Deformation and breakup of high-viscosity droplets with symmetric microfluidic cross flows

The dynamic response of highly viscous droplets to a sharp increase in the surrounding liquid velocity is experimentally investigated in a square microchannel junction. The local injection of the continuous phase from symmetric side channels onto a train of droplets produces a large velocity contrast between the front and the rear of droplets, yielding a broad range of time-dependent deformation and breakup. In particular, due to microscale confinement, the system displays a nonlinear behavior with the initial droplet size. Deformations, relaxation times, and fragmentation processes are examined as a function of flow parameters and fluids properties with emphasis on the formation of slender viscous structures such as spoon-shaped droplets, i.e., asymmetrical droplets.

Figure 1

(Color online) High-viscosity droplets in microfluidic cross flows. Top: two-step hydrodynamic sections in series. Bottom: experimental time series of typical regimes, $\chi ={\eta }_1/{\eta }_2=85$, flow rates in $\mu \text{l}/\text{min}$, time intervals in ms. (a) Relaxing ($Q_1=1$, $Q_2=2$, $Q_3=10$, and $\Delta t=160$), (b) convective breakup ($Q_1=1$, $Q_2=2$, $Q_3=15$, and $\Delta t=64$), and (c) absolute breakup ($Q_1=0.5$, $Q_2=1$, $Q_3=10$, and $\Delta t=26$).

Figure 2

Phase diagram of droplet deformation. Viscosity contrast: $\chi =17$ (△), 42 (◻), 85 (○), 169 $(▷)$, and 426 (◇). (a) Relaxing deformation (open symbols). (b) Convective breakup (filled symbols). (c) Absolute breakup (dotted symbols). Dashed lines: $d_0/h=a{\left({\alpha }_3{\text{Ca}}_3\right)}^{-0.17}$ with $a=0.5$ (i), 0.95 (ii), and 1.13 (iii).

Figure 3

Dynamics of relaxed deformation. (a) Temporal evolution of a droplet length $d/h$ during relaxing deformation where $d_0$ is the initial length, $d_{\text{max}}$ is the maximal elongation, and $d_{\text{F}}$ is the final length. Inset: evolution of $d_{\text{F}}/d_0$ vs capillary number Ca. (b) Corresponding temporal evolution of droplet nose velocity $V_{\text{N}}$ and rear velocity $V_{\text{R}}$ during deformation where $J_0$ and $J_{\text{F}}$ are the initial and final homogeneous superficial velocities. (c) Evolution of droplet lengths $d$ scaled by maximal elongation $d_{\text{max}}$ vs time $t$ scaled by viscous-capillary time $t_{\text{c}}$. Inset: temporal evolution of droplet lengths $d/h$ for different elongation $d_{\text{max}}/h$. (d) Evolution of $k_1$ vs viscosity ratio $\chi$. Dashed line: $k_1=-0.068$.

Figure 4

Maximum droplet elongation in the junction. (a) Dimensionless elongation $d_{\text{max}}/d_0$ vs superficial velocity ratio and cross-flow capillary number $\varphi {\text{Ca}}_3$ for a fixed initial droplet size $d_0/h=2.0\pm 0.1$ and $\chi =85$. Solid line: $d_{\text{max}}/d_0=1+a\varphi {\text{Ca}}_3$, where $a=17.7$. (b) Evolution of prefactor $a$ vs initial droplet size $d_0/h$ for $\chi =17$ (△), 42 (◻), 85 (○), 169 $(▷)$, and 426 (◇). Solid line: $a={\left(d_0/h\right)}^{5.7}$. (c) Evolution of droplet elongation for all fluid pairs and all initial droplet sizes. Solid line: $d_{\text{max}}/d_0=1+x^{0.65}$, with $x=\varphi {\text{Ca}}_3{\left(d_0/h\right)}^{5.7}$.

Figure 5

Spoon-shaped droplets: morphology during convective breakup. (a) Influence of Ca on droplet symmetry and confinement at the channel junction ($h=250\mu \text{m}$ and $\chi =85$), from top to bottom: $\text{Ca}=5\times {10}^{-3}$, $8\times {10}^{-3}$, and $2\times {10}^{-2}$. (b) Spoon-shaped droplets in straight square microchannel $\left(h=250\mu \text{m}\right)$, from top to bottom: “tea spoon” $\left(\chi =83\right)$, “dessert spoon” $\left(\chi =17\right)$, and “salad spoon” $\left(\chi =17\right)$. (c) Time series of a complex fragmentation process $\Delta t=10\text{ms}$ ($h=100\mu \text{m}$ and $\chi =42$).