Anomalous Microfluidic Phonons Induced by the Interplay of Hydrodynamic Screening and Incompressibility

We investigate the acoustic normal modes (“phonons”) of a 1D microfluidic droplet crystal at the crossover between 2D flow and confined 1D plug flow. The unusual phonon spectra of the crystal, which arise from long-range hydrodynamic interactions, change anomalously under confinement. The boundaries induce weakening and screening of the interactions, but when approaching the 1D limit we measure a marked increase in the crystal sound velocity, a sign of interaction strengthening. This nonmonotonous behavior of the phonon spectra is explained theoretically by the interplay of screening and plug flow.

Figure 1

Droplets of water in oil [mineral oil, viscosity $30\mathrm{mPa}\mathrm{s}$, with 2% span-80 surfactant ($\mathrm{w}/\mathrm{w}$)] were formed at a $\mathsf{T}$ junction under constant pressure. Channel height was $10\mu \mathrm{m}$. (a) Droplet formation in confinement of $\gamma =0.62$. (b) Longitudinal waves (along $x$) in $\gamma =0.58$. (c) Transversal waves (along $y$) in $\gamma =0.46$. Scale bars are $100\mu \mathrm{m}$.

Figure 2

(a)–(g) Transversal (left column) and longitudinal (right column) phonon power spectra for increasing values of the confinement parameter $\gamma$ (shown on each plot). Color code represents the logarithm of the amplitude. White curves show our theory for $\omega (k)$. Black curves indicate the corresponding theoretical $\omega (k)$ for unconfined crystals. Dashed line is $\omega (k)=-u_{d}k$. Experimental parameters are shown in Table . (h) $C_s(\gamma )/C_s(\gamma =0)$ for both polarizations. $C_s(\gamma =0)$ was calculated from theory for an unconfined crystal with the same flow parameters as the confined one. (i) Mean amplitude of the transversal modes as a function of $\gamma$.

Figure 3

(a),(b) Flow lines around a single confined droplet (dark blue) are the result of summing the dipole flow fields of its infinite array of reflections (gray). (a) When the droplet is at the center of the channel, the array has uniform spacing $W$. (b) When the droplet is off-centered, the reflections array splits into two interlacing arrays. (c),(d) Flow lines around three droplets with a longitudinal (c) and transversal (d) perturbation. (e) Dipolar flow field of an unconfined droplet.

Figure 4

Prediction for $C_s$ as a function of $\gamma$ for a few crystal densities $a/R$. The transversal (solid red) curve and the longitudinal $C_s$ (dashed blue) curve are normalized by $C_s(\gamma =0)$ of an unconfined crystal. $C_{s,x}$ and $C_{s,y}$ measured in experiments (Fig. 2) are shown by closed circles and squares, respectively.