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Knudsen Diffusion in Rough Nanopores: Discussion of a Paradox

Anton J. Dammers and Marc-Olivier Coppens. DelftChemTech, Delft University of Technology, Julianalaan 136, Delft, 2628 BL, Netherlands

Nanoporous materials play an important role in chemical engineering, key examples being porous particles used in heterogeneous catalysis and membranes for separation processes. In gas-phase applications, the intrinsic mean free path of the molecules is often considerably larger than typical pore cross-sections. Consequently, diffusive transport through the pores is dominated by molecule-wall interactions, which marks the so-called Knudsen regime.

In previous work [1][2] the effect of fractally rough wall structure on diffusion was studied by kinetic Monte-Carlo simulations, using an analytically tractable randomized Koch surface as test case. This confirmed the theoretically predicted reduction of the self-diffusion coefficient Ds [3] relative to that of a smooth pore. The underlying theoretical model is based on temporary trapping of a particle in the local labyrinth of surface corrugations, which effectively causes a delay of the particle's global motion only. The pore is modelled as a smooth channel, dressed with fjords, which carry the statistical properties of the surface roughness. A particle enters a fjord and after multiple collisions it returns at, on average, the same position. Consequently, it has no effect on the probability ft that a particle, which enters the pore at one end, leaves it at the other. Due to the equivalence of the net flux entering the pore and that related to the concentration gradient over the pore (Fick's law) the transport diffusivity is given by Dt = ft uL/4, where L is the pore length and u is the average molecular velocity. Therefore, this model predicts that Dt is independent of surface roughness. Monte-Carlo simulations [1][2] corroborated this conclusion.

However, these findings seem to contradict the common notion that in a system where interactions between particles are absent, self- and transport diffusivities should be equal. This raised a considerable amount of debate, which eventually led Russ and co-workers [4] to repeat our simulations with slightly modified wall roughness, belonging to the same family of randomized Koch surfaces as used previously. They found equivalence of Ds and Dt, at odds with our results. To explain this discrepancy, they disputed our simulation approach, as it was supposed to lead to spurious entrance effects.

We will present an analysis of these diverging results. In [4] our simulation procedure [2] was interpreted incorrectly, and in fact the curves (fig.4 in [4]), which are supposed to prove its failure, can be appropriately scaled to the correct ones. While other aspects of our methodology are still under review, we developed new theoretical insights. Translation of a rough pore into an effective smooth pore with trapping at collision points implies that real time should be replaced by operational time, accounting for the delays. We will show that this leads to an effective average molecular velocity ueff, yielding Dt = ft ueffL/4. This introduces exactly the required factor to obtain Dt = Ds, in accordance with [4]. These results will be discussed in a broader context, incorporating the Lvy walk character of the particle motion.

 

References

[1]   K. Malek and M.-O. Coppens, Phys. Rev. Lett. 87, 125505 (2001).

[2]   K. Malek and M.-O. Coppens, J. Chem. Phys. 119, 2801 (2003).

[3]   M.-O. Coppens and G.F. Froment, Fractals 3, 807 (1995).

[4]   S. Russ, S. Zschiegner, A. Bunde and J. Kaerger, Phys. Rev. E 72, 030101 (2005).