Free energy for inclusion of nanoparticles in solvated polymer brushes from molecular dynamics simulations

The inclusion of nanoparticles (NPs) into solvated polymer brushes (PBs) provides a path for designing novel nanocomposites and a multifunctional surface for wide applications. Despite intensive study over the years, the correlation between the structural properties of NPs (or PBs) and the NP-PB interactions is still to be well unveiled. Here, we present molecular dynamics simulations with the umbrella sampling method to systematically investigate the interaction between NPs and PBs, via calculating the free energy cost (U_ins, associated with the inclusion of NPs into PBs) as a function of a series of factors, such as brush grafting density (ρ_g), grafted polymer chain architecture, NPs' size, NPs' surface properties, and NPs' shape and surface structure, as well as the solvent quality. Our results show that at a fixed NP size, the inclusion free energy approximately scales with the osmotic pressure (Π) of PBs under good solvent conditions [U_ins∼Π(ρ_g)∼ρ_g ^3/2], regardless of the NPs' shape and surface properties. Once the radius of the NP (R_NP) is varied, a scaling law, U_ins∼R_NP ³, can be obtained for NPs deeply inserted in swollen PBs with a high grafting density. While for shallow inclusions, a surface tension correction of the form ∼R_NP ² plays a role. Further studies reveal that Θ and poor solvents will weaken the osmotic pressure effects of PBs and reversely enhance the surface tension effects due to the increased NP-brush repulsion. Our simulation results verify previous theoretical perspectives that the U_ins can be approximated by the sum of the volume and surface contributions from the osmotic pressure Π and surface tension γ (U_ins∼ΠR_NP ³+γR_NP ²). Our work not only helps us to understand the applicability of previous theories on the NP-PB system but also reveals the key factors that impact the NP-PB interaction in a series of probable conditions, which may provide valuable guidelines for designing and engineering novel nanomaterials based on functional NPs and PBs.