Emergence of directed motion in a crowded suspension of overdamped particles with different effective temperatures

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Emergence of directed motion in a crowded suspension of overdamped particles with different effective temperatures

Deborah Schwarcz and Stanislav Burov

Phys. Rev. Research 6, 013156 – Published 12 February 2024

Abstract

In this work, we focus on the behavior of a single passive Brownian particle in a suspension of passive particles with short-range repulsive interactions and higher effective temperature. While the forces affecting the single particle are thermal-like fluctuations and repulsion, due to other particles in the suspension, our numerical simulations show that on intermediate timescales directed motion on a single-particle level emerges. This emergent directional motion leads to a breakdown of the Einstein relation and nonmonotonic augmentation of the measured diffusion coefficient. Directional tendency increases with the density of the suspension and leads to growth of the diffusivity with the density of the suspension, a phenomenon recently observed for a system of hard spheres by Ilker, Castellana, and Joanny. Counterintuitively, the directional flow originates from the tendency of different particles to push each other out of their way. Due to such strictly repulsive interactions, nearby particles form into temporally correlated pairs and move cooperatively, thus creating a preferred direction of motion on intermediate timescales. We show that directional motion emerges when the ratio of the effective temperatures of the tracked particle and suspension constituents is below a critical value.

Received 21 March 2022
Accepted 13 October 2023

DOI:https://doi.org/10.1103/PhysRevResearch.6.013156

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Brownian motion Diffusion Nonequilibrium statistical mechanics

Living matter & active matter

Langevin equation

Statistical Physics & Thermodynamics

Authors & Affiliations

Deborah Schwarcz ^1,* and Stanislav Burov ^2,†

¹Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel
²Physics Department, Bar-Ilan University, Ramat Gan 5290002, Israel

^*deborah.schwarcz@gmail.com
^†stasbur@gmail.com

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Vol. 6, Iss. 1 — February - April 2024

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Figure 1
(a) MSD of a SP with self-diffusion coefficient $D_{b} = 0.01$ in a solution of PBPs with $D_{a} = 0.1$ , while the density $ϕ = 0.47$ . Gray circles represent simulated data, the blue line represents short period diffusion $2 D_{b} Δ$ . The pink line represents long-time diffusion $2 D_{b}^{\infty}$ , where the logarithmic time diffusion coefficient is $D_{b}^{\infty} = 0.022$ . The black line represents the fit to the MSD of an active particle, Eq. (5), where $A = 0.0245$ and $B = 0.375$ . Inset: MSD/ $Δ$ with parameters of panel (a) and semilogarithmic scale. The black line represents the fit to the MSD of an active particle. (b) $D_{b}^{\infty} / D_{b}$ as a function of the density of the solution. The squares represent the calculation of $D_{b}^{\infty}$ via the MSD of unperturbed SP [Eq. (4)]. Inset: The measured mobility $μ_{b}^{\infty} = v_{d} / F$ of the SP measured at different densities, divided by the mobility $μ_{b}$ measured at $ϕ = 0$ . $v_{d}$ is the measured terminal velocity of the SP measured for external force; the value $F = 0.1$ was used. For both panels $6 \times 10^{5}$ simulation steps were performed. Averaging over time and 40 ensembles was performed.
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Figure 2
(a) Relative angle distribution $P (θ; Δ)$ for the SP with $D = D_{b}$ . Three different solution densities $ϕ$ are presented: $Δ = 200 δ t$ , $D_{a} = 0.1$ , and $D_{b} = 0.01$ . (b) Mean relative angle $〈 θ 〉$ of the tracked SP as a function of $ϕ$ for three different ratios of $D_{b} / D_{a}$ ( $Δ = 200 δ t$ ). Values of $〈 θ 〉 < π / 2$ correspond to directed motion. (c) Phase diagram of directed vs uncorrelated motion of the tracked SP. For every $D_{b} / D_{a}$ the minimal value of $〈 θ 〉$ ( ${〈 θ 〉}_{min}$ ) was found (searched for different $Δ$ 's and $ϕ$ ). $π / 2 - {〈 θ 〉}_{min} = 0$ states that the behavior is passive, when $π / 2 - {〈 θ 〉}_{min} > 0$ directed motion is observed. When $D_{b} / D_{a} > 0.3$ the correlations were either nonexistent or negative, therefore the zero value of the parameter $π / 2 - {〈 θ 〉}_{min}$ . $D_{a}$ was always set to 0.1. For all panels, $6 \times 10^{5}$ simulation steps were recorded, and averaging over 50 ensembles was performed.
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Figure 3
(a) Weighted impact: The portion of temporal span $Δ$ during which the tracked SP simultaneously interacted with $n$ particles, as a function of $n = 1, 2, 3, 4, 5, 6$ . $D_{a} = 0.1$ , $D_{b} = 0.01$ , $ϕ = 0.475$ , and $Δ = 200 δ t$ . (b) $G (α, Δ)$ : The probability density function of the angle $α (t; Δ)$ [Eq. (7)] between the direction traced by the tracked SP during $Δ$ and the direction traced by a particle with which the tracked SP mostly interacted during $Δ$ . Such two particles form a temporally correlated pair. $Δ = 200 δ t$ and two ratios of $D_{b} / D_{a} = 1$ ( $◯$ ), $D_{b} / D_{a} = 0.2$ ( $▵$ ) are presented. (c) Average $α (t; Δ)$ ( $〈 α 〉$ ) as obtained from $G (α, Δ)$ in panel (b). $6 \times 10^{5}$ simulation steps and averaging over 40 ensembles was performed.
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Figure 4
(a) Nonmonotonic evolution of the mean relative angle $〈 θ 〉$ as a function of $Δ$ for four different densities. (b) $Δ^{*}$ is the $Δ$ for which the minimum value of $〈 θ 〉$ was obtained [see panel (a) for a given $ϕ$ and $D_{b} / D_{a}$ ]. Plotted as a function of the suspension density $ϕ$ . (c) Order parameter $π / 2 - 〈 θ 〉$ , when the mean relative angle $〈 θ 〉$ is computed at $Δ = Δ^{*}$ [panel (b)]. Higher values of the order parameter describe more pronounced directed motion. For all panels, $D_{a} = 0.1$ and $D_{b} = 0.01$ ; $6 \times 10^{5}$ simulation steps and averaging over 100 ensembles was performed.
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