Effects of microgeometry and surface relaxation on NMR pulsed-field-gradient experiments: Simple pore geometries

Abstract

We derive an expression for the magnetization M(k,\ensuremath{\Delta}) in a pulsed-field-gradient experiment for spins diffusing in a confined space with relaxation at the pore walls. Here k=\ensuremath{\gamma}\ensuremath{\delta}g, \ensuremath{\delta}= pulse width, g= gradient strength, \ensuremath{\gamma}= the gyromagnetic ratio, and \ensuremath{\Delta} is the time between gradient pulses. We show that the deviation of -ln[M(k,\ensuremath{\Delta})/M(0,\ensuremath{\Delta})] from quadratic behavior in k in experiments in porous media can be a more sensitive probe of the microgeometry (size, connectivity, size distribution, shape, etc.), than either the enhancement of 1/${\mathit{T}}_{1}$ over the bulk water value, or the macroscopic diffusion coefficient, which is derived from the slope of -ln[M(k,\ensuremath{\Delta})/M(0,\ensuremath{\Delta})] at small ${\mathit{k}}^{2}$, in the limit of large \ensuremath{\Delta}. We propose some simple models of randomly oriented tubes and sheets to interpret the k dependence of the amplitude beyond the leading small-k quadratic behavior. When the macroscopic diffusion coefficient is unobtainable, due to the decay, the present considerations should be useful in extracting geometrical information. The effective diffusion constant derived from NMR exactly equals that derived from electrical conductivity only when the surface relaxivity is zero, but can be close to each other in favorable circumstances even for finite surface relaxivity. Exact solutions with partially absorbing boundary conditions are obtained for a slab and a sphere to infer that the normalized amplitude M(k,\ensuremath{\Delta},\ensuremath{\rho})/M(0,\ensuremath{\Delta},\ensuremath{\rho}) depends only weakly on the surface relaxivity \ensuremath{\rho} for monodisperse convex-shaped pores in the parameter ranges of interest. We also obtain expressions for the mean lifetime of the amplitude in the geometries considered.