Overdamped diffusion through a channel of varying cross-section

Characterizing the transport properties of individual nanoparticles in pores is an important challenge for materials, environmental, and biological sciences. In this article, we analyze the diffusion of spherical particles within a long pore featuring a varying cross-section to demonstrate how geometric variations influence mass transport. We consider two distinct double-cone pore configurations. The first consists of two truncated cones joined at their wide ends, resulting in a pore that expands to its maximum width at the midpoint and then narrows. The second configuration involves two truncated cones joined at their narrow ends-apex to apex-producing a pore that narrows to its minimum width at the center and then widens again. Our findings reveal that the nature of effective diffusion depends on the initial geometry of the channel: when the pore initially widens, superdiffusive behavior is observed; conversely, when it initially narrows, subdiffusive behavior emerges. In addition to the analysis of the mean squared displacement, we examine the mean first passage time and its symmetries, showing that reversal of the channel slope does not change the exit time. We conduct some analytical analysis to evaluate the quality of the approximation provided by the Fick-Jacobs equation, revealing that this approach underestimates the mean first passage times. Furthermore, we demonstrate that the approximation worsens with the increasing number of spatial dimensions.