Signal attenuation of PFG restricted anomalous diffusions in plate, sphere, and cylinder

Pulsed field gradient (PFG) NMR is a noninvasive tool to study anomalous diffusion, which exists widely in many systems such as in polymer or biological systems, in porous material, in single file structures and in fractal geometries. In a real system, the diffusion could be a restricted or a tortuous anomalous diffusion, rather than a free diffusion as the domains for fast and slow transport could coexist. Though there are signal attenuation expressions for free anomalous diffusion in literature, the signal attenuation formalisms for restricted anomalous diffusion is very limited, except for a restricted time-fractional diffusion within a plate reported recently. To better understand the PFG restricted fractional diffusion, in this paper, the PFG signal attenuation expressions were derived for three typical structures (plate, sphere, and cylinder) based on two models: fractal derivative model and fractional derivative model. These signal attenuation expressions include two parts, the time part T_n(t) and the space part X_n(r). Unlike normal diffusion, the time part T_n(t) in time-fractional diffusion can be either a Mittag-Leffler function from the fractional derivative model or a stretched exponential function from the fractal derivative model. However, provided the restricted normal diffusion and the restricted time-fractional diffusion are in an identical structure, they will have the same space part X_n(r) as both diffusions have the same space derivative parameter β equaling 2, therefore, they should have similar diffractive patterns. The restricted general fractional diffusion within a plate is also investigated, which indicates that at a long time limit, the diffusion type is insignificant to the diffractive pattern that depends only on the structure and the gradient pulses. The expressions describing the time-dependent behaviors of apparent diffusion coefficient D_f,app for restricted anomalous diffusion are also proposed in this paper. Both the short and long time-dependent behaviors of D_f,app are distinct from that of normal diffusion. The general expressions for PFG restricted curvilinear diffusion of tube model were derived in a conventional way and its result agree with that obtained from the fractional derivative model with α equaling 1/2. Additionally, continuous-time random walk simulation was performed to give good support to the theoretical results. These theoretical results reported here will be valuable for researchers in analyzing PFG anomalous diffusion.