A model for the prediction of droplet size in Pickering emulsions stabilized by oppositely charged particles.

Colloidal particles irreversibly adsorb at fluid-fluid interfaces stabilizing what are commonly called "Pickering" emulsions and foams. A simple geometrical model, the limited coalescence model, was earlier proposed to estimate droplet sizes in emulsions. This model assumes that all of the particles are effective in stabilization. The model predicts that the average emulsion drop size scales inversely with the total number of particles, confirmed qualitatively with experimental data on Pickering emulsions. In recent years, there has been an increasing interest in synthesizing emulsions with oppositely charged particles (OCPs). In our experimental study, we observed that the drop size varies nonmonotonically with the number ratio of oppositely charged colloids, even when a fixed total number concentration of colloids is used, showing a minimum. We develop a mathematical model to predict this dependence of drop size on number ratio in such a mixed particle system. The proposed model is based on the hypothesis that oppositely charged colloids form stable clusters due to the strong electrostatic attraction between them and that these clusters are the effective stabilizing agents. The proposed model is a two-parameter model, parameters being the ratio of effective charge of OCPs (denoted as k) and the size of the aggregate containing X particles formed due to aggregation of OCPs. Because the size of aggregates formed during emulsification is not directly measurable, we use suitable values of parameters k and X to best match the experimental observations. The model predictions are in qualitative agreement with experimentally observed nonmonotonic variation of droplet sizes. Using experiments and theory, we present a physical insight into the formation of OCP stabilized Pickering emulsions. Our model upgrades the existing Wiley's limited coalescence model as applied to emulsions containing a binary mixture of oppositely charged particles.