In this and subsequent papers, we consider the effects of the microstructure, in particular grain orientations, and the associated anisotropy of Sn grains on electromigration-induced degradation in lead-free solder bumps. This paper investigates the effect of the anisotropy of Sn grains on current-driven self-diffusion and the resulting atomic flux divergence that is associated with material depletion at the cathode side of the bump and ultimately failure. Finite-element thermoelectric simulations of a section of a chip scale package and a refined submodel are carried out assuming single-crystal and bi-crystal solder bumps. Anisotropic material properties of β-Sn are used in the simulations. By varying crystal orientation of the solder bumps, the effect of grain orientation on the current density, temperature, and atomic flux divergence is studied and discussed. In the future papers, we will consider the coupled mechanical response including stress-driven diffusion.