Sorption of small molecules in polymeric media

We discuss the sorption of penetrant molecules from the gas phase by a polymeric medium within a model which is very close in spirit to the dual sorption mode model: the penetrant molecules are partly dissolved within the polymeric matrix, partly fill the preexisting voids. The only difference with the initial dual sorption mode situation is the assumption that the two populations of molecules are in equilibrium with each other. Applying basic thermodynamics principles we obtain the dependence of the penetrant concentration on the pressure in the gas phase and find that this is expressed via the Lambert W-function, a different functional form than the one proposed by dual sorption mode model. The Lambert-like isotherms appear universally at low and moderate pressures and originate from the assumption that the internal energy in a polymer-penetrant-void ternary mixture is (in the lowest order) a bilinear form in the concentrations of the three components. Fitting the existing data shows that in the domain of parameters where the dual sorption mode model is typically applied, the Lambert function, which describes the same behavior as the one proposed by the gas-polymer matrix model, fits the data equally well.