Topology-Driven Ordering of Flocking Matter

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Topology-Driven Ordering of Flocking Matter

Amélie Chardac, Ludwig A. Hoffmann, Yoann Poupart, Luca Giomi, and Denis Bartolo

Phys. Rev. X 11, 031069 – Published 29 September 2021

An article within the collection: Highlights in Experimental Statistical, Biological, and Soft-Matter Physics

Abstract

When interacting motile units self-organize into flocks, they realize one of the most robust ordered states found in nature. However, after 25 years of intense research, the very mechanism controlling the ordering dynamics of both living and artificial flocks has remained unsettled. Here, combining active-colloid experiments, numerical simulations, and analytical work, we explain how flocking liquids heal their spontaneous flows initially plagued by collections of topological defects to achieve long-ranged polar order even in two dimensions. We demonstrate that the self-similar ordering of flocking matter is ruled by a living network of domain walls linking all $\pm 1$ vortices and guiding their annihilation dynamics. Crucially, this singular orientational structure echoes the formation of extended density patterns in the shape of interconnected bow ties. We establish that this double structure emerges from the interplay between self-advection and density gradients dressing each $- 1$ topological charge with four orientation walls. We then explain how active Magnus forces link all topological charges with extended domain walls, while elastic interactions drive their attraction along the resulting filamentous network of polarization singularities. Taken together, our experimental, numerical, and analytical results illuminate the suppression of all flow singularities and the emergence of pristine unidirectional order in flocking matter.

Received 10 March 2021
Revised 4 June 2021
Accepted 27 July 2021

DOI:https://doi.org/10.1103/PhysRevX.11.031069

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)

Biological self-organization Collective behavior Emergence of patterns Flocking Topological defects

Colloids Living matter & active matter Microfluidic devices Self-propelled particles

Finite-element method

Condensed Matter, Materials & Applied PhysicsFluid DynamicsNonlinear DynamicsPolymers & Soft MatterStatistical Physics & Thermodynamics

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Highlights in Experimental Statistical, Biological, and Soft-Matter Physics

A collection of recent highlights in experimental Statistical, Biological, and Soft-Matter Physics showcases PRX’s superb quality and topical diversity.

Authors & Affiliations

Amélie Chardac ¹, Ludwig A. Hoffmann ², Yoann Poupart ¹, Luca Giomi ², and Denis Bartolo ^1,*

¹Université Lyon, ENS de Lyon, Université Claude Bernard, CNRS, Laboratoire de Physique, F-69342, Lyon, France
²Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, Netherlands

^*Corresponding author. denis.bartolo@ens-lyon.fr

Popular Summary

Active-matter physics describes the mesmerizing dynamics of interacting motile bodies such as bird flocks, cell colonies, and even ensembles of synthetic “swimmers.” However, after 25 years of intense research, the mechanism controlling the ordering dynamics of living and artificial flocks remains unsettled. Here, we combine experiments on active-colloid flocks, numerical simulations, and analytical analysis to show how flocks smooth over their initial multidirectional flows.

The question is deceptively simple: Starting from a homogeneous ensemble of uncoordinated motile particles, how does their velocity field, initially marred by many singularities, heal and reach pristine order at all scales? To answer this question, we needed to solve another outstanding challenge of active matter physics: describing the geometry and the dynamics of topological defects in flocking liquids. In doing so, we show that flocking matter annihilates its defects by linking the $m$ with polarization walls separating regions of incompatible flow directions. This filamentous structure mirrors the emergence of density patterns—shaped like interconnected bow ties—that stabilize long-lived domain walls. Finally, we explain how the intimate interplay between density gradients, self-advection, and elasticity heals the distortions and discontinuities of flocking matter.

We solved the problem of flock ordering in two dimensions. Future work needs to extend this analysis to three dimensions and address how the interplay between density fluctuations and self-propulsion alter the fundamental topological excitations of higher-dimensional active matter.

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Vol. 11, Iss. 3 — July - September 2021

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Figure 1
Ordering dynamics of flocking fluids: experiments and simulations. (a) Picture of an experiment showing approximately $4 \times 10^{6}$ colloidal rollers self-assembled into an active polar fluid. The picture reveals typical bow-tie patterns prior to the formation of a pristine vortex pattern at system spanning scale; see Supplemental Material, Video 1 [18]. Scale bar: 1.5 cm. (b) At the onset of flocking, the flow field of the active-colloidal fluid is plagued by a collection of $\pm 1$ topological defects. The density bow ties are centered on $- 1$ defects. $+ 1$ defects are located at the minima of the local density. The color indicates the local packing fraction. Scale bar: $250 μ m$ . (c) Series of five experimental snapshots of the density field and a map of the local strain magnitude $| \nabla p |$ . The polarization field is segmented by a fully connected network of domain walls focusing the flow distortions. Each wall coincides with a bow-tie edge in the density field. Scale bar: 2 mm. (d) Same representation as in (c) for the numerical resolution of Toner-Tu hydrodynamics, Eqs. (1) with periodic boundary conditions, starting from a homogeneous packing fraction $ρ_{0} = 0.1$ and a random velocity field ( $λ = 0.7$ , $σ = 5 {mm}^{2} s^{- 2}$ , $D = 10^{- 2} {m m}^{2} s^{- 1}$ , $α_{0} = 100 s^{- 1}$ , $β = 10 {mm}^{- 2} s$ ). See the Appendix pp2. Scale bar: 1.5 mm.
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Figure 2
$- 1$ topological charges are dressed with four domain walls and a density bow tie. (a) Top: picture of a polar liquid flowing in microfluidic channel in the shape an 8 and corresponding polarization field. The polarization vectors are colored according to their orientation and reveal an isolated $- 1$ defect. Channel width: 5 mm. Bottom: polarization and density fields in the vicinity of the $- 1$ defect center. The bow-tie structure is clearly visible in the absence of other defects. A single $- 1$ charge is stabilized by the imposed direction of the flow field at the boundaries. The density is represented by the local packing fraction. Scale bar: 1 mm. (b) Numerical simulation of an isolated $- 1$ defect. Density and polarization fields. Boundary conditions described in Appendix pp2. Same color maps as in (a). Scale bar: $150 μ m$ . (c) Angular power spectra of the polarization field computed along the circle shown in (a). The maximum of $| p_{k} |$ is reached for $k = - 1$ (vertical line in the leftmost panels). However, the polarization field does not merely feature a single angular mode as in the case of an ideal antivortex with hyperbolic streamlines. Top: experimental data. Bottom: numerical resolution of the Toner-Tu equations (1) in the geometry shown in (b). The solid line $| p_{k} | \sim k^{- 1}$ indicates our theoretical prediction. It corresponds to the defect geometry sketched in Fig. 3: Four wedge-shaped regions, where $ρ$ and $v$ are piecewise constant, are separated by four domain walls emanating from the defect center. (d) Angular power spectra of the density field computed along the circle in (a). Same power spectra and same observation as in (c). Both $| p_{k} |$ and $| ρ_{k} |$ decay algebraically revealing the singular nature of the bow-tie patterns.
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Figure 3
Bow ties and domain walls emerge from the competition between self-convection and pressure. (a) Left: sketch of the ideal $- 1$ defect configuration dressed with four domain walls. The density and flow speed are both piecewise uniform. Right: close-up on the domain wall and definition of the local tangent $T$ and normal $N$ unit vectors at the domain wall separating the acute and the obtuse regions. (b) Left: density field in a shamrock chamber. The density is represented by the local packing fraction. Scale bar: 1 mm. Right: measuring the stationary flow and density fields in the square region delimited in (b) left, we plot the magnitude of the convective term $| v \cdot \nabla v |$ against the density gradient $| \nabla ρ |$ . The slope of the solid line is given by the ratio $σ / λ$ measured independently by fitting the density profiles around the $+ 1$ defects hosted by the circular petals; see Supplemental Material [18]. The collapse of all data points demonstrates that density gradients locally balance the convective pressure to stabilize the singular bow-tie patterns. (c) Same measurements as in (b) for the numerical solution of Eqs. (1) in the geometry of Fig. 2. The straight line has a slope $σ / λ = 7.1 {mm}^{2} s^{- 1}$ ; see Appendix pp2.
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Figure 4
Self-advection and pressure gradients link topological charges with polarization walls. (a) Solving Eqs. (1), we study the dynamics of the polarization and density field around a pair of topological defects in a periodic box. At $t = 0$ , the flow field is the superposition of the ideal vortex and antivortex (hyperbolic streamlines). The flow gradient first focuses to form sharp domain walls growing from the center of the $- 1$ defect. The domain walls subsequently bend and rotate to link the $+ 1$ charge. Once the defects are linked, the $+ 1$ charge approaches and annihilates with the $- 1$ defect, thereby yielding a perfect uniaxial order. Scale bar: $500 μ m$ . (b) Corresponding evolution of the initially uniform density field around the defect pair. A bow-tie pattern emerges as the domain walls grow and eventually vanishes once the two topological charges annihilate. Scale bar: $500 μ m$ . (c) Illustration of the pressure-driven dynamics of the domain walls around a $- 1$ defect. (d) Once the four domain walls form four $π / 2$ wedges and link to the $+ 1$ charge, their opening remains fixed provided that the incoming and outgoing flows satisfy the horizontal mirror symmetry.
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Figure 5
Orientational elasticity rules a self-similar phase-ordering kinetics. (a) Snapshot of the polarization field (experiments). The red lines indicate the domain walls, the light green (resp., dark blue) circles indicate the position of the $+ 1$ (resp., $- 1$ ) topological charges. The surrounding regions form domains of incompatible uniform polarization. (b) Time evolution of the total number of topological defects. We take advantage of the numerical resolution of Eqs. (1) to vary independently the three essential hydrodynamic parameters $D$ , $σ$ , and $λ$ setting the magnitude of the orientational elasticity of the compressibility and of the self-advection coefficient of the flocking liquid. All the constant parameters are the same as in Fig. 1. The number of the defects plaguing the active flows decays as $1 / t$ for all hydrodynamic parameters. The long-time coarsening dynamics is chiefly determined by the orientational elasticity of the polar liquid, and all data collapse on a single master curve when plotted against $D t$ (left panel inset). (c) Experimental characterization of the phase-ordering kinetics in chambers of increasing size. In agreement with our simulations, the number of defects decays as $1 / t$ . (d) Equivalently, the absolute $ξ_{1}$ and curvilinear $ξ_{3}$ distances between the defects defined in (a) grow algebraically as $t^{1 / 2}$ and so does the correlation length $ξ_{2}$ of the $p$ field; see also the Supplemental Material [18]. (e) The typical size of the polarization domains $ℓ$ , however, grows ballistically in time. (f) Three snapshots of the strain field corresponding to the experiment shown in (a). When a defect pair annihilates (dashed line), two domain walls emanating from them bend and eventually vanish. As a result, defects defining the vertices of the polarization network end up living along the domain walls. The typical distance between the defects ( $ξ_{3}$ ) hence grows slower than the distance between the network vertices ( $ℓ$ ).
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