# Difference between revisions of "Image charge"

The method of image charges in commonly used to solve electrostatics problem. This entry is intended to be an extension of the ideas presented in wikipedia, as applied to colloidal suspensions.

## Motivation

Oil-water emulsions are everywhere, from food to cosmetics to oil spills. Surfactants are most commonly used to stabilise these emulsions, but using solid particles is of great interest because solid particles can be functionalised more readily than surfactants. Such Pickering emulsions (http://pubs.rsc.org/en/Content/ArticleLanding/1907/CT/ct9079102001) have been the subject of much basic research, as well as the development of interface-scaffolded self assembly, and colloidosomes for drug delivery.

## A simple system

Let us consider a charged colloidal particle near an oil water interface. The relative permittivities of the oil will be ~2, and of water ~80.

If we approximate our colloidal particle as a point charge, near an interface between two media, we get an expression for the image charge which is 'located' at an equal distance on the other side of the interface as our particle:

$Z_{image}=Z \frac{\epsilon_{1} + \epsilon_{2}}{\epsilon_{1} + \epsilon_{2}}$

### Particle in water

Figure 1. A particle starts in the water and experiences electrostatic repulsion from its image charge.

For a particle in water,

$Z_{image} = Z \frac{80 - 2}{80 + 2} \approx Z$

Note that the sign of the image charge will be the SAME as that of the particle. This means the particle will experience a repulsive interaction as it nears the interface (see Figure 1).

### Particle in oil

Figure 1. A particle starts in the oil and experiences electrostatic attraction to its image charge.

For a particle in oil,

$Z_{image} = Z \frac{2 - 80}{2 + 80} \approx -Z$

Note that the sign of the image charge will the OPPOSITE as that of the particle. This means the particle will experience an attractive interaction as it nears the interface (see Figure 2).

## Implications

Considering the image charges involved in a colloidal suspension can be critical. For instance, some particles will only stabilise emulsions if they start in the oil phase (work yet to be published, University of Rhode Island) whereas if the particles stay in the oil phase, they will stabilise the emulsion for weeks. This asymmetry can be understood if we consider image charges. Particles may not penetrate an oil-water interface if they start off in the water phase due to image charge repulsion; if they do not penetrate the interface, they are unable to stabilise the emulsion. Starting the particles off in the oil phase, however, means that particles will be attracted to the interface and hence readily stabilise the emulsion.

One way to allow particles to penetrate the interface even from the water phase is to screen out the electrostatic repulsion with the addition of salt. Introducing a spreading solvent or surfactant may also allow the particle to breach the interface.