Brownian transport in corrugated channels with inertia

P. K. Ghosh, P. Hänggi, F. Marchesoni, F. Nori, and G. Schmid

Phys. Rev. E 86, 021112 – Published 13 August 2012

Abstract

Transport of suspended Brownian particles dc driven along corrugated narrow channels is numerically investigated in the regime of finite damping. We show 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 the the channel corrugation and the drive intensity. With such a diffusion length being inversely proportional to the damping constant, transport through sufficiently narrow obstructions turns out to be always sensitive to the viscosity of the suspension fluid. The inertia corrections to the transport quantifiers, mobility, and diffusivity markedly differ for smoothly and sharply corrugated channels.

Received 29 November 2011

DOI:https://doi.org/10.1103/PhysRevE.86.021112

Authors & Affiliations

P. K. Ghosh^1,2, P. Hänggi², F. Marchesoni³, F. Nori¹, and G. Schmid²

¹Advanced Science Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan
²Institut für Physik Universität Augsburg, D-86135 Augsburg, Germany
³Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy

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Vol. 86, Iss. 2 — August 2012

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Figure 1
Sketch of a smoothly corrugated (a) and a compartmentalized 2D channel (b) directed along the $x$ axis. In both cases the channel unit cell is $x_{L}$ long and $y_{L}$ wide; the radius of the connecting bottlenecks or pores is $Δ$ .Reuse & Permissions
Figure 2
Rescaled mobility, $γ μ$ , in a smoothly corrugated channel with $r = x_{L} / y_{L} = 1$ , $Δ / y_{L} = 0.1$ , and (a) vs. $γ / γ_{F}$ for different $F$ ; (b) vs. $F / F_{T}$ for different $γ$ . The relevant scaling parameters are $F_{T} = k T / Δ$ , $γ_{T} = \sqrt{m k T} / Δ$ [Eq. (8)] and $γ_{F} = \sqrt{m F / Δ}$ [Eq. (9)]. The dashed lines represent, respectively, the fitting power laws ${(γ / γ_{F})}^{α}$ in (a) and ${(F / F_{T})}^{- α / 2}$ in (b), both with $α = 1.4$ (see Sec. 3). In (c) $γ μ$ is plotted vs. $γ / γ_{F}$ (main panel) and $F / F_{0}$ (inset) for $γ / γ_{T} = 0.8$ , $F / F_{T} = 125$ , and different cross-section ratios, $Δ / y_{L}$ . The corresponding fitting exponents $α$ are also reported in the legend. In the inset, $F$ is expressed in units of $F_{0}$ instead of $F_{T}$ for graphical reasons. The dependence of $γ μ$ on the geometry of the channel unit cell for low damping and small drives is illustrated in (d), where $γ μ$ is plotted vs. $r γ / γ_{T}$ . The predicted linear law with slope $π / 4$ [32] is represented by a dotted line [see also Eq. (12) and text following].Reuse & Permissions
Figure 3
Rescaled diffusivity, $D / D_{0}$ , vs. $F / F_{0}$ (main panel) and $D / D_{0}$ vs. $F / F_{T}$ (inset) in the corrugated channel of Eq. (2) with $r = 1$ , $Δ / y_{L} = 0.1$ , and different $γ$ . The scaling parameters introduced here are $D_{T} = k T / γ_{T}$ and $F_{0} = γ \sqrt{k T / m}$ . The solid line in the inset is the heuristic power law of Eq. (17).Reuse & Permissions
Figure 4
Transport in a compartmentalized channel with $r = x_{L} / y_{L} = 1$ : rescaled mobility, $γ μ$ , vs. $F / F_{0}$ (a) and vs. $γ / γ_{T}$ (b); diffusivity, $D / D_{0}$ , vs. $F / F_{T}$ (c). The remaining simulation parameters are reported in the legends. The relevant scaling parameters are $F_{0} = γ \sqrt{k T / m}$ , $F_{T} = k T / Δ$ , and $γ_{T} = \sqrt{m k T} / Δ$ . Inset of (a): $γ μ$ vs. $F / F_{T}$ for different $r$ and $Δ / y_{L}$ . Inset of (b): $γ μ$ vs. $γ / γ_{T}$ for different $Δ / y_{L}$ and $r$ . Inset of (c): $D / D_{0}$ vs. $Δ / y_{L}$ for $F / F_{T} = 2 \cdot 10^{3}$ . The dotted curves represent the approximate analytical expressions of Eqs. (11) and (15), respectively, for the mobility in (b) and the diffusivity in (c) (main panel and insets). In (b) the quantity ${γ μ |}_{\infty}$ was estimated from the horizontal asymptotes. Note that $γ μ_{\infty}$ (see Sec. 2) is known to be proportional to $Δ$ for $F \to \infty$ [horizontal arrows, Eq. (4)] and to $| \ln Δ |$ for $F \to 0$ [19].Reuse & Permissions
Figure 5
Channel with tunable corrugation [Eq. (16)]: rescaled mobility, $γ μ$ vs. $F / F_{T}$ for different $η$ . Other simulation parameters: $r = 1$ , $Δ / y_{L} = 0.1$ , and $γ / γ_{T} = 0.8$ . The dashed line is the power law ${(F / F_{T})}^{- α / 2}$ with $α = 1.4$ drawn in Fig. 2b for $η = 2$ .Reuse & Permissions

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