Temperature Dependence of the Permeability Upper Bound Predicted From Transition State Theory and Non-Equilibrium Lattice Fluid Theory

Robeson first reported the existence of an upper bound on the relationship between permeability and permeability selectivity (Journal of Membrane Science, 320, 390-400, 2008). The existence of such an upper bound has significant implications for the development of new polymeric materials for gas separation.

A theoretical foundation for the existence of this upper bound has been proposed by Freeman (Macromolecules, 32, 375-380, 1999; Journal of Membrane Science 341, 178-185, 2009). While this work provides good estimates of the location of the upper bound, it suggests all materials should lie along the upper bound and provides limited insight into the relationship between permeation and other physical properties.

Recently we have used non-equilibrium lattice fluid theory (NELF) to predict the existence of an upper bound for solubility. NELF provides good a priori predictions of solubility and solubility selectivity for a variety of polymer-gas systems using material parameters derived from pure component physical properties. The NELF theory subsequently was combined with transition state theory (TST) to predict permeability and permeability selectivity. The combined NELF-TST theory uses NELF to calculate the free energy change associated with forming the diffusion transition state. NELF-TST predicts the existence of an upper bound on permeability and permeability selectivity which is in good agreement with the literature. The location of the upper bound is determined primarily by the polymer interaction energy and the extent of polymer deformation required to create the transition state.

As membrane processes are pushed to temperature and pressure extremes, the dependence of the upper bound on operating conditions has attracted interest. We use the NELF-TST theory to determine how the upper bound moves with temperature.

The NELF-TST can be used to predict the temperature dependence of the upper bound by accounting for changes in polymer density with temperature using the thermal expansion coefficient obtained from the NELF equation of state or experiment. The theory provides predictions for the heat of sorption (H_S), diffusion activation energy (E_d), and permeation activation energy (E_p). The results are in good agreement with experiment for CO₂, CH₄, O₂, and N₂ transport in polycarbonate (PC) and tetramethyl polycarbonate (TMPC). The theory is used to predict how the solubility, diffusivity, and permeability upper bounds move with temperature.

Keywords: solubility, diffusivity, permeability, upper bound, transition state theory, non-equilibrium lattice fluid theory, gas penetrant, glassy polymer, thermal expansion coefficient