Phase structure of polymers by dipolar-induced spin diffusion: Model spin diffusion calculations

The evolution of the spatial inhomogeneity of magnetization induced by spin diffusion is investigated theoretically for different models of a multiphase polymer solid. It is shown that the decay of the total magnetization associated with the domains where the magnetization initially resides is affected by the distributions in the size of the domains and the separation between them. The models examined include the two-phase/multidomain (2P-MD) case, in which the system consists of two phases with an infinite number of domains in each phase, and the two-phase/four-domain (2P-4D) model, in which the system has two phases with two domains in each. The spin diffusion equation is solved for these models by using a simple two-value distribution in the domain width or the domain separation. The solutions for the magnetization of the 2P-MD model show a slower decay in the long-time regime in comparison to those of the 2P-4D case. The solutions of the latter give a reasonable approximation to the former, provided that the spread in the distribution is sufficiently smaller than the average of the distribution. The simple two-phase/two-domain (2P-2D) model, in which the two phases are represented by a single domain each, is also investigated. In this case, the effect of the distribution is incorporated by taking an ensemble average. It is shown that there is no correct way of taking the average, and the results always yield erroneous long-time behavior for the magnetization. The results of the 2P-MD, 2P-4D, and 2P-2D models are applied to the spin diffusion experiment of a semicrystalline polypropylene film.