?Transition-site? model for the permeation of gases and vapors through compact films of polymers

Abstract

The theory of the permeation of effectively spherical molecules of gases and vapors (permeants) through films of compact amorphous solids such as polymers is developed using the activated-jump model (AJM). The conventional view is that permeation has to be analyzed as the resultant of sorption (solution) and diffusion effects. By contrast, in the present article, it is proposed that permeation may be viewed as a simple or fundamental process. This is suggested by a number of experimental observations: (a) the permeation correlations of Stannett and Szwarc (1955–1956); (b) the “Permachor” concept of Salame (1961–1973); (c) the “ideal” permeation behavior of water vapor through moderately polar polymers; (d) the absence of any effect of oxidation on the water-vapor permeability of polyethylene (PE); and (e) the “isokinetic” correlations between the Arrhenius parameters for permeation. The conventional AJM for diffusion is analyzed using the principle of microscopic reversibility, which shows that the average jump is characterized by a “transition site” L at its midpoint, analogous to the transition state in chemical reactions. For amorphous solids, these transition sites would be structural features, distributed at random and with their axes pointing at random. This leads to the present transition-site model (TSM) of permeation, where, at the steady state, a certain fraction of these sites will be transiently occupied by molecules of the permeant in equilibrium with the free molecules at that level. The concentration of these free molecules corresponds to the thermodynamic activity at that point, that is, for gases and vapors, the partial pressure. The rate-determining step of the permeation process is then taken to be the release of the permeant molecule from the transition site according to classical transition-state theory. Using an idealized cubic-lattice model for the distribution of the transition sites, this is shown to lead to the observed proportionalities of the permeation rate to the area of the film, the pressure difference across it, and the reciprocal of the film thickness. It also accords with the observed Arrhenius-type dependence of the permeability coefficient on temperature, where the Arrhenius parameters relate to the thermodynamic parameters for the transfer of the permeant molecule from the gas phase and its insertion in the transition site. The Arrhenius parameters from the literature (Polymer Handbook) for 16 homopolymers—NR, PA 11, PC, PDMB, PDMS, three PEs (HDPE, LDPE, and hydrogenated polybutadiene), PETFE, PEMA, PET, PP, PTFE, PVAC, PVBZ, and PVC—with 16 “simple” permeants—H2, He, CH4, Ne, N2, CO, O2, HCl, Ar, CO2, SO2, Cl2, Kr, SiF4, Xe, and SF6 as well as H2O vapor—are used as the dataset. These Arrhenius parameters are first discussed in relation to isokinetic behavior. They are then correlated according to the TSM theory with the van der Waals molecular diameter of the permeant σG, and its absolute entropy S0. With certain exceptions, linear correlations are obtained with the 10 smaller-molecule permeants (He to CH4) that show that they use the same set of transition sites, below and above the glass transition temperature, with each polymer; the permeant molecules evidently behave here as “hard spheres,” regardless of their other chemical characteristics. This enables estimates to be made of the four characteristic parameters for the polymer: the intersite spacing λ (equivalent to the lattice parameter of the idealized model and to the jump length of the AJM); the size of the transition-site aperture, σL; the force constant θ associated with expansion of the aperture by the permeant molecule; and the entropy increment ν also associated with this expansion. For most of the systems, the site-spacing λ is of the order of 10 nm, and the aperture σL is about 200 pm. The theory provides a molecular basis for the interpretation and design of the permeation characteristics of polymers. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 79: 981–1024, 2001