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Topological mechanical metamaterials are artificial structures whose unusual properties are protected very much like their electronic and optical counterparts. Here, we present an experimental and theoretical study of an active metamaterial-composed of coupled gyroscopes on a lattice-that breaks time-reversal symmetry. The vibrational spectrum displays a sonic gap populated by topologically protected edge modes that propagate in only one direction and are unaffected by disorder. We present a mathematical model that explains how the edge mode chirality can be switched via controlled distortions of the underlying lattice. This effect allows the direction of the edge current to be determined on demand. We demonstrate this functionality in experiment and envision applications of these edge modes to the design of oneway acoustic waveguides.topological mechanics | gyroscopic metamaterial | metamaterial A vast range of mechanical structures, including bridges, covalent glasses, and conventional metamaterials, can be ultimately modeled as networks of masses connected by springs (1-6). Recent studies have revealed that despite its apparent simplicity, this minimal setup is sufficient to construct topologically protected mechanical states (7-11) that mimic the properties of their quantum analogs (12). This follows from the fact that, irrespective of its classic or quantum nature, a periodic material with a gapped spectrum of excitations can display topological behavior as a result of the nontrivial topology of its band structure (13-21).All such mechanical systems, however, are invariant under time reversal because their dynamics are governed by Newton's second law, which, unlike the Schrödinger equation, is second order in time. If time-reversal symmetry is broken, as in recently suggested acoustic structures containing circulating fluids (16), theoretical work (13) has suggested that phononic chiral topological edge states that act as unidirectional waveguides resistant to scattering off impurities could be supported. In this paper, we show that by creating a coupled system of gyroscopes, a "gyroscopic metamaterial," we can produce an effective material with intrinsic time-reversal symmetry breaking. As a result, our gyroscopic metamaterials support topological mechanical modes analogous to quantum Hall systems, which have robust chiral edge states (22)(23)(24). We demonstrate these effects by building a real system of gyroscopes coupled in a honeycomb lattice. Our experiments show long-lived, unidirectional transport along the edge, even in the presence of significant defects. Moreover, our theoretical analysis indicates that direction of edge propagation is controlled both by the gyroscope spin and the geometry of the underlying lattice. As a result, deforming the lattice of gyroscopes allows one to control the edge mode direction, offering unique opportunities for engineering novel materials.Much of the counterintuitive behavior of rapidly spinning objects originates from their large angular momentum, which endows th...

Hexagons can easily tile a flat surface, but not a curved one. Introducing heptagons and pentagons (defects with topological charge) makes it easier to tile curved surfaces; for example, soccer balls based on the geodesic domes of Buckminster Fuller have exactly 12 pentagons (positive charges). Interacting particles that invariably form hexagonal crystals on a plane exhibit fascinating scarred defect patterns on a sphere. Here we show that, for more general curved surfaces, curvature may be relaxed by pleats: uncharged lines of dislocations (topological dipoles) that vanish on the surface and play the same role as fabric pleats. We experimentally investigate crystal order on surfaces with spatially varying positive and negative curvature. On cylindrical capillary bridges, stretched to produce negative curvature, we observe a sequence of transitions-consistent with our energetic calculations-from no defects to isolated dislocations, which subsequently proliferate and organize into pleats; finally, scars and isolated heptagons (previously unseen) appear. This fine control of crystal order with curvature will enable explorations of general theories of defects in curved spaces. From a practical viewpoint, it may be possible to engineer structures with curvature (such as waisted nanotubes and vaulted architecture) and to develop novel methods for soft lithography and directed self-assembly.

Mechanical metamaterials are artificial structures with unusual properties, such as negative Poisson ratio, bistability or tunable vibrational properties, that originate in the geometry of their unit cell [1][2][3][4][5]. At the heart of such unusual behaviour is often a soft mode: a motion that does not significantly stretch or compress the links between constituent elements. When activated by motors or external fields, soft modes become the building blocks of robots and smart materials. Here, we demonstrate the existence of topological soft modes that can be positioned at desired locations in a metamaterial while being robust against a wide range of structural deformations or changes in material parameters [6][7][8][9][10]. These protected modes, localized at dislocations in deformed kagome and square lattices, are the mechanical analogue of topological states bound to defects in electronic systems [11][12][13][14]. We create physical realizations of the topological modes in prototypes of kagome lattices built out of rigid triangular plates. We show mathematically that they originate from the interplay between two Berry phases: the Burgers vector of the dislocation and the topological polarization of the lattice. Our work paves the way towards engineering topologically protected nano-mechanical structures for molecular robotics or information storage and read-out.Central to our approach is a simple insight: mechanical structures on length scales ranging from the molecular to the architectural can often be viewed as networks of nodes connected by links [15]. Whether the linking components are chemical bonds or metal beams, mechanical stability depends crucially on the number of constraints relative to the degrees of freedom. When the degrees of freedom exceed the constraints, the structure displays excess zero (potential) energy modes. Conversely, when the constraints exceed the degrees of freedom, there are excess states of self-stress-balanced combinations of tensions and compressions of the links with no resultant force on the nodes. The generalized Maxwell relation [16] stipulates that the index ν given by the difference between the number of zero modes, n m , and the number of states of self-stress, n ss , is equal to the number of degrees of freedom N df minus the number of constraints N c ν ≡ n m − n ss = N df − N c .(1) * vitelli@lorentz.leidenuniv.nlA trivial way to position a zero-energy mode in the interior of a generic rigid lattice is to remove some bonds, locally reducing the number of constraints. Consider, instead, a network that satisfies everywhere the local isostatic condition N df = N c (which precludes bond removal). In this case, zero modes can only be present in conjunction with an equal number of states of selfstress, invisible partners from the perspective of motion. Isostaticity by itself, however, does not dictate how the modes are distributed spatially. Kane and Lubensky [6] recently introduced a special class of isostatic lattices that possesses an additional feature called topolo...

We study the hydrodynamics of fluids composed of self-spinning objects such as chiral grains or colloidal particles subject to torques. These chiral active fluids break both parity and time-reversal symmetries in their non-equilibrium steady states. As a result, the constitutive relations of chiral active media display a dissipationless linear-response coefficient called odd (or equivalently, Hall) viscosity. This odd viscosity does not lead to energy dissipation, but gives rise to a flow perpendicular to applied pressure. We show how odd viscosity arises from non-linear equations of hydrodynamics with rotational degrees of freedom, once linearized around a non-equilibrium steady state characterized by large spinning speeds. Next, we explore odd viscosity in compressible fluids and suggest how our findings can be tested in the context of shock propagation experiments. Finally, we show how odd viscosity in weakly compressible chiral active fluids can lead to density and pressure excess within vortex cores.

The prospect of mimicking molecular chemistry with colloidal rather than molecular building blocks could enable unprecedented control over the properties of microstructured materials 1 . The usual absence of directionality to the interaction between colloids has limited the complexity of the structures they can spontaneously form. One way to address this is to coat spherical colloid particles with a thin layer of nematic liquid crystal 2 and functionalize 3 the unavoidable defects or bold spots that arise when nematic order is established on the surface of a sphere 4,5 . The number and arrangement of these defects can vary 2,6-16 , providing flexibility for tuning directional interactions that are more difficult to achieve by other methods [17][18][19][20][21][22][23][24][25][26] . Yet, many theoretically predicted structures have not been observed and control over defect location remains elusive. In this work, we show that varying the thickness of a nematic liquid crystal shell enables us to systematically control the number and orientation of defects formed. For thin shells, these defects can be engineered to emulate the linear, trigonal and tetrahedral geometries of sp, sp 2 and sp 3 carbon bonds, respectively. Such control opens up the possibility to engineer particles with tunable-valence and directional-binding capabilities.To fabricate spherical nematic shells, we generate double emulsions with a microcapillary device 27 ; these consist of a nematic drop that contains a smaller aqueous drop, all inside an aqueous continuous phase. Both the inner and outer water phases contain 1 wt% polyvinyl alcohol, which stabilizes the emulsion against coalescence and enforces tangential anchoring of the rod-like molecules of the nematic liquid crystal, pentylcyanobiphenyl. The resulting double-emulsion drops are characterized by an outer radius, R, of around 50 µm and an inner radius, a, that are varied to produce shells of different average thicknesses,h = R − a, as schematically shown in Fig. 1a. With this microfluidic method the thinnest shells that we can generate haveh ≈ 1 µm. However, it is possible to significantly reduce this value by increasing the volume of the inner drop once the double emulsion is formed. We achieve this by inducing a difference in osmotic pressure between the inner and outer water phases through the addition of a salt, CaCl 2 . As pentylcyanobiphenyl has a finite permeability to water, an incoming flow of water from the outer phase can be established if the inner drop contains a higher salt concentration than the outer phase. By controlling this difference, we can control the kinetics of the process and ultimately the thickness of the shells.The thinnest shells have four defects, each with a topological charge s = 1/2, reflecting the π rotation experienced by the local nematic direction along a path encircling each defect. As a result, the total topological charge on the sphere is equal to 4 × 1/2 = 2; this is consistent with a mathematical theorem due to Poincaré and Hopf, which establish...

We study harmonic and anharmonic properties of the vibrational modes in 3-dimensional jammed packings of frictionless spheres interacting via repulsive, finite-range potentials. A crossover frequency is apparent in the density of states, the diffusivity and the participation ratio of the normal modes of vibration. At this frequency, which shifts to zero at the jamming threshold, the vibrational modes have a very small participation ratio implying that the modes are quasi-localized. The lowest-frequency modes are the most anharmonic, with the strongest response to pressure and the lowest-energy barriers to mechanical failure.

Networks of rigid bars connected by joints, termed linkages, provide a minimal framework to design robotic arms and mechanical metamaterials built of folding components. Here, we investigate a chain-like linkage that, according to linear elasticity, behaves like a topological mechanical insulator whose zero-energy modes are localized at the edge. Simple experiments we performed using prototypes of the chain vividly illustrate how the soft motion, initially localized at the edge, can in fact propagate unobstructed all of the way to the opposite end. Using real prototypes, simulations, and analytical models, we demonstrate that the chain is a mechanical conductor, whose carriers are nonlinear solitary waves, not captured within linear elasticity. Indeed, the linkage prototype can be regarded as the simplest example of a topological metamaterial whose protected mechanical excitations are solitons, moving domain walls between distinct topological mechanical phases. More practically, we have built a topologically protected mechanism that can perform basic tasks such as transporting a mechanical state from one location to another. Our work paves the way toward adopting the principle of topological robustness in the design of robots assembled from activated linkages as well as in the fabrication of complex molecular nanostructures.topological matter | origami | isostaticity | jamming | active matter M echanical structures composed of folding components, such as bars or plates rotating around pivots or hinges, are ubiquitous in engineering, materials science, and biology (1). For example, complex origami-like structures can be created by folding a paper sheet along suitably chosen creases around which two nearby faces can freely rotate (2-4). Similarly, linkages can be viewed as 1D versions of origami where rigid bars (links) are joined at their ends by joints (vertices) that permit full rotation of the bars ( Fig. 1 A-C). Some of the joints can be pinned to the plane while the remaining ones rotate relative to each other under the constraints imposed by the network structure of the linkage (5). Familiar examples include the windshield wiper, robotic arms, biological linkages in the jaw and knee, and toys like the Jacob's ladder (6) and the Hoberman sphere. Moreover, linkages and origami can be used in the design of microscopic and structural metamaterials whose peculiar properties are controlled by the geometry of the unit cell (7,8).Many of these examples are instances of what mechanical engineers call mechanisms: structures where the degrees of freedom are nearly balanced by carefully chosen constraints so that the allowed free motions encode a desired mechanical function. However, as the number of components increases, more can go wrong: lack of precision machining or undesired perturbations. Robustness in this sense is a concern relevant to the design of complex mechanical structures from the microscopic to the architectural scale, typically addressed at the cost of higher manufacturing tolerances or active feedbac...