Nonequilibrium Lattice Fluids: A Predictive Model for the Solubility in Glassy Polymers Ferruccio Doghieri and Giulio C. Sarti* Dipartimento di Ingegneria Chimica, Mineraria e delle Tecnologie Ambientali, Universita ` degli Studi di Bologna, Viale Risorgimento, 2, 40136 Bologna, Italy Received September 12, 1995; Revised Manuscript Received August 15, 1996 X ABSTRACT: The calculation of sorption isotherms for gases and vapors in glassy polymers is approached through a nonequilibrium equation of state procedure. The basic peculiar feature of the system, represented by the nonequilibrium structure of the mixture, is accounted for by introducing an order parameter for an isotropic glass. By revisiting the lattice fluid model by Sanchez and Lacombe (Macromolecules 1978, 11, 1145.), an expression for the Gibbs free energy of nonequilibrium lattice fluids is obtained in which the polymer species density in the solid mixture is considered as an order parameter and it is thermodynamically treated as an internal state variable. The absence of adjustable parameters makes the resulting model entirely predictive for the solubility, once the pseudoequilibrium volumetric data are available. The comparison of the predicted isotherms with the data for CO2-poly(carbonate) systems at 35 °C, obtained by Fleming and Koros (Macromolecules 1990, 23, 1353.) under different polymer prehistories, points out the remarkably good ability of the model to represent the sorption/desorption behavior and hysteresis experimentally observed. 1. Introduction Sorption and desorption of low molecular weight species in or from glassy polymers has attracted con- siderable attention in recent years in view of the various different applications in which they play a crucial role, such as in membrane separations, in the extraction of solvents and of contaminants, and in the preparation of packaging materials. In addition, a stimulating conceptual challenge is present, associated with the nonequilibrium nature of the glassy systems, which, at a given temperature, pressure and external penetrant activity, exhibit sig- nificantly different pseudosolubilities as well as trans- port kinetics as a consequence of different concentration or thermomechanical prehistories. 2-5 Correspondingly, all the usual well-established and powerful tools based on equilibrium thermostatics become inappropriate to describe the behavior of such systems. For rubbery phases and polymer melts the thermo- dynamic properties, including the solubility of low molecular weight species, can be described properly through reliable expressions based on the equation of state (EOS) approach 1,6-8 or on the activity coefficient procedure; 9-12 the latter method can suitably be applied as long as the operating pressure is below the critical pressure of the penetrants, while the EOS approach is to be preferred at higher pressures, in order to avoid the use of hypothetical states far apart from the actually measurable physical states of the dissolved species. For the solubility of gases and vapors in glassy polymers, on the other hand, the more successful description is represented thus far by the dual-mode sorption model; 13,14 the extensive use in correlating solubility data for different systems, 15-17 the excellent success, and the simplicity of its application render the model very attractive and apparently insuperable. Yet, the heuristic way in which it is introduced originated several research efforts aiming to obtain a more fundamental background for the parameters en- tering the model itself. 18-20 Actually, dual-mode sorp- tion was introduced on a phenomenological basis, assuming that two populations of penetrant molecules are present in the polymeric solid: one is dissolved in the bulk polymer matrix and has a partial concentration (c D ) expressed as a function of pressure through Henry’s law; the second is considered as adsorbed into the surface of the microvoids which are supposed to be present into the polymer, essentially as a consequence of the excess free volume frozen into the glassy matrix. The concentration of the adsorbed molecules (c H ) is described by a Langmuir adsorption isotherm. The resulting overall concentration of the dissolved gas is thus expressed as the sum of c D and c H as follows: and its variation as a function of pressure may be calculated on the basis of the model parameters k D , c′ H and b. As already mentioned, the model can rather ac- curately correlate the concentration of gases in glassy polymers in most of the known cases and provides a successful conceptual framework for gas sorption; the idea of the coexistence of “equilibrium” and “nonequi- librium” contributions to the overall concentration of the penetrant species, described by the first and the second term of eq 1.1, respectively, appears to be physically sound. Indeed, it has been shown that the parameter c′ H can be related to the initial volume of the polymer matrix, 19 while the “equilibrium” parameter k D can be related to the dilation of the polymer due to the sorption process. 18 As has been shown very recently, 20 the “equilibrium” term can also be calculated by an EOS procedure, while the c H contribution can be correlated to the nonequilibrium properties of the mixture through the excess free volume sites distribution. Nonetheless, the dual-mode sorption shows the typical limitations of substantially empirical models: the pa- rameters must be evaluated for each polymer penetrant system at each temperature independently and, more- over, in desorption the three parameters must be given values different from the ones used in the sorption X Abstract published in Advance ACS Abstracts, October 15, 1996. c ) c D + c H ) k D p + c′ H bp 1 + bp (1.1) 7885 Macromolecules 1996, 29, 7885-7896 S0024-9297(95)01366-0 CCC: $12.00 © 1996 American Chemical Society