Odd viscosity in chiral active materials

Chiral active materials are composed of self-spinning rotors that continuously inject energy and angular momentum at the microscale. Out-of-equilibrium materials 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 materials break both parity and time-reversal symmetries in their steady states, giving rise to a dissipationless linear-response coefficient called odd viscosity in the constitutive relations. This odd viscosity, analogous to the Hall viscosity of electron fluids, causes motion in a direction transverse to applied compression and allows the creation of crank-like active metamaterials. Our results suggest how an anomalous viscous response can be engineered in active metamaterials by controlling the dynamics of their constituents in the same spirit as an unusual elastic response can be engineered in passive metamaterials by controlling their architecture.