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Materials made from active, living, or robotic components can display emergent properties arising from local sensing and computation. Here, we realize a freestanding active metabeam with piezoelectric elements and electronic feed-forward control that gives rise to an odd micropolar elasticity absent in energy-conserving media. The non-reciprocal odd modulus enables bending and shearing cycles that convert electrical energy into mechanical work, and vice versa. The sign of this elastic modulus is linked to a non-Hermitian topological index that determines the localization of vibrational modes to sample boundaries. At finite frequency, we can also tune the phase angle of the active modulus to produce a direction-dependent bending modulus and control non-Hermitian vibrational properties. Our continuum approach, built on symmetries and conservation laws, could be exploited to design others systems such as synthetic biofilaments and membranes with feed-forward control loops. Mechanical metamaterials can be engineered with properties not possible in ordinary materials. Here the authors demonstrate and study an active metamaterial with self-sensing characteristics that enables odd elastic properties not observed in passive media.

Active materials are characterized by continuous injection of energy at the microscopic level and typically cannot be adequately described by equilibrium thermodynamics. Here we study a class of active fluids in which equilibrium-like properties emerge when fluctuating and activated degrees of freedom are statistically decoupled, such that their mutual information is negligible. We analyse three paradigmatic systems: chiral active fluids composed of spinning frictional particles that are free to translate, oscillating granular gases and active Brownian rollers. In all of these systems, a single effective temperature generated by activity parameterizes both the equation of state and the emergent Boltzmann statistics. The same effective temperature, renormalized by velocity correlations, relates viscosities to steady-state stress fluctuations via a Green–Kubo relation. To rationalize these observations, we develop a theory for the fluctuating hydrodynamics of these non-equilibrium fluids and validate it through large-scale molecular dynamics simulations. Our work sheds light on the microscopic origin of odd viscosities and stress fluctuations characteristic of parity-violating fluids, in which mirror symmetry and detailed balance are broken.Active fluids exhibit properties reminiscent of equilibrium systems when their degrees of freedom are statistically decoupled. A theory for the fluctuating hydrodynamics of these fluids offers a probe of their anomalous transport coefficients.

The Stokes equation describes the motion of fluids when inertial forces are negligible compared with viscous forces. In this article, we explore the consequence of parity-violating and non-dissipative (i.e. odd) viscosities on Stokes flows in three dimensions. Parity-violating viscosities are coefficients of the viscosity tensor that are not invariant under mirror reflections of space, while odd viscosities are those which do not contribute to dissipation of mechanical energy. These viscosities can occur in systems ranging from synthetic and biological active fluids to magnetized and rotating fluids. We first systematically enumerate all possible parity-violating viscosities compatible with cylindrical symmetry, highlighting their connection to potential microscopic realizations. Then, using a combination of analytical and numerical methods, we analyse the effects of parity-violating viscosities on the Stokeslet solution, on the flow past a sphere or a bubble and on many-particle sedimentation. In all the cases that we analyse, parity-violating viscosities give rise to an azimuthal flow even when the driving force is parallel to the axis of cylindrical symmetry. For a few sedimenting particles, the azimuthal flow bends the trajectories compared with a traditional Stokes flow. For a cloud of particles, the azimuthal flow impedes the transformation of the spherical cloud into a torus and the subsequent breakup into smaller parts that would otherwise occur. The presence of azimuthal flows in cylindrically symmetric systems (sphere, bubble, cloud of particles) can serve as a probe for parity-violating viscosities in experimental systems.

Active systems composed of energy-generating microscopic constituents are a promising platform to create autonomous functional materials 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 – 16 that can, for example, locomote through complex and unpredictable environments. Yet coaxing these energy sources into useful mechanical work has proved challenging. Here we engineer active solids based on centimetre-scale building blocks that perform adaptive locomotion. These prototypes exhibit a non-variational form of elasticity characterized by odd moduli 8 , 12 , 17 , whose magnitude we predict from microscopics using coarse-grained theories and which we validate experimentally. When interacting with an external environment, these active solids spontaneously undergo limit cycles of shape changes, which naturally lead to locomotion such as rolling and crawling. The robustness of the locomotion is rooted in an emergent feedback loop between the active solid and the environment, which is mediated by elastic deformations and stresses. As a result, our active solids are able to accelerate, adjust their gaits and locomote through a variety of terrains with a similar performance to more complex control strategies implemented by neural networks. Our work establishes active solids as a bridge between materials and robots and suggests decentralized strategies to control the nonlinear dynamics of biological systems 8 , 18 , 19 , 20 , 21 – 22 , soft materials 5 , 6 , 9 , 11 , 12 , 23 , 24 – 25 and driven nanomechanical devices 7 , 26 , 27 , 28 , 29 – 30 . The development of active solids based on centimetre-scale building blocks incorporating odd elasticity shows that they can spontaneously undergo limit cycles of shape changes, leading to adaptive locomotion such as rolling and crawling.

A passive solid cannot do work on its surroundings through any quasistatic cycle of deformations. This property places strong constraints on the allowed elastic moduli. In this Article, we show that static elastic moduli altogether absent in passive elasticity can arise from active, non-conservative microscopic interactions. These active moduli enter the antisymmetric (or odd) part of the static elastic modulus tensor and quantify the amount of work extracted along quasistatic strain cycles. In two-dimensional isotropic media, two chiral odd-elastic moduli emerge in addition to the bulk and shear moduli. We discuss microscopic realizations that include networks of Hookean springs augmented with active transverse forces and non-reciprocal active hinges. Using coarse-grained microscopic models, numerical simulations and continuum equations, we uncover phenomena ranging from auxetic behaviour induced by odd moduli to elastic wave propagation in overdamped media enabled by self-sustained active strain cycles. Our work sheds light on the non-Hermitian mechanics of two- and three-dimensional active solids that conserve linear momentum but exhibit a non-reciprocal linear response. Active, non-conservative interactions can give rise to elastic moduli that are forbidden in equilibrium and enter the antisymmetric part of the stiffness tensor. The resulting solids function as distributed elastic engines that can perform work on their surroundings through quasistatic strain cycles.

Chiral fluids – such as fluids under rotation or a magnetic field as well as synthetic and biological active fluids – flow in a different way than ordinary ones. Due to symmetries broken at the microscopic level, chiral fluids may have asymmetric stress and viscosity tensors, for example giving rise to a hydrostatic torque or non-dissipative (odd) and parity-violating viscosities. In this article, we investigate the motion of rigid bodies in such an anisotropic fluid in the incompressible Stokes regime through the mobility matrix, which encodes the response of a solid body to forces and torques. We demonstrate how the form of the mobility matrix, which is usually determined by particle geometry, can be analogously controlled by the symmetries of the fluid. By computing the mobility matrix for simple shapes in a three-dimensional (3-D) anisotropic chiral fluid, we predict counterintuitive phenomena such as motion at an angle to the direction of applied forces and spinning under the force of gravity.

The interface between two tissues can have very different bioelectrical properties compared to either tissue on its own. Here we show that an interface between non-excitable tissues can be electrically excitable because of an interaction between the currents passing through the gap junctions—electrically resistive intercellular connections—and the non-linear current–voltage dependence in the ion channels on either side of the interface. Our theory shows that this topologically robust excitability occurs over a far larger range of ion channel expression levels than can support excitability in the bulk. The corresponding interfacial action potentials can cause local elevations in calcium concentration, possibly providing a bioelectrical mechanism for interface sensing. The observed topological action potentials point to the possibility of other types of topological effect in electrophysiology and at other diffusively coupled interfaces.Interfaces between non-excitable tissues can be electrically excitable, suggesting a possible bioelectrical mechanism for interface sensing.

This thesis explores three distinct topics, each requiring a generalization of a classical continuum theory in order to capture a striking phenomenon from a coarse-grained perspective.Chapter 2 revisits the time-honored subject of elasticity theory, which describes how solids exert stresses in response to deformation. Classical elasticity is strongly constrained by the assumption that stress is related to strain through gradients of a potential energy. This assumption is not generally valid for systems with internal sources of energy or that are governed by non-energetic effective interactions. In this chapter, I formulate a continuum theory known as odd elasticity, which generalizes classical elasticity to include nonconservative forces. Phenomenological consequences and experimental implications are discussed.Chapter 3 visits the topic of reaction-diffusion equations. In homogeneous media, it is known that adversarial forces (present everywhere in space) can give rise to local bistabilities, resulting in spikes and wave propagation. In this chapter, I show how global bistability can emerge and cause spikes and wave propagation in the presence of heterogeneities, e.g. boundaries and interfaces, that segregate competing forces. These surprisingly robust interfacial excitations arise in models of chemical reactions and predator-prey dynamics, as well as in recent experiments wherein localized action potentials are created at the interface of distinct, nonspiking bioelectric tissues.Finally, Chapter 4 studies the emergence of spontaneous wrinkles recently observed in novel confinement-free measurements of atomically thin films. I show that a classic thin sheet model with a minimal twist, disordered strain, is sufficient to recreate these wrinkles. Using continuum theory, I derive scaling predictions for the wrinkle morphology that are consistent with experimental observations. A theoretical analysis of indentation experiments reveals that these wrinkles have dramatic implications for the effective strength and heterogeneity of unconfined atomically thin films.

Soft materials are usually defined as materials made of mesoscopic entities, often self-organised, sensitive to thermal fluctuations and to weak perturbations. Archetypal examples are colloids, polymers, amphiphiles, liquid crystals, foams. The importance of soft materials in everyday commodity products, as well as in technological applications, is enormous, and controlling or improving their properties is the focus of many efforts. From a fundamental perspective, the possibility of manipulating soft material properties, by tuning interactions between constituents and by applying external perturbations, gives rise to an almost unlimited variety in physical properties. Together with the relative ease to observe and characterise them, this renders soft matter systems powerful model systems to investigate statistical physics phenomena, many of them relevant as well to hard condensed matter systems. Understanding the emerging properties from mesoscale constituents still poses enormous challenges, which have stimulated a wealth of new experimental approaches, including the synthesis of new systems with, e.g. tailored self-assembling properties, or novel experimental techniques in imaging, scattering or rheology. Theoretical and numerical methods, and coarse-grained models, have become central to predict physical properties of soft materials, while computational approaches that also use machine learning tools are playing a progressively major role in many investigations. This Roadmap intends to give a broad overview of recent and possible future activities in the field of soft materials, with experts covering various developments and challenges in material synthesis and characterisation, instrumental, simulation and theoretical methods as well as general concepts.

Soft materials are usually defined as materials made of mesoscopic entities, often self-organised, sensitive to thermal fluctuations and to weak perturbations. Archetypal examples are colloids, polymers, amphiphiles, liquid crystals, foams. The importance of soft materials in everyday commodity products, as well as in technological applications, is enormous, and controlling or improving their properties is the focus of many efforts. From a fundamental perspective, the possibility of manipulating soft material properties, by tuning interactions between constituents and by applying external perturbations, gives rise to an almost unlimited variety in physical properties. Together with the relative ease to observe and characterise them, this renders soft matter systems powerful model systems to investigate statistical physics phenomena, many of them relevant as well to hard condensed matter systems. Understanding the emerging properties from mesoscale constituents still poses enormous challenges, which have stimulated a wealth of new experimental approaches, including the synthesis of new systems with, e.g. tailored self-assembling properties, or novel experimental techniques in imaging, scattering or rheology. Theoretical and numerical methods, and coarse-grained models, have become central to predict physical properties of soft materials, while computational approaches that also use machine learning tools are playing a progressively major role in many investigations. This Roadmap intends to give a broad overview of recent and possible future activities in the field of soft materials, with experts covering various developments and challenges in material synthesis and characterisation, instrumental, simulation and theoretical methods as well as general concepts.