALPHAD and (iii) ab-initio methods. Mixing enthalpy (calculated by Miedema method) and valence electron concentration predicted the formation of the BCC phase (; and VEC = 5). However, the atomic mismatch factor, δ (7.17%) value being higher than the desired value, can de-stabilize the lattice to form a multi-phase structure. CALPHAD technique predicted that at room temperature three BCC and two Laves phases would form. The enthalpy of mixing (4.25 kJ/mol) calculated using Density Functional Theory (DFT) also suggests phase separation at room temperature. The phases and microstructure that evolved in the as-cast and annealed samples are studied by X-ray diffraction (XRD), scanning electron and transmission electron microscopy (TEM) techniques. Two BCC phases and a minor amount of ordered B2 and C15 type Laves phase are found to co-exist in the as-cast sample. The primary BCC phase (BCC#1) is rich in Mo and W (a = 3.17 ± 0.02 Å), while the minor BCC phase (BCC#2) is rich in Zr and Ti (a = 3.65 ± 0.02 Å). The annealed sample shows the transformation of both the disordered BCC phase to the ordered B2 phases and the increase in the amount of C15 Laves phase. The phase evolution and their relative stability in this RHEA has been discussed after analysing the experimental and theoretical results.

Thermodynamic description of Ru–Ge–Si ternary system

With the help of Miedema's method, Ru–Ge–Si ternary system has been thermodynamically described through the CALPHAD approach. Firstly, mixing enthalpy of liquid Ru–Ge alloy was calculated in light of Miedema's empirical method. Subsequently, the Ru–Ge and Ru–Si binary systems were thermodynamically assessed. Solution phases including Liquid, Hcp and Diamond were described with a substitutional model, of which the excess Gibbs energies were formulated with the Redlich–Kister polynomial. Binary intermetallic phases were treated as stoichiometric compounds. Then, combined with the reported parameters of Ge–Si system, phase equilibria in the ternary Ru–Ge–Si system were assessed where the ternary solubility in intermetallic compounds was taken into consideration. In addition, liquidus projection was further calculated.

A generalized hard-sphere model for the irradiation induced viscosity of amorphous binary alloys

The entropy of mixing of binary alloys using a generalized hard-sphere model of binary fluids is considered in an application to alloys that undergo an irradiation-induced crystalline–amorphous transformation at temperatures significantly below the melting temperature. The model is based on the observation that many amorphous materials respond in a “liquid like” manner while under irradiation. The generalization of the model to include attractive forces is obtained by adding the effect of a uniform negative background potential to the hard-sphere model. An application of the method is demonstrated for U–Si, U–Al and Al–Si irradiated alloys. Because of the scarcity of experimental information, estimates of the binary alloy formation enthalpy made using the empirical Miedema method are employed. Thermal expansion coefficients are estimated using the quantum approximate BFS method assuming solid solution. The model is used to calculate the viscosity of the alloys as a function of composition.

Heat of Formation in (Mg, X)B 2 (X = Li, Na, Ca, Al)

The recent discovery of superconductivity in MgB 2 with T c ∼ 39 K caused exciting in the solid state physics community. A semi-empirical Miedema method and the Toop’s model were applied to calculate the heat of formation (HT) for the ternary systems of B–Mg–X with X = Li, Na, Ca, Al. HT for (Mg, X)B 2 was determined from the projection of HT B–Mg–X ternary system into the level with the equivalent boron composition at x B = 2/3. The calculated results showed that a meta-stable or an unstable phase forms with X take the hole dopants of Li and Na, while a stable phase of (Mg, X)B 2 forms only with X take di-valence such as Ca or the electronic dopants, such as Al.

Prediction of Critical Compositions for Bulk Glass Formation in La-Based, Cu-Based and Zr-Based Ternary Alloys

Using a quasi-chemical method coupled with Miedema's semi-empirical method, we have estimated critical glass-forming compositions for bulk metallic glasses of metal-metal type alloys such as La-Ni-Al, Cu-Zr-Ti and Zr-Ni-Al that exhibit maximum temperature spans of their supercooled liquid region, ΔT x (= T x - T g ), reaching 37--77 K. We applied Cowley's chemical short-range order (CSRO) parameters η ij for specific constituent i-j atomic pairs calculated by the quasi-chemical method to obtain the short-range order parameter γ of the amorphous phase, which is used as a constant in the calculation of formation enthalpies by Miedema's method. In this study, the constant y is treated as a function of the composition of atomic pairs such as γ ij (η ij ). From the calculated results, it appears that the most negative value of η ij corresponds to a highly stable liquid state because of the occurrence of chemical short-range order, implying that bulk metallic glasses can be produced. Stability and instability of the liquid phase have been discussed in terms of the compositional dependence of Cowley's CSRO parameters and enthalpy difference parameters. The calculated results predict fairly well the critical glass-forming compositions for the various bulk metallic glasses, even predicting the asymmetrical composition dependence of the glass-forming composition ranges. Furthermore, the relation of η ij (= η 12 with interaction parameters e 2 1 yields the characteristics of interaction between dissimilar solutes in 0-1-2 ternary metallic solutions.

Gibbs energy coupling of phase stability and thermochemistry in the HfNb system

The thermodynamic properties of the HfNb system are little known from experimental data. There have been conflicting reports about the phase diagram, and no measurements have been presented of the thermochemical properties. As a consequence, there is a lack of information on the Gibbs energy ( G m) of its various stable phases, which is necessary in order to establish systematically the thermal properties of transition metal alloys, and to model ternary and higher order phase diagrams based on HfNb. This problem is studied in the present paper by applying a theoretical method which relies on the coupling between thermochemistry and phase diagrams. The approach involves the analysis and synthesis of selected phase equilibrium data using models for G m of bcc, hcp and liquid, and the evaluation of a set of optimum model parameters through a computer optimisation technique. The G m functions arrived at in this way are used in constructing, by calculation, a phase diagram for the HfNb system, and in analysing the composition and structure dependence of the enthalpy of formation. Detailed comparisons are reported with estimates from the Miedema method and other predictive approaches, and with the behaviour of the related systems HfTa, ZrNb and ZrTa, which have recently been analysed.

Phase diagram and thermochemical properties of the ZrTa system. An assessment based on Gibbs energy modelling

The thermochemical properties of the ZrTa system are not known from direct measurements and there is a lack of information on the Gibbs energy ( G m ) of its various phases, which is necessary in systematic analyses of cohesive and thermal properties of transition metal systems, and in thermodynamic modelling of ternary and higher-order phase diagrams based on ZrTa. This kind of information has been obtained by using a theoretical method which relies on the coupling between thermochemistry and phase diagrams. The approach involves the analysis and synthesis of selected pieces of phase equilibrium data using models for G m of bcc, hcp and liquid, and the evaluation of a set of optimum model parameters through a computer optimisation technique. The G m functions arrived at in this way are applied to construct, by calculation, a phase diagram for the ZrTa system, which is used in critical comparisons with the experimental data available. In addition, the enthalpy of formation values generated by the present study are compared with estimates based on the Miedema method and other predictive approaches.

Gibbs Energy Modelling of the Phase Diagram and Thermochemical Properties in the Hf-Ta System

The thermochemical properties of the Hf-Ta system are not known from direct measurements and there is a lack of information on the Gibbs energy G m of its various stable phases, which is necessary in establishing systematically the thermal properties of transition metal alloys, and in the modelling of ternary and higher-order phase diagrams based on Hf-Ta. This kind of information has been obtained using a theoretical method which relies on the coupling between thermochemistry and phase diagrams. The approach involves the analysis and synthesis of selected pieces of phase equilibrium data using models for G m of bcc, hcp and liquid, and the evaluation of a set of optimum model parameters through a computer optimization technique. The G m functions arrived at in this way are used in constructing, by calculation, a phase diagram for the Hf-Ta system, and in analysing the composition and structure dependence of the enthalpy of formation. Detailed comparisons with estimates from the Miedema method and other predictive approaches are also reported.

Calculation of the heat of formation of ternary compounds: P-Ga-As, N-Ga-As, N-Ga-P

Miedema method, which was proposed to calculate the heat of formation of binary alloy system, is extended for calculating the heat of formation of ternary system by using the Toop's model. The hypothetical metallic allotropes correspounding to the properties of nonmetallic elements are introduced. In our calculation, the structure-dependent energy contributions have not been taken into account. The heat of formation of GaAs1-xPx, GaAs1-xNx, and GaP1-xNx are obtained by projecting the heat of formation of ternary systems, P-Ga-As, N-Ga-As, and N-Ga-P onto the plane of the concentration of Ga, c2 = 0.5, respectively. The possibility of existence of ordered ternary compound was predicted. The dependence of direct band gap, EΓ, or indirect one, EX, for GaP1-xNx and GaAs1-xNx, on x, the concentration of N, is obtained by using the dielectric theory of electronegativity.

On the thermodynamics of the β-phase in CrTi

The possibility of amorphization of the β-phase in the CrTi system is considered using free energy calculations based on the Miedema method. The results of this calculation are also compared with those obtained by other workers using phase diagram fitting and the cluster variation method. These calculations do not suggest any possibility of amorphization, called spontaneous vitrification, as has been reported in the literature. It is also found that strain energy can make only a negligible contribution to the overall free energy and thus cannot destabilize the β-phase. The possibility of spinodal decomposition of the β-phase is also considered.

Solubility of hydrogen in Ti 3In

Hydrogen solubility isotherms in the intermetallic phase Ti 3In have been determined in the ranges 1 ⩽ P (mbar) ⩽ 1000 and 923 ⩽ T (K) ⩽ 1123 using the manometric method. The partial molar enthalpy of hydrogen solution at infinite dilution ( Δ H ̄ O h ) and the partial molar excess entropy of hydrogen solution at infinite dilution ( Δ S ̄ e,O h ) are −55.37 ± 0.26 kJ mol −1 and −53.32 ± 0.63 J K −1 mol −1 respectively. Both ¦ Δ H ̄ h ¦ and ¦ Δ S ̄ e h ¦ decrease with an increase of the hydrogen to metal atom ratio r = n H : n M . A linear correlation is found which connects Δ H ̄ O h and Δ S ̄ eo h , of the hexagonal Ti 3X ( X ≡ Al, Sn, Ga, In) alloys. Ti 3In decomposes to Ti 3In 2 and TiH 2−δ, when reacting with hydrogen. The calculation of the heat of formation ( Δ H ̄ f ) based on Miedema's method suggests that the decomposition of Ti 3In is probably due to the low heat of formation when compared with other hexagonal Ti 3X intermetallic phases.

Surface segregation in jellium binary solid solutions.

The electron-density-functional method is applied to a jellium model for the surface-segregated binary (single-phase) solid solution. The segregation potential, thickness of the segregated layer, and the segregating component are determined and the following rule is derived for a binary (single-phase) solid solution of substitutional type: Atoms of the component which has the larger Wigner-Seitz radius segregate to the surface. The present theoretical predictions are completely compatible with those determined by Miedema's method and agree reasonably well with various experimental data.

Superconductivity in ZrxNb1−2xMox alloys: Possible dominance of a soft phonon mode

Normal-state specific heat and superconducting transition temperature measurements were performed on pure polycrystalline Nb and isoelectronic bcc ZrxNb1−2xMox alloys (x = 0.02, 0.05, and 0.10). Measured T c, γ and θ D all decline nearly linearly from the pure Nb values with increasing x. The changes are 15, 10, and 6%, respectively, for x = 0.1. T cH calculated by Hopfield's theory, using data and linearly interpolated values of η and θ D /〈θ2〉linear 1/2, increases with x by 28% for x = 0.1. By relaxing the linear interpolation of θD/〈θ2〉1/2, Hopfield's theory can be used to evaluate an average phonon frequency 〈θ2〉emp 1/2 from data. It is found to be nearly constant at about 229 K, in contrast to 〈θ2〉linear 1/2, which declines by 8% at x = 0.1. We suggest that one possible explanation for this behavior is the dominance of incipient soft-phonon longitudinal modes which occur in Nb and presumably in the alloys. Miedema's method has been used to calculate the variation of both γ and T c with alloying. The calculated γ is nearly constant, while the calculated T c declines more quickly than the measured variation. The band-structure electron density of states calculated with the aid of McMillan's equation for these isoelectronic alloys is found to decline by 8% at x = 0.1, in contrast to the prediction of the rigid band model. Normal-state low-temperature specific heat measurements on a pure annealed polycrystalline sample of Nb yield γ = 7.80 ± 0.02 mJ mole−1 K−2 and θD = 276 ± 2 K, in excellent agreement with previous measurements. The break in slope of C p/T vs. T 2 at 3.2 K is discussed.