# Elasticity of compressed emulsions

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

## Reference

T. G. Mason, J. Bibette, and D. A. Weitz, *PRL* **75,** 2051 (1995).

T. G. Mason, M. D. Lacasse, G. S. Grest, D. Levine, J. Bibette, and D. A. Weitz, *PRE* **56,** 3150 (1997).

## Keywords

emulsions, osmotic pressure, rigidity percolation, shear modulus, bulk modulus, rheology, jamming

## Key Points

An emulsion is a metastable suspension of droplets of one fluid within another fluid, with the two fluids being immiscible. Emulsion droplets are stabilized against coalescence upon contact by a range of surfactants; typically, surfactants are ionic, imparting stability due to electrostatic repulsions at the droplet interfaces.

For low volume fractions, a Brownian emulsion is liquid-like; as the volume fraction of droplets increases, the viscosity of the emulsion may diverge (at the colloidal glass transition, volume fraction ~ 58%), similar to a hard-sphere suspension. However, while a disordered hard-sphere suspension can only be packed up to a maximum volume fraction of 64% (random close packing), a disordered emulsion can be packed past random close packing, due to the deformability of the emulsion droplets. When the droplets first begin to touch (at random close packing), the system 'jams': it becomes solid-like, and develops an elastic modulus.

A good deal of work in the past decade has focused on understanding this jamming transition, in a variety of ways. This set of papers were among the first to provide quantitative data motivating current ideas on jamming. In them, Mason et al. describe very systematic experiments on disordered, Brownian oil-in-water emulsions stabilized by an ionic surfactant, providing two measures of the elasticity of the emulsion (the bulk modulus, a measure of the material's resistance to uniform compression, and the shear modulus, a measure of the material's resistance to uniform shear) as the emulsions are compressed, and the volume fraction is increased from below random close packing up to nearly 100% (the limit of a biliquid foam). The bulk modulus K is measured by measuring the osmotic pressure <math>\Pi</math> of the emulsion as a function of the volume fraction (K is defined as <math>\phi\cdot d\Pi/d\phi</math>); experimentally, the osmotic pressure of a sample is set using dialysis, and the corresponding volume fraction is measured by evaporating off the water. The zero-frequency shear modulus is measured using linear rheology; Mason et al. find that the linear elastic modulus G' plateaus at low frequencies, consistent with other soft glasses, and use this value (G'p) as the zero-frequency shear modulus.

These experiments yielded a number of key results:

- Both bulk modulus and shear modulus measurements for droplets of different sizes collapsed onto a single dataset when rescaled by the Laplace pressure, <math>\sigma/r</math>, where <math>\sigma</math> is the interfacial tension between the two phases and r is the droplet radius. This shows that the emulsion elasticity is set by the Laplace pressure, and is purely due to energy storage in the interfaces of the deformed droplets.
- As a function of volume fraction <math>\phi>\phi_{RCP}</math>, <math>G'_{p}\sim \phi\cdot(\phi-\phi_{RCP})</math> and <math>K\sim \phi^2+\phi\cdot(\phi-\phi_{RCP})</math>. This has a number of important implications:
- The scaling of G' near random close packing is reminiscent of scaling near a critical point. This observation has motivated a good deal of recent theories of jamming that treat it as a critical phenomenon.
- The onset of the shear modulus at random close packing is gradual, while the onset of the bulk modulus at random close packing is very sharp. This is very surprising -- two measures of the elasticity of compressed emulsions have very different behavior as a function of the volume fraction. Mason et al. explain this with the help of numerical simulations. They suggest that the shear modulus is much smaller than the bulk modulus because (i) the droplet structure is disordered, (ii) the droplets are 'slippery' -- they slide over each other without friction. As a result, when subjected to shear, 'pockets' form in the disordered droplet structure at which most of the shear stress is relieved: this weakens the material in shear and makes the shear modulus much smaller than the bulk modulus.

While practical applications of this work are not immediately clear, the fundamental implications are profound: these experiments and simulations motivated much of the current work on jamming, and highlight the importance of inter-droplet interactions and disorder in determining the elasticity of compressed emulsions.