John W. Cahn

A linear theory of thermochemical equilibrium of solids under stress

Abstract Many multicomponent solids can be represented by a continuum model. On this basis a thermodynamic theory of multicomponent stressed solids has been constructed. After linearisation, the necessary equations for equilibrium can be separated into purely elastic and purely chemical equations. In particular, the usual Hookes law coefficients are replaced by new open system coefficients. All the parameters of these equations can be computed from standard measurements. Examples showing the use of these formulae are presented.

A Vector Thermodynamics for Anisotropic Surfaces I. Fundamentals and Application to Plane Surface Junctions

Abstract In this paper we propose a vector function ξ(n) as an alternative to the scalar function γ(n) with which the free energy of anisotropic surfaces is usually described. The vector function is chosen such that its component on the unit normal (n) represents the tendency of the surface to minimize free energy by contraction; the component of ξ off the normal represents the tendency of the surface to minimize free energy by rotation. By elementary operations one obtains from ξ both the tensile and torque components of the thermodynamic force to be balanced at any exposed edge of a planar surface. The ξ-vectors for surfaces of various orientations on a given solid are, in fact, radius vectors of the corresponding Gibbs-Wulff form for the shape of the equilibrium body. It is therefore suggested that the Gibbs-Wulff form itself is preferable to the gamma plot for representing the free energy of solid surfaces. To demonstrate the incisiveness of the vector formulation we consider equilibrium at the edges and corners of facetted crystals and at the junctions of three and four grains or phases. In particular, analysis of the point junction of four isotropic phases is simplified, while solution of the anisotropic case is achieved for the first time. We conclude that the configurations of interfaces which satisfy equilibrium at the point junction of four crystals are discrete and fixed in space, allowing no freedom in the orientation of any element.

Mechanisms of phase transformations within the miscibility gap of Fe-rich Fe-Al alloys

Abstract The coherent phase diagram of the Fe-Al system possesses a tricritical point where a line of higher-order transitions ends at a miscibility gap at about 23 at.% Al and 615°C. Rules of general applicability governing phase separation within the miscibility gap of such a system are developed. Application of the rules to the Fe-Al system results in detailed predictions about the mechanisms of decomposition and ordering reactions and their sequences. Electron microscopy is used to study the reactions experimentally and the results are in agreement with theoretical predictions.

Coherent fluctuations and nucleation in isotropic solids

Abstract By adding a suitably defined elastic energy term to the free energy, a new free energy function is obtained which, over a limited composition range, has all the properties of a thermodynamic potential for coherent processes. It may be used for instance to obtain the coherent phase diagram by the ordinary method of common tangents, and in this paper it is used to examine the coherent nucleation in isotrpic solids, expecially in the vicinity of the limit of metastability, where such nucleation is not classical.

Spinodal decomposition in age hardening copper-titanium alloys

Abstract The early stages of spinodal decomposition in age hardening Cu-Ti alloys have been studied by electron microscopy. The alloys (1.55, 3.08 and 5.17 w/o Ti) decomposed on the quench from solutionizing temperatures into Ti enriched and Ti lean regions. Superlattice reflections, at 1 5 {420} m positions as well as 1 2 {210} m reflections were observed in the diffraction patterns of the as quenched 5.17 w/o Ti alloy. The alloys continued to decompose when aged at elevated temperatures. A sequence of microstructures was used to show that continuous phase separation, and hence spinodal decomposition, was the mechanism of decomposition. The metastable two phase structure which formed from the spinodal process was aligned and periodic from the start of the process. The Ti enriched phase was ordered, with the Dla(Ni4Mo; 14/m) structure. Reversion experiments were performed to determine the position of the coherent metastable solvus. When aging treatments were performed near this solvus. heterogeneous nucleation of the metastable phase was observed.

Pairwise interactions and the ground state of ordered binary alloys

Abstract The ground state energy and superstructures of b.c.c. and f.c.c. Ising model systems with first and second neighbor interactions have been examined. A large variety of superstructures with symmetries as low as monoclinic and stoichiometries as strange as A6B are a natural consequence of the search for lowest energy. For non-stoichiometric alloys, two-phase mixtures of ordered structures of adjacent stoichiometries are often the state of lowest energy. These features seem to correspond more closely to those of real systems than would be expected from the usual first neighbor only Ising model.

The later stages of spinodal decomposition and the beginnings of particle coarsening

Abstract Several features of the later stages of spinodal decomposition leading to two phases and particle coarsening can be obtained by considering terms in the diffusion equation that originate from a composition dependent diffusion coefficient. These give rise to harmonic distortions of the composition waves typical of spinodal decomposition. Two classes of distortions are identified and related to the free energy curve through the composition dependence of the thermodynamic factor in diffusion. Asymmetric free energy curves give even harmonics that are consistent with the lever rule, i.e. a minor phase far from the average composition and a major matrix phase. Odd higher harmonics, originating from higher even derivatives of the free energy function flatten the extremes in composition and sharpen the interfaces leading to the composition distribution expected for a two-phase alloy. The initial stages of particle coarsening can also be described and two types of effects are distinguished. One mechanism occurs only in unsymmetric alloys and it is suggested that such alloys should coarsen much more rapidly than the symmetric ones.

Coherent and incoherent equilibria in iron-rich iron-aluminum alloys

Abstract The discrepancy between two recent phase diagram determinations of the Fe-Al system is resolved experimentally. Both diagrams are correct, but one displays metastable coherent equilibria. The form of the combined coherent and incoherent phase diagram for these alloys is critically discussed. A theoretical treatment of the effect of coherency stresses on lowering the miscibility gap in this system, given in an appendix, is compared with experimental observations.

Discontinuous coarsening of aligned eutectoids

Abstract The thermal stability of aligned Co-Si, Cu-In, and Ni-In euteotoids has been studied. In addition to the usual continuous coarsening within the grains, a discontinuous-coarsening process involving grain-boundary migration was observed. The effects of temperature, lamellar orientation, lamellar spacing, and lamellar perfection on this discontinuous-coarsening process were studied. The driving force and mechanism for this process are analyzed and discussed.

Spinodal decomposition: A reprise

Tiller et al. have recently attempted(1) to reformulate the theory of spinodal decomposition. Unfortunately, the equations they propose violate the second law of thermodynamics as well as displaying other incongruities. It is shown that, even if their intentions are carried out correctly, the results are at variance with existing diffusion data. It is also demonstrated that their criticisms of the mathematics of the original treatment are invalid.