Osmotic pressure and viscoelastic shear moduli of concentrated emulsions
We present an experimental study of the frequency \ensuremath{\omega} dependence and volume fraction \ensuremath{\varphi} dependence of the complex shear modulus ${G}^{*}(\ensuremath{\omega},\ensuremath{\varphi})$ of monodisperse emulsions which have been concentrated by an osmotic pressure \ensuremath{\Pi}. At a given \ensuremath{\varphi}, the elastic storage modulus ${G}^{\ensuremath{'}}(\ensuremath{\omega})=\mathrm{Re}[{G}^{*}(\ensuremath{\omega})]$ exhibits a low-frequency plateau ${G}_{p}^{\ensuremath{'}},$ dominating the dissipative loss modulus ${G}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\omega})=\mathrm{Im}[{G}^{*}(\ensuremath{\omega})]$ which exhibits a minimum. Above a critical packing fraction ${\ensuremath{\varphi}}_{c},$ we find that both \ensuremath{\Pi}(\ensuremath{\varphi}) and ${G}_{p}^{\ensuremath{'}}(\ensuremath{\varphi})$ increase quasilinearly, scaling as $(\ensuremath{\varphi}\ensuremath{-}{\ensuremath{\varphi}}_{c}{)}^{\ensuremath{\mu}},$ where ${\ensuremath{\varphi}}_{c}\ensuremath{\approx}{\ensuremath{\varphi}}_{c}^{\mathrm{rcp}},$ the volume fraction of a random close packing of spheres, and \ensuremath{\mu} 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 \ensuremath{\Pi} that are in excellent agreement with the experimental values of ${G}_{p}^{\ensuremath{'}}$ and \ensuremath{\Pi}.