Effective medium approximations for penetrant sorption in glassy polymers accounting for excess free volume

An accurate determination of a penetrant volume fraction in a swollen polymer is of crucial importance in a range of different technologies. Using optical methods, such as in-situ spectroscopic ellipsometry, it is possible to extract the thickness and refractive index of dry and swollen polymer films. The volume fraction of the penetrant can then be calculated from the change in thickness, or from the refractive index using effective medium approximations. For thermodynamically equilibrated and ideal swollen rubbery polymers, these calculations yield accurate results. However, for glassy polymers the influence of the excess free volume trapped within the polymer network during vitrification is rarely taken into account. In this work we investigate the effect of excess free volume in the calculations of penetrant volume fraction in a model glassy polymer - penetrant system. The influence of the excess free volume is included by extrapolating the properties of an equilibrium polymer matrix from above its glass transition temperature. The error between the approaches that do, and do not, take account for the non-equilibrium of the glassy polymer is quantified and the implications for other systems are discussed. The errors are shown to be very significant, especially when the dry polymer has a large excess free volume. Such materials are particularly relevant in membrane applications.