Dual-mode free volume model for diffusion of gas molecules in glassy polymers

The development of a new model for the diffusion of gas molecules in glassy polymers is presented which utilizes concepts from free volume theory and relies on a dual-mode interpretation of sorptive dilation in glassy polymers. Three assumptions are made in the development of the model. First, the free volume available for molecular transport processes is taken as constant below the glass transition temperature. Second, two populations of gas molecules are assumed to exist—one which contributes to the maintenance of an iso-free volume state upon sorptive dilation and one which does not contribute owing to sorption into regions of unrelaxed volume. Third, the former population is assumed to be mobile while the latter is not. The resulting model predicts, at constant temperature, a diffusion coefficient that is independent of solute volume fraction. This is in contrast to the widely used dual-mode sorption model with partial immobilization for gas transport in glassy polymers which leads to a diffusion coefficient that is dependent on solute mole fraction through the molar gas concentration. The new model is used to interpret gas transport data from permeation experiments for carbon dioxide, methane, and ethylene in three polycarbonates. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 1737–1746, 1997     

Effect of penetrant-induced isothermal glass transition on sorption,dilation, and diffusion behavior of polybenzylmethacrylate/CO2

The effect of a penetrant-induced isothermal glass transition on sorption, dilation, and diffusion behavior was studied in a single experimental run for CO₂ in cast polybenzylmethacrylate films. The dual-mode type sorption isotherms below the glass transition temperature of the polymer changed to linear ones above a certain concentration. Meanwhile, partial molar volume of CO₂ determined from the dilation of the films above the concentration gave a value very close to the one reported for rubbery polymers, and diffusion coefficients became less concentration-dependent. The results were conformable to the concept of unrelaxed volume in glassy polymers. © 1996 John Wiley & Sons, Inc.     

Steady-state permeation of carbon dioxide through a miscible,glassy polymer blend

Steady-state permeation measurements are reported for carbon dioxide (CO₂) through quenched, amorphous films of a miscible blend of poly(butylene terephthalate) (PBT) and a random copolyester of bisphenol-A and iso/terephthalate acids (PAr). Permeabilities were determined at 35°C on blends with up to 60 wt % PBT and for CO₂ pressures up to 300 psi (2.06 MPa). At a fixed blend composition, the permeability, P̄, decays with driving pressure, p, as described by dual-mode models for gas transport in glassy polymers. From regression fits of the data to dual-mode model predictions for P̄(p), high-and low-pressure limiting permeabilities are determined. These decrease with PBT content in a manner indicating strong, favorable energetic interactions between the PBT and PAr components in the blend. © 1996 John Wiley & Sons, Inc.     

Gas sorption in polymers based on bisphenol-A

Equilibrium gas sorption measurements for CO₂, CH₄, and N₂ were made with three polymers based on bisphenol-A, namely a polyhydroxyether, a polyetherimide, and a polyarylate. These data plus previous results for two other bisphenol-A polymers, polycarhonate and polysulfone, were analyzed using the dual-mode sorption model and the more recent gas-polymer-matrix model. The models were compared on the basis of physical interpretations of the resulting parameters. The Langmuir capacity from the dual-model model was related to the unrelaxed volume of the glassy polymer. The Henry's law sorption parameter from the dual-mode model was related to the internal pressure of the polymer and to its tensile stress at yield. The work suggests a means for estimation of gas sorption levels from thermal and mechanical properties of the polymer.     

Ending Aging in Super Glassy Polymer Membranes

Aging in super glassy polymers such as poly(trimethylsilylpropyne) (PTMSP), poly(4‐methyl‐2‐pentyne) (PMP), and polymers with intrinsic microporosity (PIM‐1) reduces gas permeabilities and limits their application as gas‐separation membranes. While super glassy polymers are initially very porous, and ultra‐permeable, they quickly pack into a denser phase becoming less porous and permeable. This age‐old problem has been solved by adding an ultraporous additive that maintains the low density, porous, initial stage of super glassy polymers through absorbing a portion of the polymer chains within its pores thereby holding the chains in their open position. This result is the first time that aging in super glassy polymers is inhibited whilst maintaining enhanced CO₂ permeability for one year and improving CO₂/N₂ selectivity. This approach could allow super glassy polymers to be revisited for commercial application in gas separations.     

Determination of dual-mode sorption and mobility parameters from permeation experiments

The precise pressure dependence of apparent diffusion and permeation coefficients was measured by using a microcomputer system for collecting and treating permeation data for CO₂ in glassy poly(ethylene terephthalate) below 1 atm between 15 and 40°C. The partial immobilization model was used to determine the dual-mode sorption and mobility parameters. The curves calculated with these parameters were in excellent agreement with experimental data. These parameters were also compared with sorption parameters obtained from measurements at 30°C. There was a small difference between the values of the parameters obtained from these permeation data and those from sorption data which we had previously obtained. Relations between this difference and the method of determination of the parameters are discussed.     

Sorption and diffusion of gases in a polyimide

Sorption and diffusion of gases (CO₂, N₂, and He) in a polyimide (PI2080) film were measured by using an apparatus which gives the sorption rate curves from the initial state to the equilibrium state. Nonlinear isotherms observed for CO₂ sorption were interpreted successfully in terms of the dual-mode model for sorption in glassy polymers. Linear isotherms observed for N₂ and He seemed to obey Henry's law. Two diffusion coefficients (D_I and D_E) were obtained using the short-time method and the long-time method for a Fickian diffusion model, together with the equilibrium solubility (C_e) from each experiment. The initial sorption rate curves agreed with the calculated curves using D_I, however near sorption equilibrium the curves are in accord with the calculated curves using D_E. These observations suggest that some relaxation process is superimposed on the diffusion process. The non-Fickian transport data were correlated successfully with a model that combines time-dependent diffusion and the Fickian model.     

Finite difference modeling of the gas transport process in glassy polymers

Finite difference modeling has been used to predict the results of gas transport experiments for a concentration-dependent diffusion coefficient. Experiments on the transport of CO₂ in poly(ethylene terephthalate) and poly(ethylene naphthalate) had previously shown a difference between the effective diffusion coefficients for absorption and desorption runs of a double-sided experiment, but this effect had not been seen for single-sided experiments. The finite difference calculations show that such results are to be expected, and the parameters included in the models that attempt to describe the diffusion process in glassy polymers, such as the dual-mode model, and which lead to concentration-dependent diffusion coefficients, can be found by fitting the experimental data for the double-sided experiment using finite difference modeling. The dependence of the effective diffusion coefficient on pressure for the single-sided experiment can be correctly predicted using results from the double-sided experiment for an identical sample. © 1996 John Wiley & Sons, Inc.     

Gravimetric study of high-pressure sorption of gases in polymers

A gravimetric method for determining precisely the solubility of gases in polymers at high pressure is described. The solubilities of N₂ and CO₂ in low-density polyethylene (LDPE); CO₂ in polycarbonate (PC); and N₂, CH₄, C₂H₆, and CO₂ in polysulfone (PSUL) have been measured as a function of pressure up to 50 atm. Most of the measured sorption isotherms agreed closely with published data, but reproducible and time-dependent hysteresis in the sorption of CO₂, C₂H₆, and CH₄ in glassy polymers, PC, and PSUL, was observed in this study for the first time. Like the well known conditioning effect of high-pressure CO₂ on the sorption capacity of glassy polymers, these hysteresis phenomena are believed to be due to the plasticizing effect of sorbed gases. On the basis of the current data, the dual-mode sorption model including the plasticization by sorbed gas is discussed and a primitive equation for the concentration of sorbed gases in a quasiequilibrium state of sorption or desorption is proposed.     

Dual-mode analysis of subatomspheric-pressure CO2 sorption and transport in Kapton H polyimide film

Sorption kinetics and equilibria as well as permeabilities and diffusion time lags for CO₂ in Kapton polyimide film have been studied at temperatures from 35 to 55°C and pressures up to 0.78 atm. The sorption/desorption cycles indicate that the diffusivity of CO₂ increases with increasing local penetrant concentration in the polymer. Both the permeability and time lag decrease with increasing upstream CO₂ pressure. All of these results are described well by theoretical expression based on the dual-mode theory of sorption and transport in glassy polymers.     

Kinetics and equilibria of sulfur dioxide sorption in kapton polyimide

The kinetics and equilibria of SO₂ sorption in Kapton polyimide film have been measured at 25°C for pressures up to 58 cm Hg using a quartz spring balance. The observed equilibria are described well by the dual-mode model for sorption of penetrants in glassy polymers. Observed hysteresis in sorption-desorption cycling indicates that the diffusivity of SO₂ increases markedly with increasing local penetrant concentration in the polymer. The concentration dependence of the effective average diffusion coefficient is described well by an expression from dual-mode theory. The Langmuir component of the sorption population has an extremely low mobility compared to that of the Henry's law component; in fact, the so-called “total immobilization” limiting case, which assigns a zero diffusion coefficient to this component, appears to be satisfactory under the conditions studied. These results, coupled with published results for other penetrants, suggest that the degree of immobilization of the Langmuir population depends to a great extent on the condensability and/or the molecular size of the penetrant. Predictions of the SO₂ diffusion time lag and permeability in Kapton as functions of the upstream SO₂ pressure are presented and discussed in terms of the dual-mode theory.     

Selective permeation of CO2 and CH4 through kapton polyimide: Effects of penetrant competition and gas-phase nonideality

Sorption isotherms for pure CO₂ and pure CH₄ in Kapton H® polymide films at 60°C are reported for pressures up to 20 atm and are analyzed in terms of the dual-mode sorption model. An experimental scheme for the measurement of steady-state permeabilities of both pure and mixed gas feeds is described. Permeabilities of Kapton to the individual components at 60°C are presented for a mixture comprised of 32.2% CO₂ in CH₄ as functions of feed pressure up to 590 psi (absolute). The permeabilities for the individual penetrants in the mixed feed are lower than the respective purecomponent values at the corresponding partial pressures. Furthermore, the permeabilities of both penetrants drop as the feed pressure is increased at constant composition. The dual-mobility transport model used to analyze the data postulates that the observed pressure and composition dependence of the permeabilities is due to competition between penetrants for a limited microvoid sorption capacity in the glassy polymer. It is demonstrated that in addition to flux depressions due to dual-mode effects, nonideality of the gas phase must be accounted for to explain the substantial flux depressions observed for the CO₂/CH₄ mixtured used in this study.     

Transient and steady-state permeation in poly(ethylene terephthlate) above and below the glass transition

Transient and steady-state permeation data are reported for CO₂ in semicrystalline poly(ethylene terephthalate) for temperatures ranging from 25 to 115°C over the pressure range from 1 to 20 atm. The pressure dependency of the time lag and permeability disappears completely above the glass transition of the polymer, and Fick's law with a concentration-independent diffusion coefficient applies. In the glassy state, a concentration-dependent diffusion coefficient is necessary to describe the data. The form of this concentration dependence is described well by the partial immobilization transport model that attributes a different mobility to each of the two populations of sorbed gas which exist in local equilibrium with each other in glassy polymers. The importance of reporting the pressure used in transport experiments involving glassy polymers is emphasized by comparing the difference in the activation energy of the apparent diffusivity calculated from the measured time lag at 1 and 20 atm. Also, the magnitude of the observed slope discontinuity at T_g in Arrhenius plots of these apparent diffusities is shown to be a function of the upstream pressure used in the experiment. The independently measured time lags are compared with the predicated values calculated from various transport models and found to be described best by the partial immobilization model.     

CO2-induced plasticization phenomena in glassy polymers

A typical effect of plasticization of glassy polymers in gas permeation is a minimum in the relationship between the permeability and the feed pressure. The pressure corresponding to the minimum is called the plasticization pressure. Plasticization phenomena significantly effect the membrane performance in, for example, CO₂/CH₄ separation processes. The polymer swells upon sorption of CO₂ accelerating the permeation of CH₄. As a consequence, the polymer membrane loses its selectivity. Fundamental understanding of the phenomenon is necessary to develop new concepts to prevent it.In this paper, CO₂-induced plasticization phenomena in 11 different glassy polymers are investigated by single gas permeation and sorption experiments. The main objective was to search for relationships between the plasticization pressure and the chemical structure or the physical properties of the polymer. No relationships were found with respect to the glass-transition temperature or fractional free volume. Furthermore, it was thought that polar groups of the polymer increase the tendency of a polymer to be plasticized because they may have dipolar interactions with the polarizable carbon dioxide molecules. But, no dependence of the plasticization pressure on the carbonyl or sulfone density of the polymers considered was observed. Instead, it was found that the polymers studied plasticized at the same critical CO₂ concentration of 36±7 cm³ (STP)/cm³ polymer. Depending on the polymer, different pressures (the plasticization pressures) are required to reach the critical concentration.     

Model of sorption of simple molecules in polymers

A model of simple molecule sorption in polymers is proposed which embraces both the glassy and rubbery regions, and incorporates the successful dual-mode model below the glass-transition temperature. Hole filling is shown to be an important sorption mechanism both above and below T_g, although saturation effects do not occur in the rubbery polymer. The model interprets the “dual-mode” Langmuir and Henry's law parameters at the molecular level, and a simple statistical mechanical analysis allows estimation of the parameter values, as well as specifying certain interrelationships between the parameters. Applications of the model to gas solubility data in five polymers are considered [polyethylene (PE), poly(ethylene terephthalate) (PET), polystyrene (PS), polymethacrylate (PMA), poly(vinyl acetate) (PVAc)] and semiquantitative agreement is obtained for PE, PET, and to a lesser extent, PS. For PMA and PVAc, the agreement is qualitative only.     

The effect of plasticization on the transport of gases in and through glassy polymers

The effects of plasticization on the transport of gases and vapors in and through glassy polymers are examined from the viewpoint of the “dual-mode” sorption model with partial immobilization. The analysis assumes the existence of two penetrant populations with different mobilities in the Henry's law and Langmuir domains of the glassy polymers. These mobilities are characterized by their mutual diffusion coefficients D_D and D_H. The plasticization of the polymer by penetrant gases is reflected in the concentration dependence of D_D and D_H. Expressions for the effective (apparent) diffusion and permeability coefficients are derived assuming that D_D and D_H are exponential functions of the penetrant concentration in the polymers. The results of this study are compared with a similar analysis which assumed the existence of a single mobile penetrant population. The present analysis provides information on the effects of plasticization on the penetrant transport in the Henry's law and Langmuir domains separately. The effects of antiplasticization or clustering of penetrant molecules on the effective diffusion and permeability coefficients are also examined.     

Model for sorption of mixed gases in glassy polymers

A model is presented for analysis of the sorption of mixed gases in glassy polymers at concentrations below which significant plasticization occurs. The well-known dual-mode sorption model comprised of a Henry's law term and a Langmuir isotherm term, which has been used extensively for interpretation of single-component gas sorption data, forms the basis for the analysis of binary mixtures discussed here. Measurements using pure gases provide dual mode parameters which can then be used to predict the resultant sorption isotherms for binary mixtures of any of the pure gases. The proposed analysis is based upon recognition that the Langmuir component of the overall sorption concentration should be governed by competition between the two penetrants for the fixed unrelaxed volume in the polymer, which is believed to be the locus of the Langmuir capacity. This effect may result in a significant depression of the measured sorption of similar penetrants competing for the limited Langmuir capacity. A numerical example is considered which illustrates the range of behavior expected for CO₂ and CH₄ in polycarbonate. Deviations from the theoretical predictions of the simple dual-mode model for binary systems are discussed in terms of plasticizing effects on the Henry's law constant and the Langmuir affinity constant. The analyses proposed here are of direct and critical interest to the applied problems of migration of trace contaminants in glassy polymers and analysis of barrier packaging for foods since all of these applied problems involve mixed-penetrant sorption. Specifically, it is predicted that the presence of residual monomers or solvents in glassy polymers can produce both anomolously low Langmuir sorption affinity constants and sorption enthalpies compared with the residual-free case.     

An application of an homogenization method to a model of diffusion in glassy polymers

The sorption of gases in polymers below their glass-transition temperature T_g is known in many cases to be described by the “dual sorption” theory, according to which the gas is held in accordance with both the Langmuir and Henry's laws. Based on this theory, expressions for the “effective diffusion coefficient” in the glassy polymers have been obtained by investigators in the past, notably by Paul and Koros.² The present analysis regards the glassy polymers as inhomogeneous with regions on which the gas sorption follows the Langmuir law. Assuming that the linear dimensions of these regions, which are often referred to as “microvoids” (although they are not space filled by vacuum), are small compared to the macroscopic length of interest but large compared to the mean free path of the penetrant gas molecules, we derive a rigorous relation between the average flux and the concentration gradient in the polymer and show that this relation can be expressed in terms of an “effective diffusion coefficient” D_eff which depends on the details of the microstructure, i.e., the size, shape and spatial distribution of the “microvoids.” This expression for D_eff is shown to reduce to that of Paul and Koros² in two situations: (1) when the “voids” consist of slabs running parallel to the concentration gradient, and (2) when the “voids” are spherical and the temperature of the polymer is not too different from T_g. The results of the present study lead to an alternative procedure for interpreting the experimental data on sorption and permeation which may have some advantages over the procedure currently employed. Finally, the analysis presented here is also applicable to polymers containing adsorptive fillers.