UNIAXIAL ELASTIC-MODULUS OF CONCENTRATED EMULSIONS

We calculate the elastic modulus, mu(phi), of a 3D concentrated oil-in-water (for example) emulsion with a simple cubic lattice for a uniaxial constant volume deformation (phi is the volume fraction of oil). We use a simple model that assumes that the deformed drop shapes are truncated spheres; we also give an exact calculation (based on recent work of Morse and Witten) valid for the regime of weakly compressed emulsions. For the truncated sphere model, mu(phi) has a discontinuity at phi = phi(0) (the volume fraction at which the drops just touch) as in the case of the 2D Princen model. The exact calculation shows mu(phi) falling off as 1/ln(phi - phi(0)). Our calculation shows that the experimental data on the shear modulus, showing a linear dependence on (phi - phi(0)), probably cannot be explained in terms of an ordered array of uniformly deformed drops.