D. de Fontaine

Cluster Approach to Order-Disorder Transformations in Alloys

Publisher Summary This chapter presents the formalism and applications of cluster expansions to the problem of ab initio calculations of thermodynamic properties of crystalline alloys. The success of the method owes much to the parallel developments of statistical and quantum mechanical methods and, of course, to the availability of powerful computational hardware and software. The technique of orthogonal expansions in cluster functions has revolutionized the treatment of configurationally disordered systems. In addition to offering the most rational and general state-of-order description of alloys, cluster methods provide the rigorous and essential link between quantum and statistical mechanical aspects of first-principles thermodynamic calculations. Cluster functions constitute a complete orthonormal set. This basis set produces a rigorous cluster algebra that is used for systems containing an arbitrary degree of configurational order (or disorder). Cluster methods are, in principle, ideally suited for calculating alloy phase equilibria.

Configurational Thermodynamics of Solid Solutions

Publisher Summary This chapter reviews current theoretical models applicable to concentrated solid solutions. It focuses on concentrated solutions because dilute ones can be considered as special cases and because the interesting effects of clustering and ordering are most apparent at high solute concentrations. The extended ranges of solubility are often found in metallic systems, and for this reason most examples selected will refer to metals and alloys, although the models presented are equally applied to off-stoichiometric compounds, metallic or inorganic, and to amorphous solids. The chapter provides description of the state of the order of the system through suitable averaging procedures. It discusses the internal energy of solutions, from both electronic and elastic standpoints. The chapter presents free energy models, such as generalized Bragg- Williamss model, Landau theory, and cluster variation method. These models are applied to the study of phase equilibrium.

The omega phase transformation in titanium alloys as an example of displacement controlled reactions

Abstract The omega phase is shown experimentally to be formed by cooling certain metastable β-Ti alloys to cryogenic temperatures. The transformation is completely reversible and is characterized by a small temperature hysteresis. The diffusionless hexagonal ω phase so formed is of the same composition as the parent b.c.c. β phase, and the gradual progression from diffuse streaking in the diffraction patterns of the β phase to the sharp ω reflections is interpreted in terms of a displacive type reaction. Computer generated plots of the elastic energy of harmonic displacement waves in the b.c.c. lattice are shown to reproduce many of the features of the diffuse intensity in the electron diffraction patterns. The complex quasi-circular streaks observed in some zone patterns are shown to result from the planar intersection of spherical surfaces of intensity in reciprocal space. It is suggested that the progressive transition from circular to rectilinear streaks correspond to a decreased coupling between {111} displacement modes responsible for the diffuse intensity. At sufficiently low temperatures, the diffuse streaks, which lie close to traces of {111} planes, contain sharp intensity peaks at positions 1 3 〈112〉, characteristic of the ω phase.

A model for diffusion on cubic lattices and its application to the early stages of ordering

Abstract A theory is given for diffusion on cubic lattices in which an allowance is made for the effect of sharp composition gradients on the driving force. The treatment is similar in intent to those previously given by Hillert and by Cahn, but we have eliminated the regular-solution assumption of the former and the continuum approximation of the latter. An expression for diffusion is derived in the form of a differential-difference equation, and a simple analytic solution is obtained for the limiting case of small composition, fluctuations. This solution is then used to determine the kinetics and morphology at the early stages of ordering as described by the time dependence of the Fourier spectrum of the composition variation. It is found that many of the characteristics of ordering are the inverse of those for spinodal decomposition.

MECHANICAL INSTABILITIES IN THE B.C.C. LATTICE AND THE BETA TO OMEGA PHASE TRANSFORMATION.

Abstract The beta to omega phase transformation in certain titanium and zirconium alloys can be accomplished formally by subjecting the b.c.c. lattice to a transverse sinusoidal displacement wave of wave vector 1 3 〈112/t>, crystallographically equivalent to a 1 3 〈111〉 longitudinal wave. The instability of this particular mode corresponds to negative eigenvalues of the dynamical matrix of the “average” b.c.c. crystal in the vicinity of the point 1 3 〈112〉 in the first Brillouin zone. By mapping out the regions of negative eigenvalues, one can reproduce the essential features of electron diffraction patterns obtained from asquenched Zr-Ti alloys. The positions and shapes of the ω reflections and the corresponding diffuse intensity are predicted correctly.

On the elastic free energy of solid solutions—I. Microscopic theory

Abstract The elasticity problem posed by a solid solution having solute and solvent atoms which differ in size is examined. Fourier methods, applied to Bravais lattices, are used to calculate the displacements and elastic free energy of an arbitrary composition modulation. For a given Fourier component of the composition modulation, the elastic free energy is found to be a function of both the direction and magnitude of the wave vector in contrast with the usual continuum formulation where it is a function of direction only. This result predicts that a macroscopically isotropic crystal will, nevertheless, be anisotropic on an atomic scale. A sample calculation is given for an isotropic b.c.c. crystal in order to show its atomistic anisotropy which is a direct consequence of the discrete nature of the crystal lattice. The calculationbased for convenience on a simple lattice force model—also shows, as expected, that the ordered β brass structure has no elastic free energy. Another consequence of the wave-vector-magnitude dependence is that a gradient energy term—analogous to the “chemical” gradient energy term introduced by Cahn and Hilliard—also exists for the elastic energy. This particular result will be discussed more fully in paper II where the theory developed here will be used to determine the elastic free energy of homogeneous, clustered, and ordered solid solutions.

The ω-phase as an example of an unusual shear transformation

The conditions leading to formation of the metastable ω-phase in Ti, Zr, and Hf alloys are described and earlier experimental observations directly related to ω -phase formation are summarized. New experimental results are presented which show that increased oxygen content suppresses athermal ω-phase formation in Ti-V alloys. A mechanism of ω-phase formation based on the formation of a linear displacement defect is described and it is shown that both the earlier and new experimental results are consistent with this mechanism. Additional experimental results on variations in electrical and physical properties in Ti alloys are also shown to be consistent with the linear displacement defect model.

Overview no. 43: Binary ordering prototype phase diagrams in the cluster variation approximation

Abstract The Tetrahedron Octahedron Cluster Variation Method (TO-CVM) has been employed to investigate f.c.c. based order-disorder phase diagrams with first (positive) and second nearest neighbor pair interactions. The investigation covers the entire range of α values, the ratio of second to first nearest neghbor pair interactions. Ground state structures are discussed in detail and phase diagrams for seven different α values are presented. Good qualitative (topological features) and reasonable quantitative (transition temperatures) agreement is obtained with available Monte Carlo results, thereby confirm the reliability of the TO-CVM phase diagram calculations. The present study concludes the first global phase diagram investigation of f.c.c.-based Ising model with first and second nearest neighbor pair interactions, and can be regarded as a precursor to more elaborate first principles phase diagram calculations.

An analysis of clustering and ordering in multicomponent solid solutions—I. Stability criteria

Abstract A microscopic, or lattice, model of multicomponent solid solutions is presented in which the free energy of the solution is expressed as a quadratic form in the local atomic concentration and atomic displacement variables. The stability of the system with respect to diffusional processes is discussed in terms of the eigenvalues and eigenvectors of the matrix of the free energy quadratic. For an n -component solid solution, stability regions in the phase diagram are separated by n − 1 surfaces, the equations of which are obtained by setting the determinant of the matrix equal to zero. The topological relationship between ‘chemical’ and ‘elastic’ stability surfaces is discussed. Two examples of ternary systems are given: the first illustrates ternary spinodal decomposition; the second illustrates ternary first and second neighbor ordering as exhibited by the AgAuZn 2 system.

Theoretical calculation of the CUAGAU coherent phase diagram

Abstract The Cluster Variation Method in the tetrahedron approximation is used to calculate the solid-state CuAgAu ternary phase diagram. The pair interaction parameters and the tetrahedron four-body interaction parameters used in the present calculation were estimated by producing the best possible fit provided by the tetrahedron approximation to the known CuAu, CuAg and AgAu binary phase diagrams. The calculated isothermal sections of the CuAgAu ternary phase diagram show complex two- and three-phase equilibria between disordered and ordered phases; the three-phase equilibria, although expected, have not been reported in experimentally determined phase diagrams. In addition, an instability analysis for the disordered solid solution is carried out and the clustering and ordering spinodals are determined.