Numerical techniques for the calculation of binary phase diagrams

Numerical techniques are presented for the calculation of binary phase diagrams. Most compositions are obtained directly through a Taylor series expansion of the composition with respect to temperature. The series is limited to the second order term and analytic expressions of the slopes and curvatures of phase boundaries have been developed for its use. A Newton-Raphson iteration technique tests the convergence of the results. The method is efficient and accommodates various formalisms for the free energy functions. The series is not applicable to very dilute solutions of a component, near the critical point of a miscibility gap or in the vicinity of the congruent transformation point of a compound; in these cases, alternate equations have been derived. Formerly Graduate Student, Department of Metallurgy and Materials Science, Carnegie-Mellon University.     

Numerical techniques for the calculation of multicomponent phase diagrams

A stepwise method has been developed for the calculation of two-phase boundaries in multicomponent phase diagrams. It is based on a direct minimization of the Gibbs free energy and places no restriction on the number of components. An alternate search technique has also been developed and is used to guard against any possible divergence of the stepwise technique. At infinite dilution of a componentm or in the case of an island miscibility gap, the method is not directly applicable. The difficulty is circumvented, however, through a study of the slopes of the phase boundaries at infinite dilution of componentm and a calculation of the critical points of the miscibility gap. Finally, a computer program is described and its applications are illustrated by examples of calculation of ternary diagrams. Formerly Graduate Student, Department of Metallurgy and Materials Science, Carnegie-Mellon University     

Numerical techniques for the calculation of multicomponent phase diagrams

A stepwise method has been developed for the calculation of two-phase boundaries in multicomponent phase diagrams. It is based on a direct minimization of the Gibbs free energy and places no restriction on the number of components. An alternate search technique has also been developed and is used to guard against any possible divergence of the stepwise technique. At infinite dilution of a componentm or in the case of an island miscibility gap, the method is not directly applicable. The difficulty is circumvented, however, through a study of the slopes of the phase boundaries at infinite dilution of componentm and a calculation of the critical points of the miscibility gap. Finally, a computer program is described and its applications are illustrated by examples of calculation of ternary diagrams. Formerly Graduate Student, Department of Metallurgy and Materials Science, Carnegie-Mellon University     

Thermodynamic calculation of phase diagrams in the iron-nitrogen system

On the basis of recent measurements of the equilibria between N₂ and NH₃ containing gas atmospheres, respectively, and of the nitrogen solved in the α, γ and ε iron phases a thermodynamic analysis of the Fe—N system has been carried out. The Gibbs free energy of the γ, ε transformation of pure iron was derived. The composition Fe_4.3N was calculated concerning the γ’ phase above 593°C. The phase boundaries were stated as temperature functions of the N₂ equilibrium pressures. The resulting phase diagram shows the stability regions of the α, γ, γ’ and ε phases and an α/γ/ε triple point which corresponds to that of the pure iron.     

Generalization of information on phase diagrams of binary hafnium systems

Prospects of applying the analogy principle when investigating binary hafnium compounds are considered. The classification of binary hafnium compounds by the main distinctive features of their phase diagrams is developed using a taxonomic approach. Seven uniform groups of compounds under consideration are distinguished and distinctive features of phase diagrams of each group are determined. The rationality of a two-dimensional metric, the application of which ensures obtaining qualitative evaluations of similarity of various diagrams and reveals the analog compounds, is substantiated. The results of classification can be used when substantiating the compositions of certain basic hafnium alloys and master alloys.     

An experimental technique for the rapid determination of binary phase diagrams: The Al-Mg system

A new experimental technique for the rapid determination of phase diagrams is described in this paper. The technique consists of producing a liquid diffusion couple of the Boltzmann-Matano type with the concentration of elements varying continuously across the specific portion of the phase diagram being investigated, heat treating the couple, and then performing microscopic and microchemical analysis to identify various phases. Since the couple contains all phase fields in the composition range, there is little ambiguity in the identification of phases. Some results on the use of this technique to resolve the confusion about the middle portion of the Al−Mg binary system are presented and discussed.     

Considerations on the Kaufman approach to binary phase diagram calculation

The Kaufman approach to phase equilibria involving primarily the fcc, bcc and hcp phases was examined outside the Group Nos. 4 to 10 range where it is customarily employed. The “stability parameters” (ΔH and ΔS of transformation) for most elements in the Group Nos. 1 to 3 region were found to fit satisfactorily the correlation curves of stability parametervs group no.; some of the parameters for Al, Be, Mg and Ti, however, did not. The rare earth parameters fit well in the Group No. 3.5 position they were expected to occupy. A sample phase diagram calculated between two Group 1 elements was in good agreement with experiment. Phase boundaries of fcc +bcc regions adjacent to terminal solid solutions in several Group 1/Group 2 and Group 1/Group 3 systems, on the other hand, were quite unsatisfactory. This difficulty was traced to the high, positive regular solution constants calculated for both phases. Such constants were shown to result from the downward concavity of a plot of enthalpy of vaporizationvs Group No. in the Nos. 1 to 3 region; in the Nos. 4 to 10 range this plot is concave upward.     

Considerations on the Kaufman approach to binary phase diagram calculation

The Kaufman approach to phase equilibria involving primarily the fcc, bcc and hcp phases was examined outside the Group Nos. 4 to 10 range where it is customarily employed. The “stability parameters” (ΔH and ΔS of transformation) for most elements in the Group Nos. 1 to 3 region were found to fit satisfactorily the correlation curves of stability parametervs group no.; some of the parameters for Al, Be, Mg and Ti, however, did not. The rare earth parameters fit well in the Group No. 3.5 position they were expected to occupy. A sample phase diagram calculated between two Group 1 elements was in good agreement with experiment. Phase boundaries of fcc +bcc regions adjacent to terminal solid solutions in several Group 1/Group 2 and Group 1/Group 3 systems, on the other hand, were quite unsatisfactory. This difficulty was traced to the high, positive regular solution constants calculated for both phases. Such constants were shown to result from the downward concavity of a plot of enthalpy of vaporizationvs Group No. in the Nos. 1 to 3 region; in the Nos. 4 to 10 range this plot is concave upward.     

Monte carlo study of the phase diagrams of binary alloys with face centered cubic lattice structure

We describe the results of Monte Carlo computations of the coherent phase diagram (in the temperature-composition plane) of ordering binary alloys on an f.c.c. lattice. Results on long- and short-range order parameters as well as ordering energies are also given. We consider the system with nearest neighbor interaction in the grand-canonical ensemble (equivalent to an Ising antiferromagnet in a magnetic field) as well as in the canonical ensemble (fixed composition). The close agreement between both approaches serves as a severe test of the good accuracy obtained, and so does the fact that differences between results for lattices with 2048 sites and with 16384 sites are found to be negligibly small. Results with next-nearest neighbor interaction are also given, and for both models a comparison with other available predictions is made, particularly with the cluster-variation method. While the latter is found to be quite accurate at stoichiometric composition, it appears to do less well in the more general case. The tetrahedral approximation of the cluster variation method predicts a topology of the phase diagram, in the case of nearest neighbor interaction different from the computer simulations. Some consequences for the interpretation of the behavior of copper-gold alloys are indicated.     

集团变分法(CVM)及其在合金相图计算中的应用

介绍了集团变分法的集团概率变量、配置熵等统计分析的理论基础及Helmholtz自由能的描述模型.提出了巨势及相对化学势的几何分析理论.讨论了应用集团变分法和巨势分析理论进行相平衡计算的过程,并分析了同结构平衡相图计算的若干实例.     

On binaryP-T phase diagrams

A thermodynamic analysis of the concentration and temperature dependencies of oxygen pressure in metal-metal oxide systems is presented. The resulting binaryP-T phase diagrams show the various species in the context of relative reducibility not only for pure elements and compounds but also for the solutions and intermediate compounds which form. It is not necessary to presume a model for the thermodynamic behavior of solid and/or liquid solutions in this treatment. The Cr-Cr₂O₃ system is presented as an example. Verification is found in the previously reported data on the V₂O₄V₂O₅ system.