Emergent Hyperuniformity in Periodically Driven Emulsions

We report the self-organization of microfluidic emulsions into anomalously homogeneous structures. Upon periodic driving confined emulsions undergo a first-order transition from a reversible to an irreversible dynamics. We evidence that this dynamical transition is accompanied by structural changes at all scales yielding macroscopic yet finite hyperuniform structures. Numerical simulations are performed to single out the very ingredients responsible for the suppression of density fluctuations. We show that, as opposed to equilibrium systems, the long-range nature of the hydrodynamic interactions are not required for the formation of hyperuniform patterns, thereby suggesting a robust relation between reversibility and hyperuniformity which should hold in a broad class of periodically driven materials.

Figure 1

Close-up of the emulsion at the beginning of the 500th cycle. The active droplets are shown as open orange circles, the passive droplets as filled blue circles. (a) For a driving amplitude $\mathrm{\Delta }/{\mathrm{\Delta }}^{\star }=0.65$, the dynamics is reversible. (b) For a driving amplitude $\mathrm{\Delta }/{\mathrm{\Delta }}^{\star }=1.54$, the dynamics is not reversible anymore. Note also the markedly different structure in the two cases: for low amplitudes the structure is more homogeneous.

Figure 2

(a) Fraction of active particles $f_a$ in the steady state at various oscillation amplitudes. At $\mathrm{\Delta }={\mathrm{\Delta }}^{\star }$ reversibility abruptly breaks down. Inset: Variations of the mean-squared strobed displacement in the flow direction $⟨\delta x^2⟩$ plotted as a function of $\mathrm{\Delta }$. (b) Centered and normalized probability density distribution of the strobed displacements of the droplets, $|\delta x|$, in the flow direction for the experiments (top curves) and numerics (bottom curves). The numerical curves have been shifted for the sake of clarity. The color indicates the mean fraction of active particles. See color bar in (a). (c) Structure factor $S(k)$ at various amplitudes [Experiments: thin lines, simulations: Thick lines, Same color code as in (b)]. For sake of clarity the curves corresponding to $\mathrm{\Delta }<{\mathrm{\Delta }}^{\star }$ have been shifted down by a constant value. The dashed line is a guide and corresponds to $k^{0.45}$.

Figure 3

(a),(b), and (c) Normalized density fluctuations for experiments (a), simulations with long-range interactions (b), and short-range interactions (c). The curves are offset for the sake of clarity. At high amplitudes (red curves) the density fluctuations are normal, whereas in the reversible regime, at low amplitudes (blue curves), the density fluctuations are suppressed. (d) Fitted power-law exponents $\lambda$ from the curves plotted in (a),(b), and (c). The lines are guides to the eye. Density fluctuations are only suppressed for $\mathrm{\Delta }\lesssim {\mathrm{\Delta }}^{\star }$. (e) Extent of the hyperuniform regions ${\ell }_{\mathrm{HU}}$. Filled symbols: experiments, open symbols: simulations. The lines are guides to the eye. The system is only hyperuniform up to length scales of $\sim 10a$ ($\sim 15a$) in the experiments (in the simulations).