A Fractional Calculus Approach to Adsorbate Dynamics in Nanoporous Materials

Abstract

Using NMR–techniques one can study the dynamics of various liquid adsorbates in diverse nanoporous media like porous silica glasses, polymers or biological tissue. Anomalous adsorbate dynamics in disordered media can be interpreted either in terms of a random walk approach or on the basis of a fractional diffusion equation. Fractional diffusion may be characterized by the variance $< \Delta {{r}^{2}} > \mathop{{here}}\limits_{ = } < {{r}^{2}}(t) > \sim {{t}^{{2/{{d}_{w}}}}} = {{t}^{\alpha }}$ where the anomalous diffusion exponent d _w = 2/α is a measure for the deviation from the classical Einstein relation result d _w = 2 or α = 1 Analyzing experimental NMR–data provides evidence for superdiffusive surface displacements with $1 \leqslant {{d}_{w}} = 2 or 1 \leqslant \alpha \leqslant 2$. The theoretical interpretation of the NMR–measurements is based on a time–fractional diffusion equation which leads to a propagator represented by a Fox’s H–function.