Geometrically Mediated Breakup of Drops in Microfluidic Devices

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Zach Wissner-Gross (February 9, 2009)


Howard Stone and coworkers describe two techniques for breaking up droplets in an emulsion using microfluidic devices. The first method makes use of a "T-junction," in which a droplet flows down a channel that abruptly bifurcates into two channels in an orthogonal direction. The second method involves placing an obstacle in the droplet's path. Also notable is the article's analysis of the conditions under which a droplet encountering a T-junction will actually break up.

Using T-junctions

The process of creating droplets in the first place was previously described by Steve Quake and coworkers [1], who mixed two immiscible liquids in a similar T-junction microfluidic circuit, made from PDMS using soft lithography [2]. Stone and coworkers used oil (hexadecane) and water to make the emulsion, including a surfactant to lower the surface tension between the two liquids in order to stabilize droplets.

Water droplets in the oil emulsion were flowed toward the T-junction, where they encountered a stagnation point at the intersection (Figure 1). Under certain conditions, the droplets would simply flow into one of the two side channels -- the authors mention that unbroken droplets in fact precisely alternate which channel they choose, although this phenomenon appears non-trivial to me. Perhaps when a given droplet selects one channel, it increases the fluidic resistance of that channel so that the next droplet favors the other channel. But more interestingly, under different conditions, oil flowing behind the droplet will pinch closed the surface, thus breaking the drop into two smaller droplets.

Figure 1: The top set of images (a-e) shows a droplet that will remain intact, while the bottom set (f-j) shows a droplet that will ultimately break up.

When breakup occurs

The real physics of this paper occurs when the authors attempt to discern the conditions under which water droplets break up at the T-junction. Breakup takes place due to Rayleigh-Plateau instability [3], which predicts the breakup of a cylindrical column of liquid when its length exceeds its circumference. The authors define this geometric ratio as the extension <math>\epsilon</math> of a droplet:

<math>\epsilon=\frac{l}{\pi w}</math>

where <math>l</math> and <math>w</math> are the length of the drop and width of the channel that constrains it, respectively. Thus, when <math>\epsilon>1</math>, a drop will typically bifurcate. The authors state that a water drop with initial extension <math>\epsilon_0>1</math> (defined by the initial length <math>l_0</math> and width <math>w_0</math>) moving toward the T-junction will not break up due to the stabilizing effects of the surrounding walls -- but such stability disappears when the drop reaches the stagnation point at the T-junction.

Due to the deformation that occurs at the T-junction, drops that would originally have been stable can deform to the point that their extension reaches a maximum value of <math>\epsilon_{\text{e}}>1</math> (with length <math>l_{\text{e}}</math> and width <math>w_{\text{e}}</math>). The authors then derive whether a water drop will break up as a function of <math>\epsilon_0</math> and the properties of the emulsion and fluid flow as follows:

First they define the capillary number <math>C</math> of the flow as the dimensionless ratio of the viscous stress to the interfacial surface tension <math>\sigma</math> in the emulsion:

<math>C=\frac{\eta v}{\sigma}</math>

where <math>\eta</math> is the viscosity of the surrounding medium (oil in this case) and <math>v</math> is the velocity of the flow toward the T-junction. Citing the previous work of Yiftah Navot [4], the authors state that

<math>C\propto (l_{\text{e}}-l_0)^2</math>

Now they wish the find the critical capillary number <math>C_{\text{cr}}</math> at which the extended droplet becomes unstable, i.e., when

<math>\epsilon_{\text{e}}=\frac{l_{\text{e}}}{\pi w_{\text{e}}}=1</math>

with the additional constraint of conservation of volume

<math>l_0 w_0^2=l_{\text{e}} w_{\text{e}}^2 </math>.

We can combine the last two relations to see that

<math>l_{\text{e}}=(\pi^2 l_0 w_0^2)^{1/3}=l_0 \epsilon_0^{-2/3}</math>

and hence that

<math>C_{\text{cr}} \propto l_0(\epsilon_0^{-2/3}-1)^2 \propto \epsilon_0(\epsilon_0^{-2/3}-1)^2</math>.

Finally, the authors show a rather remarkable plot of <math>\epsilon_0</math> versus <math>C</math> that they created by using different droplet sizes and different velocities (Figure 2). The transition between drops that break and those that do not behaves very much as the theory predicted (with a constant of proportionality in the above near unity!).

Wiki figure2.png

Using isolated obstacles