Showing papers in "Calphad-computer Coupling of Phase Diagrams and Thermochemistry in 2002"

Thermo-Calc & DICTRA, computational tools for materials science

Abstract: Software for calculation of phase diagrams and thermodynamic properties have been developed since the 1970's. Software and computers have now developed to a level where such calculations can be used as tools for material and process development. In the present paper some of the latest software developments at Thermo-Calc Software are presented together with application examples. It is shown how advanced thermodynamic calculations have become more accessible since: - A more user-friendly windows version of Thermo-Calc, TCW, has been developed. - There is an increasing amount of thermodynamic databases for different materials available. - Thermo-Calc can be accessed from user-written software through several different programming interfaces are available which enables access to the thermodynamic software from a user-written software. Accurate data for thermodynamic properties and phase equilibria can then easily be incorporated into software written in e.g. C++, Matlab and FORTRAN. Thermo-Calc Software also produces DICTRA, a software for simulation of diffusion controlled phase transformations. Using DICTRA it is possible to simulate processes such as homogenization, carburising, microsegregation and coarsening in multicomponent alloys. The different models in the DICTRA software are briefly presented in the present paper together with some application examples.

FactSage thermochemical software and databases

Abstract: This paper presents a summary of the FactSage thermochemical software and databases. FactSage was introduced in 2001 and is the fusion of the FACT-Win/F∗A∗C∗T and ChemSage/SOLGASMIX thermochemical packages that were founded over 25 years ago. The FactSage package runs on a PC operating under Microsoft Windows® and consists of a series of information, database, calculation and manipulation modules that enable one to access and manipulate pure substances and solution databases. With the various modules one can perform a wide variety of thermochemical calculations and generate tables, graphs and figures of interest to chemical and physical metallurgists, chemical engineers, corrosion engineers, inorganic chemists, geochemists, ceramists, electrochemists, environmentalists, etc. In this article emphasis is placed on the calculation and manipulation of phase diagrams. However the reputation of FactSage has been established mainly in the field of complex chemical equilibria and process simulation where the software has unique capabilities. Some of these capabilities are also shown in this paper.

The Alloy Theoretic Automated Toolkit: A User Guide

Abstract: Although the formalism that allows the calculation of alloy thermodynamic properties from first-principles has been known for decades, its practical implementation has so far remained a tedious process. The Alloy Theoretic Automated Toolkit (ATAT) drastically simplifies this procedure by implementing decision rules based on formal statistical analysis that frees the researchers from a constant monitoring during the calculation process and automatically “glues” together the input and the output of various codes, in order to provide a high-level interface to the calculation of alloy thermodynamic properties from first-principles. ATAT implements the Structure Inversion Method (SIM), also known as the Connolly-Williams method, in combination with semi-grand-canonical Monte Carlo simulations. In order to make this powerful toolkit available to the wide community of researchers who could benefit from it, this article present a concise user guide outlining the steps required to obtain thermodynamic information from ab initio calculations.

MTDATA - thermodynamic and phase equilibrium software from the National Physical Laboratory

Abstract: In the past, the complexity of the chemical and phase equilibria established during many industrial processes has prevented the kind of in-depth understanding of their thermodynamics necessary for successful and efficient process control. Predictive thermochemistry, as embodied within MTDATA, makes such an understanding possible. MTDATA allows equilibria to be calculated for multicomponent systems of practical interest, containing many different types of phase, from critically assessed data for their component binary and ternary subsystems. Very complicated calculations can be undertaken with as many as thirty different components and 500 phases being considered simultaneously. A number of modules are incorporated for critically assessing, manipulating and retrieving the data, making various types of calculation and plotting binary, ternary, multicomponent, and predominance area diagrams. Facilities are also available for users to link the complexities of phase equilibrium calculations within MTDATA to their own software or to third party commercial software packages enabling users to simulate unit operations within an industrial plant or to integrate kinetic effects. Special derivatives of MTDATA are also now being developed to provide the sophisticated capabilities within MTDATA to non-expert users for special applications using carefully designed graphical user interfaces, which are both easy to use, and which provide output specifically designed for the particular industrial need.

The PANDAT software package and its applications

Abstract: PANDAT is a software package for multicomponent phase diagram calculation. Given a set of thermodynamic parameters for all phases in a system and a set of user constraints, PANDAT automatically calculates the stable phase diagram without requiring either prior knowledge of the diagram or special user skills. The features of PANDAT are discussed and some application examples presented. In addition to PANDAT, its calculation engine, PanEngine, is also discussed.

Thermodynamic database MALT for Windows with gem and CHD

Abstract: We have revised the thermodynamic database MALT to match the window's environments and also to make it bilingual (English and Japanese). The main characteristic features are described with an emphasis on those functions which will be available in the new version. To facilitate the application of thermodynamic analysis on materials development and process analysis, the database MALT provides generalized programs for calculating chemical equilibria and constructing chemical potential diagrams. MALT encourages users to utilize thermodynamic data in the databases as well as their own in thermodynamic analysis on their own materials/processes problems. To facilitate it, MALT provides the function called "MALT Direct," which can make access to the data users prepare, and can utilize them in their own programs written in Pascal and compiled in Delphi. Some functions of the generalized calculation programs, gem and CBD, are also described in this report. 0 2002 Published by Elsevier Science Ltd. The thermodynamic database MALT( l-31 (Materials-oriented Little Thermodynamic database) was developed by a thermodynamic database task group, which was organized initially in the Japan Society for Calorimetry and Thermal Analysis and later became independent from the society. This task group started its activity with an aim at facilitating the utilization of the high-quality thermodynamic data in practical materials problems. In particular, its industrial use was highly encouraged. The present paper describes the main characteristic features of a new version of MALT together with related computer programs, gem (4) and

Thermodynamic modeling of the ZrO system

Abstract: In this study, the complete zirconium-oxygen system has been critically assessed at 1 at. from 300°C to liquidus temperatures. Thermochemical measurements and phase diagram data were used to model the Gibbs free energies of seven phases. Additionally, the ordered interstitial HCP-based solutions were included and considered as simple line compounds. By using the PARROT module in Thermo-Calc, it was possible to optimize the parameters of the models used to describe the Gibbs free energies of the HCP, BCC, Liquid, γ ZrO2−x,β ZrO2−x and α ZrO2−x phases. The Gas phase was considered to behave ideally. Although phase diagrams including the stoichiometric zirconia phases have been assessed, this is the first time, to the best of our knowledge that a complete assessment of this system is published.

Ab-initio calculations of the formation energies of BCC-based superlattices in the FeAl system

Abstract: First-principles calculations of the total energies of A2 iron and aluminum, B2 (FeAl), B32 (FeAl) and D0 3 (Fe 3 Al and FeAl 3 ) compounds were performed in the frame of density functional theory (DFT) using the Full Potential - Linear Augmented Plane Wave method (FP-LAPW). These results have been used to obtain formation energies of the respective ground states. The calculated formation energies of the D0 3 (Fe 3 Al) and B2 (FeAl) compounds show excellent agreement with available calorimetric data on standard enthalpies of formation of FeAl alloys up to 50 at.% aluminum. As the FeAl system has a controversial magnetic behavior when described by ab-initio methods in the DFT, this agreement is remarkable.

Evaluation of the stable and metastable Cu-Co-Fe phase diagrams

Abstract: The thermodynamic description of the Cu-Co-Fe system was modified on the basis of experimental data obtained by electromagnetic levitation. Using the modified database, the solidus and liquidus temperatures were calculated. Very good agreement was obtained between the calculated and experimental data. The compositions of the metastable miscibility gap in the liquid were found to agree well with the observed values.

Phase stability of the Hf-Nb system: From first-principles to CALPHAD

Abstract: Hf-Nb is representative of a large number of alloy systems where the phase diagram is known but experimentally measured thermodynamic data is unavailable. This motivated us to investigate solid-state phase stability of this system employing ab-initio techniques based on electronic density functional theory (DFT). The total energy calculations are performed at the generalized gradient approximation level to account for non-locality of the exchange-correlation functional within DFT. The zero-temperature heat of formation of bcc-, hcp-and fcc-based ordered phases (virtual) and the finite temperature heat of mixing of bcc, hcp and fcc solid solutions are calculated. The latter properties are obtained by the cluster expansion technique coupled with Monte-Carlo simulations. In an effort to build multicomponent thermodynamic and kinetic databases containing Hf and Nb, we demonstrate a hybrid approach to integrate the ab-initio results with CALPHAD formalism to obtain thermodynamic data for the Hf-Nb system. The similarities and differences between previous CALPHAD assessment and our results are discussed.

Thermodynamic description of the CuAlZn and CuSnZn systems in the copper-rich corner

Abstract: Thermodynamic descriptions of the ternary CuAlZn and CuSnZn systems in the copper-rich corner are presented. The thermodynamic parameters of the binary sub-systems, CuAl, CuSn, CuZn, AlSn, AlZn and SnZn, are based on earlier SGTE-based assessments and those of the ternary CuAlZn and CuSnZn systems are optimized in this study by using the experimental thermodynamic and phase equilibrium data. The present ternary descriptions are valid for aluminum contents up to xAl=0.33, tin contents up to xSn=0.25 and zinc contents up to xZn=0.7.

Contribution of the Rietveld method to non-stoichiometric phase modeling. Part II: α-Tl5Te3 and μ Nb-Ni as experimental examples

Abstract: Two detailed examples are considered to show the ability of powder diffraction data analyzed with the Rietveld method to get insight into the detailed structure of non-stoichiometric phases. Both the γ-phase in Tl-Te system and μ-phase in Nb-Ni system were investigated as a function of composition across their whole homogeneity domain. In the case of the γ Tl-Te phase, the excess Tl relative to the stoichiometric composition is accommodated by Tl substitution on one of the two Te sites. In the μ Nb-Ni, three of the five sites of the structure are partially disordered for all the compositions studied. The results are analyzed in view of Calphad modeling of both phases.

Assessment of the Cr-B system and extrapolation to the Ni-Al-Cr-B quaternary system

Abstract: The Cr-B system is re-assessed treating boron as an interstitial element in the disordered solid solution fcc and bcc phases. The binary assessment is in good agreement with the available experimental phase diagram data and thermochemical data. The Cr-B binary assessment is then used in the assessment of the Ni-Cr-B and Al-Cr-B systems. The ternary assessments are in reasonable agreement with the limited experimental data for the systems. Finally, the ternary assessments are combined with the previous ternary assessment of the Ni-Al-B system to extrapolate to the quaternary Ni-Al-B-Cr system.

Thermodynamics of formation of ammonium, sodium and potassium alums and chromium alums

Abstract: The Pitzer ion-interaction model has been used for thermodynamic simulation of the binary Cr 2 (SO4) 3 H2O, KCr(SO4) 2 H2O, and ternary (NH 4 ) 2 SO 4 Al 2 (SO 4 ) 3 H 2 O, Na 2 SO 4 Al 2 (SO 4 ) 3 H 2 O, K 2 SO 4 Al 2 (SO 4 ) 3 H 2 O, (NH 4 ) 2 SO 4 Cr 2 (SO 4 ) 3 H 2 O, Na 2 SO 4 Cr 2 (SO 4 ) 3 H 2 O, and K 2 SO 4 Cr 2 (SO 4 ) 3 H 2 O systems at T=298.15 K. The optimum values of the binary parameters of ionic interactions for chromium solutions have been calculated using osmotic data for unsaturated binary solutions. The mixing parameters have been chosen on the basis of the compositions of saturated ternary solutions and data on the binary solubility of the alums and chromium alums in water. Good agreement between experimentally determined and calculated solubilities has been found. Important thermodynamic characteristics (thermodynamic solubility product, standard molar Gibbs energy of formation) of the solid phases (simple and double salts) crystallizing in the systems under consideration are determined.

First-principles calculation of L10-disorder phase diagram in FePt system within the first and second nearest neighbor pair interaction energies

Abstract: FLAPW total energy calculations are performed for four kinds of ordered compounds in addition to pure Fe and pure Pt. Cluster expansion on the set of these total energies extracts effective nearest and second nearest neighbor pair interaction energies in addition to multibody interaction energies up to regular tetrahedron interactions. The entropy term is evaluated within the Tetrahedron-Octahedron approximation of the Cluster Variation Method. Thereby, the second nearest neighbor pair interaction and correlation are fully taken into account in the resultant phase diagram. As compared with the phase diagram obtained previously within the nearest neighbor pair interaction, the transition temperature is raised. This is ascribed not only to the enhanced ordering tendency due to the second nearest neighbor pair interaction but also to the destabilization of a homogeneous solid solution.

Thermodynamic study of quaternary systems with participation of ammonium and sodium alums and chromium alums

Abstract: The Pitzer ion-interaction model has been used for thermodynamic simulation of the ternary NaCl-NH&l-H,O, NaBr-NH,Br-H,O, Na,SO,-(NH,),SO,-H,O, NiSO,-Al,(SO,),-H,O, and CuSO,-Al,(SO,),- H,O systems at T=29815 K The mixing parameters have been chosen on the basis of the compositions of saturated ternary solutions taking into account the unsymmetrical mixing terms % and EO' The sodium- ammonium systems have been simulated simultaneously with a view to preserving a constant value of the binary 6Na,~, mixing parameter It was established that the zero values of &Me and \vM,M~,SO, describe the properties of the ternary aqueous mixtures of 2-2 and 3-2 electrolytes with a sufficient accuracy Important thermodynamic characteristics (thermodynamic solubility product, standard molar Gibbs energy of formation) of the solid phases (simple and double salts) crystallizing in the systems under consideration are determined The calculated thermodynamic functions have been used for a thermodynamic study of the quatemary NqSO,- (NH&SO,-Cr,(SOJ,-H,O, and Na$O,-NiSO,-Al,(SO,),-H,O systems Good agreement between experimentally determined and calculated solubilities has been found 0 2002 Elsevier Science Ltd All rights reserved

Phase diagram calculation in CoCr system using Ab initio determined lattice instability of sigma phase

Abstract: The calculations of phase equilibria in the CoCr system were performed using the CALPHAD method on the basis of a new two-sublattices model of sigma phase. This model enables us to utilise the results of ab initio calculations of total energy differences between the sigma phase structure and the Standard Element Reference (SER) structures of pure metal at the relaxed lattice parameters (Δ0Eiσ−SER). Total energies were calculated by Full-Potential Linear Augmented Plane Waves (FLAPW) method in the General Gradient Approximation (GGA). The entropy contribution to the Gibbs energy of the pure elements in the sigma phase structure, and the excess Gibbs energy of mixing of the sigma phase were adjusted to the experimental phase equilibrium data.

A thermodynamic assessment of the CuMgNi ternary system

Abstract: A complete thermodynamic description of the CuMgNi ternary system was carried out based on the CALPHAD method. Ternary solubilities of the binary phases Mg 2 Cu, Cu 2 Mg, Mg 2 Ni and Ni 2 Mg were considered. The Gibbs energies of these semi-stoichiometric compounds were expressed by the compound-energy formalism. Liquid and solid solution phases were described using the Redlich-Kister-Muggiani formalism. The model parameters were evaluated using ThermoCalc software, utilizing the experimental data available in the literature including experimental thermodynamic functions and previously established portions of the phase diagram, such as isopleths and isotherms. Reasonable agreement was obtained between calculated results and experimental information.

Cluster variation method in the computational materials science

Abstract: Cluster Variation Method (CVM) has been very successful in the computations of alloy phase diagrams as well as in many problems of the materials science related to the phase transitions. Originally, CVM was developed in the framework of the so-called rigid lattice approximation, but it has recently been extended to include continuous atomic displacements due to thermal lattice vibration and local atomic distortion due to size mismatch of the constituent atoms. In the present study, we focus our attention on the latter continuous displacement treatment of CVM. The continuous displacement (CD) formulation of the CVM is applied to study the phase stability of the binary alloys. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r . The effects of continuous atomic displacement on the thermodynamic quantities and phase transitions of binary alloys are investigated in detail. We also discuss the extension of the CD treatment of CVM to the calculations of solid-liquid and gas liquid phases transitions.

Remarks on crystallochemical aspects in thermodynamic modeling

Abstract: The problem of the thermodynamic modeling has been widely discussed and approached along different lines. From a crystallochemical point of view two groups of questions are worthwhile to be mentioned: 1. a) The crystallographically sound thermodynamic description of a complex phase, having a structure corresponding to several sets of equipoints (or Wyckoff positions). 2. b) The subdivision of the atoms pertaining, in a certain space group, to a specific Wyckoff position into two or more groups of equipoints of lower multiplicity, when some ordering takes place, i.e. on passing from a structure to its derivative ordered variants (cF4-Cu → cP4-AuCu 3 , for inst.). It may be remarked that a coherent crystallographic approach is very useful in modeling thermodynamic properties of the involved phases. In [97Ans] various important structure types (σ, μ, χ, Laves) have been analysed and suggestions given for a convenient thermodynamic modeling, coherent with a correct crystallographic description. It is well known, on the other hand, that alternative, complementary descriptions of the different structure types are available. In selected systematic collections, critically revised descriptions of the structure types are given which include, for each prototype, not only Pearson symbol, space group, lattice parameters and occupied Wyckoff positions, but also a description of the coordination (coordination polyhedra and next neighbour hystograms). On the other hand the role of the coordination, described in terms of “atomic environment types” (AET) has been discussed and used for a systematic presentation of the different prototypes, their possible classification and grouping in inter-related families. These aspects are certainly relevant while discussing phase (and alloy system) properties and possibly while looking for general criteria for their thermodynamic modeling in the framework of the Compound Energy Formalism. Examples of their use in the thermodynamic modeling of a few selected phases are presented and discussed.

Experimental and calculated Ti-rich corner of the Al-Si-Ti ternary phase diagram

Abstract: The phase equilibria between β(bcc), α(hcp), Ti 3 Al-α 2 (hcp) and Ti 5 Si 3 (hcp) in the Ti-Al-Si system have been investigated in the temperature range of 700 to 1200°C. Isothermal sections of the ternary phase diagram have been assessed employing thermodynamic software, which uses the compound-energy model to describe mathematically the phase. Available experimental phase equilibria results on the Ti-Al-Si system were used to calculate missing binary and ternary interaction parameters and assess isothermal phase diagrams. Extrapolations in the resulting tie-triangles indicate the existence of three eutectoid reactions in the Ti-rich corner of the ternary diagram: β −⪢ α + Ti 5 Si 3 , α −⪢ α 2 + Ti 5 Si 3 and β −⪢ α 2 + Ti 5 Si 3 . Additionally, extrapolations in the β + α 2 + α tie-triangle observed at 1100°C indicate that two possibilities arise to represent a peritectoid reaction involving α, β and α 2 phases: β + α 2 −⪢ α and β + α −⪢ α 2 , depending on the alloy composition and on the effect of temperature on the solubility of Si in α phase.

A thermodynamic analysis of the Mo-V and Mo-V-C system

Abstract: A new thermodynamic evaluation of the binary Mo-V system and the ternary Mo-V-C system using thermodynamic models for the Gibbs energy of individual phases is presented. The CALPHAD method has been used, with predictions of unknown thermodynamic quantities, to optimize a set of thermodynamic parameters taking related experimental information into consideration. The predictions are based on regularities in bonding properties and vibrational entropy of 3d-transition metal carbides. The results are summarized in tables of thermodynamic parameters, calculated binary phase diagrams and isothermal sections of the ternary phase diagram compared with experimental information. Finally the influence of ternary interaction parameters, especially in the fcc phase, on calculations of equilibria in multicomponent systems is discussed.

Thermodynamic description of BaO-SrO-TiO2 system

Abstract: The thermodynamic descriptions of the BaO-SrO-TiO2 pseudo-ternary and its lower order sub-systems have been made by critically evaluating all available experimental data. The temperature-dependent Gibbs energies of perovskite structural phases were evaluated to be relative to the standard element reference state (SER). The substitutional solution model for liquid and line compounds, Neumann-Kopp Rule for stoichiometric compounds were adopted. A set of self-consistent parameters reproducing most of the corresponding experimental data has been obtained.

Phenomenological calculation of phase equilibria in the FeNi system

Abstract: In order to seek inexpensive substituting elements for Pd in an FePd based L10 ordered phase, a phenomenological calculation based on the cluster variation method combined with a Lennard-Jones type pair-wise atomic interaction energies has been attempted for a FeNi binary system. The phase equilibria for FeNi3-disorder and FeNi-disorder are investigated. In particular, a tetragonal distortion is explicitly considered for the FeNi L10 ordered phase. The calculated FeNi3-disorder phase boundary shows good agreements with an experimental result. An FeNi L10 ordered phase which does not appear in a conventional FeNi binary phase diagram is predicted as a stable phase, implying that Ni is a potential substituting element for Pd in an FePd based L10 ordered phase.

Determination of Lennard-Jones type potential for FePd phase

Abstract: Lennard-Jones type pair-wise atomic interaction energies for FeFe, FePd and PdPd are derived from extrapolated thermodynamic and structural data of pure elements and an ordered compound. In order to estimate the values at zero Kelvin, extrapolation was performed with the aid of thermodynamic database. Also, additional measurement of ordering energy was attempted by Differential Scanning Calorimetry to determine the L1 0 -disorder transition temperature. The resultant pair interactions are combined with the Cluster Variation Method and the concentration dependency of heats of formation at 1565K was calculated. In addition, the phase boundary for L1 0 -disorder was obtained. Fairly good agreements with experimental results are achieved.

“ExTHERM”: the interactive support package of experimental thermodynamics

Abstract: The presented program package “ExTHERM” consists of three parts: (i) cETD , a simple interactive open data bank module covering all types of metal alloys (over 150 binary phases and some ternaries), (ii) cMod , a module for algebraic conversion among different representations of excess data, and (iii) cM3_(KMS/ EMF/HTC) , three interactive evaluation units suitable for overall best fits of experimental investigations expandable to mass spectrometric (KMS), EMF, and calorimetric investigations (HTC). The concept is based upon the multi-component TAPS ( T hermodynamically A dapted P ower S eries) concept of excess quantities covering all types of solutions, includingn liquid alloys showing short range ordering. The data bank is based upon the Hultgren data of binary systems and on new experimental data. Individual supplements may be performed easily. The conversion module employs a modular concept procedure which makes the conversions independent of the actual system data. For all software modules contact the author.

Contribution of the Rietveld method to non-stoichiometric phase modeling. Part I: generalities

Abstract: While being the most efficient way of analyzing a powder diffraction diagram for evidencing an additional phase, refining lattice parameters and performing quantitative phase analysis, the Rietveld method is also able to perform structure refinement including accurate determination of atomic positions and atom occupancies at the different sites. Owing to technological improvements of both powder diffractometers, and Rietveld analysis software, the method is now able to compete with single crystal analysis. It is actually the only alternative when single crystals do not exist or when no single crystal diffractometer is available. The Rietveld method makes possible the determination of vacancy ratio or relative occupancies when atoms of different kinds can occupy the same sites. It allows the measurement of the order parameter in partially ordered structures and the determination of the mechanism for accommodating the excess or deficient atoms in non-stoichiometric phases. This makes the Rietveld method the ideal tool to use in conjunction with the sublattice model especially if experimental site fractions can be taken into account in phase diagram assessment software. Powder diffraction associated with Rietveld analysis should therefore be widely used as an efficient experimental tool for thermodynamic optimization.

Calculation of the vibrational contribution to the Gibbs energy of formation for Al3Sc

Abstract: The ground-state energies were calculated for the stoichiometric compound Al 3 Sc in the L1 2 structure, and its constituents (fcc-Al and hcp-Sc) using the Vienna ab initio simulation package (VASP). The enthalpy of formation for Al 3 Sc at 0 K was obtained. On the basis of the calculated equation of state (EOS) at 0 K, the vibrational contribution to the Gibbs energy of formation at finite temperatures was estimated using the Mean-Field Potential (MFP) method, and compared with the phonon spectrum calculation and the existing CALPHAD assessments. The calculated entropy of formation can serve as a reasonable start value in a CALPHAD assessment although it is not as accurate as the enthalpy of formation calculated at 0 K. Both high-temperature x-ray scattering measurements and first-principles calculation show that Al 3 Sc is almost perfectly ordered up to its melting point and thus the configurational contribution can be neglected.

Systematic trends in structural and thermodynamic properties of ternary oxides in the systems Ln-Pd-O (Ln = lanthanide element)

Abstract: Isothermal sections of the phase diagrams for the systems Ln-Pd-O (Ln = lanthanide element) at 1223 K indicate the presence of two inter-oxide compounds Ln(4)PdO(7) and Ln(2)Pd(2)O(5) for Ln = La, Pr, Nd, Sm, three compounds Ln(4)PdO(7), Ln(2)PdO(4) and Ln(2)Pd(2)O(5) for Ln = Eu, Gd and only one compound of Ln(2)Pd(2)O(5) for Ln = Tb to Ho. The lattice parameters of the compounds Ln(4)PdO(7), Ln(2)PdO(4) and Ln(2)Pd(2)O(5) show systematic nonlinear variation with atomic number. The unit cell volumes decrease with increasing atomic number. The standard Gibbs energies, enthalpies and entropies of formation of the ternary oxides from their component binary oxides (Ln(2)O(3) and PdO) have been measured recently using an advanced version of the solid-state electrochemical cell. The Gibbs energies and enthalpies of formation become less negative with increasing atomic number of Ln. For all the three compounds, the variation in Gibbs energy and enthalpy of formation with atomic number is markedly non-linear. The decrease in stability with atomic number is most pronounced for Ln(2)Pd(2)O(5), followed by Ln(4)PdO(7) and Ln(2)PdO(4). This is probably related to the repulsion between Pd2+ ions on the opposite phases Of O-8 cubes in Ln(2)Pd(2)O(5), and the presence of Ln-filled O-8 cubes that share three faces with each other in Ln4PdO7. The values for entropy of formation of all the ternary oxides from their component binary oxides are relatively small. Although the entropies of formation show some scatter, the average value for Ln = La, Pr, Nd is more negative than the average value for the other lanthanide elements. From this difference, an average value for the structure transformation entropy of Ln(2)O(3) from C-type to A-type is estimated as 0.87 J.mol(-1).K-1. The standard Gibbs energies of formation of these ternary oxides from elements at 1223 K are presented as a function of lanthanide atomic number. By invoking the Neumann-Kopp rule for heat capacity, thermodynamic properties of the inter-oxide compounds at 298.15 K are estimated. (C) 2002 Elsevier Science Ltd. All rights reserved.

Application of statistical moment method to thermodynamic quantities of metals and alloys

Abstract: The thermodynamic quantities of metals and alloys are studied using the moment method in the quantum statistical mechanics, going beyond the harmonic approximation, and fully taking into account the anharmonicity of lattice vibrations. Including the power moments of the atomic displacements up to the fourth order, the free energy, specific heats C v and C p , mean square relative displacements and thermal lattice expansion coefficients are given explicitly in terms of the second and fourth order vibrational coupling constants. The effects of anharmonicity of lattice vibration on the long range order (LRO) parameter and order-disorder transitions of the binary alloys are investigated in detail. The changes in the root mean square atomic displacements upon the order-disorder phase transition are evaluated and compared with the available experimental results. We also discuss the melting transitions of metals and alloys within the framework of the present statistical moment method, and estimate the melting temperatures through the limiting temperature of the crystalline stability.