Scaling and scaling crossover for transport on anisotropic fractal structures

Diffusion into fibrous anisotropic structures can exhibit a variety of crossover phenomena. Scaling of amount adsorbed versus time in such structures is studied by standard renormalization methods as a function of anisotropy for several kinds of discrete models. Total mass adsorbed as a function of time from a reservoir attached at a single point exhibits different power laws in different logarithmic ranges separated by crossover times. For example, one expects a transition from scaling characteristic of a one-dimensional channel to that of an effective isotropic medium as adsorbed material spreads out over successively longer length scales. In the models studied, there is an easy diffusion pathway imbedded in a medium having a much lower diffusivity. The easy-diffusion subspace can have fractal dimension below that of the background. Different types of crossovers are identified. Power-law exponents for mass sorption are controlled by interplay between effective source dimension and fractal dimension of the active diffusion space. Exponents characterizing scaling of crossover times as a function of anisotropy are largely independent of the fractal dimension of the easy-diffusion pathways.