Modeling size controlled nanoparticle precipitation with the co-solvency method by spinodal decomposition

The co-solvency method is a method for the size controlled preparation of nanoparticles like polymersomes, where a poor co-solvent is mixed into a homogeneous copolymer solution to trigger precipitation of the polymer. The size of the resulting particles is determined by the rate of co-solvent addition. We use the Cahn-Hilliard equation with a Flory-Huggins free energy model to describe the precipitation of a polymer under changing solvent quality by applying a time dependent Flory-Huggins interaction parameter. The analysis focuses on the characteristic size R of polymer aggregates that form during the initial spinodal decomposition stage, and especially on how R depends on the rate s of solvent quality change. Both numerical results and a perturbation analysis predict a power law dependence $R \sim s^{-1/6}$, which is in agreement with power laws for the final particle sizes that have been reported from experiments and molecular dynamics simulations. Hence, our model results suggest that the nanoparticle size in size-controlled precipitation is essentially determined during the spinodal decomposition stage.