Transport of Brownian particles in confined geometries - Steps beyond the Fick-Jacobs approach

Abstract

In this work, we investigate the transport of Brownian particles in confined geometries where entropic barriers play a decisive role. The commonly used FickJacobs approach provides a powerful tool to capture many properties of entropic particle transport. Unfortunately, its applicability is mainly limited to the overdamped motion of point-like objects in weakly corrugated channels. We perform asymptotic perturbation analysis of the probability distribution in terms of an expansion parameter specifying the channel corrugation. With this methodology, exact solutions of the associated stationary Smoluchowski equation are derived. In particular, we demonstrate that the leading order of the series expansion is equivalent to the Fick-Jacobs approach. By means of the higher expansion orders, which become significant for strong channel corrugation, we obtain corrections to the key particle transport quantities in the diffusion dominated limit. In contrast to the commonly used Lifson-Jackson formula, these corrections can be calculated exactly for most smooth and discontinuous boundaries, and they provide even better agreements with simulation results. Going one step further, we overcome the limitation of the Fick-Jacobs approach to curl-free forces (scalar potentials). For this purpose, we study entropic transport caused by force fields containing curl-free and divergence-free (vector potential) parts. Based on our methodology, we develop a generalized Fick-Jacobs approach leading to a one-dimensional, energetic description. As an exemplary application, we consider the prevailing situation in microfluidic devices, where Brownian particles are subject to external constant forces and pressure-driven flows. The analysis of particle transport leads to the interesting finding that the vanishing of the mean particle current is accompanied by a significant suppression of diffusion, yielding the effect of hydrodynamically enforced entropic trapping. This effect offers a unique opportunity to efficiently separate particles of the same size. Since separation and sorting by size is a main challenge in basic research, we intend to incorporate the particle size into the Fick-Jacobs approach. Finite particle size inevitably causes additional forces, e.g., hydrodynamic particle-particle and particle-wall interactions. We identify the limits for the ratio of particle size to pore size and the mean distance between particles, for which these forces can safely be disregarded in experiments. Moreover, we demonstrate that within these limits the analytic expressions for the key transport quantities, derived for point-like particles, can be applied to extended objects, too. We study the impact of the solvent’s viscosity on entropic transport. If the time scales separate, adiabatic elimination results in an effective, kinetic description for particle transport in the presence of finite damping. The possibility of such description is intimately connected with equipartition and vanishing correlation between the particle’s velocity components. Numerical simulations show that this approach is accurate for moderate to strong damping and for weak forces. For strong external forces, equipartition may break down due to reflections at the boundaries. This leads to a non-monotonic dependence of particle mobility on the force strength. Finally, we study the impact of boundary conditions on entropic transport. We show numerically that perfectly inelastic particle-wall collisions can rectify entropic transport. In summary, this work shows how experimentally relevant issues such as strong channel corrugation, sophisticated external force fields, particle size, particle inertia, and the solvent’s viscosity can be incorporated into the Fick-Jacobs approach.

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Cite this paper

@inproceedings{Martens2013TransportOB, title={Transport of Brownian particles in confined geometries - Steps beyond the Fick-Jacobs approach}, author={Steffen Martens and J Olbertz and S. R. Klapp}, year={2013} }