Molecular transport through nanometer sized channels is a topic of great interest to chemical engineering. Typical examples are inorganic porous particles used for heterogeneous catalysis and membranes for separation processes. A proper choice of material requires insight into the relation between structure and composition on the one hand and functionality on the other. As a result of the rapidly growing capabilities of controlled synthesis of nanoporous materials, a thorough understanding of molecular transport processes has become even more important, as revolutionary novel designs are coming within reach.

Biological trans-membrane pores constitute a class of channels which may serve as a source of inspiration for design of technologically relevant structures. As an example, the outer membrane of Escherichia Coli contains various types of water-filled channels, “porins”, which act as molecular sieves, showing high selectivity for hydrophobic, hydrophilic or ionic species. This is one particular instance of effects of heterogeneity on diffusion in nanopores [1]. Previous studies [2] revealed detailed information on diffusion behaviour of water in OmpF. The local diffusion coefficient D in the direction of the channel (z) axis, determined from the linear relation between the mean squared displacement and time t, the Einstein relation <z(t)²> = 2D t, appeared to vary by as much as a factor 5. This is caused by geometrical (cross section) and chemical heterogeneity. It is not clear, however, how this information can be translated into the diffusive transport properties of the full channel. A first approach would be to monitor molecules over distances comparable to the channel length. However, this implies that parts of the trajectories extend into the bulk liquid, so we do not obtain a purely channel-related characterisation. On the other hand, if we were to cut off trajectories in the pore-bulk transition region, finite size effects would influence the results, even for an otherwise homogeneous channel.

In order to overcome this fundamental complication, we analysed our simulation results in terms of a first passage time formalism [3], which implicitly accounts for the finite size of the channel. This amounts to solving the diffusion equation with absorbing boundaries at the channel ends. For a homogeneous channel, transient solutions can be readily obtained as a sum of exponentials. The permeation time distribution of the system, where only particles crossing the full channel are monitored, is of particular importance, as it characterizes the transport through the channel. The cumulative distribution of permeation times appears to be strongly dominated by a single exponential exp(-t/τ), where τ ~ D^-1. While this prediction is strictly valid for a homogeneous (effectively one-dimensional) channel only, we observe similar dominant mono-exponential decay for the strongly heterogeneous OmpF channel as well. The time constant can then be used to define an effective diffusion coefficient D_eff. Other characteristics of the OmpF data, however, clearly reveal the heterogeneity of the channel. We discuss these features by comparing the results with those of a “virtual channel” in bulk water, and a cylindrical channel with a carbon-nanotube-like structure.

References

[1]   M.-O. Coppens and A.J. Dammers, Fluid Phase Equilibria 241, 308-316 (2006).

[2]   D.P. Tieleman and H.J.C. Berendsen, Biophys. J. 74, 2786-2801 (1998).

[3]   S. Redner, “A Guide to First-Passage Processes”, Cambridge University Press (2001).