Self-Consistent Hopping Theory of Activated Relaxation and Diffusion of Dilute Penetrants in Dense Crosslinked Polymer Networks

We generalize and apply a microscopic force-level statistical mechanical theory of the activated dynamics of dilute spherical penetrants in glass-forming liquids to study the influence of permanent crosslinking in polymer networks on the penetrant relaxation time and diffusivity over a wide range of temperature and crosslink density. Calculations are performed for model parameters relevant to recent experimental studies of an nm-sized organic molecule diffusing in crosslinked PnBA networks. The theory predicts the penetrant alpha relaxation time increases exponentially with the crosslink fraction ($f_n$) dependent glass transition temperature, $T_g$, which grows roughly linearly with the square root of $f_n$. Moreover, $T_g$ is also found to be proportional to a geometric confinement parameter defined as the ratio of the penetrant diameter to the mean network mesh size. The decoupling ratio of the penetrant to polymer Kuhn segment alpha relaxation times displays a complex non-monotonic dependence on crosslink density and temperature that can be well collapsed based on the variable $T_g(f_n)/T$. The microscopic mechanism for activated penetrant relaxation is elucidated and a model for the penetrant diffusion constant that combines activated segmental dynamics and entropic mesh confinement is proposed which results in a significantly stronger suppression of mass transport with degree of effective supercooling than predicted for the penetrant alpha time. This behavior corresponds to a new polymer network-based type of decoupling of diffusion and relaxation. In contrast to the diffusion of larger nanoparticles in high temperature rubbery networks, our analysis in the deeply supercooled regime suggests that for the penetrants studied the mesh confinement effects are of secondary importance relative to the consequences of crosslink-induced slowing down of glassy activated relaxation.