Multicomponent Phase Diagrams Research Articles

Calculation of the Liquidus and Solidus Surface of Al Rich Corner of Al-Mg-Si Alloy System by ESTPHAD Method

A simple but thermodinamically well-founded method is described for the recalculation of liquidus surfaces and partition coefficients of multicomponent equilibrium phase diagrams used a data file of the liquidus temperature and partition coefficients calculated by CALPHAD method. The applicability of the method is shown in case of Al-Mg-Si ternary alloy systems.

Geometrical approach to reaction schemes of multicomponent phase diagrams

A unique method for describing and establishing the reaction scheme of a multicomponent phase diagram is proposed in terms of graph theory. A geometrical representation of the connections between invariant reactions is considered as the basis to introduce the matrix formula of a graph that models a reaction scheme; then the number of all possible reaction schemes is enumerated. With a matrix operation, the most probable (full or partial) reaction scheme(s) can be selected within available experimental data. A ternary phase diagram was chosen as an example to show how to relate with experimental data. It is also shown that the number of invariant reactions in a ternary phase diagram can be expressed as a function of the numbers of binary and ternary phases.

Application of the cluster/site approximation to fcc phases in Ni–Al–Cr system

The cluster/site approximation (CSA) has been used to model the face-centered cubic (fcc) phases (disordered γ with A1 structure and ordered γ′ with L1 2 structure) in the Ni–Al–Cr ternary system. The CSA takes into account short-range order (SRO), which is essential to satisfactorily describe the thermodynamics of order/disorder transitions such as occur between the fcc phases in the Ni–Al–Cr system. It possesses computational advantages over the cluster variation method (CVM) while offering comparable accuracy in the calculation of multi-component phase diagrams. This makes the CSA a practical method for carrying out calculations on real alloy systems. These points are illustrated in the application of the CSA to fcc phases in the Ni–Al–Cr system. The CSA-calculated phase diagrams, which use fewer model parameters than used in previous descriptions, based on the point approximation, are in good accord with the experimental data. In addition, the fcc metastable phase diagrams calculated by using the CSA show reasonable ordering/disordering behavior and phase relationships.

A Quaternary Solution Model for White Micas Based on Natural Coexisting Phengite–Paragonite Pairs

A thermodynamic model for the quaternary white mica solid solution with end-members muscovite–Mg-celadonite–paragonite–Feceladonite (Ms–MgCel–Pg–FeCel) is presented. The interaction energies for the MgCel–Pg join, the FeCel–Pg join and the ternary interactions were obtained from natural coexisting phengite– paragonite pairs. Phengite–paragonite pairs were selected based on the criteria that their chemical compositions may be represented as a linear combination of the model end-member compositions and that the respective formation conditions (350–650 � C, 4–21kbar) are accurately known. Previously published excess free energy expressions were used for the Ms–Pg, Ms–MgCel and Ms–FeCel binaries. The suggested mixing model was tested by calculating multicomponent equilibrium phase diagrams. This proved to be particularly well suited to reproduce compositional variations of white micas from amphibolite-facies metapelites.

Critical assessment of computational thermodynamics in the alloy design of PM high speed steels

Abstract The experimental powder M42HVIG is used to illustrate how the adequate selection of the constituent phases involved in the calculation of multicomponent phase diagrams is a key factor for success in the computer-aided alloy design of highly sinterable high speed steel compositions.

Weldability Approach to Duplex Stainless Steels Using Multicomponent Phase Diagrams

Welding is a non-equilibrium process. However, some weldability issues, as the extension of the heat-affected zone (HAZ) can be addressed using equilibrium phase diagrams. The 70 wt% Fe-Cr-Ni pseudo-binary phase diagram is commonly used to establish the phase transformations during welding of duplex stainless steels. The predicted results are assumed to be reasonably good for most of the duplex stainless steels. Thermodynamic calculations were used to determine multicomponent phase diagrams and volumetric fraction of phases present as a function of temperature several commercial duplex stainless steels. Results showed that simplified pseudobinary phase diagram approach is valid to estimate welded joint microstructures only for the low alloy duplex stainless steels as UNS S32304, but phase transformations and mainly solidification paths of high alloy duplex stainless steels should predicted only using a multi-component phase diagram.

Prediction of macrosegregation during the solidification involving a peritectic transformation for multicomponent steels

Micro-macro segregation models are based on the multi-scale approach of the segregation, i.e., on the coupling of a macrosegregation model, which describes transport phenomena at the ingot scale, with a microsegregation model, which describes solidification at the scale of the primary or secondary dendrite arm spacing. The goal of this work is to illustrate the influence of the microsegregation phenomena on the macrosegregation. A microsegregation model for multicomponent steel alloys has been developed which takes into account solutal diffusion in solid and liquid phases, a correct description of the multicomponent phase diagram, and the variation of the local average solute concentration (open system). The peritectic transformation has been tackled and specific attention has been drawn on the accurate determination of the δ/γ and γ/liquid interface movements during solidification. Examples of micro-macro calculations obtained with this microsegregation model on ingot cases are presented and original results which illustrate specific effects of the peritectic transformation on macrosegregation are discussed.

Application of the cluster/site approximation to the calculation of multicomponent alloy phase diagrams

The cluster/site approximation (CSA) offers computational advantages, without loss of accuracy, over the cluster variation method (CVM) in the calculation of multicomponent phase diagrams. Its ease of use and advantages are illustrated in the calculation of a prototype Cu–Ag–Au fcc order/disorder phase diagram previously calculated using the CVM.

The Chemical Modeling of Electronic Materials and Interconnections

Thermodynamic and kinetic modeling, together with careful experimental work, is of great help for developing new electronic materials such as lead-free solders, their compatible metallizations and diffusion-barrier layers, as well as joining and bonding processes for advanced electronics manufacturing. When combined, these modeling techniques lead to a rationalization of the trial-and-error methods employed in the electronics industry, limiting experimentation and, thus, reducing significantly time-to-market of new products. This modeling provides useful information on the stabilities of phases (microstructures), driving forces for chemical reactions, and growth rates of reaction products occurring in interconnections or thin-film structures during processing, testing, and in longterm use of electronic devices. This is especially important when manufacturing advanced lead-free electronics where solder joint volumes are decreasing while the number of dissimilar reactive materials is increasing markedly. Therefore, a new concept of local nominal composition was introduced and applied together with the relevant ternary and multicomponent phase diagrams to some solder/conductor systems.

The PANDAT software package and its applications

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.

Computational and experimental investigation of microsegregation in an Al-rich Al–Cu–Mg–Si quaternary alloy

A new micromodel was developed to predict the microstructure and microsegregation in multicomponent alloys during dendritic solidification. The micromodel was directly coupled with multicomponent phase diagram calculations using a user friendly and robust phase diagram calculation engine—PanEngine. Solid back diffusion, undercooling and coarsening effects were included in this model, and the experimentally measured cooling curves were used as the inputs to carry out the microsegregation calculations. Microsegregation in Al–4.5 wt%Cu–1 wt%Si–0.5 wt%Mg alloy was experimentally investigated from directional solidification and electron probe microanalysis. Calculated results using this model are in accord with the experimental data, while results from the Scheil model deviate significantly from the experimental data.

Visualization of High-Dimensional Systems via Geometric Modeling with Homogeneous Coordinates

A framework is presented for the visualization of high-dimensional systems, particularly multicomponent phase diagrams. It is based on geometric modeling with homogeneous coordinates of the transformations among geometric varieties in high-dimensional space. By reduction of the dimensionality of the system through such transforms, the resulting linear cuts and projections, in their aggregate, provide a mental picture of the system in its entirety. A general way to perform cuts and projections is presented, including how to calculate the transforms and how to define the projective space. Specifically, a procedure is presented for the development of a set of canonical coordinates for the projective space with reduced dimensions. Also, it is shown that the seemingly different transformed coordinates for reactive systems in the literature can be unified under our framework. Examples are provided to highlight the procedure and to demonstrate the visualization of phase diagrams for nonreactive and reactive systems.

The superconducting phases ofUPt3

The heavy-fermion compound ${\mathrm{UPt}}_{3}$ is the first compelling example of a superconductor with an order parameter of unconventional symmetry. To this day, it is the only unambiguous case of multiple superconducting phases. Twenty years of experiment and theory on the superconductivity of ${\mathrm{UPt}}_{3}$ are reviewed, with the aim of accounting for the multicomponent phase diagram and identifying the superconducting phases. First, the state above the superconducting critical temperature at ${T}_{c}=0.5\mathrm{K}$ is briefly described: de Haas--van Alphen and other measurements demonstrate that this state is a Fermi liquid, with degeneracy fully achieved at ${T}_{c}.$ This implies that the usual BCS theory of superconductivity should hold, although the strong magnetic interactions suggest the possibility of an unconventional superconducting order parameter. The role of the weak antiferromagnetic order below ${T}_{N}=5\mathrm{K}$ in causing phase multiplicity is examined. A comprehensive analysis of which superconducting states are possible is given, and the theoretical basis for each of the main candidates is considered. The behavior of various properties at low temperature $(T\ensuremath{\ll}{T}_{c})$ is reviewed. The experiments clearly indicate the presence of nodes in the superconducting gap function of all three phases. In particular, the low-temperature low-field phase has a gap with a line node in the basal plane and point nodes along the hexagonal $c$ axis. The phase diagram in the magnetic-field--temperature plane has been determined in detail by ultrasound and thermodynamic measurements. Experiments under pressure indicate a coupling between antiferromagnetism and superconductivity and provide additional clues about the order parameter. Theoretically, Ginzburg-Landau theory is the tool that elucidates the phase diagram, while calculations of the temperature and field dependence of physical quantities have been used to compare different order parameters to experiment. On balance, the data point to a two-component order parameter belonging to either the ${E}_{1g}$ or the ${E}_{2u}$ representation, with degeneracy lifted by a coupling to the symmetry-breaking magnetic order. However, no single theoretical scenario is completely consistent with all the data. The coupling of superconductivity and magnetism may be the weakest link in the current picture of ${\mathrm{UPt}}_{3},$ and full understanding depends on the resolution of this issue.

Liquidus surface of FeO-Fe2O3-SiO2-CaO slag containing Al2O3, MgO, and Cu2O at intermediate oxygen partial pressures

The liquidus surface of the FeO-Fe2O3-SiO2-CaO system is of special importance in general pyrometallurgy. It determines several industrial slags, such as “fayalite” slag, calcium ferrite slag, acidic and basic steelmaking slags, as well as the newly proposed “ferrous calcium silicate slags” (FCS), which are all nominally based on this system. Nevertheless, the liquidus surface of this system has been studied only in equilibrium with metallic iron or in air, and the data at intermediate oxygen partial pressures are almost missing. In reductive smelting processes, the aforementioned slags were based on the liquidus surface of the FeO-Fe2O3-SiO2-CaO system in equilibrium with metallic iron. In oxidative matte smelting and in today’s modern trends of nonferrous smelting, such as continuous converting or white metal production, the characterization of the liquidus surface of this system at intermediate oxygen partial pressures becomes imperative. Furthermore, the effect of minor oxides on the liquidus surface of this system has not been studied at higher oxygen potentials and confusing conclusions are found in literature even for low oxygen potentials. In this work, a quantitative description of the liquidus surface of FeO-Fe2O3-SiO2-CaO slag containing Al2O3, MgO, and Cu2O is carried out by means of a thermodynamic model at various intermediate oxygen partial pressures. Through a new type of easy-to-understand multicomponent phase diagram, it is shown that important differences exist between the liquidus surfaces of this system in reductive and oxidative conditions. It is also shown that minor components can have fundamentally different effects in reductive and oxidative conditions.

Simultaneous calculation of mechanical properties and phase equilibria

Many formulas have been developed to describe the mechanical properties of various microstructures, but there have been relatively few attempts to combine such formulations directly with multicomponent phase diagram calculations obtained by the CALPHAD technique. This paper describes some of the techniques used in developing such a combined approach as part of a more comprehensive materials property program (MPP). The basic requirement is to describe the mechanical properties of the overall alloy by determining the effect of each solute, and the grain size, on the mechanical properties of the various phases in a multicomponent system. As some of the coefficients are not readily available, methods have to be evolved that echo problems already faced by the CALPHAD community with respect to characterizing the thermodynamics of metastable phases. It will be shown how the inter-relation of various mechanical properties, such as yield strength and hardness, can also be used to produce self-consistent data. The promising results obtained for dual-phase materials can be considered a firm foundation for future work on more complex systems.

Calculated phase diagrams of aluminum alloys from binary Al–Cu to multicomponent commercial alloys

More than three decades have passed since the publication of Alan Prince’s book on multicomponent phase equilibria. The most significant development in this time has been the use of a combined computational/experimental approach to calculate multicomponent phase diagrams. This has led to important advances in the design and processing of structural and functional materials for practical applications. In this paper, we present a few examples focusing on aluminum alloys from the classical Al–Cu binary to multicomponent alloys with a view toward practical applications.

Use of multicomponent phase diagrams for predicting phase evolution in solder/conductor systems

Although the complete phase equilibrium is never reached in interconnection applications, the assumption of local equilibrium at the interfaces is generally valid in most systems composed of dissimilar materials. Therefore, the tie lines in ternary (or multicomponent) phase diagrams—together with the relevant stability diagrams and the mass balance requirements—can be used for predicting the phase sequences (i.e. diffusion paths) formed, for example, in solder/conductor joints. Generally, binary phase diagrams alone cannot provide sufficient information on the phase formation in solder/conductor systems because they do not bear any information on the relative stabilities between different binary phases in multicomponent systems. As examples, the formation of intermetallic compounds in several solder/conductor systems with Au- or Cu-metallization was studied with the help of ternary phase diagrams as well as experimentally. The phase diagrams were calculated using thermodynamic methods. The experimental results confirmed that the dependence of formation of intermetallic compounds on temperature and solder composition is clearly represented by the phase diagrams supplemented with the stability diagrams.

A Tool for Modelling of Microsegregation: An Approximation Method for Partition Coefficients in Experimentally Determined Multicomponent Phase Diagrams

A Tool for Modelling of Microsegregation: An Approximation Method for Partition Coefficients in Experimentally Determined Multicomponent Phase Diagrams

Alpha/Gamma equilibria in the Fe-Co phase diagram

It is possible to treat the equilibrium between α and γ phases and obtain a good representation of the phase boundaries in the Fe-Co system using the cluster variation method. In this study, calculations were performed on quaternary Fe+Fe-Co+Co- by considering both chemical and magnetic interactions. The A2/B2 order-disorder transition and the α/γ phase equilibria agree well with the experimental data. These results were obtained with very few adjustable parameters; however, the study is a first step toward calculating multicomponent phase diagrams based on the Fe-Co system.

Supersolidus liquid phase sintering of high speed steels: Part 3: computer aided design of sinterable alloys

Calculated multicomponent phase diagrams were used to identify high speed steel (HSS) type alloys having the potential to exhibit enhanced sinter ability. The requirement was for an extensive austenite + carbide + liquid phase field. Of the six tungsten and molybdenum based systems studied, Fe–14Mo–C + 4Cr–8Co systems were potentially the most promising. Appropriate compositions were water atomised and additional alloys prepared by blending annealed powders with graphite powders. Powders were compacted to green densities of about 70% theoretical and then vacuum sintered. Sinterability was assessed in terms of sintered densities and microstructures. Alloys containing Fe–13Mo–1·3C, Fe–14Mo–4Cr–1·3C, and Fe–14Mo– 8Co–4Cr–1·4C were sintered to full density at temperatures as low as 1170°C, 70–150 K lower than for existing HSSs. Sintering windows were 20– 30 K, a significant improvement on existing HSSs. As sintered microstructures consisted of angular M6 C carbides dispersed in martensitic matrixes, which is typical for correctly sintered HSS. Heat treatment response and cutting performance for the sinterable grades were assessed and found to be comparable to existing HSS. The cutting performance of Fe–14Mo– 8Co–4Cr–1·4C tools at 45 and 52·5 m min-1 was superior to both cast wrought M2 and T1 tools of identical geometry. Lower carbon contents resulted in an increase in sintering temperature and a reduction in the width of the sintering window. Higher carbon contents destroyed sinterability, since they led to the formation of M2 C eutectic structures in the undersintered condition. Alloy sinterability was correlated to differential thermal analysis data obtained during heating of powders. The variations in sinterability with alloy composition are discussed with reference to phase diagrams; the degradation in sinterability observed at carbon contents above 1·4% is attributed to the presence of ternary eutectic phase fields. The commercial implications of the relationship between sinterability and alloy composition are discussed.