# Formation of Droplets and Bubbles in a Microfluidic T-junction - Scaling and Mechanism of Break-up.

Original entry: Pratomo Putra (Tom) Alimsijah, APPHY 226, Spring 2009

## Contents

# Formation of Droplets and Bubbles in a Microfluidic T-junction - Scaling and Mechanism of Break-up

Garstecki, P., Fuerstman, M. J., Stone, H. A. & Whitesides, G. M., Lab on a Chip, 2006, 6, 437-446.

## Soft Matter Keywords

Capillary number, shear stress, Laplace pressure

## Abstract

(From Paper)

## Soft Matter Discussion

The authors studied the scaling characteristics and mechanisms of fluid streams in T-junctions over a range of viscosities and flow rate. This is probably the most popular of microfluidic devices used for the generation of droplets. The experiments done are mainly at low capillary numbers since the flow rates typically used in microchannels (characteristic dimensions of 100micron), have such properties. They found that at these capillary numbers (Ca<10^-2), the dynamics of break-up of immiscible threads in T-junctions is dominated by the pressure drop across the droplet or bubble as it forms.

At low capillary number, the interfacial forces dominate the shear stresses, thus the shear stresses exerted on the interface of the emerging droplet are not sufficient to distort it significantly. Thus, the droplet blocks almost the entire cross-section of the main channel. This leads to an increase of pressure upstream of the emerging droplet which leads to the squeezing of the neck of the immiscible thread.

They also observed that there is a critical value of capillary number (Ca ~ 10^-2) above which shear stresses start to play an important role. In this regime (Ca>10^-2), the system operates in a similar manner to the dripping regime in an unbounded, co-flowing liquid.

The schematics of the T-junction are showed here:

The previous papers postulated that droplets are sheared off from the stream of the discontinuous fluid. Thorsen et al. believed that the size of the droplets is determined by the competition between the Laplace pressure and the shear stress exerted on the interface by the continuous fluid. The authors did experiments to characterize this role of shear stress. The result is shown below:

The results showed that even when the values of shear stress are changed over two orders of magnitude, there are not the approximated two orders of magnitude of change in the size of droplets. These observations at Ca Є (8*10^-5, 8*10^-3) asserts that shear stress is not the primary contributor to the break-up mechanisms at low enough Ca values and suggests that interfacial tension dominates shear stress.

The author then considered the three forces acting on the tip of the discontinuous phase during break-up: surface tension, shear-stress, and flow resistance.

The surface tensions is associated with the Laplace pressure jump <math>\Delta p_{L}</math> across static interface,

<math>\Delta p_{L}=\gamma (r_{a}^{-1}+r_{r}^{-1})</math>

where r_a is the axial curvature in the plane of the device and r_r is the radius of the radial curvature in the cross section of the neck joining the inlet for the discontinuous fluid with the tip. This is shown below:

The author finds that the interface at the downstream side of the thread acts on the liquid inside the thread with a stress

<math>p_{L}\approx -\gamma (\frac{2}{w}+\frac{2}{h})</math>

While the interface located upstream acts on the discontinuous liquid with a stress:

<math>p_{L}\approx \gamma (\frac{1}{w}+\frac{2}{h})</math>

The sum of the two, multiplied by the cross-section of the channel gives the estimate of the surface tension force:

<math>F_{\gamma }\approx -\gamma h</math>

acting upstream and stabilizing the tip.

The author then postulates that the dynamics of break-up in a typical T-junction is dominated by the balance of pressures in the discontinuous and continuous phases at the junction.

<math>F_{\gamma }\approx -\gamma h</math>

The author identifies the four stages of formation of a droplet:
1. The tip of the discontinuous phase enters the main channel
2. The growing droplet spans almost the entire cross-section of the main channel
3. The droplet elongates in the downstream direction and its neck thins
4. The neck breaks, and the disconnected droplet flows downstream and the tip of the discontinuous phase recoils back towards the inlet.
This then repeats.

Between the first and second stage, the Laplace pressure decreases because the radial curvature is still bound by the height of the channel and the axial curvature of the interface located upstream of the tip decreases. This continues until stage 3 where the radial curvature grows and the Laplace pressure increases until the droplet breaks. This is shown here:

The experiments done by the authors are in agreement with this.