Quantification of pore size distribution using diffusion NMR: experimental design and physical insights.

Pulsed field gradient (PFG) diffusion NMR experiments are sensitive to restricted diffusion within porous media and can thus reveal essential microstructural information about the confining geometry. Optimal design methods of inverse problems are designed to select preferred experimental settings to improve parameter estimation quality. However, in pore size distribution (PSD) estimation using NMR methods as in other ill-posed problems, optimal design strategies and criteria are scarce. We formulate here a new optimization framework for ill-posed problems. This framework is suitable for optimizing PFG experiments for probing geometries that are solvable by the Multiple Correlation Function approach. The framework is based on a heuristic methodology designed to select experimental sets which balance between lowering the inherent ill-posedness and increasing the NMR signal intensity. This method also selects favorable discrete pore sizes used for PSD estimation. Numerical simulations performed demonstrate that using this framework greatly improves the sensitivity of PFG experimental sets to the pores' sizes. The optimization also sheds light on significant features of the preferred experimental sets. Increasing the gradient strength and varying multiple experimental parameters is found to be preferable for reducing the ill-posedness. We further evaluate the amount of pore size information that can be obtained by wisely selecting the duration of the diffusion and mixing times. Finally, we discuss the ramification of using single PFG or double PFG sequences for PSD estimation. In conclusion, the above optimization method can serve as a useful tool for experimenters interested in quantifying PSDs of different specimens. Moreover, the applicability of the suggested optimization framework extends far beyond the field of PSD estimation in diffusion NMR, and reaches design of sampling schemes of other ill-posed problems.