Pattern and Scale In Intra-Granular Diffusion.

Some observations of diffusive contaminant release from intra-granular pores deviate significantly from analytical solutions for diffusion out of a sphere.  This research examines how a specific phenomenon – low connectivity of the intra-granular porespace – may cause that deviation.  Our initial modeling used a pore-scale network model of a single grain, with diffusion modeled using random walk (RW) methods.  We examined how contaminant release, starting from equilibrium, was affected by low pore connectivity.  As connectivity decreased, the release curve shifted to later times, as expected.  But as connectivity approached the percolation threshold, the shape of the release curve also changed; becoming more like the observed “deviant” curves.  Specifically, the low-connectivity curves showed increased early-time release, and much slower late-time release.  The pore-scale networks were mapped pore-by-pore to examine reasons for the observed release patterns.  At connectivities near the percolation threshold, accessible porosity and tortuosity varied strongly with both distance from the grain’s exterior, and even small changes in connectivity.  These changes in both accessible porosity and tortuosity agreed with scaling relationships of percolation theory.  To test our understanding of the observed patterns, we developed a finite difference (FD) model.  The FD model uses percolation scaling relationships to parameterize the accessible porosity and diffusivity as functions of both distance from the exterior, and the connectivity’s proximity to the percolation threshold.  It also handles the infinite cluster separately from the edge-accessible finite clusters; this increases complexity and time, but improves the match.  Over a wide range of connectivities and non-equilibrium points, the FD model closely matched the RW model.  The model’s pattern of increased accessible porosity near the edge, faster initial solute release, and extremely slow late solute release, provide a coherent explanation for observed deviations from the expected analytical behavior.