Modeling diffusional coarsening in microelectronic solders
- Dreyer, Wolfgang
- Müller, Wolfgang H.
2010 Mathematics Subject Classification
- 76R50 82B26 82B24 35K30 35K35
- Diffusion, phase transisions, interface problems, initial value problems for higher order parabolic equations, boundary value problems for higher order parabolic equations, solders, lead, tin, lead-free, coarsening, aging
This paper presents a detailed numerical simulation of the coarsening phenomenon observed in microelectronic solder materials that are subjected to high homologeous temperatures in combination with thermo-mechanical stresses. The simulations are based on a phase field model which, for simplicity, is explicitly formulated for a binary alloy. To this end, the thermomechanical stresses originating within a Representative Volume Element (RVE) of the solder material are calculated first. This is achieved by means of a closed-form solution of the Navier equations resulting in explicit expressions for the displacements of an anisotropic, heterogeneous, thermally stressed elastic medium in discrete Fourier space. Inverse discrete Fourier transforms are then applied to these expressions in order to obtain the local stresses in real space. These in turn are inserted into an extended expression for the diffusion flux, which, in addition to the classical driving force of a concentration gradient takes the influence of different surface tensions between the solder phases as well as the local strain energy into account. The equations are evaluated numerically for the exemplary case of eutectic SnPb solder, for which all material constants are known explicitly. A comparison with aging experiments is performed.