Ratchet effect in an asymmetric two-dimensional system of Janus particles

Pavol Kalinay and František Slanina

Phys. Rev. E 108, 014606 – Published 25 July 2023

Abstract

We consider a disk-like Janus particle self-driven by a force of constant magnitude $f$ , but an arbitrary direction depending on the stochastic rotation of the disk. The particle diffuses in a two-dimensional channel of varying width $2 h (x)$ . We applied the procedure mapping the $2 + 1$ -dimensional Fokker-Planck equation onto the longitudinal coordinate $x$ ; the result is the Fick-Jacobs equation extended by the spatially dependent effective diffusion constant $D (x)$ and an additional effective potential $- γ (x)$ , derived recursively within the mapping procedure. Unlike the entropic potential $\sim ln h (x)$ , $γ (x)$ becomes an increasing or decreasing function also in periodic channels, depending on the asymmetry of $h (x)$ and thus it visualizes the net force driving the ratchet current. We demonstrate the appearance of the ratchet effect on a trial asymmetric channel; our theory is verified by a numerical solution of the corresponding Fokker-Planck equation. Isotropic driving force $f$ results in the monotonic decrease of the ratchet current with a growing ratio $α = D_{R} / D_{T}$ of the rotation and the translation diffusion constants; asymptotically going $\sim 1 / α^{2}$ . If we allow anisotropy of the force, we can observe the current reversal depending on $α$ .

Received 17 March 2023
Accepted 28 June 2023

DOI:https://doi.org/10.1103/PhysRevE.108.014606

Physics Subject Headings (PhySH)

Brownian motion Nonequilibrium statistical mechanics

Brownian ratchet Self-propelled particles

Series expansions & exact enumeration

Statistical Physics & Thermodynamics

Authors & Affiliations

Pavol Kalinay ¹ and František Slanina²

¹Institute of Physics, Slovak Academy of Sciences, Dúbravska cesta 9, 84511, Bratislava, Slovakia
²Institute of Physics, Czech Academy of Sciences, Na Slovance 2, CZ-18200, Praha, Czech Republic

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Vol. 108, Iss. 1 — July 2023

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Images

Figure 1
Scheme of the considered device. An active (red) spot on the disk drives the particle in the opposite direction by the force $f$ . The particle diffuses in a channel bounded by $- h (x) < y < h (x)$ and also rotates stochastically in the angle $ϕ$ .
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Figure 2
(a) Inverse effective diffusion coefficient $1 / D (x)$ according to Eq. (2.20) (solid line) for the trial shaping function (3.4), compared to the Reguera-Rubí formula (2.24) (dotted line) and its first order (Zwanzig) approximation (dashed line). (b) Correction $γ (x)$ (solid line) to the entropic potential $\sim ln h (x)$ (dashed line), obtained by integration of Eqs. (2.11), (2.21), and (2.22) for the same function and parameters. The dotted line marks decrement of $γ (x)$ over one period.
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Figure 3
Potential decrement $Δ γ$ over one period of the channel (3.4) (the left scale) and the corresponding ratchet current $J$ , Eq. (3.3) for one particle per period (the right scale), depending on the ratio $α = D_{R} / D_{T}$ of the rotation and translation diffusion constants of the Janus particle. The long dashed line describes the results up to the second order according to Eqs. (2.21) and (2.22); the next order is added in the solid line. The dotted and the short dashed lines correspond to the asymptotics (3.2) and with Eq. (A6) added, respectively. The numerical solutions of the fully dimensional problem are depicted by the orange triangles ( $f = 0.1$ , 0.2, 0.5), green diamonds ( $f = 1$ ), and blue disks ( $f = 2$ ).
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Figure 4
Potential decrement $Δ γ$ over one period and the ratchet current $J$ vs. $α = D_{R} / D_{T}$ for $ε = 1$ (blue), 0.25 (orange), 0.04 (green), and 0.01 (red), calculated according to Eqs. (2.21) and (2.22) (dashed and solid lines). The asymptotics (3.2) is described by the dotted line. The symbols (red triangles, green diamonds, orange squares, and blue disks) depict the numerical data for the corresponding values of $ε$ .
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Figure 5
(a) Contour plot of the 2D density $\bar{ρ} (x, y)$ averaged over the orientation of the force (color scale) in the upper half-channel shaped by $h (x)$ , Eq. (3.4), $ε = 0.25$ and $g = f / 2$ . The streamlines depict the corresponding averaged current density $\vec{j} (x, y)$ . (b) The same plot for the channel shaped by $h (x) / 2$ , but with $ε = 1$ and $g = f$ . Black arrows here depict polarization. The net flux in both cases $J = - 6.33 \times 10^{- 6} f^{2}$ (c) Deflection $δ$ of the polarization from the normal to the boundary (in radians, dashed line), gradient of the averaged density along the boundary, ${\bar{ρ}}^{'} = d \bar{ρ} [x, h (x)] / d x$ (dotted line), and the current density $j$ at the boundary (solid line).
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Figure 6
The ratchet current $J$ vs. $α$ for an anisotropic self-propulsion; $g = 0.1 > \sqrt{ε} f = 0.05$ . Full line represents the leading terms, given by Eqs. (2.21), (2.22), the dashed line contains the next order corrections added. The dots depict the numerical solution of the problem.
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