Multipath diffusion: A general numerical model

The effect of high-diffusivity pathways on bulk diffusion of a solute in a material has been modeled previously for simple geometries such as those in tracer diffusion experiments, but not for the geometries and boundary conditions appropriate for experiments involving bulk exchange. Using a coupled system of equations for simultaneous diffusion of a solute through two families of diffusion pathways with differing diffusivities, a general 1-D finite difference model written in FORTRAN has been developed which can be used to examine the effect of high-diffusivity paths on partial and total concentration profiles within a homogeneous isotropic sphere, infinite cylinder, and infinite slab. The partial differential equations are discretized using the θ-method/central-difference scheme, and an iterative procedure analogous to the Gauss-Seidel method is employed to solve the two systems of coupled equations. Using Fourier convergence analysis, the procedure is shown to be unconditionally convergent. Computer simulations demonstrate that a multipath diffusion mechanism can enhance significantly the bulk diffusivity of a diffusing solute species through a material. The amount of solute escaping from a material is dependent strongly on the exchange coefficients, which govern the transfer of solute from the crystal lattice to the high-diffusivity paths and vice versa. In addition, the exchange coefficients (κ{script}₁, and κ{script}₂) seem to control not only the amount of solute that is lost, but also the shape of the concentration profile. If |K₁| < |K₂|, concentration profiles generally are non-Fickian in shape, typically having shallow concentration gradients near the center (radius r = 0) and steep gradients towards the outer boundary of the material (r = R). When |K₁| ≥ |K₂| a concentration profile is generated which resembles a Fickian (volume) diffusion profile with an apparent bulk diffusivity between that of the crystal lattice and that of the high-diffusivity pathways. Because the input parameters are general, this model may have widespread applicability in any area of earth sciences where diffusion considerations are important.