New relation between diffusion and free volume: II. Predicting vacancy diffusion

It has been suggested that physical aging is a result of vacancies (cavities, holes or pores) travelling to the external surface where they disappear and therefore cause density increase. The Doolittle relation D = A exp(−B/f) where f is the fractional free volume and A and B are constants, has been used in previous work to describe the diffusion coefficient D (diffusivity) for vacancy transport. In part I a new empirically determined relation has been suggested as an alternative to the Doolittle relation, and this new relation has been shown to accurately model gas permeability data over a wide range of polymeric free volume. This new relation takes the form D = α exp(βf) with α and β as constants. Here we show that when the Doolittle relation is replaced with the new relation an exact analytical solution exists to the differential equation that governs the fractional free volume behaviour throughout the sample during physical aging leading to an Empirically derived Vacancy Diffusion (EVD) model for physical aging. An approximate analytical solution based on the exact solution is then compared to experimental data and other popular models such as the Kovacs, Aklonis, Hutchinson and Ramos phenomenological model and the Curro, Lagasse and Simha vacancy diffusion model. This EVD model is also combined with a lattice contraction model to form a dual lattice contraction and vacancy diffusion model which is compared with McCaig, Paul and Barlowʹs experimental results, showing a good correlation. Further support for the new EVD model is revealed by its similarity with the early established constitutive kinetic equation. Previous aging models are complicated and difficult to implement, therefore, a model that is easy to implement and physically meaningful such as this EVD model is highly sought after. An application of the model reveals that vacancy diffusion to the external surface is the dominant aging mechanism within polysulfone thin films (<1 μm), but not within the thick films (>1 μm).