Труды российских конференций

1. Introduction

The problem of diffusion in a tube of varying cross-section arises in various contexts, including particle transport in porous solids, translocation of ions through channels in biomembranes, and development of technologies for controllable transfer on the nanoscale. The essential physics of the problem is associated with a spatial dependence of the dif-fusing particle entropy, induced by a variation in the tube cross-sectional area along the propagation direction. The most common approach consists in reducing the essentially 3D (or 2D) geometrically restricted Brownian motion to an effective 1D diffusion along the tube axis. The resulting kinetic equation for the effective 1D distribution is known as the Fick-Jacobs equation, which is the Smoluchowski equation with the entropic potential that accounts for changes in the space accessible for the diffusing particle [1]. The con-ventional approach is applicable to systems with smooth enough variations in the con-fining cross-section, but fails at strong forcing.

We study drift and diffusion of a point particle moving under the action of uniform force in a tube of radius  with periodic zero-thickness partitions dividing the cylin-drical tube into identical compartments of length  Each partition bears a circular open-ing of radius  through which the particle can go from one compartment to the other. With this setup, the Fick-Jacobs approach is inapplicable.

The focus is on the dependence of the effective mobility and diffusivity  on the driving force  We find, and this is our main result [2], that the behavior of  and  is qualitatively different from that previously reported in studies of driven transport in tubes and 2D channels of varying cross-section (see Ref. [3] and references therein): (i) monotonically decreases with  while usually it grows between two limiting values; (ii)  diverges as , while usually it remains finite. Thus, entropic effects in diffusive transport in the tube with orthogonal cross-walls are enhanced with the driving force.

2. Two scenarios of the particle motion

Our consideration is based on two different scenarios, which are deduced from the analysis of statistics of the particle’s transition times between neighboring openings, using Brownian dynamics simulations. At zero or small F, a transition from one compartment to the next one is a rare event in the sense that the passage time is much greater than all other characteristic times of the problem. This scenario, which we call homogeneous, suggests using a coarse-grained approach, making the problem analytically treatable. With this approach, 3D motion of the particle in the tube is mapped onto a 1D continuous-time random walk and then the approximation called “boundary homogenization” is invoked.

At strong forcing, one gets a new hierarchy of times. In particular, the time character-rizing the overwhelming majority of intercompatment transitions is much smaller than all other characteristic times. In addition there are very rare, slow transitions associated with the particle diffusive motion along the partition wall, which are however very significant due to their dominating contribution to the transition time average. In other words, the driving force induces intermittency [4] in the particle transitions between neighboring compartments, which is manifested most clearly in a progressive growth of the corres-ponding statistical moments with respect to their order. Thus in this regime, an alternative scenario for the particle motion occurs, which we call intermittent.

3. Results for the effective mobility and diffusivity

Based on the qualitative picture of the two opposite scenarios, homogenous and intermittent, the limiting small-F and large-F behaviors of the effective mobility and diffusivity are derived analytically. In particular, we have shown that  diverges like  as . The formulas obtained are in a good agreement with Brownian dynamics simulations, which are also used to find  and at intermediate values of the driving force for various compartment lengths and opening radii. Our results clearly show that the particle transport through a tube formed by identical compartments may be qualitatively different depending on the compartment shape.

Acknowledgments

Yu.A.M. and V.Yu.Z. acknowledge support by RFBR (Grant No. 10-03-00393). Yu.A.M. was partially supported by DFG (Grant 436 RUS 113/722). A.M.B. was supported by the Intramural Research Program of the NIH, Center for Information Technology. L.V.B. was partially supported by a Leverhulme Research Fellowship.

References

[1]     R. Zwanzig, J. Phys. Chem. 96 (1992) 3926.

[2]     Yu.A. Makhnovskii, A.M. Berezhkovskii, L.V. Bogachev, V.Yu. Zitserman (Manuscript under preparation).

[3]     P.S. Burada, P. Hänggi, F. Marchesoni, G. Schmid, P. Talkner, ChemPhysChem 10 (2009) 45.

[4]     Ya.B. Zeldovich, S.A. Molchanov, A.A. Ruzmaikin, D.D. Sokolov, Usp. Fiz. Nauk 152 (1987) 3 [Sov. Phys. Usp. 30 (1987) 353].