Figure 1

(a) Top view of the experiment. Tetradecane (black) and water labeled with fluorescein (white) meet in the T junction, emitting droplets. (b) Sketch of the actuation system (1) working channel; (2) actuation channel; textured regions such as (3) PDMS; (4) glass substrate. The working and the actuation channels are separated by a membrane. (c) Typical Fourier power spectrum for a T junction producing droplets without actuation.Reuse & Permissions

Figure 2

Fourier power spectra obtained at different forcing frequencies, with pressure driven flows, in a $40\mu \mathrm{m}$ deep microchannel; the natural frequency $f_0$ is 2.37 Hz in cases (a), (b), and (c). (a) $f_f=2.4\mathrm{Hz}$ ($1/1$ resonance). (b) $f_f=7.5\mathrm{Hz}$ ($1/3$ resonance). (c) $f_f=3.76\mathrm{Hz}$ ($2/3$ resonance). (d) (Here, the natural frequency is 6.7 Hz.) $f_f=12.8\mathrm{Hz}$ (quasiperiodic regime).Reuse & Permissions

Figure 3

(a) Synchronized (gray zones) and quasiperiodic regimes (white zones) shown on a diagram where the ordinate is $P$ and the abscissa is the ratio of the natural frequency $f_0$ over the forced frequency $f_f$. Syringe pumps are used, imposing oil and water flow rates of, respectively, 4 and $0.3\mu \mathrm{l}/\min$. The microchannel depth is $20\mu \mathrm{m}$. Droplets are shown in some cases. (b) Evolution of the winding numbers $W$ with the ratio of the natural frequency over the forcing frequency. The microchannel depth is $40\mu \mathrm{m}$ in this case, and the oil and water flow rates are 2 and $0.40\mu \mathrm{l}/\min$, respectively. The plateaus, which correspond to specific rational numbers, reveal synchronized states; outside the plateaus, the regimes are quasiperiodic and the winding number varies with the forcing frequency. This plot is often called “devil staircase.”Reuse & Permissions

Figure 4

Regions of synchronization and quasiperiodicity obtained for an oil flow rate of $6.7\mu \mathrm{l}/\min$ and an actuation pressure $P=2\mathrm{bar}$. The range of variation of the water flow rate extends from 0.3 to $4\mu \mathrm{l}/\min$. The susceptibility $S$ is determined, without forcing, from the experimental curve $f_0=f(Q_w)$. The symbols appearing on the plot signal the various synchronized regimes obtained at 1 (crosses), $1/2$ (circles), or $1/3$ (diamonds) of the forcing frequency. (a) Evolution with $f_0/f_f$ of the ratio of the droplet size $\lambda$ (as compared to ${\lambda }_0$, obtained without forcing. (b) Evolution with $f_0/f_f$ the relative dispersion ${\sigma }_r$ of the droplet sizes. On this plot, the crosses correspond to QP regimes and the disks to synchronized states. Plots (a) and (b) are obtained for $S=550\mu {\mathrm{l}}^{-1}$.Reuse & Permissions

Figure 5

Evolution of the droplet volumes in a function of the actuation frequency, in a $20\mu \mathrm{m}$ deep microchannel; oil and water flow rates are, respectively, 4 and $0.1\mu \mathrm{l}/\min$, and the actuation pressure $P$ is 1.7 bar. The pictures represent droplets at various frequencies. The inset shows the evolution of the winding number with $f_0/f_f$, and indicates that the devil staircase includes one single step.Reuse & Permissions