Phonons in a One-Dimensional Microfluidic Crystal at Very Low Re

The development of a general theory for the behavior of a crystal driven far from equilibrium has proved difficult. Microfluidic crystals of water-in-oil droplets provide a convenient means to explore and develop models for non-equilibrium dynamics. Owing to the fact that these systems operate at low Reynolds number (Re), in which viscous dissipation dominates inertial effects, vibrations are expected to be over-damped. Against such expectations, we report the emergence of collective normal vibrational modes (equivalent to acoustic `phonons') in a 1D microfluidic crystal of droplets at Re$\sim $10-4. These phonons propagate at ultra-low sound velocity of $\sim $100$\mu $m/s and frequencies of a few Hz, 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 the symmetry-breaking flow field that induces long-range inter-droplet interactions, similar in nature to those observed in other systems including dusty plasma crystals, vortices in superconductors and active membranes.\newline Nature Physics 2, 743-748 (2006).