The combined application of electronic and thermodynamic perturbation theories allows for an ab-initio calculation of the phase diagrams of binary simple-metal alloys.

The physical picture of cohesion and compound formation provided by parameter-free, self-consistent, energy-band calculations will be described. For transition-metal elements, the calculations allow us to “see” which electrons are holding the solid together and which are holding it apart. For compounds, calculated heats of formation agree well with available measurements and are in general agreement with those given by Miedema's empirical formula. (The agreement with Miedema is paradoxical, in that Miedema's conception of the formation process differs qualitatively from ours.) Preliminary, but very encouraging, results of efforts to extend the analysis to disordered materials and to the calculation of phase diagrams are described.

The chemical, ordering and magnetic properties of binary alloys at the surface are examined in terms of a model of pairwise interacting atoms. Both the case of segregating alloys and the case of alloys which form intermetallic compounds are examined. Some surprising results about surface concentration and surface ordering are found. Applications to a number of systems, including ferromagnets, are presented.

The prediction of terminal solubilities for a given metal solute and solvent is a formidable problem, and it remains a major unsolved question in metallurgy. In this paper we describe the relative solubilities of the transition metals in one another. After discussing the problem of defining an appropriate measure of solubility for the transition-metal alloys which have any intermediate phases, we present a map of relative solubilities for the transition metals. From this map, we delineate the trend which occurs. Both the Engel-Brewer theory and d-band transition-metal alloy theory are consistent with this trend.

An electronic theory for the total energy of binary alloys is presented. It treats the entire range of concentration and short-range order. The method in not ab initio, but requires only information about the pure elemental constituents of the alloy. Results for two very different metal series, monovalent metals and 4-d transition metals, are calculated and compared to experiment and other models. The model calculation allows us to examine the physical basis for experimental trends.

Lattice energy functions have been used to predict various properties. The energies and volumes of formation of alloy phases, as well as the relative stabilities of competing phases are provided by a model based on Mie type interatomic potentials. The elastic constants of alloy phases are treated using a model sensitive to the quadrupole distortion of a repulsion spheroid. The properties of defects are treated using a model based on both a volume dependent contribution and an interatomic potential contribution to the lattice energy. Good predictability is obtained for all three types of lattice energy functions in their respective applications.

The types of information required for the calculation of phase diagrams are discussed by considering the computation of typical ternary sections from the constituent binary systems. Such calculations require a knowledge of the Gibbs energy of transformation (lattice stabilities) and Gibbs energies of mixing of wholly metastable, as well as the stable phases in binary systems. Similarly, the stabilities of metastable compounds such as Fe7C3 would be required for computations in the C-Cr-Fe system.

These requirements are compared to the information provided by solid-state theoreticians. Essentially such calculations provide enthalpy values at 0 K (or some unspecified temperature for semi-empirical models); however the lattice dynamics and configurational entropy of simple phases have been included in some recent computations. The importance of predicting the entropy and thus heat capacity of metallic phases - particularly metastable phases - is therefore emphasized. Identification of those contributions to the heat capacity which are responsible for the differences between metal polymorphs is discussed, particularly the formalism for magnetic and atomic ordering phenomena. Predictions of ordering temperatures and magnetic moments as a function of composition would be of considerable help for phase diagram calculations.

Ab-initio calculations already have considerable success in predicting molar volumes of both stable and metastable phases, so that such information will undoubtedly be of considerable value in studying alloy behaviour at high pressures.

We suggest a new method of calculating structural properties of metals. The method utilizes the data base of “firstprinciple” charge-density distributions in pure metals accumulated in recent years. If, for a metal, the solution to the Schrodinger equation within the atomic cell is known, the electron charge-density distribution in an orbitrary point of the crystal may be “reconstructed” within the framework of a model. Using the Density-Functional Formalism and the virial theorem, the total energy of the crystal may be found in terms of an effective interatomic potential. The latter may be used for the calculating of other structural properties, such as phonon spectra and elastic moduli. The extension of this method to alloys is briefly discussed.

Alloy properties are often summarized on two-dimensional maps. For instance, Darken and Gurry related the terminal solubilities of alloy systems to two coordinates (the differences in electronegativities and in atomic radii of the alloy constituents). Maps are considered here which correlate the structures in which the compounds form as well as providing some indication of whether a compound forms at some given composition in the first place. This class of map involves one coordinate which is the difference in an atomic parameter (here it will be taken to be the electronegativity) while the other is an average (in this case the d band hole count). This contrasts with Darken Gurry maps and Giessen's maps related to glass-forming ability (elsewhere in this volume) where both coordinates are differences. The situation for 50/50 transition metal alloys is reviewed and results are presented for systems off 50/50 composition.

In the beginning there were ideal solutions (Geneva 1966) [1] and lattice stability values were deduced for the transition elements. Next regular solution interaction parameters for second and third row transition elements were predicted from simple models (Cambridge 1970)[2] combined with the lattice stability values via computer methods to calculate binary and ternary phase diagrams. Line compounds were introduced for comparison with experimental phase diagram data. At Brunel (1971)[3] and Munster (1972)[4] the regular solution models were expanded to include subregular and higher order terms and since the origin of CALPHAD (1973)[5]many coupled sets of thermochemical and phase diagram descriptions have been derived and published. Examples of multicomponent phase diagram calculations applied to solution of industrial problems have been illustrated in the subject areas of transformations and stability in steels and superalloys, uranium alloys, SIALON materials, metallic and fluoride glasses, electronic materials, and novel materials refining and processing methods. [6-19]

In addition to theoretical models which have been developped to interpret the interactions in metallic systems, polynomials of different types have been used to represent the thermodynamic properties of solution phases. Ternary and higher order systems have been described by adding the properties of the limiting binary systems, using various summation methods ; interaction parameters specific to the multicomponent systems can be derived from sufficiently precise experimental measurements.

A thermodynamic model, based on the regular solution approximation is presented and a formalism, suitable for phases with an arbitrary number of elements and sublattices is described. A new concept, the component array, is introduced in order to simplify the analytical expressions for the Gibbs energy. The mathematical complexity of the model has been solved by the implementation of the model as a part of a general software system for thermochemical and phase diagram calculations by computer.

The sublattice model is suitable for describing phases where the difference in size, charge or electronegativity causes a deviation from random mixing of the atoms in a manner which can be described as long range order. This is the case for interstitial solutions and many intermetallic phases, but also for ionic liquids and slag systems. The application of the model to several alloy system will be described.

The relative sizes of the atoms and their electronic configurations are important factors in determining the thermodynamic properties of metallic phases. The regular solution model provides for an asymmetric excess Gibbs energy when there is a size difference, but the actual degree of asymmetry does not correspond to crystallographic sizes. This is due in part to the effect of electronic configurations upon the asymmetry of the Gibbs energy. The Hume–Rothery Rules indicate that the bcc, hcp, and ccp structures are associated with characteristic outer–shell electron per atom concentrations. The electronic factors require modification of the usual definition of internal pressure as used in the regular solution theory. Rather than using the energy of vaporization to the ground atomic state, the energy of vaporization to the valence state atom, which has the same electronic configuration as in the condensed state, is a more appropriate measure of the degree of cohesion and the internal pressure. The problems of providing mathematical expressions that accomodate the contributions of internal pressure, electronic factors, and size factors are discussed.

The use of simple nearest neighbour pair-wise interaction models for the description of the thermodynamic properties of ordered alloys is reviewed and extended to the treatment of phases containing several sublattices. Employing individual sublattice occupation parameters to define atomic distributions, enthalpies corresponding to these are described by the summation of pair-wise interaction energies over all the resulting first co-ordination shell neighbours. Invariant like and unlike bond energies are assumed, their respective values being estimated using heats of vaporisation of the elements and a heat of formation of the phase at a single composition. Combination of the enthalpies with corresponding configurational entropies yields an expression for the free energy of the phase which may be minimised with respect to variation of the sublattice occupation parameters. This leads to the prediction of the stable atomic distributions and the variation of these and the thermodynamic properties with composition. The application of the model to sigma phases and other multi-sublattice structures is discussed

Coherent phase diagrams are defined as pertaining to equilibria between phases which differ from one another merely by the distribution of different types of atoms on fixed crystallographic sites. Resulting ordered structures must then all be superstructures of one parent lattice, and corresponding phase diagrams are isomorphous to those of the Ising model with non–vanishing magnetic field. Rather complex phase diagrams can then be obtained from a single free energy functional by means of the cluster variation method. Calculated phase diagrams will be shown for the case of the fcc parent lattice with various positive and negative ratios of the values of second-to-first-neighbor pair interactions, this ratio being the only parameter which enters the calculation. Results will be compared to those of Monte Carlo calculations. The possibility of performing these first-principle calculations of coherent phase diagrams will be briefly touched upon.

The concentration and the temperature dependence of thermodynamic mixing functions of liquid alloys with compound formation tendency, which often exhibit large deviations from a regular behavior, can be calculated according to an association model using only a few parameters which have a definite physical significance. The results obtained for binary and ternary alloy melts with one ore more, simultaneously occurring, binary or ternary associates are in good accordance with the experimental values. For the calculation of phase diagrams, the association model enables a correct extrapolation into concentration and temperature regions for which no experimental results are available. The occurrence and the borderline of miscibility gaps in liquid alloys with strong compound forming tendency can be quantitatively described.

It can be concluded from experiments that atomic (also called “chemical”) and magnetic ordering reactions exert a mutual influence on each other. In bcc Fe-Al alloys a strong depression of the Curie temperature is observed with the onset of atomic long range order (lro) as compared to the extrapolation of Tc from the random alloys, see Fig. 1. In fcc Co-Pt alloys a depression of the critical temperature of atomic lro L12 →A1 is observed in the Co-rich alloys which are ferromagnetic, as compared to the Pt-rich counterparts which are paramagnetic, see Fig. 2. In this instance it should be noticed that this assymetry is not observed in Ni-Pt allovs where the Curie temperature is inferior to the critical temperatures of atomic lro [1].

Magnetic ordering and phase separation in binary alloys with one magnetic component are studied within the mean field approximation. The influence of ferromagnetic ordering on the phase separation temperature (TCS) is examined in detail. In general it is found that magnetism increases TCS and causes strong asymmetries in the phase diagram. In particular, it is found that in systems with only ferromagnetic interactions, the Curie temperature (TCM) and TCS are the same. Results for the phase diagrams and for the temperature dependence of the magnetic and spatial order parameters are presented.

Many features in the vicinity of critical points in phase diagrams can be illustrated using a Landau type free energy expansion as a power series in one or more order parameters and composition. This simple approach can be used with any solution model. It also predicts limits to metastability, and is useful for understanding mechanisms of phase change. The theory is applied to all the critical points that can occur in binary systems according to a Landau theory: critical consolute points, order-disorder transitions, tricritical points, critical end points, as well as to systems in which two transitions such as chemical and magnetic ordering occur.

The phase boundaries of the C-Cr-Nb-Ni-W system are calculated from the available thermodynamic properties of the limiting binary systems. The methods of calculation and the models used in the description of the ternary and quaternary systems are presented as well as the influence of tungsten on the solubility of the M23C6 phase in the nickel rich alloys.