Toward a better understanding of morphology changes in solders using phase field theories: Quantitative modeling and experimental verification

This paper concentrates on open problems regarding recently published work in the field of modeling micromorphological changes in lead-free solders using phase field theories of the Cahn-Hilliard type. Specific examples will be eutectic SnPb, SnAg, and AgCu. After a short review of the phenomenology all the relevant equations will be set up, which are required for describing the state of stress and the temporal/spatial distribution of the atomic species in these alloys. The numerical procedures used for solution of these equations will be briefly touched upon. Examples of simulations will be given and the material parameters used therein will be discussed in detail. In particular the available data for the so-called higher gradient coefficients (HGCs) will be carefully examined. In order to obtain reliable HGC data an atomistic point-of-view is proposed. To this end an atomistic interpretation of the Gibbs free energy of a binary alloy is presented and, by comparison with a phenomenological Redlich-Kistler ansatz, as well as data for the compressibility and for the sublimation energy of the alloy the Lennard-Jones potentials of all participating atomic species are determined. These in turn allow to numerically obtain the gradient energy coefficients through summation w.r.t. nearest and higher neighbor interactions. Deviations to the classical theory as outlined in the seminal paper by Cahn and Hilliard are also presented, various other higher gradient terms for the extended diffusion flux are derived, and their influence on microstructural development is assessed. All steps and procedures involved are outlined and evaluated numerically for the special cases of eutectic AgCu solder showing pronounced phase separation and coarsening behavior.