Polymer grafted Nanoparticles in homopolymer solvents or PGNs have become increasingly popular in mechanical, optical and electrical applications due to their ability to improve the properties of the host matrix. Dispersion of PGNs in host matrix is necessary to achieve the desired properties in these hybrid nanomaterials. These systems transition into mixed (dispersed) and demixed (phase separated) state depending on the molecular design and interactions between the grafted polymer and host matrix, with a marked difference in properties between these two states. To establish whether a PGN system will undergo a mixed to demixed transition one needs to calculate the free energy difference between the dispersed and aggregated states. In this work, we have utilized mesoscale modelling to calculate the free energy difference associated with the mixed to demixed transition in PGN and homopolymer system. To this end, we first use conventional Thermodynamic Integration (TI) to obtain the free energy difference along a temperature driven path. Since, a temperature driven transition path may not always be reversible, and the free energy calculation can be prone to hysteresis, we verify our results using umbrella sampling calculations, wherein we model the order parameter as a coarse-grained number density of one component in the system. We use a harmonic biasing field, based on this coarse-grained number density to sample configurations in both the mixed and demixed regions of the phase space to obtain a free energy landscape.
To validate our method, we first obtain the free energy of mixed-demixed transition for a binary LJ fluid system at conditions where the phase behavior is already established by previous studies and we find that our predictions for the most favorable system state agree with those in literature. Next, we use this method to calculate the free energy of transition in PGN system. From the free energy landscape, we find that the energy associated with a mixed to demixed transition in this system is large, making the mixed state as the stable system state for the conditions studied. We find that our predictions, consistent with experimental observations, rule out the possibility of any stable demixed states in the systems studied. We also study the viscoelastic behavior of PGN systems with attractive solvent and grafted chain interactions. Our preliminary results indicate that these dispersed systems show a richer viscoelastic behavior characterized by higher viscosity and storage and loss moduli on increasing loading of nanoparticles. Modifying the interaction parameters to model systems with rich physical properties ranging from waxes and gels at high loading to low viscosity fluids at low loading will provide useful insights on the viscoelastic behavior PGN systems.