Single-file diffusion through inhomogeneous nanopores

Strict one-dimensional diffusion, due to geometrical confinement in a nanopore, of an assembly of particles forbids overtaking by each other, giving rise to single-file diffusion (SFD). Smooth carbon nanotube is the epitome of SFD. However, natural nanoporous materials are far from smooth; morphologically, the nanopores' inner surface may provide an inhomogeneous environment for diffusion to occur, giving rise to subnormal diffusion even for an isolated particle diffusing through this fractal landscape. The realm of fractional diffusion (FD) falls under this paradigm. In order to understand the characteristics of SFD through inhomogeneous nanopores, here, we introduce a fractional SFD (FSFD) formalism that deals with a combination of these two phenomena, namely, SFD of particles, each of which are moving subdiffusively in one dimension. For an infinite system, we obtain the mean square displacement (MSD) of the combined entity and our analysis is based on FD equation for particles moving in concert where the single-file correlation is established through reflection principle. For a finite system, we calculate the transport probabilities based on continuous time random walk model. While both the diffusion mechanisms (SFD and FD) acting separately are responsible for slow dynamics at long times, their combined effect leads to ultraslow diffusion. For example, while the long time asymptote of MSD of SFD scales as root t, that for FSFD is root t(alpha), where alpha is the measure of the extent of inhomogeneity. These findings, which are believed to occur in a natural inhomogeneous nanopore, is also important for design and fabrication of nanofluidic devices through which the fluid delivery can be engineered.