Phonons in a one-dimensional microfluidic crystal

The development of a general theoretical framework for describing the behaviour of a crystal driven far from equilibrium has proved difficult1. Microfluidic crystals, formed by the introduction of droplets of immiscible fluid into a liquid-filled channel, provide a convenient means to explore and develop models to describe non-equilibrium dynamics2,3,4,5,6,7,8,9,10,11. Owing to the fact that these systems operate at low Reynolds number (Re), in which viscous dissipation of energy dominates inertial effects, vibrations are expected to be over-damped and contribute little to their dynamics12,13,14. Against such expectations, we report the emergence of collective normal vibrational modes (equivalent to acoustic ‘phonons’) in a one-dimensional microfluidic crystal of water-in-oil droplets at Re∼10−4. These phonons propagate at an ultra-low sound velocity of ∼100 μm s−1 and frequencies of a few hertz, exhibit unusual dispersion relations markedly different to those of harmonic crystals, and give rise to a variety of crystal instabilities that could have implications for the design of commercial microfluidic systems. First-principles theory shows that these phonons are an outcome of the symmetry-breaking flow field that induces long-range inter-droplet interactions, similar in nature to those observed in many other systems including dusty plasma crystals15,16, vortices in superconductors17,18, active membranes19 and nucleoprotein filaments20.