A model of Sn whisker growth by coupled plastic flow and grain boundary diffusion

Abstract

Finite element simulations are used to calculate the rate of whisker growth due to intermetallic formation in a Sn film with columnar grain structure on a Cu substrate. The simulations account for plastic flow by dislocation motion within the grains, as well as diffusion along grain boundaries. Grains with a slightly lower yield stress than their neighbors are shown to act as sinks for material and are progressively extruded from the film. A simple analytical model is developed to estimate the resulting whisker growth rate, and is shown to be in good agreement with experimental measurements.

Modeling tin whisker growth.

“…The model also does not provide a clear method to determine the outer Radius R, which helps control growth as much as β and tan θ . The model of Buchovecky et al [4] determines the outer radius R and tells us that weak grains will grow, but not what weak grains are.…”

“…Notably, the work of Tu [32,33] and Buchovecky et al [4] have introduced a set of models for the growth of tin whiskers based on the idea of a "weak" grain. Tu [32] originally proposed that the nucleation site was associated with a crack in the tin oxide layer.…”

“…This basic model has been modified by several other authors [4,27] to account for the effects of plasticity or surface grains. One of the key aspects of this model is the choice of the outer Radii R. Tu assumes that the spacing is controlled by cracks or weak spots in the oxide.…”

“…The earliest theories associated with tin whisker growth involve dislocation theory [10,13,14,28,29]; a theory that it is now discounted by most researchers. Most modern theories of whisker growth [4,21,24,27,32,33] are associated with diffusion, usually through the grain boundaries, with stress or stress gradients as the driving force.…”

“…This simple model has been modified by Buchovecky et al [4] to include a mass generation term, which accounts for the stress generation rate in the film, to match their finite element simulations. The authors assume that the film is plastically deforming in areas outside the whisker and have a yield stress of σ y while the whisker grain has a yield stress of σ w .…”