Moving particles inside drops with electric fields

The goal

Particles trapped on the surface of a drop are hard to move. The aim here is to gather them with an electric field, then remove them, and to find the conditions under which that works.

Fig. 1. The idea. A field gathers the particles at the drop's poles, and a stronger field then ejects them.

A drop in a uniform field

When a drop and the liquid around it have different dielectric properties, an applied field polarizes the drop. There is no net force, so the drop does not move, but it stretches along the field. Above a critical field strength it breaks up or tip-streams, shedding fluid from its ends. A smaller drop resists stretching because surface tension dominates.

The drop also distorts the field around it. Even when the applied field is uniform, it is strongest at the drop's poles and weakest at its equator. That has two consequences: the electric stress on the drop's surface is uneven, which deforms it, and a particle on or near the surface sits in a non-uniform field.

Contour plot of electric field strength around a circular drop: high values in lobes above and below the drop, low values at its sides. — **Fig. 2.** Computed field around a drop whose dielectric constant (1.0) is twice that of the surrounding fluid (0.5), at an electric Weber number of 0.3. The field peaks at the poles, above and below the drop, and dips at the equator.

Dielectrophoresis

A particle in a non-uniform field feels a dielectrophoretic (DEP) force along the gradient of the field squared. In the point-dipole approximation, for a particle of radius R:

F_DEP = 2πε₀ε_cR³ β′ ∇E²

ε₀ is the permittivity of free space and ε_c that of the surrounding liquid. The Clausius–Mossotti factors of the particle and of the drop,

β′ = Re ε_p* − ε_c*ε_p* + 2ε_c*, β = Re ε_d* − ε_c*ε_d* + 2ε_c*

depend on the frequency-dependent complex permittivities of the particle, drop and liquid. When β′ > 0 a free particle moves toward the field maximum (positive DEP); when β′ < 0 it moves toward the minimum (negative DEP).

For a particle on the surface of a drop of radius a in an applied field E₀, the tangential force works out to

F_DEP,θ = −12π R³a ε₀ε_cE₀² β′β(2 + β) cos θ sin θ

Direction
The sign of ββ′ decides where particles go. If ββ′ > 0 they gather at the poles; if ββ′ < 0, at the equator.
Drop size
The force grows as the drop radius a shrinks, so particles on smaller drops are easier to move.

Break up or sort first?

Two things happen as the field rises. At some strength the drop breaks up or tip-streams. At another, the DEP force overcomes the particles' buoyant weight and they start to move. Sorting is only possible if the second comes first. Comparing the electric Weber number with a gravity parameter gives a critical drop radius:

a_crit = 3 (γβ′(2/β + 1) We_crit2(ρ_p − ρ_f)g)^1/2

Particles can be moved only on drops smaller than a_crit. The condition does not depend on the field strength or on the particle size, only on the drop radius and the physical properties. For a density difference of 500 kg/m³ and a surface tension of 0.02 N/m, the drop must be smaller than about 1 mm.

Experimental setup

We built a small transparent cell with electrodes on its side walls and a Teflon-coated hydrophobic base, so drops sit on it without spreading. Drops loaded with particles are placed in the cell with a microsyringe, a high-voltage amplifier drives the electrodes, and cameras on microscopes image the drop from above and below.

Fig. 3. The cell, seen from the side. Electrodes on the side walls apply the field across the liquid, and the drop rests on a hydrophobic base that keeps it from spreading. Cameras view the drop from above and below.

Particles at the poles

A 180 µm water drop sat in corn oil, carrying hollow glass particles 16 to 20 µm across. Glass has a dielectric constant of about 2.5, lower than both water and corn oil, so both Clausius–Mossotti factors are negative and their product is positive. At around 3000 V, the particles moved to the poles. The drop kept its shape, because at this size surface tension wins over the electric stress.

Bottom of the drop before the field: glass particles scattered across the surface. — **Fig. 4.** A 180 µm water drop in corn oil with 16–20 µm hollow glass particles, imaged at its bottom and top surfaces before and after the field is applied. The particles collect at the poles.

Top of the drop before the field: particles spread over the middle. — **Fig. 4.** A 180 µm water drop in corn oil with 16–20 µm hollow glass particles, imaged at its bottom and top surfaces before and after the field is applied. The particles collect at the poles.

Cleaning a drop

To remove the particles, we first gathered them at the poles and then raised the field abruptly. In an 805 µm water drop in corn oil carrying extendospheres about 50 µm across, 3000 V collected the particles at the poles. A jump to 4000 V stretched the drop into Taylor cones that tip-streamed and ejected the collected particles. With the field switched off, the drop was clean.

Round drop with a cluster of dark particles in its middle. — **Fig. 5.** Cleaning an 805 µm water drop in corn oil. The particles gather at the poles at 3000 V, then tip-streaming at 4000 V ejects them. After the field is switched off, the drop is free of particles.

Drop elongated slightly, with particles gathered at its upper pole. — **Fig. 5.** Cleaning an 805 µm water drop in corn oil. The particles gather at the poles at 3000 V, then tip-streaming at 4000 V ejects them. After the field is switched off, the drop is free of particles.

Giant vesicles

We tried the same with giant unilamellar vesicles (GUVs) in corn oil. The particles inside formed chains and moved toward the vesicle poles, and the vesicles themselves lined up into chains. Particles then passed from one vesicle to the next along the chain. Under the same field, vesicles deformed far less than drops of similar size, and inside them the particle chains attached to the vesicle walls.

A large vesicle filled with small particles, with a cluster near its center. — **Fig. 6.** Giant unilamellar vesicles carrying particles in corn oil under an applied field. The particles chain up inside the vesicles, the vesicles chain together, and particles transfer between neighboring vesicles.

A small vesicle full of particles joined to neighboring vesicles in a chain. — **Fig. 6.** Giant unilamellar vesicles carrying particles in corn oil under an applied field. The particles chain up inside the vesicles, the vesicles chain together, and particles transfer between neighboring vesicles.

The results point to a way to concentrate and remove particles from drops without touching them, as long as the drop is below the critical radius.