Effect of impermeable interfaces on apparent diffusion coefficient in heterogeneous media

It is known that the short-time behavior of the diffusion coefficient, which is measurable by nuclear magnetic resonance (NMR), provides an estimate of the specific surface of porous samples filled with an NMR-detectable fluid. This method is not directly applicable to the exploration of structure of impermeable restrictions in mixtures such as living tissues, in which the inherent microscopic structure in the bulk medium results in the same pattern of the temporal dynamics of the diffusion coefficient as the impermeable restrictions studied. Here, an approach to describe diffusion and the boundary effect in heterogeneous media is developed in the framework of a cumulant expansion of the NMR signal. The leading term of this expansion is determined by the velocity auto-correlation function which is expressed in terms of properties of microscopic transport in the medium. Given these properties, the apparent diffusion coefficient as measured by NMR can be found by a straightforward integration. Calculations are performed in one spatial dimension.