Polymer nanocomposite (PNC) T$_{g}$ from the perspective of percolation theory

A number of researchers have speculated on the connections between the liquid to glass transition and percolation concepts. Most recently, the film thickness dependence of T$_{g} $ exhibited by polymers has been quantified in terms of the change in percolation threshold when transitioning from a 3D to 2D system. Here, we extend these concepts to develop a model for the behavior of PNC T$_{g}$. The placement of impurities (nanoparticles) in a lattice (polymer matrix) changes the percolation threshold of the system, which we claim can be related to a change in T$_{g}$. Specific treatments of the impurities can result in either increases or decreases in T$_{g} $ relative to the pure system, and the magnitude of the changes in T$_{g}$ are related to the size and shape of the lattice impurities. We discuss the details of our results for such effects and compare them to experimental observations reported for PNCs.