Non-orientable order and non-commutative response in frustrated metamaterials.

From atomic crystals to animal flocks, the emergence of order in nature is captured by the concept of spontaneous symmetry breaking1-4. However, this cornerstone of physics is challenged when broken symmetry phases are frustrated by geometrical constraints. Such frustration dictates the behaviour of systems as diverse as spin ices5-8, confined colloidal suspensions9 and crumpled paper sheets10. These systems typically exhibit strongly degenerated and heterogeneous ground states and hence escape the Ginzburg-Landau paradigm of phase ordering. Here, combining experiments, simulations and theory we uncover an unanticipated form of topological order in globally frustrated matter: non-orientable order. We demonstrate this concept by designing globally frustrated metamaterials that spontaneously break a discrete [Formula: see text] symmetry. We observe that their equilibria are necessarily heteregeneous and extensively degenerated. We explain our observations by generalizing the theory of elasticity to non-orientable order-parameter bundles. We show that non-orientable equilibria are extensively degenerated due to the arbitrary location of topologically protected nodes and lines where the order parameter must vanish. We further show that non-orientable order applies more broadly to objects that are non-orientable themselves, such as buckled Möbius strips and Klein bottles. Finally, by applying time-dependent local perturbations to metamaterials with non-orientable order, we engineer topologically protected mechanical memories11-19, achieve non-commutative responses and show that they carry an imprint of the braiding of the loads' trajectories. Beyond mechanics, we envision non-orientability as a robust design principle for metamaterials that can effectively store information across scales, in fields as diverse as colloidal science8, photonics20, magnetism7 and atomic physics21.

Model-free characterization of topological edge and corner states in mechanical networks.

Topological materials can host edge and corner states that are protected from disorder and material imperfections. In particular, the topological edge states of mechanical structures present unmatched opportunities for achieving robust responses in wave guiding, sensing, computation, and filtering. However, determining whether a mechanical structure is topologically nontrivial and features topologically protected modes has hitherto relied on theoretical models. This strong requirement has limited the experimental and practical significance of topological mechanics to laboratory demonstrations. Here, we introduce and validate an experimental method to detect the topologically protected zero modes of mechanical structures without resorting to any modeling step. Our practical method is based on a simple electrostatic analogy: Topological zero modes are akin to electric charges. To detect them, we identify elementary mechanical molecules and measure their chiral polarization, a recently introduced marker of topology in chiral phases. Topological zero modes are then identified as singularities of the polarization field. Our method readily applies to any mechanical structure and effectively detects the edge and corner states of regular and higher-order topological insulators. Our findings extend the reach of chiral topological phases beyond designer materials and allow their direct experimental investigation.

Emergence of dynamic vortex glasses in disordered polar active fluids.

In equilibrium, disorder conspires with topological defects to redefine the ordered states of matter in systems as diverse as crystals, superconductors, and liquid crystals. Far from equilibrium, however, the consequences of quenched disorder on active condensed matter remain virtually uncharted. Here, we reveal a state of strongly disordered active matter with no counterparts in equilibrium: a dynamical vortex glass. Combining microfluidic experiments and theory, we show how colloidal flocks collectively cruise through disordered environments without relaxing the topological singularities of their flows. The resulting state is highly dynamical but the flow patterns, shaped by a finite density of frozen vortices, are stationary and exponentially degenerated. Quenched isotropic disorder acts as a random gauge field turning active liquids into dynamical vortex glasses. We argue that this robust mechanism should shape the collective dynamics of a broad class of disordered active matter, from synthetic active nematics to collections of living cells exploring heterogeneous media.

Curvature Heterogeneities Act as Singular Perturbations to Smooth Laplacian Fields: A Fluid Mechanics Demonstration.

In this Letter, we use a model fluid mechanics experiment to elucidate the impact of curvature heterogeneities on two-dimensional fields deriving from harmonic potential functions. This result is directly relevant to explain the smooth stationary structures in physical systems as diverse as curved liquid crystal and magnetic films, heat and Ohmic transport in wrinkled two-dimensional materials, and flows in confined channels. Combining microfluidic experiments and theory, we explain how curvature heterogeneities shape confined viscous flows. We show that isotropic bumps induce local distortions to Darcy's flows, whereas anisotropic curvature heterogeneities disturb them algebraically over system-spanning scales. Thanks to an electrostatic analogy, we gain insight into this singular geometric perturbation, and quantitatively explain it using both conformal mapping and numerical simulations. Altogether, our findings establish the robustness of our experimental observations and their broad relevance to all Laplacian problems beyond the specifics of our fluid mechanics experiment.

Emergence of scale-free smectic rivers and critical depinning in emulsions driven through disorder.

During the past 60 min, oil companies have extracted 6 trillion liters of oil from the ground, thereby giving a striking illustration of the impact of multiphase flows on the world economy. From a fundamental perspective, we largely understand the dynamics of interfaces separating immiscible fluids driven through heterogeneous environments. In stark contrast, the basic mechanisms ruling the transport of fragmented fluids, such as foams and emulsions, remain elusive with studies mostly limited to isolated droplets and bubbles. Here, we demonstrate that the mobilization of emulsion driven through model disordered media is a critical plastic depinning transition. To elucidate this collective dynamics, we track the trajectories of hundreds of thousands of microfluidic droplets advected through random lattices of pinning sites. Their dynamics reveals that macroscopic mobilization only requires the coordinated motion of small groups of particles and does not involve any large-scale avalanches. Criticality arises from the interplay between contact and hydrodynamic interaction, which channel seemingly erratic depinning events along smectic river networks correlated over system spanning scales. Beyond the specifics of emulsion transport, we close our article discussing the similarities and profound differences with the plastic depinning transitions of driven flux lines in high-T c superconductors, charged colloids, and grain transport in eroded sand beds.

When Soft Crystals Defy Newton's Third Law: Nonreciprocal Mechanics and Dislocation Motility.

The effective interactions between the constituents of driven soft matter generically defy Newton's third law. Combining theory and numerical simulations, we establish that six classes of mechanics with no counterparts in equilibrium systems emerge in elastic crystals challenged by nonreciprocal interactions. Going beyond linear deformations, we reveal that interactions violating Newton's third law generically turn otherwise quiescent dislocations into motile singularities which steadily glide though periodic lattices.

Colloidal transport in bacteria suspensions: from bacteria collision to anomalous and enhanced diffusion.

Swimming microorganisms interact and alter the dynamics of Brownian particles and tend to modify their transport properties. In particular, dilute colloids coupled to a bath of swimming cells generically display enhanced diffusion on long time scales. This transport dynamics stems from a subtle interplay between the active and passive particles that still resists our understanding despite decades of intense research. Here, we tackle the root of the problem by providing a quantitative characterisation of the single scattering events between a colloid and a bacterium, a smooth running E. coli. Based on our experiments, we build a minimal model that quantitatively predicts the geometry of the scattering trajectories, and enhanced colloidal diffusion at long times. This quantitative confrontation between theory and experiments elucidates the microscopic origin of enhanced transport. Collisions are solely ruled by stochastic contact interactions and the ratio of the drag coefficients of the colloid and the bacteria. Such description accounts both for genuine anomalous diffusion at short times and enhanced diffusion at long times with no ballistic regime at any scale.

Sounds and hydrodynamics of polar active fluids.

Spontaneously flowing liquids have been successfully engineered from a variety of biological and synthetic self-propelled units1-11. Together with their orientational order, wave propagation in such active fluids has remained a subject of intense theoretical studies12-17. However, the experimental observation of this phenomenon has remained elusive. Here, we establish and exploit the propagation of sound waves in colloidal active materials with broken rotational symmetry. We demonstrate that two mixed modes, coupling density and velocity fluctuations, propagate along all directions in colloidal-roller fluids. We then show how the six material constants defining the linear hydrodynamics of these active liquids can be measured from their spontaneous fluctuation spectrum, while being out of reach of conventional rheological methods. This active-sound spectroscopy is not specific to synthetic active materials and could provide a quantitative hydrodynamic description of herds, flocks and swarms from inspection of their large-scale fluctuations18-21.

Emergence of macroscopic directed motion in populations of motile colloids.

From the formation of animal flocks to the emergence of coordinated motion in bacterial swarms, populations of motile organisms at all scales display coherent collective motion. This consistent behaviour strongly contrasts with the difference in communication abilities between the individuals. On the basis of this universal feature, it has been proposed that alignment rules at the individual level could solely account for the emergence of unidirectional motion at the group level. This hypothesis has been supported by agent-based simulations. However, more complex collective behaviours have been systematically found in experiments, including the formation of vortices, fluctuating swarms, clustering and swirling. All these (living and man-made) model systems (bacteria, biofilaments and molecular motors, shaken grains and reactive colloids) predominantly rely on actual collisions to generate collective motion. As a result, the potential local alignment rules are entangled with more complex, and often unknown, interactions. The large-scale behaviour of the populations therefore strongly depends on these uncontrolled microscopic couplings, which are extremely challenging to measure and describe theoretically. Here we report that dilute populations of millions of colloidal rolling particles self-organize to achieve coherent motion in a unique direction, with very few density and velocity fluctuations. Quantitatively identifying the microscopic interactions between the rollers allows a theoretical description of this polar-liquid state. Comparison of the theory with experiment suggests that hydrodynamic interactions promote the emergence of collective motion either in the form of a single macroscopic 'flock', at low densities, or in that of a homogenous polar phase, at higher densities. Furthermore, hydrodynamics protects the polar-liquid state from the giant density fluctuations that were hitherto considered the hallmark of populations of self-propelled particles. Our experiments demonstrate that genuine physical interactions at the individual level are sufficient to set homogeneous active populations into stable directed motion.

Spatiotemporal order and emergent edge currents in active spinner materials.

Collections of interacting, self-propelled particles have been extensively studied as minimal models of many living and synthetic systems from bird flocks to active colloids. However, the influence of active rotations in the absence of self-propulsion (i.e., spinning without walking) remains less explored. Here, we numerically and theoretically investigate the behavior of ensembles of self-spinning dimers. We find that geometric frustration of dimer rotation by interactions yields spatiotemporal order and active melting with no equilibrium counterparts. At low density, the spinning dimers self-assemble into a triangular lattice with their orientations phase-locked into spatially periodic phases. The phase-locked patterns form dynamical analogs of the ground states of various spin models, transitioning from the three-state Potts antiferromagnet at low densities to the striped herringbone phase of planar quadrupoles at higher densities. As the density is raised further, the competition between active rotations and interactions leads to melting of the active spinner crystal. Emergent edge currents, whose direction is set by the chirality of the active spinning, arise as a nonequilibrium signature of the transition to the active spinner liquid and vanish when the system eventually undergoes kinetic arrest at very high densities. Our findings may be realized in systems ranging from liquid crystal and colloidal experiments to tabletop realizations using macroscopic chiral grains.

Mixing by Unstirring: Hyperuniform Dispersion of Interacting Particles upon Chaotic Advection.

We show how to achieve both fast and hyperuniform dispersions of particles in viscous fluids. To do so, we first extend the concept of critical random organization to chaotic drives. We show how palindromic sequences of chaotic advection cause microscopic particles to effectively interact at long range, thereby inhibiting critical self-organization. Based on this understanding we go around this limitation and design sequences of stirring and unstirring which simultaneously optimize the speed of particle spreading and the homogeneity of the resulting dispersions.

Forced Imbibition in Porous Media: A Fourfold Scenario.

We establish a comprehensive description of the patterns formed when a wetting liquid displaces a viscous fluid confined in a porous medium. Building on model microfluidic experiments, we evidence four imbibition scenarios all yielding different large-scale morphologies. Combining high-resolution imaging and confocal microscopy, we show that they originate from two liquid-entrainment transitions and a Rayleigh-Plateau instability at the pore scale. Finally, we demonstrate and explain the long-time coarsening of the resulting patterns.

Braiding a Flock: Winding Statistics of Interacting Flying Spins.

When animal groups move coherently in the form of a flock, their trajectories are not all parallel, the individuals exchange their position in the group. In this Letter, we introduce a measure of this mixing dynamics, which we quantify as the winding of the braid formed from the particle trajectories. Building on a paradigmatic flocking model we numerically and theoretically explain the winding statistics and show that it is predominantly set by the global twist of the trajectories as a consequence of a spontaneous symmetry breaking.

Emergent Hyperuniformity in Periodically Driven Emulsions.

We report the self-organization of microfluidic emulsions into anomalously homogeneous structures. Upon periodic driving confined emulsions undergo a first-order transition from a reversible to an irreversible dynamics. We evidence that this dynamical transition is accompanied by structural changes at all scales yielding macroscopic yet finite hyperuniform structures. Numerical simulations are performed to single out the very ingredients responsible for the suppression of density fluctuations. We show that, as opposed to equilibrium systems, the long-range nature of the hydrodynamic interactions are not required for the formation of hyperuniform patterns, thereby suggesting a robust relation between reversibility and hyperuniformity which should hold in a broad class of periodically driven materials.

Geometric capture and escape of a microswimmer colliding with an obstacle.

Motivated by recent experiments, we consider the hydrodynamic capture of a microswimmer near a stationary spherical obstacle. Simulations of model equations show that a swimmer approaching a small spherical colloid is simply scattered. In contrast, when the colloid is larger than a critical size it acts as a passive trap: the swimmer is hydrodynamically captured along closed trajectories and endlessly orbits around the colloidal sphere. In order to gain physical insight into this hydrodynamic scattering problem, we address it analytically. We provide expressions for the critical trapping radius, the depth of the "basin of attraction," and the scattering angle, which show excellent agreement with our numerical findings. We also demonstrate and rationalize the strong impact of swimming-flow symmetries on the trapping efficiency. Finally, we give the swimmer an opportunity to escape the colloidal traps by considering the effects of Brownian, or active, diffusion. We show that in some cases the trapping time is governed by an Ornstein-Uhlenbeck process, which results in a trapping time distribution that is well-approximated as inverse-Gaussian. The predictions again compare very favorably with the numerical simulations. We envision applications of the theory to bioremediation, microorganism sorting techniques, and the study of bacterial populations in heterogeneous or porous environments.

Revisiting the Saffman-Taylor experiment: imbibition patterns and liquid-entrainment transitions.

We revisit the Saffman-Taylor experiment focusing on the forced-imbibition regime where the displacing fluid wets the confining walls. We demonstrate a new class of invasion patterns that do not display the canonical fingering shapes. We evidence that these unanticipated patterns stem from the entrainment of thin liquid films from the moving meniscus. We then theoretically explain how the interplay between the fluid flow at the contact line and the interface deformations results in the destabilization of liquid interfaces. In addition, this minimal model conveys a unified framework which consistently accounts for all the liquid-entrainment scenarios that have been hitherto reported.

Emergent spatial structures in flocking models: a dynamical system insight.

We show that hydrodynamic theories of polar active matter generically possess inhomogeneous traveling solutions. We introduce a unifying dynamical-system framework to establish the shape of these intrinsically nonlinear patterns, and show that they correspond to those hitherto observed in experiments and numerical simulation: periodic density waves, and solitonic bands, or polar-liquid droplets both cruising in isotropic phases. We elucidate their respective multiplicity and mutual relations, as well as their existence domain.

Tailoring the interactions between self-propelled bodies.

We classify the interactions between self-propelled particles moving at a constant speed from symmetry considerations. We establish a systematic expansion for the two-body forces in the spirit of a multipolar expansion. This formulation makes it possible to rationalize most of the models introduced so far within a common framework. We distinguish between three classes of physical interactions: i) potential forces, ii) inelastic collisions and iii) non-reciprocal interactions involving polar or nematic alignment with an induced field. This framework provides simple design rules for the modeling and the fabrication of self-propelled bodies interacting via physical interactions. A class of possible interactions that should yield new phases of active matter is highlighted.

Hydrodynamics of confined active fluids.

We theoretically describe the dynamics of swimmer populations in rigidly confined thin liquid films. We first demonstrate that hydrodynamic interactions between confined swimmers depend solely on their shape and are independent of their specific swimming mechanism. We also show that, due to friction with the nearby rigid walls, confined swimmers do not just reorient in flow gradients but also in uniform flows. We then quantify the consequences of these microscopic interaction rules on the large-scale hydrodynamics of isotropic populations. We investigate in detail their stability and the resulting phase behavior, highlighting the differences with conventional active, three-dimensional suspensions. Two classes of polar swimmers are distinguished depending on their geometrical polarity. The first class gives rise to coherent directed motion at all scales, whereas for the second class we predict the spontaneous formation of coherent clusters (swarms).

Hydrodynamic fluctuations in confined particle-laden fluids.

We address the collective dynamics of non-Brownian particles cruising in a confined microfluidic geometry and provide a comprehensive characterization of their spatiotemporal density fluctuations. We show that density excitations freely propagate at all scales, and in all directions even though the particles are neither affected by potential forces nor by inertia. We introduce a kinetic theory which quantitatively accounts for our experimental findings, demonstrating that the fluctuation spectrum of this nonequilibrium system is shaped by the combination of truly long-range hydrodynamic interactions and local collisions. We also demonstrate that the free propagation of density waves is a generic phenomenon which should be observed in a much broader range of hydrodynamic systems.