# Pillar-induced droplet merging in microfluidic circuits

Original entry: Tony Orth, APPHY 226, Spring 2009

Xize Niu, Shelly Gulati, Joshua B. Edel, and Andrew J. deMelloa, Lab On a Chip (2008), 8, 1837-1841 [1]

### Abstract

"A novel method is presented for controllably merging aqueous microdroplets within segmented ﬂow microﬂuidic devices. Our approach involves exploiting the difference in hydrodynamic resistance of the continuous phase and the surface tension of the discrete phase through the use of passive structures contained within a microﬂuidic channel. Rows of pillars separated by distances smaller than the representative droplet dimension are installed within the ﬂuidic network and deﬁne passive merging elements or chambers. Initial experiments demonstrate that such a merging element can controllably adjust the distance between adjacent droplets. In a typical scenario, a droplet will enter the chamber, slow down and stop. It will wait and then merge with the succeeding droplets until the surface tension is overwhelmed by the hydraulic pressure. We show that such a merging process is independent of the inter-droplet separation but rather dependent on the droplet size. Moreover, the number of droplets that can be merged at any time is also dependent on the mass ﬂow rate and volume ratio between the droplets and the merging chamber. Finally, we note that the merging of droplet interfaces occurs within both compressing and the decompressing regimes."

### Surface Tension Phenomena

The goal of this study is to engineer a device which passively merges droplets and achieves some control over the number of droplets to be merged. The authors construct a tapered pillar region which is wider than the channel feeding into it. The pillars are the same height of the channel (ie. single exposure photolithography is used to create the master for the PDMS device) and are arranged two in two rows on either side of the channel as shown in figure 1. The pillars are wider further down the channel, hence a regular "tapered" pattern of pillars. When a single drop with dimensions comparable to the spacing between the rows of pillars enters he expanded channel/pillar region, it is trapped. The authors explain this by looking the the hydrodynamic and Laplace pressures at play. The pressure drop across the the interface at the tail - subscript "t" (head - subscript "h") of the droplet is <math>\Delta P_{t,h} = \gamma (1/r_{t,h} + 2/H)</math> where <math>r</math> is the radius of the appropriate end of the drop and H is the height of the channel so that H/2 is the radius of curvature in the direction orthogonal to the micrograph. Alternatively, one can replace one of the radii of curvature with the taper angle <math>\alpha</math> as defined in figure 2. The pressure drop from the region just outside the drop tail to just outside the drop head is: <math>\Delta P_d = \gamma L tan\alpha/(r_1(r_1 - Ltan\alpha))</math>. There is also a hydrodynamic pressure drop associated with pressure driven flow. The authors denote this also as <math>\Delta P_d</math> and se tit equal to a sum of two terms, one of which is proportional to a flow rate into each of the small channels between the pillars. For <math>n</math> pillars, this flow rate is <math>Q_1 = Q/n</math> where Q is the unperturbed channel flow rate. The hydrodynamic pressure drop associated with this term is <math>\propto Q_1 \propto 1/n</math> so that as the drop moves into the pillar chamber, the effective number of pillars <math>n</math> increases and so this pressure differential decreases. The authors then note the competition between this hydrodynamically induced pressure drop and the Laplace pressure drop. As the drop enters the chamber, the two become closer in magnitude until they are equal and the drop is stuck.

The authors report that this trapping arrangement is very stable; drops can be made to sit in the merging chamber for hours upon end. When another drop enters the chamber, it coalesces with the first droplet if the two together roughly fill the chamber. Up to 5 drops have been made to merge in this fashion. Unfortunately, the authors do not discuss the merging dynamics to a great extent; the drops merge in less than 40 <math>\mu s</math> in some cases. Mixing occurs very rapidly (see figure 3) which is advantageous as it is usually quite difficult to efficiently mix different substances within drops. Presumably the energy released by the rupture of the interface of the drops is partially transferred into kinetic energy of the fluid causing chaotic advection within the drop.

Questions: Why does the interfacial pressure drop act in a direction opposite to that resulting from the pressure driven flow (figure 1). Shouldn't there be a bigger pressure drop across the head interface so that the fluid outside the drop head is at a lower pressure than that just outside the tail? This would result in a force in the same direction as the flow which does not counter the hydrodynamic pressure drop.