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Article Dans Une Revue Physical Review E Année : 2020

Speed-dispersion-induced alignment: A one-dimensional model inspired by swimming droplets experiments

Charlotte de Blois
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Yang Liu
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  • IdHAL : yi-liu
Marjolein van Der Linden
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Olivier Dauchot

Résumé

We investigate the collective dynamics of self-propelled droplets, confined in a one dimensional micro-fluidic channel. On one hand, neighboring droplets align and form large trains of droplets moving in the same direction. On the other hand, the droplets condensates, leaving large regions with very low density. A careful examination of the interactions between two "colliding" droplets demonstrates that local alignment takes place as a result of the interplay between the dispersion of their speeds and the absence of Galilean invariance. Inspired by these observations, we propose a minimalistic 1D model of active particles reproducing such dynamical rules and, combining analytical arguments and numerical evidences, we show that the model exhibits a transition to collective motion in 1D for a large range of values of the control parameters. Condensation takes place as a transient phenomena which tremendously slows down the dynamics, before the system eventually settles into a homogeneous aligned phase. Collective dynamics in systems of active particles have been the topic of a fantastic amount of work [1-3]. Both the transition to collective motion [1, 4-6] and the Motil-ity Induced Phase Separation (MIPS) [7] are now well understood. The interplay of alignment and crowding effects have been investigated more recently [8-12]. The majority of these studies have however been conducted in two dimensional space and much less is known about active systems in one dimension. Yet the physics of active system in 1D is relevant as soon as the confinement breaks the continuous symmetry of the order parameter describing collective motion. It is the case in systems of highly confined bacteria [13], pedestrians [14] or molecular motors [15]. Also, from a more theoretical point of view, 1D systems often exhibit peculiar dynamics [16, 17], resulting from the presence of strong correlations, as exemplified by single-file diffusion [18, 19], 1D inelastic dynamics [20-22] or 1D exclusion processes [23]. In the context of active matter, kinetic theory results [6, 24] or generic arguments relying on the non conservation of momentum [25] do not easily generalize in 1D because of the discrete symmetry of the polar order parameter. Despite this limitation, a few models were put forward to describe 1D active systems. The "active Ising model" [5, 26], a stochastic lattice gas model, has been decisive in our current understanding of the transition to collective motion in terms of a bona fide liquid-gas phase transition. It however does not include steric interactions. Conversely, several models were proposed to describe the clustering transition in assemblies of excluding run-and-tumble particles, but do not include alignment [27-30]. Finally, hydrodynamic limits were derived exactly provided that the different processes have appropriate scal-ings [31], but this approach still lacks the combined effect of volume-exclusion and alignment. FIG. 1. Collective dynamics of swimming droplets in a micro-fluidic serpentine (a) Part of the setup ; (b,c) Spatio-temporal diagram of the dynamics for packing fraction φ = N a/L = [0.082; 0.122] (blue and red colors code for the direction of motion; a trajectory changes color when a droplet reverses its direction; time in s and space in droplet diameter). (d,e) Polarisation m and participation ratio r (see main text) vs. time for φ = 0.082 (black) and φ = 0.122 (red). In this Letter we first report on the observation of collective alignment and spatial condensation in a one dimensional system of swimming droplets (Fig. 1): 'trains of droplets' spontaneously form and move coherently. To the best of our knowledge, this is the first experimental realization and observation of the onset to collective motion in a one-dimensional active system. The analysis of the short time dynamics, resulting from the interaction of two droplets, reveals that the local alignment results from the interplay between the dispersion of the droplet speeds and the absence of Galilean invariance. Wether such a mechanism leads to a large scale transition to collective motion, especially in 1D, is far from obvious. We propose a minimal model based on these observations, and, combining analytical arguments and numerical evidences, we show that a system of active particles reproducing the arXiv:1910.00525v2 [cond-mat.soft]
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Dates et versions

hal-03001327 , version 1 (13-11-2020)

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Pierre Illien, Charlotte de Blois, Yang Liu, Marjolein van Der Linden, Olivier Dauchot. Speed-dispersion-induced alignment: A one-dimensional model inspired by swimming droplets experiments. Physical Review E , 2020, 101 (4), ⟨10.1103/PhysRevE.101.040602⟩. ⟨hal-03001327⟩
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