Pseudo-equilibrium polymer density

For the solubility in glassy phases, the situation is substantially different since the polymer density does not match its equilibrium value p, but it finally reaches an asymptotic value determined by the kinetic constraints acting on the glassy molecules, and is substantially dependent on the past history of the polymer sample. Thus the penetrant concentration in the polymeric phase reflects the pseudo-equilibrium state reached by the system. In view of the NET-GP results, such pseudo-equilibrium condition corresponds to the minimum Gibbs free energy for the system, under the constraint of a fixed value (the pseudo-equilibrium value) of the polymer density in the condensed phase ... 

In equation 2.14, the non-equilibrium solute chemical potential is calculated through the use of equation 2.12 and of an appropriate EoS for the polymer-penetrant system under consideration. The pseudo-equilibrium penetrant content in the polymer, can be easily calculated whenever the value of the pseudo-equilibrium polymer density Pp i is known. Such a quantity represents, obviously, a crucial input for the non-equiUbrium approach, since it labels the departure from equilibrium it must be given as a separate independent information, and cannot be calculated simply from temperature and pressure since it depends also on the thermomechanical history of the sample. 

The pseudo-equilibrium state of the glassy mixture is desaibed by the usual state variables (temperature, pressure and composition) plus the polymer density p2 that accounts for the departure from equilibrium frozen into the glass. 

The results obtained in the previous section permit calculation of penetrant solubility in a glassy polymer as a function of temperature, penetrant fugacity and polymer density. Contrary to the situation for the equilibrium solubility calculation, the polymer density under pseudo- uilibrium conditions must be known in order to calculate the corresponding penetrant content. For pure penetrants, the comparison of model predictions with experimental isotherms is rather satisfactory for the cases in which volume dilation data are also available (i, 13),... 

The requirement to specify the polymer density may represent a serious limitation for the practical application of the NELF model. Indeed, the dilation of the polymer matrix at high penetrant pressure could be significant and difficult to estimate without specific experimental data. On the other hand, the use of the non-equilibrium polymer (tensity is actually a powerful tool to represent complex non-equilibrium phenomena. It has been shown, for example, that it allows a description of sorpdon-desorption hysteresis (7) as well as the influence of pretreatments on the solubility isotherms of gases in glassy polymers. For such cases, the different pseudo-equilibrium solubility values at the same prevailing temperature and penetrant fugacity are satisfactorily accounted for by considering die different pseudo-equilibrium polymer densities. 

In the calculations considered in the following sections, the dilation of the polymer matrix is ignored since no direct experimental evidence is available for the volume dilation associated to the sorption of the mixture considered. The solubility of penetrants in the solid polymer will then be calculated by assuming that the polymer density at pseudo-equilibrium conditions is equal to the pure polymer density at the same temperature. 

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