Isostaticity, auxetic response, surface modes, and conformal invariance in twisted kagome lattices

Model lattices consisting of balls connected by central-force springs provide much of our understanding of mechanical response and phonon structure of real materials. Their stability depends critically on their coordination number $z$. $d$-dimensional lattices with $z=2d$ are at the threshold of mechanical stability and are isostatic. Lattices with $z<2d$ exhibit zero-frequency "floppy" modes that provide avenues for lattice collapse. The physics of systems as diverse as architectural structures, network glasses, randomly packed spheres, and biopolymer networks is strongly influenced by a nearby isostatic lattice. We explore elasticity and phonons of a special class of two-dimensional isostatic lattices constructed by distorting the kagome lattice. We show that the phonon structure of these lattices, characterized by vanishing bulk moduli and thus negative Poisson ratios and auxetic elasticity, depends sensitively on boundary conditions and on the nature of the kagome distortions. We construct lattices that under free boundary conditions exhibit surface floppy modes only or a combination of both surface and bulk floppy modes; and we show that bulk floppy modes present under free boundary conditions are also present under periodic boundary conditions but that surface modes are not. In the the long-wavelength limit, the elastic theory of all these lattices is a conformally invariant field theory with holographic properties, and the surface waves are Rayleigh waves. We discuss our results in relation to recent work on jammed systems. Our results highlight the importance of network architecture in determining floppy-mode structure.Comment: 12 pages, 7 figure

Odd viscosity in chiral active fluids

Chiral active fluids are materials composed of self-spinning rotors that continuously inject energy and angular momentum at the microscale. Out-of-equilibrium fluids with active-rotor constituents have been experimentally realized using nanoscale biomolecular motors, microscale active colloids, or macroscale driven chiral grains. Here, we show how such chiral active fluids break both parity and time-reversal symmetries in their steady states, giving rise to a dissipationless linear-response coefficient called odd viscosity in their constitutive relations. Odd viscosity couples pressure and vorticity leading, for example, to density modulations within a vortex profile. Moreover, chiral active fluids flow in the direction transverse to applied compression as in shock propagation experiments. We envision that this collective transverse response may be exploited to design self-assembled hydraulic cranks that convert between linear and rotational motion in microscopic machines powered by active-rotors fluids

Optimal power and efficiency of odd engines

Odd materials feature antisymmetric response to perturbations. This anomalous property can stem from the nonequilibrium activity of their components, which is sustained by an external energy supply. These materials open the door to designing innovative engines which extract work by applying cyclic deformations, without any equivalent in equilibrium. Here, we reveal that the efficiency of such energy conversion, from local activity to macroscopic work, can be arbitrarily close to unity when the cycles of deformation are properly designed. We illustrate these principles in some canonical viscoelastic materials, which leads us to identify strategies for optimizing power and efficiency according to material properties, and to delineate guidelines for the design of more complex odd engines.Comment: 6 pages, 2 figure

Hydrodynamic correlation functions of chiral active fluids

The success of spectroscopy to characterize equilibrium fluids, for example the heat capacity ratio, suggests a parallel approach for active fluids. Here, we start from a hydrodynamic description of chiral active fluids composed of spinning constituents and derive their low-frequency, long-wavelength response functions using the Kadanoff-Martin formalism. We find that the presence of odd (equivalently, Hall) viscosity leads to mixed density-vorticity response even at linear order. Such response, prohibited in time-reversal invariant fluids, is a large-scale manifestation of the microscopic breaking of time-reversal symmetry. Our work suggests possible experimental probes that can measure anomalous transport coefficients in active fluids through dynamic light scattering