Modified Free Volume Theory for Self-Diffusion of Small Molecules in Amorphous Polymers

Abstract

Diffusion of small molecules in amorphous polymers near the glass transition is a topic of interest for a variety of technological applications. Within this study, the free volume theory (FVT) of Vrentas and Duda as a predictive tool for estimating self-diffusion coefficients is extended by suggesting modifications to certain aspects of this model. These modifications encompass the description of the free volume of the mixture, the viscosity–diffusivity interplay, and the energetics of the diffusion. To support the derived modifications, they are exercised on the diffusion of three active ingredients in polyvinylpyrrolidone. The results are partly compared with experimental data based on dielectric spectroscopy as well as secondary ion mass spectrometry (adopted from the literature). Concerning the specific hole free volume, an approach for describing nonideal mixtures achieved by means of estimating the mixtures’ Vogel temperature at each composition is proposed. Moreover, a new fragility-dependent parameter, ξ1, is introduced into the main equation of the FVT model, which brings the model in harmony with the fractional Stokes–Einstein (F-SE) relation at the pure diffusant limit, hence improving its predictions at higher diffusant mass fractions. Furthermore, the F-SE equation is employed to describe the pre-exponential factor as well as the effective molar jump energy at the pure diffusant extreme. The resulting molar energy is subsequently combined with an empirically obtained effective energy at the pure polymer limit and weighted according to a quadratic expression analogous to the cohesive energy of a solution. In conclusion, the modified FVT model discussed in this study presents a fully predictive platform for describing diffusion in polymers and offers an extended temperature and compositional applicability range compared to the original version of FVT. This model is expected to be applicable to the diffusion of small molecules, which are commensurate in size with typical segments of the polymer.