Anomalous diffusion coefficient in disordered media from NMR relaxation

TL;DR

This paper derives a relationship between the fractional diffusion coefficient and NMR relaxation data, enhancing the use of NMR techniques to measure anomalous diffusion in complex media.

Contribution

It provides a detailed, pedagogical derivation linking the fractional diffusion coefficient to experimentally observable NMR relaxation data, facilitating experimental measurement.

Findings

Derived a formula connecting fractional diffusion coefficient with NMR relaxation data

Demonstrated NMR relaxation as a powerful tool for measuring anomalous diffusion

Compared NMR relaxation and PFG NMR methods for studying complex media

Abstract

Application of fractional calculus to the description of anomalous diffusion and relaxation processes in complex media provided one of the most impressive impulses to the development of statistical physics during the last decade. In particular the so-called fractional diffusion equation enabled one to capture the main features of anomalous diffusion. However the price for this achievement is rather high - the fractional diffusion coefficient becomes an involved function of a characteristic of the media (e.g., that of the radius of pores in the case of the porous one). Revealing this dependence from the first principles is one of the main problems in this field of science. Another one still remains that of extracting this dependence from the experiment. The latter problem is tackled in the present paper. Our aim is to provide detailed and pedagogical deriving the relationship of the fractional diffusion coefficient with experimentally observable value from nuclear magnetic resonance (NMR) spin-lattice relaxation data. The result obtained promotes the NMR relaxation method to become a powerful tool in solving the problem of experimental measuring the fractional diffusion coefficient. Also the merits and limitations of NMR relaxation method and pulsed-field gradient (PFG) NMR for the research of anomalous diffusion are compared and discussed.