Diffusion equation under swelling stresses

Abstract

Water diffusing into silica glass surfaces reacts with the SiO2 structure under hydroxyl generation. This reaction must cause a volume expansion that is already reported in literature. As a consequence of volume swelling and restriction of free expansion by the bulk material, swelling stresses are caused. Since the diffusivity is a function of stress, the consequence is a diffusivity that depends on the local water concentration. Then the solution of the diffusion equation is complicated and makes numerical computations necessary. Disadvantage of numerical computations is the fact that the used extend of the depth range must be finite and, consequently, the semi-infinite body can only be approximated. In the following considerations we will discuss analytical diffusion solutions for constant diffusivity as well as under swelling conditions. Based on this, exact and semi-analytical solutions for diffusion problems in the half-space are given. Further, the effect of swelling affected diffusivity on resulting concentration profiles is shown. We used an analytical solution of the diffusion problem that is known from literature by Gardner and Singh, where the concentration-dependent diffusivity had to be approximated by an exponential function. In case of swelling stresses in silica the diffusivity is given by an exponential function, so that the obtained solution is exact. Due to the compressive swelling stresses, steeper diffusion profiles appear compared to the stress-free state. In the discussion section it will be shown that the water-affected diffusivity correctly predicts the astonishing time-dependency of effective diffusivities observed in literature on silica.