Permeation of gases in polymers: parameter identification and nonlinear regression analysis

In numerous previous studies it has been shown that the permeability of gases in polymers depends strongly on the polymer structure, on the gas type, as well as on the conditions of temperature and pressure. The theoretical models that have been proposed and developed in the literature, describe the transport mechanism of molecular species in polymers by diffusion involving a concentration-dependent diffusion coefficient. However, the exact form of this dependency and exact information on the diffusion coefficient and on the solubility coefficient are not always available in the literature, in particular in extreme conditions like in the case of protective polymer coatings for flexible offshore pipes where the temperature and pressure of the diffusing gas can be very high. Using experimental data from permeation experiments on particularly developed experimental devices at the Institut Français du Pétrole (IFP) we are able to study this dependency and have produced a method which allows to identify the parameters in the model from this data. Mathematically, this leads to a nonlinear least-squares optimisation problem with constraints in the form of partial differential equations (PDEs), which we will concentrate on in this paper. In particular, we will focus on the statistical analysis of the results and give confidence intervals for the estimated parameters. We will also present tests to check whether the assumptions on our model are appropriate and to identify whether the introduction of a concentration-dependent diffusion coefficient is necessary for particular pairs of gas and polymer and at certain temperatures and pressures. We will demonstrate the performance of our method and the statistical analysis of the results on some data obtained with the experimental devices at the IFP.