Effective interactions between grafted nanoparticles in polymer melts: challenging full-scale simulations, effect of entanglements and morphology of clusters

We employed an extended slip-spring model treating explicitly grafted polymer chains to study the effective potential of mean force (PMF) between two nanoparticles (NPs) embedded in an entangled polymer matrix as a function of NP grafting density, length of free and grafted chains, and surface affinity between polymers and NPs. While good agreement is found between the ensemble-averaged PMFs in entangled and non-entangled cases at an iso-chain length, we show that a considerable trial-to-trial variability of the measured PMF (of around several <i>k</i>_B<i>T</i>) emerges in entangled systems, likely related to a long relaxation of entanglements between grafted chains. We used the obtained PMFs, modeled by a set of potentials in the entangled case, to simulate much larger NP systems over extended timescales. We first assessed the quality of the effective coarse-grained representation by direct comparison with full-scale slip-spring simulations with an explicit polymer matrix at a high filler volume fraction. While the resulting cluster sizes were captured robustly at a two-body PMF level for a variety of simulated conditions, some discrepancies appeared in the detailed cluster structure, especially at low grafting densities, suggesting the necessity of incorporating many-body interactions. The NP phase diagram was then explored systematically, reproducing the transition between percolated and well-dispersed states upon increasing grafting density or polymer-NP attraction strength. We found that the enhanced PMF amplitude dispersion in entangled systems impacts the transition lines between percolated and well-dispersed systems, typically limiting the observed cluster sizes.