Using NMR to Measure Fractal Dimensions

TL;DR

This paper discusses how NMR measurements can be used to determine the fractal dimensions of surfaces by analyzing the time-dependent diffusion coefficient, challenging previous assumptions and providing new analysis methods.

Contribution

It introduces a new theoretical relationship for diffusion on fractal surfaces and applies preliminary NMR data to validate this approach.

Findings

The diffusion coefficient follows a specific power law for fractal surfaces.

Preliminary NMR data supports the proposed fractal diffusion model.

The usual diffusion law is invalid for fractal geometries.

Abstract

A comment is made on the recent PFG NMR measurements by Stallmach, et al. on water-saturated sands [Phys. Rev. Lett. 88, 105505 (2002)]. It is pointed out that the usual law for the time-dependent diffusion coefficient D(t) used by these authors is not valid for a fractal surface. It is shown that (1-D(t)/D0) \~ t^[(3-Ds)/2] at short times for a surface of fractal dimension Ds, where D0 is the bulk diffusion coefficient. Preliminary PFG NMR data on water saturated limestone and plastic beads are presented to illustrate this analysis.