Hydrodynamically Enforced Entropic Trapping of Brownian Particles

We study the transport of Brownian particles through a corrugated channel caused by a force field containing curl-free (scalar potential) and divergence-free (vector potential) parts. We develop a generalized Fick-Jacobs approach leading to an effective one-dimensional description involving the potential of mean force. As an application, the interplay of a pressure-driven flow and an oppositely oriented constant bias is considered. We show that for certain parameters, the particle diffusion is significantly suppressed via the property of hydrodynamically enforced entropic particle trapping.

Figure 1

A segment of the corrugated channel at cross section $z=0$, confining the overdamped motion of pointlike Brownian particles. The wiggling profiles are given by periodic functions ${\omega }_{\pm }(x)$, see Eq. (11); a unit cell is marked by the separating dashed lines. The quantities $\Delta \omega$ and $\Delta \Omega$ denote the minimal and maximal channel widths, respectively. The field lines depict a typical external force field $\mathbf{F}(\mathbf{r})$, acting on the particles which are the result of a competition of a constant bias $f$ and an oppositely oriented divergence-free force from the driven solvent induced by a pressure $p(\mathbf{r})$ with pressure change $\Delta p$ along a unit cell. This results both in vortices and stagnation points, note the dots.

Figure 2

Comparison of Brownian dynamics simulations (markers) based on Eqs. (1, 2) with the Fick-Jacobs approximation (lines) for a corrugated channel with the profiles given by Eq. (11) for $\Delta \Omega =0.5$ and $\Delta \omega =0.1$. Panel (a): Mean particle velocity $⟨\dot{x}⟩$ versus pressure drop $\Delta p$ for different force magnitudes $f$. The solid line corresponds to $f=0$, Eq. (12), the dashed lines represent Eq. (7) for $f\ne 0$. Panel (b): The dependence of $f_{\mathrm{cr}}$ on $\Delta p_{\mathrm{cr}}$ for different $\delta$ is depicted [lines: Eq. (14)]. Panel (c): Effective diffusion $D_{\mathrm{eff}}$ as a function of $\Delta p$ for different $f$ [lines: Eq. (8)] and the horizontal dashed-dotted line corresponds to $D_{\mathrm{eff}}=D_0$. Panel (d): Stationary joint PDF $P(x,y)$ (color coding) obtained via direct simulation of Eqs. (1, 3, 10), and force field $\mathbf{F}(\mathbf{r})$ (lines) for $f={10}^2$ and $\Delta p=6.5\times {10}^4$, showing hydrodynamically enforced entropic trapping.

Figure 3

Snapshot of marginal PDF $P(x,t)$ at $t=100$ for different force strengths $f$ in units of $f_{\mathrm{cr}}=100$ in a corrugated channel with $\Delta \omega =0.1$. The width of $P(x,t)$ is several magnitudes smaller compared to the case of unbounded geometry $\propto D_{0}t$. Inset: Péclet number $|⟨\dot{x}⟩|/D_{\mathrm{eff}}$ versus $f/f_{\mathrm{cr}}$ for $\delta =0.2$ and $\delta =1$ (straight channel). The maximum width $\Delta \Omega =0.5$ and $\Delta p=6.5\times {10}^4$ are kept fixed.