Derivation of Multi-Exponential Magnetic Resonance Relaxation Equations in Simple Pore Geometries

Abstract The common interpretation of magnetic resonance relaxation time distribution of liquids in porous media assumes a one-to-one relationship between the pore size and the relaxation time constants. This common conviction may not be correct in many microporous materials. Each pore size may be associated with more than one peak in the NMR relaxation time distributions: a single dominant peak and also possibly one or a few minor peaks. The appearance of minor peaks is due to the non-vanishing nonground eigenvalues of the diffusion–relaxation equation. Brownstein and Tarr (Phys Rev A 19:2446, 1979) described these features, but their solutions at conditions beyond the fast-diffusion regime are not widely adopted. We provide the derivation of Brownstein–Tarr equations for multi-exponential magnetic resonance relaxation decay for liquids in simple pore geometries. General solutions are presented for planar, cylindrical, and spherical pores—as well as two limiting cases of fast and slow diffusion for each geometry. Similar solutions are also relevant to first-order dilute reactions in porous media in heterogeneous reaction–diffusion systems. We hope that the availability of these derivations helps wider adoption of more realistic interpretation of magnetic resonance relaxation in porous media in the light of the multi-exponential Brownstein–Tarr model.