Modeling nonlinear moisture diffusion in inhomogeneous media

While moisture diffusion in microelectronic device and packaging has been studied for decades, the problems involving complex nonlinear moisture diffusion in multi-material assembly have not been fully studied. This paper has developed a general nonlinear diffusion model by adopting water activity, a continuous state variable, as the field variable. The generalized solubility is introduced, which is temperature- and water activity-dependent. The effective diffusivity is defined and derived in terms of generalized solubility and water activity. By comparing the water activity-based model with the existing various normalized models, the present theory can unify and generalize the current approaches. More importantly, the present model can solve both linear and nonlinear moisture diffusion in inhomogeneous material system without normalization. The commercial finite element software has been applied to solve the nonlinear generalized moisture diffusion problem using the analogy of water activity and temperature. A source code of user-defined subroutines in ABAQUS has been provided in the Appendix of the paper. The mathematical formulation and the numerical implementation method presented in this paper can be applied to any nonlinear sorption or diffusion problems in inhomogeneous material system.