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 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 anultra-low sound velocity of∼100μms−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. To investigate many-body effects of one-dimensional (1D) hydrodynamic crystals, we built a microfluidic water-in-oil droplet generator (Fig. 1a, Methods section and Supplementary Information, Movie S1). Water droplets formed at a T-junction between water and oil channels under continuous flow, emanating at a constant rate with uniform radii R (10–15 μm) and fixed interdroplet distances a (10–200 μm). The thin channel (h = 10 μm) deformed the droplets into discs, confining their motion to 2D and exerting friction with the floor and ceiling (Fig. 1b). Owing to friction, the droplets were dragged by the oil at a velocity ud (150−800 μms−1) that was slower than that of the oil (uoil ∼ 5ud). Symmetry was broken by the relative motion of the oil with respect to the droplet crystal. Thus, we obtained a flowing 1D crystal of droplets that can move in 2D. The crystal exhibited visible longitudinal and transversal fluctuations, which were reminiscent of solid-state phonons (we henceforth term these normal modes ‘phonons’) (Fig. 1c,d and Supplementary Information, Movies S2–S4). We explored these modes by measuring their wave dispersion relations (Fig. 1e–h). This was done by tracking the positions of droplets in time and applying a Fourier transform to obtain the power spectrum of vibrations in terms of the wavevector k and frequency ω (see the Methods section). We then extracted the dispersion relations of waves in the crystal, ω(k). Surprisingly, the dispersion relations reveal the existence of acoustic phonons that propagate in the crystal at ultra-low frequencies of a few hertz. Manifestly, at Re∼10−4, collectivemodes at such low frequencies cannot arise from inertial effects and are likely to be due to hydrodynamic interactions within the crystal. Themain feature of the dispersion is an unusual sine-like curve that spans the Brillouin zone (0≤ k≤ π/a) and has unique properties (Fig. 1). The linear behaviour of the curve ω(k) = Csk around k= 0 shows that these waves are acoustic and propagate at a sound velocity of Cs = (∂ω/∂k)k→0 ≈ 250 μms−1. This velocity is some six orders of magnitude slower than sound in common liquids. Close to the edge of the Brillouin zone, k=π/a, the acoustic waves travel in the opposite direction at a velocity−Cs/2, with a crossover between positive and negative group velocities. The longitudinal (Fig. 1e,g) and transversal (Fig. 1f,h) modes are identical in form and magnitude but travel in opposite directions ωy(k)=−ωx(k). Dispersion of modes in the hydrodynamic crystal of droplets is markedly different from that of a harmonic crystal, where each wavevector has both symmetric forward and backward waves, ω(k) = ω(−k), with standing waves at the edge of the Brillouin zone. In the hydrodynamic crystal, this symmetry is broken by the flow field of the oil such that waves travel only in one direction per wavevector. In addition, a standing wave appears at the groupvelocity crossover point within the Brillouin zone. A secondary feature of ω(k) is a straight line, ω = −udk, where ud is the velocity of droplets relative to the channel. This simply stems from droplet deflections owing to stationary defects along the channel that appear as if moving backwards at−ud, as the camera is moving in frame with the droplets. As we now explain theoretically, the unusual dispersion of the moving crystal arises from hydrodynamic interactions between droplets induced by the symmetry-breaking flow field. The motion of each droplet perturbs the flow of the surrounding oil. This perturbation affects the other droplets in