Modeling of the coelescence of micro-voids under the influence of elastic stresses and electromigration

Void growth and coalescence usually cause reliability concerns to solder joints in integrated circuits. In this paper, a diffuse interface model is developed to simulate the coalescence of micro-voids under the influence of elastic stresses and electromigration. A variable c stands for the composition of vacancies, with c=1.0 indicating the void phase and c=1.0×10⁴ indicating the bulk phase. A double well potential is used to describe the chemical free energy density of the thermodynamic system. In this model, a modified Cahn-Hilliard type equation was solved with a nonlinear multi-grid method to determine the composition field c; the Cauchy-Navier equations for the elastic displacements u and v were solved with a linear multi-grid method to determine the elastic energy density; a steady state equation of electric potential was solved to determine the effective driving force for electromigration. Simulation results show that the applied stresses have significant effects on the coalescence of micro-voids. For the given initial configuration, the tensile stresses accelerate the coalescence of micro-voids; on contrast, the shear and the combined stresses interrupt the coalescence of micro-voids and steady states are eventually achieved under the influence of these stresses. Under the influence of tensile stresses, a simulated “failure” which leads to an opening is also shown.