Capillary Action in Scalar Active Matter

Adam Wysocki and Heiko Rieger

Phys. Rev. Lett. 124, 048001 – Published 28 January 2020

Abstract

We study the capacity of active matter to rise in thin tubes against gravity and other related phenomena like wetting of vertical plates and spontaneous imbibition, where a wetting liquid is drawn into a porous medium. This capillary action or capillarity is well known in classical fluids and originates from attractive interactions between the liquid molecules and the container walls, and from the attraction of the liquid molecules among each other. We observe capillarity in a minimal model for scalar active matter with purely repulsive interactions, where an effective attraction emerges due to slowdown during collisions between active particles and between active particles and walls. Simulations indicate that the capillary rise in thin tubes is approximately proportional to the active sedimentation length $λ$ and that the wetting height of a vertical plate grows superlinear with $λ$ . In a disordered porous medium the imbibition height scales as $⟨ h ⟩ \propto λ ϕ_{m}$ , where $ϕ_{m}$ is its packing fraction. These predictions are highly relevant for suspensions of sedimenting active colloids or motile bacteria in a porous medium under the influence of a constant force field.

Received 9 August 2019

DOI:https://doi.org/10.1103/PhysRevLett.124.048001

Physics Subject Headings (PhySH)

Collective behavior Surface & interfacial phenomena

Living matter & active matter

Monte Carlo methods

Polymers & Soft Matter

Authors & Affiliations

Adam Wysocki^* and Heiko Rieger ^†

Department of Theoretical Physics and Center for Biophysics, Saarland University, Saarbrücken 66123, Germany

^*a.wysocki@lusi.uni-sb.de
^†h.rieger@mx.uni-saarland.de

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Vol. 124, Iss. 4 — 31 January 2020

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Figure 1
(a) Sketch of the four subprocesses defining the stochastic dynamics of the ALG: symmetric diffusion, self-propulsion, sedimentation, and tumbling with rates $D$ , $V_{0} / n$ , $V_{g} / n$ , and $α / n^{2}$ , respectively. (b)–(c) Capillary rise of active lattice gas at active Péclet number ${Pe}_{a} = 10$ , gravitational Péclet number ${Pe}_{g} = 0.2$ , capillary width $δ x / l = 1$ and capillary height $δ y / l = 12.5$ . The total size of the system is $L_{x} / l = 60$ and $L_{y} / l = 120$ , which corresponds to $N_{x} \times N_{y} = 1200 \times 2400$ lattice sites used in the Monte Carlo simulation. (b) Mean total density $ρ (x, y) \in [0, 1]$ . (c) Absolute value $| m | (x, y)$ of the mean normalized polarization field $m$ , see Eq. (2). (d) Phase $φ (x, y) \in [- π, π]$ of $m = | m | \cdot (\cos φ, \sin φ)$ . A phase $ϕ = {0, π / 2, {π, - π}, - π / 2}$ corresponds to a polarization along ${\hat{x}, \hat{y}, - \hat{x}, - \hat{y}}$ or ${right, up, left, down}$ , respectively.
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Figure 2
(a)–(c) The height of the meniscus $Δ h$ for different tube heights $δ y$ and at a fixed capillary width $δ x / l = 1$ . (a) $Δ h$ vs activity ${Pe}_{a}$ at fixed ${Pe}_{g} = 1$ and (b) $Δ h$ vs gravity ${Pe}_{g}$ at fixed ${Pe}_{a} = 10$ . (c) A master curve merging data from (a) and (b). $Δ h$ is plotted as a function of active gravitational length $λ$ and $Δ h$ is scaled by $δ y^{0.24}$ . For every parameter pair $({Pe}_{a}, {Pe}_{g})$ we estimate the corresponding $λ$ from a fit of the density profile in the bulk. Note that, at finite activity $λ$ is larger than the gravitational length in equilibrium $λ_{eq} = D / V_{g}$ . (d) $Δ h$ as a function of the capillary width $δ x$ for different parameter pairs $({Pe}_{a}, {Pe}_{g})$ and at fixed tube height $δ y / l = 17.5$ . The height of the meniscus $Δ h$ is scaled by $λ^{0.9}$ .
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Figure 3
Wetting of a vertical plate. (a) Total density $ρ (x, y) \in [0, 1]$ for ${Pe}_{a} = 10$ and ${Pe}_{g} = 1$ . A cutout of a larger system ( $L_{x} / l = 60$ and $L_{y} / l = 120$ ) is shown. Black line indicates the isodensity at $ρ = 0.6$ , which is our definition of the interface between the dense and the dilute phase. The height of the wetting layer $Δ H$ is marked by a circle. (b) Height of the wetting layer $Δ H$ as a function of the active gravitational length $λ$ . Two datasets are shown: $2 \leq {Pe}_{a} \leq 20$ at fixed ${Pe}_{g} = 1$ and $0.1 \leq {Pe}_{g} \leq 2$ at fixed ${Pe}_{a} = 10$ .
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Figure 4
Imbibition of a porous media by the active lattice gas. The porous media consists of nonoverlapping hard discs with uniformly distributed diameters in the range $σ / l \in [0.5, 2]$ ; the minimum gap size is 0.325 in units of $l$ . The total density $ρ (x, y) \in [0, 1]$ is shown for ${Pe}_{a} = 15$ , ${Pe}_{g} = 1$ and a system size $L_{x} / l = 60$ and $L_{y} / l = 100$ . The porosity is $1 - ϕ_{m} = 0.2$ , where $ϕ_{m}$ is the packing fraction of the porous matrix.
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Figure 5
(a) Density profiles $ρ (y)$ of the active lattice gas within the porous matrix (the horizontal average was performed over the void region only) for different porosities $1 - ϕ_{m}$ (solid lines) and the corresponding $ρ (y)$ in the bulk region (dashed line) are shown. The position of the interface is defined as the height, where $ρ = 0.6$ and is indicated by circles for the porous matrix and by a square in the bulk. The parameters are $({Pe}_{a}, {Pe}_{g}) = (15, 1)$ . (b) Mean interface height $⟨ h ⟩$ scaled by the active gravitational length $λ$ as a function of the packing fraction of the porous matrix $ϕ_{m}$ for different parameter pairs $({Pe}_{a}, {Pe}_{g})$ .
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