A self-consistent model for predicting interaction parameters in multicomponent alloys

Several models have been proposed to predict interaction parameters in multicomponent alloys from the physical characteristics of constituent elements. But all these models are not self-consistent, i.e., they do not meet the basic requirement of the reciprocal relation, ε _i ^j =ε _j ⁱ , automatically. In this article, a new and self-consistent model is proposed by coupling Chou’s geometric solution model with Miedema’s model on the enthalpy of mixing for binary alloys. The new model is applied to the calculation of interaction parameters in a large number of iron-based alloys, and the agreement between calculation and experimental data is found to be reasonable.     

A comparison of the molecular interaction volume model with the subregular solution model in multicomponent liquid alloys

The molecular interaction volume model (MIVM) is a two-parameter model, meaning it can predict the thermodynamic properties in a multicomponent liquid alloy system using only the coordination numbers calculated from the ordinary physical quantities of pure liquid metals and the related binary infinite dilute activity coefficients, γ _∞ ⁱ and γ _∞ ^j , which avoids somewhat adjustable fitting for the binary parameters of B _ji and B _ij. In comparison with the subregular solution model (SRSM), the prediction effect of the MIVM is of better stability and safety because it has a good physical basis.     

Fundamental theories and concepts for predicting thermodynamic properties of high temperature ionic and metallic liquid solutions and vapor molecules

Modern concepts and theories have created the ability to predict the thermodynamic properties of high-temperature liquid solutions (molten salts, metals, and slags) and vapors. These advances have made it possible to calculate thermodynamic properties and total chemistries for many technologically and scientifically important systems. Specific theories include (1) a cycle for accurately calculating the solubility products of relatively insoluble salts in reciprocal molten salt systems, (2) the coordination cluster theory, which allows one to predict the temperature and concentration dependence of the activities of a dilute solute in a multicomponent system, (3) the conformal ionic-solution theory, which predicts the properties of reciprocal and additive multicomponent molten salt systems, (4) the modified quasi-chemical theory, which predicts the properties of multicomponent silicate (and other polymeric) systems, (5) a simple extension of polymer theory, which leads to methods for predicting the sulfide capacities (as well as capacities for PO ₄ ³⁻ , SO ₄ ²⁻ , Cl⁻, Br⁻, I⁻, etc.) in molten silicates and other polymeric solvents, and (6) a dimensional theory for the prediction of nonelectronic entropies and free-energy functions of vapor molecules. These accomplishments have helped to create computer programs which can calculate realistic total chemistries of complex systems and have provided a method of extending the scope of fundamental thermodynamic databases of vapors. He was a Group Leader and Senior Scientist at the Argonne National Laboratory until 1996 when he retired. This article is based on a presentation made at “The Milton Blander Symposium on Thermodynamic Predictions and Applications” at the TMS Annual Meeting in San Diego, California, on March 1–2, 1999, under the auspices of the TMS Extraction and Processing Division and the ASM Thermodynamics and Phase Equilibrium Committee.     

An experimental and thermodynamic study of the Fe-Cr-C-N system at 1273 K

The thermodynamic properties of the Fe-Cr-C-N system at 1273 K (1000 °C) have been evaluated using old and new information. The binary systems are well established. The Fe-Cr-C system is fairly well established, but some experiments were performed in this study in order to establish theα/γ/M₂₃C₆ equilibrium. The Fe-Cr-N system was evaluated in a parallel study. In the Fe-C-N system the properties of theα andγ phases are well established. No direct information from the Cr-C-N system was used. In order to establish the properties of the quaternary system some experiments were made by equilibrating a set of Fe-Cr-C-N alloys at 1273 K, using a sealed capsule technique. After quenching, the carbon and nitrogen activities were evaluated by analyzing the specimens, which were completely austenitic. Phase equilibria in other specimens were studied by microprobe measurements and X-ray phase identification. When the quaternary system was evaluated thermodynamically, it was found that all the experimental information could be reasonably well accounted for without introducing new parameters for the quaternary system. However, it was necessary to evaluate the properties of the metastableε-Cr₂C phase in order to fit the quaternary experimental information. The phase diagram was calculated from the evaluated thermodynamic properties. A number of sections are presented for direct comparison with the experimental data. Formerly with Royal Institute of Technology, Stockholm.     

Interaction coefficients in the iron-carbon-titanium and titanium-silver systems

A liquid Fe-C-Ti system was studied by establishing an iso-titanium-activity state for ternary samples at 1600 °C through the medium of a bath of liquid silver which permits diffusion of titanium only. From the two iso-titanium-activity lines obtained, the self-interaction coefficients of titanium and interaction coefficients of carbon on titanium in liquid iron were estimated:ε _Ti ^Ti = 4.67, ρ_Ti ^Ti = 0.32, ε_Ti ^C = −11.94, ρ_Ti ^C = −4.52, ρ_Ti ^Ti,C = −9.96 An experimental study has been made of the distribution of titanium between liquid silver and liquid iron at 1600 °C. By the use of the interaction coefficients of titanium and r_Ti ^o in liquid iron, the thermodynamic parameters of titanium in liquid silver were determined asr _Ti ^o _Ag = 2.44 X 10⁻³, (ε_Ti ^Ti _Ag = −6.17, (ρ_Ti ^Ti)_Ag = −16.3     

Thermodynamic calculation for alloy systems

The formulas for calculating the activity coefficient, γ _i, in the binary system and the activity coefficient of a solute at infinite dilution, γ _i ⁰ , as well as interaction parameters, ε _i ⁱ and ε _i ^j , in a metallic melt have been proposed on the basis of Miedema et al.’s model for the estimation of the heat of formation of binary alloys. The formulas can be used in both solid and liquid solutions and have been used in the calculation of both binary and ternary alloy systems taken from 75 elements. The results were compared with the experimental values.     

The activity coefficient of oxygen in binary liquid metal alloys

The Wagner model with one energy parameter,h, for describing the effect of alloying elements on the activity coefficients of nonmetallic solutes in liquid metals is extended to have two energy parameters,h ₁andh ₂. The validity of both the Wagner one-parameter equation and the newly derived two-parameter equation is tested using data available in the literature for twelve ternary metal-oxygen systems. In order to have consistent thermodynamic data, all the relevant binary, as well as the twelve ternary metal-oxygen systems are evaluated using the same thermodynamic values for the reference materials which were used in carrying out the experimental measurements. It is found that the twoparameter equation is capable of quantitatively accounting for the compositional dependences of the activity coefficients of oxygen in all twelve ternary systems while the Wagner one-parameter equation is not. A correlation between the Wagner parameter,h, and the thermodynamic properties of the respective binary metal-oxygen and binary metals systems is found, from which the value of this parameter may be predicted without referring to any ternary data. Accordingly, the two-parameter equation is more useful in evaluating ternary experimental data while the Wagner one-parameter equation in connection with the correlation betweenh and binary data is capable of predicting ternary data without any experimental investigation in the ternary region. Based on the one-parameter and the two-parameter equations, theoretical equations for the first-order and second-order free energy interaction parameters,(∈ ₀ ^j )_sand(ρ ₀ ^j )_s, are derived in terms of the model parameters. The values of(∈ ₀ ^j )_s and(ρ ₀ ^j )_s for all the systems are derived and are found to vary linearly with the reciprocal of temperature. Furthermore, linear relationships between these two interaction parameters and their slopes with 1/T are found, from which the temperature dependence of the interaction parameters may be estimated in the absence of experimental data.     

The activity coefficient of nitrogen in binary liquid metal alloys

Literature data for the limiting thermodynamic properties of nitrogen in liquid Fe, Co, Ni, and Cr, as well as in the six binary alloys consisting of two of the above elements, are critically evaluated. The ternary data are evaluated in terms of the Wagner model for quasi-interstitial liquid alloys and values of the Wagner parameter h are obtained for each system. The correlation formulated by Chiang and Chang between the Wagner parameter h and the relevant binary thermodynamic properties for oxygen in binary alloys has been found to be equally valid for nitrogen in binary alloys. From this correlation, a value of h may be predicted in the absence of any ternary experimantal data. Utilizing the Wagner model and the value of h obtained for each system, values for the first-order and second-order Gibbs energy interaction parameters ∈N(s) and ρsujN(s) are derived.. The parameters are expressed as a linear function of the reciprocal temperature, i.e. ∈^jN(s) =α + β/T and.ρ^jN_(s) = α′ + β′/T. A linear correlation found by Chipman and Corrigan between β and ∈^jN(s) for systems where Fe is the solvent is also valid for systems with non-ferrous solvents. A linear correlation between β′ and ρ^jN_(s) is also found in the present study.     

A Derivation of a Quadratic Activity Coefficient <Emphasis Type="Italic">vs</Emphasis> Composition Relationship in a Quaternary System, <Emphasis Type="Italic">A-B-C-D</Emphasis>

In the literature, no direct derivation exists of the quadratic activity coefficient vs composition relationships for a quaternary system with high solute concentrations. Such relations for a ternary system (1-2-3) were derived by Darken by extending the results of a binary system (1-2), introducing a new concept of “hypothetical system” (2-3). To present a better scheme to find the activity coefficient-composition relations for multicomponent systems, derivations are made for a quaternary system A-B-C-D in the current work. Using a MacLaurin series expansion, the (Raoultian) activity coefficient, ln γ _i , of each component is equated with a quadratic expression of mole fractions (x), involving the activity coefficient at zero concentration ( g_i⁰ ) \left( {\gamma_{i}^{0} } \right) and nine interaction coefficients (ε). Subsequently, with the help of a Gibbs–Duhem equation, followed by a comparison of coefficients, most preceding 9 × 4, i.e., 36 interaction coefficients are eliminated, leaving behind only three self- and three ternary interaction coefficients, which are enough to express the activity coefficient vs composition relationships for the solutes B, C, and D, as well as for the solvent A. Setting the mole fraction x _D  = 0, the preceding expressions establish the same relations as proposed by Darken for the ternary system A-B-C. The derivation also clarifies how the quadratic concentration terms accompany the first-order interaction coefficients, not the second-order ones. Applications of the derived relations to determine simultaneously the activity coefficients g_i⁰ \gamma_{i}^{0} and the interaction coefficients ε in a new way in some iron- and steelmaking systems are presented. A new data on interaction coefficients in liquid iron at 1873 K (1600 °C), e_\textV^\textV = - 6. 1, \varepsilon_{\text{V}}^{\text{V}} = - 6. 1, has been generated through such an application.     

Isothermal Phase Transformation in Biomedical Co-29Cr-6Mo Alloy without Addition of Carbon or Nitrogen

The isothermal phase transformation behavior in a biomedical Co-29Cr-6Mo alloy without carbon or nitrogen was investigated during aging at temperatures between 973 K and 1273 K (700 °C and 1000 °C) for up to 90 ks. Transformation from the γ to the ε phase did not occur at 1273 K (1000 °C) as the γ phase was more stable than the ε phase, and the σ phase precipitated at the γ grain boundaries. At 1173 K (900 °C), a γ → ε ₁ phase transformation occurred by massive precipitation. Prolonged annealing at 1173 K (900 °C) led to a lamellar structure of ε ₂ and σ phases at ε ₁/ε ₁ boundaries by a discontinuous/cellular reaction, expressed by the reaction equation ε ₁ → ε ₂ + σ. After decreasing the aging temperature to 973 K (700 °C), transformation from the γ to the ε phase occurred mainly by isothermal martensitic transformation, but a lathlike massive ε ₁ phase and ε ₂/σ lamellar colonies were also observed at the original γ-grain boundaries. It is likely that not adding carbon results in the promotion of the massive transformation and the precipitation of the σ phase during isothermal aging in the Co-29Cr-6Mo alloy system, whose composition corresponds to the ASTM F75 standard for metallic materials for surgical implantation. The resultant isothermal transformation behavior of the present alloy is described on the basis of thermodynamic calculations using Thermo-Calc.     

Symmetry-breaking transitions in equilibrium shapes of coherent precipitates: Effect of elastic anisotropy and inhomogeneity

We examine the symmetry-breaking transitions in equilibrium shapes of coherent precipitates in two-dimensional (2-D) systems under a plane-strain condition with the principal misfit strain components ε* _xx and ε* _yy . For systems with cubic elastic moduli, we first show all the shape transitions associated with different values of t=ε* _yy /ε* _xx . We also characterize each of these transitions, by studying its dependence on elastic anisotropy and inhomogeneity. For systems with dilatational misfit (t=1) and those with pure shear misfit (t=−1), the transition is from an equiaxed shape to an elongated shape, resulting in a break in rotational symmetry. For systems with nondilatational misfit (−1<t<1; t ≠ 0), the transition involves a break in mirror symmetries normal to the x- and y-axes. The transition is continuous in all cases, except when 0<t<1. For systems which allow an invariant line (−1≤t<0), the critical size increases with an increase in the particle stiffness. However, for systems which do not allow an invariant line (0<t≤1), the critical size first decreases, reaches a minimum, and then starts increasing with increasing particle stiffness; moreover, the transition is also forbidden when the particle stiffness is greater than a critical value.     

Driving force for γ→ε martensitic transformation and stacking fault energy of γ in Fe-Mn binary system

A regular solution model for the difference of the chemical free energy between γ and ε phases during γ→ε martensitic transformation in the Fe-Mn binary system has been reexamined and partly modified based on many articles concerning the M _s and A _s temperatures of Fe-Mn alloys. Using the regular solution model, the measured M _s temperatures, and a thermodynamic model for the stacking fault energy (SFE) of austenite (γ), the driving force for γ→ε martensitic transformation, and the SFE of γ have been calculated. The driving force for γ→ε martensitic transformation increases linearly from − 68 to − 120 J/mole with increasing Mn content from 16 to 24 wt pct. The SFE of γ decreases to approximately 13 at. pct Mn and then increases with increasing Mn content, which is in better agreement with Schumann’s result rather than Volosevich et al.’s result.     

The effect of electron concentration on the thermodynamic properties of two alloy phases in the Al-Zn-Ag system

A reversible electrochemical cell was used to determine the thermodynamic activity of aluminum in a series of α-phase Al-Zn and Al-Zn-Ag alloys, and a dew-point technique was used to determine the thermodynamic activity of zinc in a series of ε-phase Ag-Zn-Al alloys. The compositions investigated for both systems included those at which previous authors had related the stability of the phase to electronic factors. The data are presented in the form of graphs showing the excess relative partial molal Gibbs free energy of aluminum, (Δ•G_A1 ^xs), and zinc, (Δ•G_Zn ^xs), as a function of both mole fraction(X _i ) and electron-to-atom ratio (e/a). The latter curves are interpreted to be indications that the effect of electron concentration upon alloy phase stability has been experimentally isolated. Equations are developed which predict the effects of ternary additions in these alloy systems. The rigid-band model is shown to be incapable of predicting the direction of the effect and an argument based upon electron screening of ion-ion interactions is offered. Formerly Graduate Students, Metallurgy Department, Washington State University, Pullman, Wash. This paper constituted a portion of the theses submitted by RONALD E. MILLER and JERRY L. STRAALSUND in partial fulfillment of the requirements of the degree of Doctor of Philosophy in Engineering Science at Washington State University.     

Tetragonal distortion field of hydrogen atoms in iron

The difference between the chemical potentialμ _σ of hydrogen atoms producing a nonspherical symmetry strain in a solid sample under stress and that μ₀ corresponding to the state without stress has been calculated. It is shown that μ_σ - μ₀ = -VΣiσ_iε _ii ^′ =U whereσ _i stands for principal stress,ε _ii ^′ for the strain component along the direction of the principal stress,V for volume, andU is the interaction energy between the strain field and external stress field. The hydrogen atoms producing the tetragonally symmetric strain are preferentially ordered in samples under stress. As a result, the variation of hydrogen concentration with tensile stress σ will be different from that with compressive stressσ*. For a general polycrystal the formulas are, respectively,C _ten =C ₀ exp[(0.70089ε₁₁ + 0.2991lε₂₂)Vσ/RT]andC _com =C ₀ exp[(0.14956ε₁₁ + 0.85044ε₂₂)Vσ*/RT], whereC _ten.,C _com., andC ₀ are, respectively, the hydrogen concentrations under tensile stress, compressive stress, and without stress; R stands for the gas constant andT for absolute temperature. Hence, ε₁₁/ε₂₂ may be determined in terms ofC _ten/C _com which can be obtained by hydrogen permeation measurement. For example, according to Bockris' data ε₁₁/ε₂₂ = 1.27 at temperature of 27 °C which implies that the strain field of hydrogen atoms inα-Fe is nonspherical symmetry. For a torsional stressτ,U _tor = -0.55133V _τ(ε₁₁ ε₂₂. The interaction can result in the enrichment of hydrogen atoms on and hydrogen induced delayed cracking along the planes inclined at an angle ofα = 45 deg to torsional axis, which was observed in precharged smooth torsional or type III cracked specimens made of ultra-high strength steel.     

Thermodynamics of phosphorus in molten Si-Fe and Si-Mn alloys

The thermodynamics of phosphorus in molten Si-Fe and Si-Mn alloys has been investigated at 1723 K by equilibrating the alloys in a controlled phosphorus partial pressure. The activity coefficient of phosphorus in each alloy shows a maximum value at a certain composition due to a strong interaction between silicon and iron and between silicon and manganese. Interaction coefficients between phosphorus and iron in molten silicon were found to be ε _P ^Fe =7.43 and ρ _P ^Fe =−16.4 (0≦X _Fe≦0.65), and those between phosphorus and manganese were ε _P ^Mn =12.0 and ρ _P ^Mn =−22.2 (0≦X _Mn≦0.5). Further discussion has revealed that the Si-Fe-P and Si-Mn-P systems approximately conform to a regular solution within the composition ranges investigated in the present work.     

The unified interaction parameter formalism: Thermodynamic consistency and applications

In a recent article, we proposed modifications to the standard interaction parameter formalism. The modified formalism, known as the “Unified Interaction Parameter Formalism,” is discussed in the present article with respect to thermodynamic consistency at finite concentrations in binary, ternary, and multicomponent systems. A new method, which is independent of integration paths, is proposed to derive the equations of the formalism by differentiation of the integral Gibbs energy expression. It is shown that the formalism is thermodynamically exact in both dilute and nondilute composition regions. It is also shown that the formalism reduces to Wagner’s formalism at infinite dilution and to Darken’s quadratic formalism in dilute solutions. Examples are presented and methods are discussed for determining the parameters of the formalism from thermodynamic data.