Dynamics of spontaneous emulsification

We present a model for spontaneous emulsification resulting from a transiently negative interfacial tension between water and oil regions, which may be achieved under conditions of strong adsorption of surfactant molecules to the interface. While our approach builds on a linear stability analysis, it addresses the essential non-linear coupling of surface growth to the diffusion flux of surfactants to the interface. We consider a large drop of oil of radius R embedded in a dilute surfactant solution and predict that undulations develop with a characteristic wavelength $\lambda^{*}$, which at long times t obeys $\lambda^{*}\sim t^{1/3}$. This suggests that the size of the droplets created spontaneously at the interface scales as $\xi_{\rm o}^{1/3}$, where $\xi_{\rm o}$ is a diffusion length which is comparable to R under steady state diffusion conditions. We discuss the regimes of applicability of our results to various experimental systems.