Three-dimensional water diffusion in impermeable cylindrical tubes: Theory versus experiments

Characterizing diffusion of gases and liquids within pores is important in understanding numerous transport processes and affects a wide range of practical applications. Previous measurements of the pulsed gradient stimulated echo (PGSTE) signal attenuation, E(q), of water within nerves and impermeable cylindrical microcapillary tubes showed it to be exquisitely sensitive to the orientation of the applied wave vector, q, with respect to the tube axis in the high-q regime. Here, we provide a simple three-dimensional model to explain this angular dependence by decomposing the average propagator, which describes the net displacement of water molecules, into components parallel and perpendicular to the tube wall, in which axial diffusion is free and radial diffusion is restricted. The model faithfully predicts the experimental data, not only the observed diffraction peaks in E(q) when the diffusion gradients are approximately normal to the tube wall, but their sudden disappearance when the gradient orientation possesses a small axial component. The model also successfully predicts the dependence of E(q) on gradient pulse duration and on gradient strength as well as tube inner diameter. To account for the deviation from the narrow pulse approximation in the PGSTE sequence, we use Callaghan's matrix operator framework, which this study validates experimentally for the first time. We also show how to combine average propagators derived for classical one-dimensional and two-dimensional models of restricted diffusion (e.g. between plates, within cylinders) to construct composite three-dimensional models of diffusion in complex media containing pores (e.g. rectangular prisms and/or capped cylinders) having a distribution of orientations, sizes, and aspect ratios. This three-dimensional modeling framework should aid in describing diffusion in numerous biological systems and in a myriad of materials sciences applications.