Theory of Localization and Activated Hopping of Nanoparticles in Cross-Linked Networks and Entangled Polymer Melts

We develop a microscopic, force-level statistical dynamical theory for the localization and activated hopping dynamics of dilute spherical particles, both hard and soft, in polymer networks and entangled melts. The main factor controlling localization is the confinement a particle experiences from polymer entanglements and/or chemical cross-links as characterized by the ratio of the effective nanoparticle diameter to the mechanical mesh length, 2R(eff)/d(T). Dynamical localization occurs when the latter ratio is of order unity, and the activated hopping mobility drastically decreases as the confinement parameter even modestly increases past threshold. Local packing correlations slightly enhance the tendency to localize for hard particles and dramatically enhance mobility for soft repulsive particles. The concept of an effective nanoparticle size largely, but not completely, collapses the diverse dynamical results for hard and soft repulsive particles. For hard spheres in entangled polymer melts we find the exponentially slow hopping diffusivity is generally much smaller than transport via reptation-driven entanglement network dissolution, except for a narrow window of confinement parameters of 2R/d(T) approximate to 1.5-2 and long enough chains.