A proposed 2D framework for estimation of pore size distribution by double pulsed field gradient NMR

Reconstructing a pore size distribution of porous materials is valuable for applications in materials sciences, oil well logging, biology, and medicine. The major drawback of NMR based methods is an intrinsic limitation in the reconstruction which arises from the ill-conditioned nature of the pore size distribution problem. Consequently, while estimation of the average pore size was already demonstrated experimentally, reliable evaluation of pore size distribution remains a challenging task. In this paper we address this problem by analyzing the mathematical characteristics that create the difficulty and by proposing an NMR methodology and a numerical analysis. We demonstrate analytically that an accurate reconstruction of pore size distribution is problematic with the current known strategies for conducting a single or a double pulsed field gradient (s-PFG, d-PFG) experiment. We then present a method for choosing the experimental parameters that would significantly improve the estimation of the size distribution. We show that experimental variation of both q (the amplitude of the diffusion gradient) and ϕ (the relative angle between the gradient pairs) is significantly favorable over single and double-PFG applied with variation of only one parameter. Finally, we suggest a unified methodology (termed Concentric d-PFG) that defines a multidimensional approach where each data point in the experiment is characterized by ϕ and q. The addition of the angle parameter makes the experiment sensitive to small compartment sizes without the need to use strong gradients, thus making it feasible for in-vivo biological applications.