Droplet Breakup Against an Isolated Obstacle

TL;DR

This work studies how droplets break up when flowing around an isolated obstacle in a quasi-two-dimensional microfluidic chamber. By combining experiments with a two-dimensional deformable-particle model and a geometric breakup criterion, the authors map when breakup occurs as a function of collision symmetry, capillary-driven stresses, droplet size, and confinement. They introduce a breakup number $Bk ~ Ca$ and show a rapid transition from no-breakup to breakup as $Bk$ crosses unity, with $Bk ~ S^{4/3}$ and a symmetry-driven length scale $h ~ R$, enabling collapse across six orders of magnitude. These results provide a predictive framework for droplet breakup in simplified porous-like microfluidic geometries and suggest avenues for extending the approach to three dimensions, coalescence, and substrate wetting effects in future work.

Abstract

We describe combined experiments and simulations of droplet breakup during flow-driven interactions with a circular obstacle in a quasi-two-dimensional microfluidic chamber. Due to a lack of in-plane confinement, the droplets can also slip past the obstacle without breaking. Droplets are more likely to break when they have a higher flow velocity, larger size (relative to the obstacle radius R), smaller surface tension, and for head-on collisions with the obstacle. We also observe that droplet-obstacle collisions are more likely to result in breakup when the height of the sample chamber is increased. We define a nondimensional breakup number Bk ~ Ca, where Ca is the Capillary number, that accounts for changes in the likelihood of droplet break up with variations in these parameters. As Bk increases, we find in both experiments and discrete element method (DEM) simulations of the deformable particle model that the behavior changes from droplets never breaking (Bk << 1) to always breaking for Bk >> 1, with a rapid change in the probability of droplet breakup near Bk = 1. We also find that Bk ~ S^(4/3), where S characterizes the symmetry of the collision, which implies that the minimum symmetry required for breakup is controlled by a characteristic distance h ~ R.

Figures (13)

• Figure 1: (Left) Images of droplets moving from left to right through the array of obstacles with collisions highlighted by red circles in each image pair. (Top Row) A head-on collision (with symmetry $S=1.0$) causes a large droplet with velocity $v=1.8$ mm/s to break up; (Middle Row) An asymmetric collision with $S=0.5$ between a large droplet with $v=1.4$ mm/s causes droplet break up; and (Bottom Row) An asymmetric collision with $S=0.5$ between a small droplet with $v=1.1$ mm/s does not lead to break up. The time between the left and right images in each row is determined by the terminal velocity $v_t=400~\mu$m.(Right) Microfluidics design of the sample chamber, which is $\approx 22$ mm long. Droplets form at the middle top region and exit through the central channel into the wider region below. In the wider region, droplets collide with small obstacles before exiting the chamber at the bottom outlet.
• Figure 2: Schematic of a droplet (shaded pink) with area $A$ interacting with an obstacle (shaded gray) with radius $R$. The inset highlights the vertices that define the droplet surface in the deformable particle model. We also define vertex center-to-center distance $r_{ij}$ and the arc length $s_{ij}$ between vertices $i$ and $j$. The droplet neck thickness, $d_{\rm neck}$, is identified using the method described in Sec. \ref{['mesoscale']}.
• Figure 3: Illustration of the definition of the symmetry parameter $S$. When a droplet first contacts an obstacle (shaded gray), a dividing line (dashed line) is drawn through the droplet to form two regions with areas $A_\text{large}$ (shaded blue) and $A_\text{small}$ (shaded yellow). The dividing line is parallel to the center of mass velocity vector of the droplet (large white arrow) and passes through the center of the obstacle.
• Figure 4: The symmetry parameter $S$ plotted versus ${\rm Ca} \widetilde{A} \widetilde{z}$ for all droplet collisions separated into those for which the droplets (Left) break up and (Right) do not break up. The separating line (black dashed line) is given in eqn (\ref{['finalform']}) with power-law scaling exponent $\beta = -0.74$.
• Figure 5: Results from the deformable particle model (DPM) simulations (stars) showing droplets that break up (upper right; green) and do not break up (lower left; purple) as a function of $S$ and ${\rm Ca} \widetilde{A}\widetilde{z}$ overlaid on the experimental data from Fig. \ref{['fig:Main']}. We set $\widetilde{z}=1$ for the simulation data. The best fit lines that separate the droplets that break up and do not break up have slopes $-0.72$ (black dashed; simulations) and $-0.74$ (orange solid; experiments).