Exploring diffusion across permeable barriers at high gradients. II. Localization regime

We present an analytical solution of the one-dimensional Bloch-Torrey equation for diffusion across multiple semi-permeable barrier. This solution generalizes the seminal work by Stoller, Happer, and Dyson, in which the non-Gaussian stretched-exponential behavior of the pulsed-gradient spin-echo (PGSE) signal was first predicted at high gradients in the so-called localization regime. We investigate how the diffusive exchange across a semi-permeable barrier modifies this asymptotic behavior, and explore the transition between the localization regime at low permeability and the Gaussian regime at high permeability. High gradients are suitable to spatially localize the contribution of the nuclei near the barrier and to enhance the sensitivity of the PGSE signal to the barrier permeability. The emergence of the localization regime for three-dimensional domains is discussed.