The impact of a complex microstructure on polycrystalline diffusion is investigated using both numerical and analytical methods. In particular, the diffusion equation is numerically integrated using the finite-difference method to obtain the concentration profile for a diffusant in a simplified microstructural representation. The methodology is first validated for an idealized model of diffusion in a prototypical, single grain-boundary system and then applied to a Voronoi model of a microstructure resulting from homogeneous nucleation and growth. The diffusive behavior is quantified by obtaining uptake curves as a function of time for different ratios of grain boundary to lattice diffusivities. Such curves can be used to estimate an unknown grain-boundary diffusivity, given certain microstructural assumptions. Finally, approximate analytical equations describing a diffusant uptake in polycrystalline microstructural models are developed and found to agree well with the numerical results.
It was convenient to use both the IMSL numerical libraries (e.g., DGMRES) and MATHEMATICA to solve the systems of equations arising here.