# Dielectrophoretic manipulation of drops for high-speed microfluidic sorting devices

Original entry: Scott Tsai, APPHY 226, Spring 2009

"Dielectrophoretic manipulation of drops for high-speed microfluidic sorting devices"

Keunho Ahn and Charles Kerbage, Tom P. Hunt, R.M. Westervelt, Darren R. Link, and D.A. Weitz

Applied Physics Letters **88**, 024104 (2006)

## Soft Matter Keywords

Droplets, microfluidics, PDMS, dielectrophoresis, droplet sorting, Stokes' Drag

## Overview

Refer to abstract of paper

## Soft Matter Examples

It has already been shown that droplets in microfluidic devices can be used as micro-reactors. In one of the configurations of this system, the droplets are water-in-oil emulsions. For these droplets to work effectively as a means of directed evolution, accurate and fast screening is required. One example of this is the sorting of a typical library of <math>10^8 - 10^9</math> genes, which requires a throughput of 1kHz so that they can be finished in a practical amount of time.

In this paper, the authors present a high-throughput microfluidic droplet sorting device that uses dielectrophoresis for actuation. Dielectrophoresis is the manipulation of dielectric particles using electric fields. The force acting on the particle is directly proportional to the gradient of the electric field, and is perpendicular in direction.

The authors describe their microfluidic device with a droplet generator, a Y-junction, and an electrode (Fig 1a). The water droplets are formed at the generator, where their size is a function of the velocity of the water stream and of the velocity of the transverse oil stream. After the droplets are formed, they flow down to the Y-junction, where without an applied electric field, they enter the shorter channel (waste stream) (Fig. 1b).

The droplets naturally flow into the shorter channel because it has a lower hydrodynamic resistance than the longer channel. So to pull the droplets into the longer channel, the electrode is charged, and an electric field is produced (Fig. 1c).

The transferse motion of the droplet here is described by a balance of forces. In the direction of the electrode, the dielectrophoretic force on the drop is <math> \vec{F} = \vec{m} \cdot \operatorname{grad} \vec{E} </math>. Where <math> \vec{m} </math> is the dipole moment of the particle and <math> \vec{E} </math> is the electric field. For a spherical particle, the dipole moment is <math> \vec{m} = 4 \pi \epsilon_{oil} Re[CM( \omega )]r^3 \vec{E} </math>. <math> Re[CM( \omega) ] </math> is the Claussius-Mossotti facto, and <math> \epsilon_{oil} </math> is the oil's dielectric permittivity.

This force is offset by the Stokes drag force <math> Fs = 6 \pi \eta_{oil} r \vec{v} </math>.

So, the balance of forces gives:

<math> m \frac{d \vec{v}} {dt} = 4 \pi \epsilon_{oil} Re[CM( \omega )]r^3 \vec{E} \cdot \vec{E} - 6 \pi \eta_{oil} r \vec{v}</math>.

Since the time for the drops to attain terminal velocity is approximately <math> t = v/a = \frac{v}{[F/(\delta \rho 4 \pi / 3 r^3)]} = 2 \delta \rho r^2 / 9 \eta </math>, so for 1nN force acting on a 12 <math> \mu m</math> drop, it will accelerate to its terminal velocity in 5 <math> \mu s </math>. This time scale is much shorter than others in this system, so it is possible to neglect the inertia term.

So, <math> 4 \pi \epsilon_{oil} Re[CM( \omega )]r^3 \vec{E} \cdot \vec{E} - 6 \pi \eta_{oil} r \vec{v} = 0</math>.

Solving this equation, the authors obtain <math> \vec{v} = \epsilon_{oil} r^2 k V^2 / 3 \eta_{oil} </math> (The Claussius-Mossotti factor is 1 for water drops in oil and for frequencies less than several MHz).

With this model for velocity, the authors determined that for a 12 micron sized drop at 1kV, the force acting on the drop would be approximately 10nN, so the resuling maximum drop velocity is around 1 cm/s. This they also verified experimentally (Fig. 2).