Phonons in a one-dimensional microfluidic crystal

Abstract

The development of a general theoretical framework for describing the behaviour of a crystal driven far from equilibrium has proved difficult(1). 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 dynamics(2-11). 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 dynamics(12-14). 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 similar to 10(-4). These phonons propagate at an ultra-low sound velocity of similar to 100 mu ms(-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 crystals(15,16), vortices in superconductors(17,18), active membranes(19) and nucleoprotein. laments(20).