H.I. Aaronson

The elastic strain energy of coherent ellipsoidal precipitates in anisotropic crystalline solids

The elastic strain energy of coherent ellipsoidal precipitates (ellipsoids of revolution) in anisotropic crystalline solids has been calculated as a function of ellipsoid aspect ratio using the method of Eshelby. When the precipitate is eithermuch softer or harder, elastically, than the matrix, the results are similar to those previously obtained using isotropic elasticity. When this condition is not met, however, anisotropic elasticity can yield quite different results which vary markedly with the orientation relationship between precipitate and matrix. When the precipitate has a non-cubic crystal structure, the elastic strain energy often passes through a maximum or a minimum at shapes which are neither thin discs nor spheres. During this study, the isotropic elasticity result that the strain energy associated with a disc-shaped precipitate is independent of the matrix elastic constants was also shown to hold under the conditions of anisotropic elasticity, and in such circumstances it depends only on the elastic properties of the precipitate in the direction of the principal directions of the disc. Incorporation of the anisotropic elastic strain energy into the calculation of ΔG*, the free energy of activation for the formation of a critical nucleus for the basic case of homogeneous nucleation with boundary-orientation independent interfacial energy, showed that the ratio of the strain energy to the volume free energy change must usually be somewhat larger than 3/4 in order to cause the shape of the critical nucleus to differ from that of a sphere.

Influence of crystallography on aspects of solid-solid nucleation theory

Expressions for the major variables in the general rate equation for solid-solid nucleation were developed on the basis of various models of the critical nucleus shape during homogeneous and heterogeneous nucleation. These models are based upon spheres, but in some a facet was incorporated at one boundary orientation to represent the presence of a partially or fully coherent structure. Gibbs’ relationship for the critical radius is applicable to all of the models. The other variables in the nucleation rate equation are affected by the model and by faceting. Reduction of AG* by faceting is concluded to be the primary cause for the presence of reproducible lattice orientation relationships and for the existence of transition phases during precipitation from solid solutions.

Surface concentration profile and surface energy in binary alloys

Abstract An improved formulation for the orientation-dependence of multi-layer surface segregation is developed in terms of bond enthalpy and strain energy minimization. Inclusion of this result in the Gibbs adsorption isotherm yields the surface energy. Sample calculations of the surface energy isotherm for solid Auue5f8Cu and liquid Cuue5f8Ni alloys are in good agreement with published experimental data. The conditions for the existence of an extremum in the surface energy isotherm, originally derived by Defay et al. from a monolayer model, are re-examined on the basis of a multi-layer model. Their results are found applicable to this more general situation in the absence of strain energy. An orientation-dependence of the extremum surface energy, but not of the composition at which the extremum surface energy occurs, is demonstrated.

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.

Two families of analytic γ-plots and their influence upon homogeneous nucleation kinetics

Abstract The equilibrium shapes corresponding to two different families of γ-plots are constructed. One γ-plot family comprises continuous variations from a sphere to an oblate ellipsoid. This set of γ-plots yields a sharp edge in the corresponding equilibrium shape when its aspect ratio is less than 1 √2 . The other family consists of nephroids of revolution, varied in cross-sectional form from two slightly overlapping near-circles to an ellipse-like morphology. This family exhibits a facet at one boundary orientation in the equilibrium shape. For the analytical expression of the equilibrium shape, the ξ-vector formalism of Cahn and Hoffman is used and found to give results identical to those from the Euler-Lagrange method. The effects of the variations in equilibrium shape within the two families treated upon the principal parameters in the general equation for the time-dependent rate of nucleation are assessed in order to ascertain their relative influence on nucleation kinetics.