# Limited coalescence

Original entry: Sujit S. Datta, APPHY 225, Fall 2009.

## Reference

S. Arditty, C. P. Whitby, B. P. Binks, V. Schmitt, F. Leal-Calderon, *Eur. Phys. J. E* **11,** 273 (2003).

## Keywords

emulsion, Pickering emulsion, interface, coalescence

## Key Points

Unlike "conventional" surfactant-stabilized emulsions (dispersions of droplets of one immiscible fluid in another), "Pickering" emulsion droplets are stabilized by solid particles (typically much smaller than the droplet size) adsorbed at the droplet interface. (Note: here, "stabilized" means "stabilized against coalescence". While the mechanisms behind this are not fully understood, and depend on the nature of the components of the particular system under consideration, Pickering emulsion stabilization is generally thought to result from a combination of steric hindrance and the formation of a thin film within the pore space of the particles, whose draining properties will depend on the structure and mechanical properties of the particle network.)

The particle wetting properties determine the bulk properties of the emulsion in two crucial ways:

- The continuous phase of the emulsion tends to be the phase that preferentially wets the particles. This can be understood in analogy to surfactants - a particle at the interface between the two fluid phases will sit deeper in the wetting phase, and the effective "packing shape" (Israelachvili, page 381) will be similar to a cone, with tapered end inside the non-wetting phase. That is to say, the majority of the particle volume prefers to be immersed in the wetting phase.

- Unless particles very significantly prefer one phase over the other, they will tend to be irreversibly adsorbed at the interface between the two fluids. This potential energy well is due to the reduced bare surface area in contact between the two fluids, and is given by <math>\pi R^{2} \gamma_{12}(1+cos\theta)^{2}</math>, where R is the particle radius, <math>\gamma</math> is the interfacial tension between the two fluids, and <math>\theta</math> is the contact angle at the particle surface. For contact angles between ~20-160 degrees, this is many times larger than kT.

Because they are stabilized by irreversibly-adsorbed solid particles, Pickering emulsions show a unique phenomenon - that of limited coalescence. While this phenomenon has been known dating back to the 1950's, this work by Arditty et al. explores it in depth. Simply put, Pickering emulsion droplets only coalesce to a limited extent, increasing the total droplet surface area until their surfaces are completely covered with solid particles. This gives rise to tremendous stability, as well as droplet dispersions of relatively low size polydispersity. The average droplet size is tuned by the concentration of stabilizing particles - this is a key result of this paper. The derivation of this is straightforward (in the following, r = droplet radius, a = particle radius, <math>\phi</math> = particle volume fraction with respect to dispersed phase):

Droplets will coalesce until the total surface area of droplets is equal to the total cross-sectional area of stabilizing particles (here, we assume that the effects of polydispersity or a contact angle different from 90 degrees are small). Letting N be the total number of droplets in the system, we have

<math>N\times 4\pi r^{2}=N\times\phi\pi r^{3}/a</math>, and so <math>r=4a/\phi</math>.