Detectable inertial effects on Brownian transport through narrow pores

We investigate the transport of suspended Brownian particles dc driven along corrugated narrow channels in a regime of finite damping. We demonstrate that inertial corrections cannot be neglected as long as the width of the channel bottlenecks is smaller than an appropriate particle diffusion length, which depends on both the temperature and the strength of the dc drive. Therefore, transport through sufficiently narrow constrictions turns out to be sensitive to the viscosity of the suspension fluid. Applications to colloidal systems are discussed. Copyright c © EPLA, 2012 Brownian transport in narrow corrugated channels is a topic of potential applications to both natural [1,2] and artificial devices [3]. Depending on the amplitude and geometry of the wall modulation, corrugated channels fall within two distinct categories: i) smoothly corrugated channels, typically modeled as quasi–one-dimensional (1D) periodic channels with axial symmetry and unit cells delimited by smooth, narrow bottlenecks, also called pores [4–12]; ii) compartmentalized channels [13–16], formed by identical compartments separated by thin dividing walls and connected by narrow pores centered around the channel axis. Brownian transport in such sharply corrugated channels must be treated as an irreducible twoor three-dimensional diffusion problem [17]. More importantly, for both categories of corrugated channels most analytical results only apply under the condition of very narrow pores [2,5,17]. Corrugated channels are often used to model transport of dilute mixtures of small particles (e.g., biomolecules, colloids or magnetic vortices) in confined geometries [3]. Each particle is subjected to thermal fluctuations with temperature T and large viscous damping γ, and a homogeneous constant bias of strength F parallel to the channel axis. Such a dc drive is applied by coupling the particle to an external field (e.g., by attaching a dielectric or magnetic dipole, or a magnetic flux to the particle), without inducing drag effects on the suspension fluid. Interparticle and hydrodynamic interactions are thus ignored for simplicity (these assumptions are discussed in ref. [4]). In this paper we investigate the relevance of inertial effects due to the viscosity of the suspended particle. As is often the case with biological (and most artificial) suspensions [3], the Brownian particle dynamics in the bulk can be regarded as overdamped. This corresponds to i) formally setting the mass of the particle to zero, m= 0, or, equivalently, to making the friction strength γ tend to infinity, and ii) assuming F smaller than the thermal force F0 = γ √ kT/m (Smoluchowski approximation) [18]. The current literature on corrugated channels invariably assumes such an overdamped limit. But how large is an “infinite” γ (or how small can be a “zero” m)? The answer, of course, depends on the geometry of the channel. Our main conclusion is that the overdamped dynamics assumption for Brownian diffusion through pores of width Δ subjected to a homogeneous drive F , applies only for γ≫ √ mkT/Δ and γ≫ √ mF/Δ, irrespective of the degree of corrugation. This means that the inertial effects cannot be neglected as long as the Brownian diffusion is spatially correlated on a length (lT = √ mkT/γ at small dc drive, or lF =mF/γ 2 at large dc drive) of the order of, or larger than, the pore width Δ. Therefore, for sufficiently narrow pores or sufficiently large drives, inertia always comes into play by enhancing the blocking