Calculation of physicochemical properties in a ternary system with miscibility gap

For aternary system with a complete miscible area, one may use geometrical models, such as the Kohler model, Muggianu-Jacob model, Toop model, Hillert model, Lück-Chou model, etc. to calculate thermodynamic properties. However, for the ternary system with a partial miscible area, if that area does not touch the binary edges in a ternary composition triangle, in principle, the geometric model cannot give an accurate estimation for thermodynamic properties due to the absence of accurate binary information at the given temperature. In this article, a special method has been proposed for calculating thermodynamic properties and other physicochemical properties for the partial miscible area in a ternary system if those properties along the miscible boundary are known. Two examples have shown that this method works very well.     

Multicomponent diffusion simulation based on finite elements

A numerical model is presented to treat multicomponent, multiphase diffusion problems. Unlike other recent approaches that are based on the finite-difference method, analytical solutions, or particular thermodynamic models, a general procedure based on the finite-element technique is applied. The suggested formalism is based on the solution of the integral statement of the generalized diffusion equation. This treatment allows for a simple implementation of particular boundary conditions and can easily be extended from a one- to a multidimensional analysis. A brief overview of the formal representation of multicomponent diffusion coefficients, as suggested by Andersson and Agren, is given. The finite-element diffusivity matrices are evaluated for a one-dimensional bar and a two-dimensional triangular element. The model is applied to some classical examples in diffusion simulation in both one and two dimensions. The results are compared to available analytical solutions or experimental data.     

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.     

A new generation solution model for predicting thermodynamic properties of a multicomponent system from binaries

During the past 3 decades, all solution models that were used to predict thermodynamic properties of a multicomponent system from binaries improperly assumed that the selected binary compositions in a model are independent of the practical system to be treated. This assumption causes problems both in symmetrical and asymmetrical models. In this article, a new solution model has been suggested, which gets rid of this traditional way and assumes that the selected binary compositions should be closely related to the system considered. After introducing a new concept, the “similarity coefficient,” the relation between the selected binary compositions and the composition of the multicomponent system is established and a new model is generated. This new generation model is more reasonable in theoretical considerations, more reliable in practical use, and more realistic in computerization for estimating thermodynamic properties and calculating phase diagrams in a multicomponent system.     

Thermodynamic description for concentrated metallic solutions using interaction parameters

The integral molar excess Gibbs energy of a multicomponent system is expressed in terms of interaction parameters, from which the analytical formulae of the activity coefficients of the solutes and solvent, as Eqs. [23] and [24], were deduced. This approach, named the ε approach, is able to describe quantitatively the thermodynamic properties of multicomponent systems. It features thermodynamic consistency, high accuracy, and a rather small influence of the higher-interaction parameters on the thermodynamic properties of metallic solutions. A simple modification to the first-order interaction parameters extends the ε approach, to be applicable to systems with strong interactions between components at both low and concentrated levels.     

多元系B2相的热力学描述

B2相广泛存在于铁基、钴基和镍基合金中.B2相的析出对于合金组织和性能具有重要影响.本文指出了合理描述B2相的热力学模型,并就如何简单有效地描述多元合金系中的B2相,以及如何处理在此过程中可能出现的各种问题,进行了深入的探讨和研究.     

The thermodynamics of dilute solutions of hydrogen in palladium and its substitutional alloys

A review has been given of the thermodynamic and kinetic behavior of hydrogen in dilute solid solution in palladium and in binary solvents consisting of palladium containing a substitutional solute element. The properties of the binary Pd-H system can be understood in terms of simple mixing statistics and classical rate theory. The situation with respect to the ternary systems is not so unequivocable. A large volume of thermodynamic and kinetic data has been analyzed in terms of the cell model for such ternary solid solutions. In some cases the statistical theory is in impressive agreement with both sets of experimental data. In other cases, although the diffusion and thermodynamic behavior of the H-atoms can be described by modified forms of the cell theory, the modifications required are somewhat arbitrary in nature. Modern developments in fundamental calculations of the interactions between H-atoms and both ideal crystals and lattices containing defects have been briefly reviewed. Impressive agreement between the calculated H-atom binding energies to lattice defects and experimental values have been achieved.     

Thermodynamic evaluation and optimization of the LiCl-NaCl-KCl-RbCl-CsCl-MgCl2-CaCl2 system using the modified quasi-chemical model

A complete critical evaluation of all available phase-diagram and thermodynamic data has been performed for all condensed phases of the LiCl-NaCl-KCl-RbCl-CsCl-MgCl₂-CaCl₂ system, and optimized model parameters have been found. The model parameters obtained for binary and ternary subsystems can be used to predict thermodynamic properties and phase equilibria for the multicomponent system. The modified quasi-chemical model for short-range ordering was used for the molten salt phase. Particularly in solutions with MgCl₂ and KCl, RbCl, or CsCl, the calculations indicate a large dregree of ordering on the cationic sublattice, with Mg-Alkali second-nearest-neighbor pairs being favored.     

Thermodynamic evaluation and optimization of the LiF-NaF-KF-MgF2-CaF2 system using the modified quasi-chemical model

A complete critical evaluation of all available phase diagram and thermodynamic data has been performed for all condensed phases of the LiF-NaF-KF-MgF₂-CaF₂ system and optimized model parameters have been found. The model parameters obtained for binary and ternary subsystems can be used to predict thermodynamic properties and phase equilibria for the multicomponent system. The modified quasi-chemical model for short-range ordering (SRO) was used for the molten salt phase. For solutions with NaF or KF together with MgCl₂ or CaCl₂, the calculations indicate a large degree of ordering on the cationic sublattice, with Mg-alkali and Ca-alkali second-nearest-neighbor (SNN) pairs being favored.     

Simplified Thermodynamic Model for Pro-Eutectoid Ferrite Formation in Multicomponent Structural Steel

AnumberofthermodynamicmodelsforFe Cbinarysystemhavebeendeveloped ,namelytheKRC′streatment[1] whichbelongstogeometryre pulsivemodel ,theLFG′streatment[2 ] andtheMD′streatment[3 ] whichareofatominteractivemodel .Afterwards,ShifletGJetal[4 ] correctedandim provedthesemodels .Inordertoinvestigatethephasetransformationinmulticomponentsteels ,AaronsonHIetal[5] firstproposedthesuperelementmodelfollowingtheZenerC′sworks[6 ] .Atpresent ,thesuperelementmodelhasbeenwidelyappliedtothermodynamic…     

Critical thermodynamic evaluation and optimization of the FeO-Fe<Subscript>2</Subscript>O<Subscript>3</Subscript>-MgO-SiO<Subscript>2</Subscript> system

A complete literature review, critical evaluation, and thermodynamic modeling of the phase diagrams and thermodynamic properties of all oxide phases in the FeO-Fe₂O₃-MgO-SiO₂ system at 1 bar total pressure are presented. Optimized model equations for the thermodynamic properties of all phases are obtained, which reproduce all available thermodynamic and phase-equilibrium data within experimental error limits from 25 °C to above the liquidus temperatures at all compositions and oxygen partial pressures. The complex phase relationships in the system have been elucidated, and discrepancies among the data have been resolved. The database of the model parameters can be used along with software for Gibbs-energy minimization in order to calculate all thermodynamic properties and any type of phase-diagram section. The modified quasichemical model was used for the liquid-slag phase. Sublattice models, based upon the compound-energy formalism, were used for the olivine, spinel, pyroxene, and monoxide solid solutions. The use of physically reasonable models means that the models can be used to predict thermodynamic properties and phase equilibria in composition and temperature regions where data are not available.     

Critical evaluation and optimization of the thermodynamic properties of liquid tin solutions

Thermodynamic and phase equilibrium data for the following 18 elements in molten Sn were collected and critically evaluated: Al, Ca, Ce, Co, Cr, Cu, Fe, H, Mg, Mo, Na, Ni, O, P, S, Se, Si, and Ti. Binary and ternary data were optimized to give polynomial expressions for the excess Gibbs energies as functions of temperature and composition. For some solutes, the optimized expressions are valid over the entire composition range 0 ≤ X_Sn ≤ 1. In other cases, the expressions apply to Snrich solutions. Solute-solute interaction terms were estimated where data were not available. The optimized Gibbs energy expressions are also presented in the form of interaction parameters, and the equivalence between the polynomial and interaction parameter formalisms is discussed. Through the Kohler equation, or the modified interaction parameter formalism, the thermodynamic properties of the multicomponent solution of 18 elements in Sn can be calculated. The database is suitable for computer storage and manipulation.     

Mathematical representation of thermodynamic properties in binary systems and solution of Gibbs-Duhem Equation

The storage, retrieval, and manipulation of thermodynamic data with the aid of a computer requires accurate analytical representation of thermodynamic properties of solutions. In the present paper, a critical assessment is made of simple power series expansions and their limitations in representing thermodynamic properties over the entire composition range of a binary system. The advantages of certain orthogonal series as an alternative method of representation is also discussed. Particular emphasis is placed upon series representations which use Legendre polynomials due to their simplicity and the fact that their functional form is consistent with empirical observations of solution behavior. Since the coefficients of orthogonal series are independent of each other when the entire composition range of a binary system is represented, any thermodynamic property can be fitted to any desired degree of accuracy by a finite number of terms without the necessity of storing a large number of significant digits. Also, because the coefficients are uncorrelated, they are amenable to mathematical interpolation and extrapolation as well as to physical interpretation. The relationships between coefficients of series expansions of all partial and integral properties for the general case have been derived using the Gibbs-Duhem equation.     

Optimization of binary thermodynamic and phase diagram data

An optimization technique based upon least squares regression is presented to permit the simultaneous analysis of diverse experimental binary thermodynamic and phase diagram data. Coefficients of polynomial expansions for the enthalpy and excess entropy of binary solutions are obtained which can subsequently be used to calculate the thermodynamic properties or the phase diagram. In an interactive computer-assisted analysis employing this technique, one can critically analyze a large number of diverse data in a binary system rapidly, in a manner which is fully self-consistent thermodynamically. Examples of applications to the Bi-Zn, Cd-Pb, PbCl₂-KCl, LiCl-FeCl₂, and Au-Ni binary systems are given.     

Contribution of the school of V. N. Eremenko to the investigation of phase equilibrium diagrams and the thermodynamics of metallic systems

Information is given regarding the metallic systems investigated by V. N. Eremenko and his school over the decades 1960–1990. His. approach to predicting the phase equilibrium diagrams and thermodynamic properties of unknown systems, based on the generalization of experimental data obtained in this period, is discussed. The prediction level (methods of successive comparison, interpolation or extrapolation, correlation, thermodynamic modeling) depends not only on the amount and reliability of data generalized, but also on the thermodynamic properties of the systems under consideration. The necessity of using a combined approach, in which experimental and calculated phase diagram and thermodynamic data are used together, is emphasized. Mathematical optimization and evaluation of such data made it possible to obtain information on the thermodynamics and phase diagrams of previously unknown systems, and to establish a methodological base for predicting the phase diagrams of multicomponent systems of any degree of complexity.Institute of Materials Science Problems, Ukrainian Academy of Sciences, Kiev. Translated from Poroshkovaya Metallurgiya, Nos. 7–8, pp. 9–26, July–August, 1996.     

Computer simulation of diffusion in multiphase systems

A general model to treat multicomponent diffusion in multiphase dispersions is presented. The model is based on multicomponent diffusion data and basic thermodynamic data and contains no adjustable parameters. No restriction is placed on the number of components or phases that take part in the calculations, as long as the necessary thermodynamic and kinetic data are available. The new model is implemented into the DICTRA software, which makes use of THERMO-CALC to handle the thermodynamics. The model is applied to carburization of Ni alloys and heat treatment of welded joints between dissimilar materials. In both cases, the diffusion is accompanied by carbide formation or dissolution. A good agreement between experiments and calculations is found, despite the fact that no adjustable parameters are needed.