Fig. 1.

Three regimes of NP–NP interactions are found in the simulations. In regime I, where the NPs have zero or low charge densities, the interaction is attractive due to depletion interactions mediated by the clustering of ions. In regime II, where the charge is sufficiently large, we find a repulsion that is stronger and longer range than the double-layer force predicted by the DLVO theory. In regime III, where the NPs have very high charge densities resulting in strong counterion condensation on the NP surfaces, the potential shows a strong short-range repulsion and a deep long-range attractive well. van der Waals interactions between NPs are not included in our model.