The development of a general theoretical framework for describing the behaviour of a crystal driven far from equilibrium has proved difficult. 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. 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. 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 crystals, vortices in superconductors, active membranes and nucleoprotein filaments.
Bibliographical noteFunding Information:
We thank G. Falkovich, S. Fleishman, A. Libchaber, E. Moses, J. Prost, S. Safran, V. Steinberg and D. Weitz for useful suggestions and discussions. R.B.-Z. is an incumbent of the Beracha career development chair. This work was supported by a grant from the Israel Science Foundation (Bikura) to R.B.-Z. Correspondence and requests for materials should be addressed to R.B.-Z. Supplementary Information accompanies this paper on www.nature.com/naturephysics.