Size-distribution of droplets in emulsions by statistical mechanics calculation

This report is concerned with theoretical demonstration of the spontaneous emulsification which has been observed in a soft contact of nitrobenzene with water without surfactant [K. Aoki, M. Li, J. Chen, T. Nishiumi, Electrochem. Commun. 11 (2009) 239]. The demonstration is based on the model of spherical oil droplets with any size in equilibrium. The droplets are composed of the smallest droplets, the total number of which is given. An assembly of small droplets has larger surface energy than that of large ones because the surface energy is proportional to the surface area. The former has larger configurational entropy than the latter because the number of small droplets is bigger than that of the large ones. Since the free energy is determined by the competition between the surface energy and the entropy, it is not clear which assembly has lower free energy. This question was solved numerically here by statistical mechanics calculation of the size distributions, which contained only a parameter of the surface energy. The results of the computation at small number of droplets were used for deriving approximate equations for extremely large number of droplets. The size distribution was localized both to the smallest and the largest droplets. The diameter of the largest droplet was estimated from the dynamics in which coalescence by diffusion of droplets is disturbed by gravitational convection. The size then predicted was of the order of micrometer, being close to experimental values.