Osmotic pressure and viscoelastic shear moduli of concentrated emulsions

We present an experimental study of the frequency ω dependence and volume fraction φ dependence of the complex shear modulus $G^{*}(\omega ,\phi )$ of monodisperse emulsions which have been concentrated by an osmotic pressure Π. At a given φ, the elastic storage modulus $G^{\prime }\left(\omega \right)=\mathrm{Re}\left[G^{*}\right(\omega \left)\right]$ exhibits a low-frequency plateau $G_p^{\prime },$ dominating the dissipative loss modulus $G^{\prime \prime }\left(\omega \right)=\mathrm{Im}\left[G^{*}\right(\omega \left)\right]$ which exhibits a minimum. Above a critical packing fraction ${\phi }_c,$ we find that both Π(φ) and $G_p^{\prime }\left(\phi \right)$ increase quasilinearly, scaling as $(\phi -{\phi }_c{)}^{\mu },$ where ${\phi }_c\approx {\phi }_c^{\mathrm{rcp}},$ the volume fraction of a random close packing of spheres, and μ is an exponent close to unity. To explain this result, we develop a model of disordered droplets which interact through an effective repulsive anharmonic potential, based on results obtained for a compressed droplet. A simulation based on this model yields a calculated static shear modulus $G$ and osmotic pressure Π that are in excellent agreement with the experimental values of $G_p^{\prime }$ and Π.