Published in JOM 49 (12) (1997) 14-19

Thermodynamic Modeling of Multicomponent Phase Equilibria

Ursula R. Kattner
Metallurgy Division
National Institute of Standards and Technology
Gaithersburg, MD 20899

Author's Note: Commercial products are identified as examples. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that they are necessarily the best available for the purpose.


The present paper gives an overview of the CALPHAD method and recent progress made. A brief history is given then the scope of phase diagram calculations is described. Thermodynamic descriptions most commonly used in the Calphad method are described and the methods used to obtain the numerical values for these descriptions are outlined. Extrapolation to higher-component systems is explained and recent progress in the quality of assessments is demonstrated. A brief overview of available computer software tools and databases is given. Finally, several applications of phase diagrams calculations are demonstrated.


Phase diagrams are visual representations of the state of a material as a function of temperature, pressure and concentrations of the constituent components and are, therefore, frequently hailed as basic blueprints or roadmaps for alloy design, development, processing and basic understanding. The importance of phase diagrams is also reflected by the publication of such handbooks as "Binary Alloy Phase Diagrams"1, "Phase Equilibria, Crystallographic and Thermodynamic Data of Binary Alloys"2, "Phase Equilibrium Diagrams"3 which continues "Phase Diagrams for Ceramists"4, "Handbook of Ternary Alloy Phase Diagrams"5 and "Ternary Alloys"6.

The state of a two-component material at constant pressure can be presented in the well known graphical form of binary phase diagrams. For three-component materials an additional dimension is necessary for a complete representation. Therefore, ternary systems are usually presented by a series of sections or projections. Due to    their multidimensionality the interpretation of the diagrams of more complex systems can be quite cumbersome for an occasional user of these diagrams. For systems with more than 3 components, the graphical representation of the phase diagram in a useful form becomes not only a challenging task but is also hindered by the lack of sufficient experimental information. However, the difficulty of graphically representing systems with many components is irrelevant for the calculation of phase diagrams. Such calculations can be customized for the materials problem of interest.


While it is only modern developments in modeling and computational technology that have made computer calculations of multicomponent phase equilibria a realistic possibility today, the correlation between thermodynamics and phase equilibria was established more than a century ago by J.W. Gibbs. Hertz7 has summarized the ground breaking work of Gibbs. Although the mathematical foundation had been laid, more than 30 years passed before J.J. van Laar8 published his mathematical synthesis of hypothetical binary systems. To describe the solution phases van Laar used concentration dependent terms which Hildebrand9 called regular solutions. More than 40 years have had passed when J.L. Meijering published his calculations of miscibility gaps in ternary10 and quaternary solutions11. Shortly afterwards Meijering applied this method to the thermodynamic analysis of the Cr-Cu-Ni system12. Simultaneously Kaufman and Cohen13 applied thermodynamic calculations in the analysis of the martensitic transformation in the Fe-Ni system. Kaufman continued his work on the calculation of phase diagrams, including the pressure dependence. In 1970 Kaufman and Bernstein14 summarized the general features of the calculation of phase diagrams and also gave listings of computer programs for the calculation of binary and ternary phase diagrams, thus laying the foundation for the CALPHAD method (CALculation of PHAse Diagrams). In 1973 Kaufman organized the first project meeting of the international CALPHAD group. Since then the CALPHAD group grown consistently larger.

Another important paper on the calculation of phase equilibria was also published in the fifties. In his paper Kikuchi15 described a method to treat order/disorder phenomena. This method later became known as the "Cluster Variation Method" (CVM) and is extensively used in conjunction with first principles calculations. Although these calculations are computationally very intensive, enormous progress in algorithms and computer speed has been made in recent years. The predicted phase diagrams are generally topologically correct but they currently still lack sufficient accuracy for practical applications. de Fontaine16 gives an extensive review of these calculations.

Scope of Phase Diagram Calculations

In order to overcome the problem of the multidimensionality posed by a system with many components, alternate methods are frequently used to represent the necessary phase diagram information. With stainless steel alloys, for example, the complexity is frequently reduced by expressing the compositions of the ferrite-stabilizing elements as "Cr equivalents" and the austenite-stabilizing elements as "Ni equivalents"17. The sums of the Cr and Ni equivalents are used to predict the phases expected in the final alloy. It should be noted that approximations like these are limited to the composition regime for which they were derived. Another example is the PHACOMP method18 used to predict detrimental TCP (topological close packed) phases in superalloys. This method is based on the theory that each element has a specific electron hole number and that the average electron hole number is correlated to the TCP phases in an alloy. Although this method works very well for Ni-based superalloys, special corrections are required with other superalloys, and it may not be easily applied to other alloy families. The CALPHAD method, on the other hand, is based on the minimization of the free energy of the system and is, thus, not only completely general and extensible, but also theoretically meaningful.

The experimental determination of phase diagrams is a time consuming and costly task. This becomes even more pronounced as the number of components increases. The calculation of phase diagrams reduce the effort required to determine equilibrium conditions in a multicomponent system. A preliminary phase diagram can be obtained from extrapolation of the thermodynamic functions of constituent subsystems. This preliminary diagram can be used to identify composition and temperature regimes where maximum information can be obtained with minimum experimental effort. This information can then be used to refine the original thermodynamic functions.

Numerical phase diagram information is also frequently needed in other modeling efforts. Even though phase diagrams represent thermodynamic equilibrium, it is well established that the phase equilibria can be applied locally (local equilibrium) to describe the interfaces between phases. In such cases only the concentrations at this interface are assumed to obey the requirements of thermodynamic equilibrium. Thermodynamic modeling of phase diagrams and kinetic modeling have been successfully coupled for a variety of processes, such as carburizing/nitriding19,20, diffusion couples21-23, dissolution of precipitates24,25 and solidification26,27. Phase equilibrium calculations can not only give the phases present and their compositions but also provide numerical values of enthalpy contents, temperature and concentration dependence of phase boundaries for coupling of microscopic and macroscopic modeling. Banerjee et al.28 give an example of such a coupling of phase equilibria calculations and solidification micromodels in a macroscopic heat and fluid flow analysis of a casting.

In recent years the expression "computational thermodynamics" is frequently used in place of "calculation of phase diagrams". This reflects the fact that the phase diagram is only a portion of the information that can be obtained from these calculations.

Thermodynamic Descriptions and Models

For the calculation of phase equilibria in a multicomponent system, it is necessary to minimize the total Gibbs energy, G, of all the phases that take part in this equilibrium:
where ni is the number of moles and Gi the Gibbs energy of phase i.

A thermodynamic description of a system requires assignment of thermodynamic functions for each phase. The CALPHAD method employs a variety of models to describe the temperature, pressure and concentration dependencies of the free energy functions of the various phases. The contributions to the Gibbs energy of a phase can be written:

where GT(T,x) is the contribution to the Gibbs energy by the temperature, T, and the composition, x, Gp(p,T,x) is the contribution of the pressure, p, and Gm(TC,0,T,x) is the magnetic contribution of the Curie or Néel  temperature, TC, and the average magnetic moment per atom, ß0.

The temperature dependence of the concentration terms of GT is usually expressed as a power series of T:

where a, b, c and dn are coefficients and n are integers. To represent the pure elements, the n are typically 2, 3, -1 and 7 or -929. This function is valid for temperatures above the Debye temperature. In each of the equations in the following models describing the concentration dependence, the G coefficients on the right hand side can have such a temperature dependence. Frequently only the first two terms are used for the representation of the excess Gibbs energy. Dinsdale29 also gives expressions for the effects of pressure and magnetism on the Gibbs energy. However, pressure dependence for condensed systems at normal pressures is usually ignored.

For multicomponent systems it has proven useful to distinguish three contributions from the concentration dependence to the Gibbs energy of a phase, G:

The first term, G0, corresponds to the Gibbs energy of a mechanical mixture of the constituents of the phase, the second term, Gideal, corresponds to the entropy of mixing for an ideal solution and the third term, Gxs, is the so-called excess term. Since Hildebrand9 introduced the term "regular solution" to describe interactions of different elements in a random solution, a series of models have been proposed for phases which deviate from this "regularity", i.e. show a strong compositional variation in their thermodynamic properties, to describe the excess Gibbs energy, Gxs. For example, an ionic liquid model30 or associate model31, among others, have been proposed for liquid phases. For ordered solid phases, Wagner and Schottky32 introduced the concept of defects on the crystal lattice in order to describe deviations from stoichiometry. A description of order/disorder transformations proposed by Bragg and Williams33. Since then many other models have been proposed. Today the most commonly used models (listed in the order of increasing complexity) are those for stoichiometric phases, regular solution type models for disordered phases, and sublattice models for ordered phases having a range of solubility or exhibiting an order/disorder transformation. The following examples give descriptions of models for binary phases and can easily be expanded for ternary and higher order phases.

The Gibbs energy of a binary stoichiometric phase is given by:

where xA0 and xB0 are mole fractions of element A and B and are given by the stoichiometry of the compound, GA0 and GB0 are the respective reference states of element A and B, and Gf is the Gibbs energy of formation. The first two terms correspond to G0 and the third term to Gxs in Eq. (4). Gideal of Eq. (4) is zero for a stoichiometric phase since there is no random mixing.

Binary solution phases, such as liquid and disordered solid solutions, are described as random mixtures of the elements by a regular solution type model:

where xA and xB are the mole fractions and GA0 and GB0 the reference states of elements A and B, respectively. The first two terms correspond to G0 and the third term, from random mixing, to Gideal in Eq. (4). The Gi of the fourth term are coefficients of the excess Gibbs energy term, Gxs, in Eq. (4). The sum of the terms (xA - xB)i is the so-called Redlich-Kister polynomial34 which is the most commonly used polynomial in regular solution type descriptions. Although other polynomials have been used in the past, in most cases they can be converted to Redlich-Kister polynomials35.

The most complex and general model is the sublattice model frequently used to describe ordered binary solution phases. The basic premise for this model is that a sublattice is assigned for each distinct site in the crystal structure. For example, the CsCl (B2) structure consists of two sublattices, one of which is occupied predominantly by Cs atoms and the other by Cl atoms. An ordered binary solution phase with two sublattices that exhibits substitutional deviation from stoichiometry can be described by the expression:

where yA1, yB1, yA2 and yB2 are the species concentrations of element A and B on sublattices 1 and 2 with a1 yA1 + a2 yA2 = xA, a1 yB1 + a2 yB2 = xB and yA1 + yB1 = 1, yA2 + yB2 = 1. a1 and a2 are the site fractions of the sublattices 1 and 2 and are given by the number of sites in the unit cell. The first two terms correspond to G0 and the third term corresponds to Gideal in Eq. (4). The remaining terms are the excess Gibbs energy term, Gxs, in Eq. (4). The coefficients GAA0 , GAB0 , GBA0  and GBB0 can be visualized as the Gibbs energies of the end-member phases. The end-member phases are formed when each sublattice is occupied only by one kind of species and can be either real (Aa1Ba2: A atoms on sublattice 1 and B atoms on sublattice 2) or hypothetical (Aa1Aa2, Ba1Aa2 and Ba1Ba2). The remaining terms of Gxs describe interactions between the atoms on one sublattice similar to regular solution type models for disordered solution phases. This model description was first introduced by Sundman and Ågren36 and later refined by Andersson et al.37. For the treatment of order/disorder transformations with this model, the coefficients in Gxs are not independent of each other. For example, Ansara et al.38 derived dependencies for the order/disorder transformation of fcc/L12. This model was later modified by Ansara et al.39 to allow independent evaluation of the thermodynamic properties of the disordered phase. Chen et al.40 have proposed another model for the treatment of ordered phases.

It should be noted that Eqs. (5) and (6) are in fact special cases of Eq. (7). Equation (7) reduces to Eq. (6) if only one sublattice is considered or Eq. (5) if only one species is considered on each of the two sublattices. The generality of the sublattice description allows formulation of a general description for multi-component phases that can easily be computerized. Lukas et al.35 give an example of such a description.

From the condition that the Gibbs energy at thermodynamic equilibrium reveals a minimum for given temperature, pressure and composition, J.W. Gibbs derived the well known equilibrium conditions that the chemical potential, µn, of each component, n, is the same in all phases, :

The chemical potentials are related to the Gibbs energy by the well known equation:
Eq. (8) results in n non-linear equations which can be used in numerical calculations. All of the CALPHAD-type software tools use methods like the two-step method of Hillert41 or the one step method Lukas et al.35 to minimize the Gibbs energy. The equations obtained from these methods are usually non-linear and are solved numerically using a Newton-Raphson technique.

Determination of the Coefficients

The coefficients of the Gibbs energy functions are determined from experimental data for each system. In order to obtain an optimized set of coefficients, it is desirable to take into account all types of experimental data, e.g. phase diagram, chemical potential and enthalpy data. The coefficients can be determined from the experimental data by a trial and error method or mathematical methods. The trial and error method is only feasible if few different data types are available. This method becomes increasingly cumbersome as the number of components and/or number of data types increases. In this case mathematical methods, such as the least squares method of Gauss42, the Marquardt method43 or Bayesian estimation method44, are more efficient. The determination of the coefficients is frequently called "assessment" or "optimization" of a system.

Higher-Component Systems

A higher-component system can be calculated from thermodynamic extrapolation of the thermodynamic excess quantities of the constituent subsystems. Several methods exist to determine the weighting terms used in such an extrapolation formula. Hillert45 analyzed various extrapolation methods and recommended the use of Muggianu's method46 since it can easily be generalized. The Gibbs energy of a ternary solution phase determined by extrapolation of the binary energies using Muggianu's method is given by:

where the parameters, Gijk, have the same values as in Eq. (6) for each of the binary systems. If necessary, a ternary term, xA xB xC GABC(T,x), can be added to describe the contribution of three element interactions to the Gibbs energy.

The usual strategy for assessment of a multicomponent systems is shown in Fig. 1. First, the thermodynamic descriptions of the constituent binary systems are derived. Thermodynamic extrapolation methods are then used to extend the thermodynamic functions of the binaries into ternary and higher order systems. The results of such extrapolations can then be used to design critical experiments. The results of the experiments are then compared to the extrapolation and, if necessary, interaction functions are added to the thermodynamic description of the higher order system. As mentioned previously, the coefficients of the interaction functions are optimized on the basis of these data. In principle, this strategy is followed until all 2, 3, ... n constituent systems of a n-component system have been assessed. However, experience has shown that, in most cases, no, or very minor, corrections are necessary for reasonable prediction of quaternary or higher component systems. Since true quaternary phases are rare in metallic systems, assessment of most of the ternary constituent systems is often sufficient to describe a n-component system.
Figure 1: CALPHAD methodology. The assessed excess Gibbs energies of the constituent subsystems are for extrapolation to higher component system.

Improved Capabilities

One goal of the CALPHAD group is to generate descriptions of binary, ternary and quaternary systems that can be used for the construction of thermodynamic databases. Thermodynamic databases of multicomponent system require consistency of the model descriptions and the parameters used. With the constant improvement of computational technology, the use of more realistic models, such as the sublattice model description, becomes feasible. This allows more accurate descriptions of complex systems and makes it desirable to reassess systems which have been previously assessed.

The progress that has been made with these reassessments is shown in Fig. 2 for the Al-Ni system, a basic system for superalloys. In the first assessment of Kaufman and Nesor47, shown in Fig. 2(a), the phases were either described as disordered solution phases (liquid, (Al), (Ni) and AlNi) or stoichiometric compounds (Al3Ni, Al3Ni2 and AlNi3). The (Al) and (Ni) phases were described as one phase since they both have the fcc structure. Although the general topology of the experimentally determined phase diagram48 (Fig. 2(d)) is reproduced, major differences occur for the equilibria involving the Al3Ni2 and AlNi phases. These differences are at least partially a result of ignoring the homogeneity range of the Al3Ni2 phase and not considering the fact that AlNi is an ordered phase with CsCl structure.
  a   b
  c   d
Figure 2: Different assessments of the Al-Ni system showing the progress made with the CALPHAD method. (a) 1978 assessment by Kaufman and Nesor [78Kau], (b) 1988 assessment by Ansara et al. [88Ans], (c) 1997 assessment by Ansara et al. [97Ans] and (d) the evaluated experimental diagram [93Oka].
In the second assessment by Ansara et al.38, shown in Fig. 2(b), the sublattice model description was introduced for the ordered phases with noticeable homogeneity ranges (Al3Ni2, AlNi and AlNi3). The disordered fcc phase ((Al) and (Ni)) and the ordered L12 phase (AlNi3) were described with a single free energy function as one phase which undergoes an order/disorder transformation. While the phase diagram calculated from these improved analytical descriptions shows better agreement with the observed diagram, some noticeable disagreement still remains. The range of the (Al) solid solution is overestimated and the region of single phase AlNi3 slants to the Ni-rich side at lower temperatures. Both problems likely result from describing all these phases with the single function. It should be also noted that the region of single phase Al3Ni2 is overestimated at higher temperatures and underestimated at lower temperatures. This may be caused by the substitutional sublattice model description used in this assessment. It has been experimentally observed that on the Ni-rich side of the nominal stoichiometry Ni atoms fill structural vacancies and on the Al-rich side Ni atoms are substituted by Al. This has been considered in the most recent assessment by Ansara et al.39 which is shown in Fig. 2(c). Ansara et al. also modified the model for the description of the order/disorder transformation mentioned above. This assessment also includes a description of the Al3Ni5 phase as a stoichiometric compound, though its homogeneity range has been ignored. The phase diagram obtained from this assessment is in very good agreement with the observed diagram, Fig. 2(d). It should be noted that the calculated phase diagram not only reproduces the experimentally observed phase diagram but also provides the thermodynamic functions for extrapolation into higher order systems or for use in modeling of, for example, casting solidification.

A disadvantage of this iterative process with improved descriptions is that the descriptions used in previous assessments may be incompatible with newer assessments based on recently developed model descriptions. Despite this, significant progress has been made in recent years and an increasing number of databases have become available for multicomponent systems.

Computer Software Tools and Databases

A variety of software packages can be used for the calculation of phase diagrams making it is impossible to list all of them. Frequently used software packages are ChemSage49, the so-called Lukas programs35,42, MTDATA50 and Thermo-Calc51. Although, all of these software packages can be used for the calculation of phase equilibria, their features and user interfaces differ. Most of the model descriptions used for alloy and ceramic systems are common to all these programs. However, not every package has other specific model descriptions, for example, models for aqueous or polymer solutions. Another important feature of these software packages is the availability of a module for the optimization of the Gibbs energy functions. Such optimizing modules are available with ChemSage52, the Lukas programs42 and Thermo-Calc53.

The development of increasingly user friendly computer interfaces, very often in conjunction with programs for special tasks, such as the ETTAN PC-Windows interface54 for Thermo-Calc or the SCHEIL and LEVER programs55, makes phase diagram information more accessible for the non-expert user. For these applications the user needs only to supply a bulk composition and temperature limits for the calculation and the programs generate the remaining conditions that are needed for the calculation.

For the incorporation of phase equilibria calculations into micromodeling, such as the modeling of diffusion processes, an interface must be created in which the important variables are transferred from one computer code segment to another. For the simulation of diffusional reactions, the software package Thermo-Calc51 has been interfaced with the package DICTRA56. A general interface (TQ interface) is available for Thermo-Calc and ChemSage57. Banerjee et al.28 used another, fairly simple interface for solidification micromodeling.

Several thermodynamic databases have been constructed from the assessments of binary, ternary and quaternary systems. For the description of commercial alloys is quite likely that at least a dozen elements need to be considered. The number of constituent subsystems of a n-component system is determined by the binomial coefficient (nk), where k is the number of components in the subsystem. A 12-component systems consists accordingly of 66 binary, 220 ternary and 495 quaternary subsystems. These numbers suggest that is impossible to obtain descriptions of all the subsystems in reasonable time. However, as mentioned previously, only rarely are quaternary excess parameters needed. If the database is for base element X, it is sufficient to consider only the X-based ternary systems, considerably reducing the number of needed assessments. Also, if more than one element occurs only in fairly small quantities in the alloy family of interest then assessments for binary systems containing only these elements or ternary systems with two or three of these elements are generally not very important for obtaining correct predictions. Based on this, databases have been developed for various commercial alloy systems58,59. However, because software packages assume different computer file formats for the databases, care must be taken to insure compatibility between database and program package.

A review of fully integrated thermochemical database systems which were available in 1990 is given by Bale and Eriksson60. Since then, their review has been complemented by a web site61.

Application Examples

In recent years, the application of phase diagram information obtained from calculations to practical processes has increased significantly. Extensive collections of examples can be found in books: "User Applications of Alloy Phase Diagrams"62, "User Aspects of Phase Diagrams"63 and "The SGTE Casebook, Thermodynamics at work"64. In the following, a few examples will be given for solidification processes.

As has been already mentioned, extrapolation to higher-component systems is one of the staples of the CALPHAD since it provides information where otherwise only educated guesses could be used. When alloys of the Sn-Ag-Bi system were considered as candidate alloys for lead-free solders no phase diagram information for the liquid phase could be found. Kattner and Boettinger65 extrapolated the descriptions of the binary systems to calculate the solidus and liquidus surfaces of the Sn-rich corner, Fig. 3. It can be seen from Fig. 3 that the Ag-rich side of the eutectic troughs should be avoided because the liquidus temperature increases significantly with increasing Ag-concentration. Fig. 3(a) and Fig. 3(b) can be used to identify composition regimes where the freezing range is suitable for solder applications.
  a    b
Figure 3: Sn-rich corner of the Sn-Bi-Ag system with isotherms. (a) Liquidus surface, the dashed lines are the boundaries of the three phase equilibria at the eutectic temperature. (b) Solidus surface.
Two simple models describe the limiting cases of solidification behavior. First, for solidification obeying the lever rule at each temperature during cooling, complete diffusion is assumed in the solid as well as in the liquid. Thus, all phases are assumed to be in thermodynamic equilibrium at all temperatures during solidification. In comparison, solidification following the Scheil path, where diffusion in the solid is forbidden and thermodynamic equilibrium exists only as local equilibrium at the liquid/solid interface, produces worst case microsegregation with the lowest final freezing temperature. Modeling of real solidification behavior requires a kinetic analysis of microsegregation and back diffusion. However, for most alloys, the predictions of the Scheil model are close to reality.
  a   b
Figure 4: Temperature vs. calculated fraction solid curves for six alloys with the composition Sn - 3.5 wt.% Ag - x wt.% Bi. (a) Lever rule calculations. (b) Scheil calculations.
Scheil and lever rule calculations were carried out for 6 alloy compositions in the solder alloy Ag-Bi-Sn system. The results are shown in Fig. 4. It can be seen that the formation of eutectic due to segregation in the Scheil solidification increases the freezing range drastically. A comparison of Fig. 4(a) and Fig. 4(b) shows that, as the equilibrium freezing range is increased (by adding Bi), the Scheil solidification curve begins to deviate from that of the lever rule solidification at smaller values of solid fraction formed. This is an indication that non-equilibrium solidification has a smaller impact on the actual freezing range of an alloy with a small equilibrium freezing range than an alloy with a large equilibrium freezing range. Since it is a practical requirement that solders have a limited freezing range, this is important information for the design of new solder alloys.
  a   b
Figure 5: Phase fraction vs. temperature curves for solidification of an alloy with the composition 0.21 wt.% Si, 0.23 wt.% Fe, 4.44 wt.% Cu, 0.55 wt.% Mn 1.56 wt.% Mg, 0.05 wt.% Zn and the remainder Al. (a) Lever rule calculation. (b) Scheil calculation.
A phase fraction diagram vs. temperature diagram is a very useful form of graphically presenting multi-component alloys. Such diagrams are shown in Fig. 5 for lever rule and Scheil calculations of an alloy which is close in composition to the commercial Al-alloy 243. The composition used for this calculation is 0.21 wt.% Si, 0.23 wt.% Fe, 4.44 wt.% Cu, 0.55 wt.% Mn 1.56 wt.% Mg, 0.05 wt.% Zn with the remainder Al. For the calculations the Al-DATA database59,66 was used. The difference between lever rule and Scheil solidification becomes most noticeable toward the end of solidification. For both solidification paths the solidification begins
with the precipitation of (Al) and Al6Mn. The lever solidification, shown in Fig. 5(a), continues with the formation of the -phase, Al20Cu2Mn3, Al7Cu2Fe, decomposition of Al6Mn and the -phase and the formation of Mg2Si. Finally, after solidification is complete, precipitation of A2CuMg begins. The final microstructure consists mainly of (Al) and small amounts of Al20Cu2Mn3, Al7Cu2Fe, Mg2Si and Al2CuMg. The Scheil solidification, shown in Fig. 5(b), continues with the formation of Al7Cu2Fe, Mg2Si, Al20Cu2Mn3 (this phase is not shown in Fig. 5(b) since the amount is extremely small), Al2CuMg and Al2Cu. The microstructures obtained from these solidification paths are quite different which might result in different mechanical properties. The actual microstructure found in castings can be compared to diagrams like those shown in Fig. 5 and casting parameters can be adjusted to obtain the desired microstructure by following a solidification path somewhere between these extremes.
Figure 6: Fraction solid vs. local temperature curves for the two nodes from the casting simulation compared to the curves obtained from Scheil and lever rule solidification calculations.
Major progress in the application of phase diagram information has been made in the implementation of such calculations in casting simulation software. Thermodynamic calculation of phase equilibria of a multicomponent alloy was interfaced with a micromodel for computing the change of fraction solid and temperature, given a specified change in enthalpy during the liquid-solid transformation. This coupling was incorporated into a finite element package developed for modeling solidification of castings. The simulation was carried out for a step wedge part and a Ni - 15 at.% Al - 2 at.% Ta alloy. Further details of the simulation are described by Banerjee et al.28. Two nodes (points on the finite element mesh) were selected to demonstrate the effect of the different cooling histories. One node (node 63) cooled approximately twice as fast as the other one (node 1). The fraction solid vs. local temperature curves were calculated for these two nodes during runtime and are compared to those of the limiting Scheil and lever rule curves in Fig. 6 with the curve of the slower cooling node revealing a slightly less pronounced Scheil behavior, and thus less segregation, than the faster cooling node. These differences are expected to appear in the final casting and to be reflected in changing microstructure and properties varying throughout the casting.


Enormous progress has been made in the calculation of phase diagrams during the past 30 years. This progress will continue as model descriptions are improved and computational technology advances. The progress of the recent years can be summarized:

The model descriptions used in the CALPHAD method are constantly improved, allowing assessments which reproduce even complex diagrams well.

A large number of systems have been assessed, allowing the construction of databases for calculating phase diagrams of complex commercial alloys.

The user interfaces of the computer programs are becoming more user friendly, allowing the non-expert user easy access to phase diagram information.

The calculation of phase diagrams has been successfully coupled with the modeling of kinetic processes.


The different versions of the Al-Ni phases diagram in Fig. 2 were calculated with the Thermo-Calc package. The remaining calculations were carried out with the original Lukas programs (Fig. 3) or with programs that were using modified code (Figs. 4-6). The author thanks H.L. Lukas, Max-Planck-Institut of Metallforschung (Stuttgart, Germany) for providing his computer programs and N. Saunders, ThermoTech Ltd. (Surrey, United Kingdom) for providing the Al-DATA database used for the calculation shown in Fig. 5.


1.   "Binary Alloy Phase Diagrams", 2nd Ed., Vol. 1-3, Ed. in Chief: T.B. Massalski, ASM International, Materials Park, OH, 1990

2.    B. Predel, "Phase Equilibria, Crystallographic and Thermodynamic Data of Binary Alloys", Vol. 5, Subvol. a-g, Ed. in Chief O. Madelung, Landolt-Börnstein, New Series, Springer, Berlin, Germany, 1991-1997; onward

3.    "Phase Equilibria Diagrams", Vol. IX-XII, Compiled in the Ceramics Division of NIST, The American Ceramic Society, Westerville, OH, 1992-1996; onward

4.    "Phase Diagrams for Ceramists", Vol. I-VIII, Compiled in the Ceramics Division of NIST, The American Ceramic Society, Westerville, OH, 1964-1990

5.    P. Villar, A. Prince and H. Okamoto, "Handbook of Ternary Alloy Phase Diagrams", Vol. 1-10, ASM International, Materials Park, OH, 1995

6.    "Ternary Alloys: A comprehensive Compendium of Evaluated Constitutional Data & Phase Diagrams", Vol. 1-15, Eds.: G. Petzow and G. Effenberg, VCH Verlagsgesellscahft, Weinheim, Germany, 1988-1995; onward

7.    J. Hertz, J. Phase Equilibria, 13 (1992) 450-458

8.    J.J. van Laar, Z. phys. Chem., 63 (1908) 216-253, 64 (1908) 257-297

9.    J.H. Hildebrand, J. Amer. Chem. Soc., 51 (1929) 66-80

10.    J.L. Meijering, Philips Res. Rep., 5 (1950) 333-356, 6 (1951) 183-210

11.    J.L. Meijering and H.K. Hardy, Acta Metall., 4 (1956) 249-256

12.    J.L. Meijering, Acta Metall., 5 (1957) 257-264

13.    L. Kaufman and M. Cohen, J. Metals, 8 (1956) 1393-1401 (Trans. AIME, 206)

14.    L. Kaufman and H. Bernstein, "Computer Calculation of Phase Diagrams with Special Reference to Refractory Metals", Academic Press, New York, NY, 1970

15.    R. Kikuchi, Phys. Rev., 81 (1951) 988-1003

16.    D. de Fontaine, Solid State Physics, 47 (1994) 33-176

17.    E.A. Schoefer, Weld. J., Res. Suppl., 39 (1974) s10-s12

18.    C.T. Sims, "Prediction of Phase Composition" in "Superalloys II", Eds.: C.T. Sims, N.S. Stoloff and W.C. Hagel, John Wiley & Sons, New York, NY, 1987, 217-240

19.    A. Engstrom, L. Höglund, and J. Ågren, Metall. Mater. Trans. A, 25A (1994) 1127-1134

20.    H. Du and J. Ågren, Metall. Mater. Trans. A, 27A (1996) 1073-1080

21.    M. Kajihara, C. Lim. and M. Kikuchi, ISIJ Inter., 33 (1993) 498-507

22.    A. Engstrom, J. E. Morral and J. Ågren, Acta mater., 45 (1997) 1189

23.    T. Helander and J. Ågren, Metall. Mater. Trans. A, 28A (1997) 303-308

24.    J. Ågren, Scand. J. Met., 19 (1990) 2-8

25.    Z.-K. Liu, L. Höglund, B. Jönsson and J. Ågren, Metall. Trans. A, 22A (1991) 1745-1752

26.    T. Kraft, M. Rettenmayr and H.E. Exner, Modelling Simul. Mater. Sch. Eng. 4 (1996) 161-177

27.    N. Saunders, Proc. 4th Decennial Int. Conf. Solidification Processing, Eds.: J. Beech and H. Jones, University of Sheffield, UK, 1997, 362-366

28.    D.K. Banerjee, M.T. Samonds, U.R. Kattner and W.J. Boettinger, Proc. 4th Decennial Int. Conf. Solidification Processing, Eds.: J. Beech and H. Jones, University of Sheffield, UK, 1997, 354-357

29.    A.T. Dinsdale, CALPHAD, 15 (1991) 317-425

30.    M. Hillert and L.-I. Staffansson, Acta Chem. Scand., 24 (1970) 3618-3626

31.    F. Sommer, Z. Metallkd., 73 (1982) 72-76

32.    C. Wagner und W. Schottky, Z. phys. Chem., B11 (1930) 163-210

33.    W.L. Bragg and E.J. Williams, Proc. Royal Soc. A, London, 145 (1934) 699-730; 151 (1935) 540-566

34.    O. Redlich and A.T. Kister, Indust. Eng. Chem., 40 (1948) 345-348

35.    H.L. Lukas, J. Weiss and E.-Th. Henig, CALPHAD 6 (1982) 229-251

36.    B. Sundman and J. Ågren, J. Phys. Chem. Solids, 42 (1981) 297-301

37.    J.-O. Andersson, A. Fernández Guillermet, M. Hillert, B. Jansson and B. Sundman, Acta metall., 34 (1986) 437-445

38.    I. Ansara, B. Sundman and P. Willemin, Acta metall., 36 (1988) 977-982

39.    I. Ansara, N. Dupin, H.L. Lukas and B. Sundman, J. Alloys Compd., 247 (1997) 20-30

40.    S.-L. Chen, C.R. Kao and Y.A. Chang, Intermetallics, 3 (1995) 233-242

41.    M. Hillert, Physica 103B (1981) 31-40

42.    H.L. Lukas, E.-Th. Henig and B. Zimmermann, CALPHAD 1 (1977) 225-236

43.    D.W. Marquardt, J. Soc. Indust. Appl. Math., 11 (1963) 431-441

44.    E. Königsberger, CALPHAD, 15 (1991) 69-78

45.    M. Hillert, CALPHAD, 4 (1980) 1-12

46.    Y.-M. Muggianu, M. Gambino et L.P. Bros, J. Chim. Phus. 72 (1975) 85-88

47.    L. Kaufman and H. Nesor, CALPHAD, 2 (1978) 325-348

48.    H. Okamoto, J. Phase Equilibria, 14 (1993) 257-259

49.    G. Eriksson and K. Hack, Metall. Trans.B, 21B (1990) 1013-1023

50.    R.H. Davies, A.T. Dinsdale, J.A. Gisby, S.M. Hodson and R.G.J. Ball, in "Applications of Thermodynamics in the Synthesis and Processing of Materials", Eds. P. Nash and B. Sundman, TMS, Warrendale, PA, 1995, 371-384

51.    B. Sundman, B. Jansson and J.-O. Andersson, CALPHAD, 9 (1985) 153-190

52.    E. Königsberger and G. Eriksson, CALPHAD, 19 (1995) 207-214

53.    B. Sundman, in "User Aspects of Phase Diagrams" Ed. F.H. Hayes, Institute of Metals, London, UK 1991, 130-139

54.    B. Sundman, "Thermo-Calc Newsletter No. 18", Royal Institute of Technology, Stockholm, Sweden 1995

55.    U.R. Kattner, W.J. Boettinger, S.R. Coriell, Z. Metallkd. 87 (1987) 522-528

56.    J. Ågren, ISIJ International, 32 (1992) 291-296

57.    G. Eriksson, H. Sippola and B. Sundman, CALPHAD, 18 (1994) 345-345

58.    SGTE Solution Database: SGTE, St. Martin d'Hères, France

59.    Al-DATA, Fe-DATA, Ni-Data, Ti-DATA: ThermoTech Ltd., Surrey, UK

60.    C.W. Bale and G. Eriksson, Canad. Metall. Quarterly, 29 (1990) 105-132

61.    C.W. Bale, "Web Sites in Inorganic Chemical Thermodynamics",

62.    "User Applications of Alloy Phase Diagrams", Ed. L. Kaufman, ASM International, Metals Park, OH, 1987

63.    "User Aspects of Phase Diagrams", Ed. F.H. Hayes, The Institute of Metals, London, UK, 1991

64.    "The SGTE Casebook, Thermodynamics at work", Ed. K. Hack, The Institute of Metals, London, UK, 1996

65.    U.R. Kattner and W.J. Boettinger, J. Electron. Mater., 23 (1994) 603-610

66.    N. Saunders, Materials Science Forum, 217-222, (1996), 667-672