# A Design for Mixing Using Bubbles in Branched Microfluidic Channels

"Design for mixing using bubbles in branched microfluidic channels"

Piotr Garstecki, Michael A. Fischbach, and George M. Whitesides

Applied Physics Letters **86**(24) 244108 (2005)

## Contents

## Soft Matter Keywords

microfluidic, bubbles, laminar mixing, Peclet

## Summary

This paper details experimental work and simple supporting theory regarding mixing in microfluidic channels. For most microfluidic systems, the Reynolds number remains small (less than 1000), so turbulence is absent and mixing only occurs via diffusion. Typical Peclet numbers in microfluidic channels are on the order of 1e5, indicating that mixing to homogeneity requires length scales on the order of 10 meters. These lengths are not easily achieved on microfluidic devices due to finite substrate limits for fabrication and large pressure drops in the long channels, so the authors propose a novel method of mixing that aid the diffusion process. Using bubbles to fold two liquid streams into one another, greater contact area between the fluids is created, aiding in diffusion.

## Practical Application of Research

A drawback to continuous flow microfluidics has been the inability to homogeneously mix streams on-chip. This limits the application of continuous flow microfluidics, as it is difficult to introduce additional reagents or samples to a flow. This passive, on-chip mixing scheme open a new realm of experiments that can take advantage of introduction of precise amounts of fluid at a given spatial or temporal point in a flow.

## Microfluidic Mixing Using Bubbles

As shown in Figure 1, the two liquids to be mixed are combined with a stream of air at a microfluidic flow focussing junction. The liquid streams pinch off air bubbles, which then flow downstream. This break up is mediated primarily by conservation of mass. As the thread of air enters the junction, it restricts flow of the two outer liquids. Pressure builds up in the outer liquid lines, causing the air/liquid interface to deform. Eventually, the interface detaches from the walls, becomes unstable, and breaks, forming a bubble [1].

Once in the device, sequential air bubbles alternate arms of the branched channels they flow into (see Figure 2). As an air bubble enters on branch, it causes the hydrodynamic resistance in that side of the branch to increase. This causes a greater percentage of the new incoming flow to be diverted down the other branch. This causes the next air bubble to flow through the branch that does not contain the first bubble. As the authors mention in the paper, this is an additive effect and will work for any number of droplets in a series of branches. The incoming droplet will end up being passively steered by the majority of the fluid streamlines to the channel with the smallest hydrodynamic resistance.

In the case where no bubbles are present, the fluidic resistances of the two arms are equivalent and the fluid streamlines will split evenly left and right. This means that if the two inlet fluids are introduced at equal flow rates, the fluid that starts on the left of the center of the main channel will always take the left branch, while the fluid on the right of center will always take the right branch (see Figure 3(a)). The only mixing that occurs is when the two fluids are in contact with each other just before each branch point. Thus, diffusion will take a long distance to homogeneously mix the two.

However, the presence of the bubbles leads to the aforementioned variations in fluidic resistances and thus the fluid streams do not split down the middle at each branch. At times, more liquid flows to the left branch, so some of the liquid that starts on the right half of center ends up in the left branch, and vice versa (Figure 3(b)). In this manner, the two fluids are folded into each other every time they pass one of the branch points. This creates more interfaces between the two fluids and decreases the characteristic length over which diffusion has to occur. The other way to think of this is that the gradients are increased (since length to diffuse is decreased), leading to a greater driving force for diffusive mixing.

To characterize their experimental results a little, the authors prepare a simple theoretical model to estimate the number of branch points required to completely mix the two fluids. The authors compare the average distance between two interfaces with the diffusion length scale. Assuming each branch point doubles the interface area between the two fluids and the overall increase is exponential in the distance travelled downstream, the average distance between interfaces is defined as <math>d_{inter}=w2^{l/a}</math>. <math>w</math> is the channel width, <math>l</math> is the length travelled downstream, and <math>a</math> is the arc length of the arm in the branching section. The diffusion length scale is <math>d_{diff}=(tD)^{1/2}</math>, where <math>t</math> is the time it takes to traverse a certain length of channel <math>\left(t=lw^2/Q\right)</math>, and <math>D</math> is the diffusivity. Homogeneity is assumed to be achieved when <math>d_{inter}=d_{diff}</math>. Equating the two expressions, solving for <math>l/a</math> and plugging in typical values gives <math>l/a</math> on the order of 10. From Figure 4, this prediction is in nice agreement with experimental evidence, as the standard deviation drops nearly to zero after the tenth branch, indicating near homogeneity across the channel.

[1] P. Garstecki, H.A. Stone, and G.M. Whitesides, Physical Review Letters **94** 164501 (2005)