Statistical Mechanical Theory of Penetrant Diffusion in Polymer Melts and Glasses

We generalize our microscopic, force-level, self-consistent nonlinear Langevin equation theory of activated diffusion of a spherical particle in a dense hard sphere fluid to treat molecular penetrant diffusion in homopolymer melts and nonaging glasses. A coarse-grained mapping is developed where polymer chains are modeled as disconnected, noninterpenetrating Kuhn scale hard spheres (diameter, σ), and the penetrant is modeled as an effective hard sphere (diameter, d) which can be attracted to the polymer segment. The polymer mapping is a priori carried out by enforcing the effective hard sphere fluid reproduces the specific polymer liquid or glass long wavelength dimensionless collective density fluctuation amplitude. The theory predicts that penetrant diffusivity exhibits supra-Arrhenius temperature dependence in supercooled polymer melts and (near) Arrhenius temperature dependence in quenched nonequilibrium polymer glasses. Polymer-penetrant attraction slows down penetrant diffusivity to a degree that is strongly enhanced as penetrants become smaller. By treating d/σ as the only adjustable material-specific parameter, the theory is in good agreement with experimental diffusivity data spanning more than 10 decades for a wide range of penetrants (from small gas to large organic molecules), amorphous polymers, and temperatures. Optimal d/σ values are consistent with a priori physical estimations of effective space-filling molecular and Kuhn segment diameters. Through comparative studies, two different a priori choices of penetrant-matrix attraction strength are established for small gas and large organic penetrants. System parameter transferability is examined. The theory represents a microscopic-based statistical mechanical approach for penetrant diffusion in polymers and provides a foundation for treating time-dependent penetrant diffusivity in aging polymer glasses, collective effects induced by finite penetrant loading, and diffusion in heterogeneous polymeric materials.