Random processes are generally described by Gaussian statistics as formulated by the central limit theorem. However, there exists a large number of exceptions to this rule that can be found in a variety of fields. Diffusion processes are often analyzed by the scaling law $⟨r^2⟩\sim t^{2\beta }$, where the second moment of the diffusion propagator or molecular mean square displacement, $⟨r^2⟩$, in the case of Gaussian diffusion is proportional to $t$, i.e., $\beta =1∕2$. A deviation from Gaussian behavior may be either superdiffusion $(\beta >1∕2)$ or subdiffusion $(\beta <1∕2)$. In this paper we demonstrate that all three diffusion regimes may be observed for the surfactant self-diffusion, on the length scale of ${10}^{-6}\mathrm{m}$ and the time scale of $0.02–0.8\mathrm{s}$. in a system of wormlike micelles, depending on small variations in the sample composition. The self-diffusion is followed by pulsed gradient NMR where one not only measures the second moment of the diffusion propagator, but actually measures the Fourier transform of the full diffusion propagator itself. A generalized diffusion equation in terms of fractional time derivatives provides a general description of all the different diffusion regimes, and where $1∕\beta$ can be interpreted as a dynamic fractal dimension. Experimentally, we find $\beta =1∕4$ and $3∕4$, in the regimes of sub- and superdiffusion, respectively. The physical interpretation of the subdiffusion behavior is that the dominating diffusion mechanism corresponds to a lateral diffusion along the contour of the wormlike micelles. Superdiffusion is obtained near the overlap concentration where the average micellar size is smaller so that the center of mass diffusion of the micelles contributes to the transport of surfactant molecules.

• Received 5 June 2006

DOI:https://doi.org/10.1103/PhysRevE.74.031403