Toward quantitative Modeling of morphology changes in solids with phase field theories:: Atomistic arguments for the determination of higher gradient coefficients

This paper starts with an outline of the potential of phase field theories of the Cahn-Hilliard type when applied to studies of micromorphological change in solids. Various examples are presented ranging from ceramics to metals. The focus will be on a discussion of the material parameters involved during the modeling and how these can be specified in terms of quantitative data. A particular emphasis will be on the so-called Higher Gradient Coefficients (HGC's), which are essential for proper modeling of the shape of the developing microstructures. For this purpose an atomistic interpretation of the Gibbs free energy of binary systems is presented which, by comparison with a phenomenological Redlich-Kister Ansatz, provides a quantitative means to numerically determine the HGC's. Deviations to the classical theory as outlined in the seminal paper by Cahn and Hilliard are discussed, various other higher gradient terms for the extended diffusion flux are derived, and their influence on microstructural development is assessed quantitatively. All steps and procedures involved are demonstrated explicitly and evaluated numerically for the special case of eutectic AgCu showing pronounced phase separation and coarsening behavior.