Diffusion in a bicrystal: effect of water uptake through grain boundary on the degradation of halide perovskites

In this work, we investigate the diffusion in a bicrystal strip with a constant concentration of solute atoms/molecules on free surfaces to mimic the diffusion of water molecules in a bicrystal of halide perovskites. The effect of grain boundary diffusion is incorporated in the analysis, and the equation of mass transport in the grain boundary is derived without a time-derivative term. Using the equation of mass transport in the grain boundary, a closed-form solution of the spatiotemporal evolution of the concentration of solute atoms/molecules in the bicrystal is derived. Numerical analysis of the uptake of water in a methylammonium lead iodide (MAPbI₃) bicrystal is performed. The degree of degradation of the MAPbI₃ bicrystal due to the uptake of water is defined as the ratio of the diffusion length of water in one of the crystal in the bicrystal to the half width of the bicrystal. The numerical results reveal that the degree of degradation of the MAPbI₃ bicrystal increases with the increase of the diffusion time and there exists effect of grain size on the degree of degradation of the MAPbI₃ bicrystal. The time to reach the same degree of degradation due to the uptake of water is proportional to the width of the MAPbI₃ bicrystal of the same thickness, in accord with the experimental results reported in literature. This result points to the need of producing halide perovskite films of large grain sizes in order to improve the structural stability and performance of perovskite-based photovoltaic cells by limiting the structural degradation induced by water diffusion.