Accurate estimation of the polymer coverage of hairy nanoparticles

Understanding and predicting the mechanisms underpinning the self-assembly of polymer-grafted nanoparticles (PGNPs) are important for controlling the engineering applications of these novel materials. The self-assembly of these materials is driven by their surfactancy, i.e., by the fact that the (inorganic) nanoparticles energetically dislike the (organic) polymer tethers. In previous work we developed a model in which a grafted polymer chain was treated as a rigid equivalent sphere (ES) which was impenetrable to the NPs, but completely penetrable to other ESs. This description, along with a geometric analogy with patchy particles, allowed us to facilely explain the self-assembly of PGNPs. However, since we model an ES as being completely penetrable to other ESs but impenetrable to the NPs the physical correspondence between a "real" grafted polymer and an ES is not clear. The application of the ES model to experiments and to computer simulations has therefore seen limited success, and only qualitative agreement has been obtained. In this paper, we develop a more realistic description, termed the modified ES (mES) model, based on the work of Daoud and Cotton on curved polymer brushes, which takes the impenetrability of the individual chain monomers into account. While this approach increases the complexity of our formalism, we find that the resulting mES model quantitatively captures computer simulation results on the structure of the PGNPs and also quantitatively explains their self-assembly over a broad range of conditions.