Qualitative estimation of the depth dependence of glass transition temperature of polymer surface using schematic mode-coupling theory

The phenomenon of glass transition near a polymer surface is significantly different from that of the bulk. This is due to the fact that the chains near the free surface are more mobile as there are more free volume. But how can we, at least qualitatively, describe the relationship between the motion of the chain and the free volume around it in glassy and melt states? In this work, we aim at verifying this line of reasoning from the first principles so that a tractable expression governing the glass transition can be obtained. First, the surface density profile of the polymer was calculated using the Euler–Lagrange equation of the square gradient theory. The numerical results show that density at each layer of the surface is a function of the depth from the surface into the bulk. Second, at each layer, the intermolecular radial distribution function in the Fourier space was computed using polymer reference interaction site model. Finally, with these, the structure factor and the correlation length can be known, allowing us to determine the transition point in schematic mode-coupling theory. In line with experimental observations and simulation results reported in the literature, our model points to the fact that the presence of larger free volume near the surface layer contributes to a reduction in the glass transition temperature of the polymer film, even though the surface layer is more compressible than the bulk.