Comparison of three models for permeation of CO2/CH4 mixtures in poly(phenylene oxide)

Two models for the permeability of pure gases have been extended to include binary gas mixtures. The first is an extension of a pure gas permeability model, proposed by Petropoulos, which is based on gradients of chemical potential. This model predicts the permeability of components in a gas mixture solely on the basis of competition for sorption sites within the polymer matrix. The second mixed gas model follows an earlier analysis by Barrer for pure gases which includes the effects of saturation of Langmuir sites on the diffusion as well as the sorption processes responsible for permeation. This generalized “competitive sorption/diffusion” model includes the effect of each gas component on the sorption and diffusion of the other component in the mixture. The flux equations from these two models have been solved numerically to predict the permeability of gas mixtures on the basis of pure gas sorption and transport parameters. Both the mixed gas Petropoulos and competitive sorption/diffusion model predictions are compared with predictions from the earlier simple competitive sorption model based on gradients of concentration. An analysis of all three models is presented for the case of CO₂/CH₄ permeability in poly(phenylene oxide) (PPO). As expected, the competitive sorption/diffusion model predicts lower permeability than either of the models which consider only competitive sorption effects. The permeability depression of both CO₂ and CH₄ predicted by the competitive sorption/diffusion model is roughly twice that predicted by the competitive sorption model, whereas the mixed gas Petropoulos model predictions for both gases lie between the other two model predictions. For the PPO/CO₂/CH₄ system, the methane permeability data lie above the predictions of all three models, whereas CO₂ data lie below the predictions of all models. Consequently, the competitive sorption/diffusion model gives the most accurate prediction for CO₂, while the simple competitive sorption model is best for methane. The effects of mixed gas sorption, fugacity, and CO₂-induced dilation were considered and do not explain the inaccuracies of any of the models. The relatively small errors in mixed gas permeability predictions using either of the three models are likely to be related to “transport plasticization” of PPO owing to high levels of CO₂ sorption and its effect on polymer segmental motions and gas diffusivity.     

Sorption of small molecules in various polycarbonates and Kapton-H

Pressure-composition isotherms were determined at 20°C for CO₂ in Kapton and various substituted polycarbonates and for H₂O, Ar, N₂, CH₄, and acetone in bisphenol-A-polycarbonate. The isotherms are described by two parameters an average free energy of sorption and a width of a Gaussian distribution of free sorption energies. Within the framework of a recent model these parameters can be calculated assuming an elastic distortion of the polymer caused by the incorporation of solute atoms in preexisting holes. By comparing experimental values with predictions of the model the experimental width of the free energy distribution is only 30% smaller than the theoretical one. Functional relationships are obeyed between the sorption parameters on the one hand and glass transition temperature, average hole volume, and molecular volume of the solute on the other hand. Deviations occur for larger molecules like acetone and ethylene which are attributed to a viscoelastic distortion of the polymer. Comparing free energies of solution for the rubbery and glassy state of the polymer reveals more negative values for the glassy polymers despite their extra elastic distortion energy. This discrepancy is overcome by taking into account that the occupied volume has to be re-formed in the case of the rubbery or liquid polymer. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 483–494, 1998     

Evaluation of gas sorption parameters and prediction of sorption isotherms in glassy polymers

A model of continuous‐site distribution for gas sorption in glassy polymers is examined with sorption data of CO₂ and Ar in polycarbonate. A procedure is presented for determining from a measured isotherm the number of sorption sites in a polymer, an important parameter that previously had to be assumed. With this parameter value and solubility data obtained at zero pressure, the model can reasonably predict sorption isotherms of CO₂ in glassy polycarbonate for a wide temperature range. The number of sorption sites and the average site volume evaluated from CO₂ sorption isotherms are employed for the prediction of Ar sorption isotherms with zero‐pressure solubility data and the independently measured partial molar volume of Ar. A reasonable fit to the measured isotherms of Ar is achieved. With the proposed procedure, the continuous‐site model shows several advantages over the conventional dual‐mode sorption model. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 883–888, 2000     

On the crystallinity effect on the gas sorption in semicrystalline linear low density polyethylene (LLDPE)

This article describes the solubility of carbon dioxide, ethylene and propane in 1‐octene based polyethylene of 0.94, 0.92, 0.904, and 0.87 densities. The isotherms obtained in the gas sorption experimental device display a sorption behavior similar to that of glassy polymers. We apply the dual model to semicrystalline polymers assuming that Henry's sites are related to the amorphous phase, which decreases when the crystallinity percentage increases, whereas the surface of the crystalline phase acts as a Langmuir site with higher gas‐polymer affinity than glassy polymers. The good concordance of the calculated k_D values, using the Flory‐Huggins theory of polymer diluent mixtures, with the experimental results suggest that Henry's gas sorption fulfills this theory and, therefore, it may be a suitable way to estimate polymer‐gas enthalpic interactions. Particularly, the variation of k_D with the crystallinity fraction is exponential and the proportionality of the total sorption with the amorphous content seems only apparent. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1798–1807, 2007     

Analysis of gas sorption in glassy polymers with the GAB model: An alternative to the dual mode sorption model

The suitability of the Guggenheim–Anderson–De Boer (GAB) model for the parameterization of gas sorption isotherms and their dependences on temperature is explored. The GAB model implies that molecules adsorb on inner surfaces of the polymer in multilayers, which contrasts with the assumptions of the classical Dual Mode Sorption (DMS) model which implies the simultaneous occurrence of Henry‐like dissolution and Langmuir's case I adsorption. The GAB model shows similar efficacy of the parameterization of the gas sorption isotherms in polymers as the DMS model. The isosteric heat of adsorption shows clear dependence on relative surface coverage for carbon dioxide sorption in cellulose acetate, polyethylene terephthalate, and the first polymer of intrinsic microporosity (PIM‐1), thus allowing for the occurrence of adsorption multilayers. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2014 , 52, 1490–1495     

Sorption of methanol/MTBE and diffusion of methanol in 6FDA–ODA polyimide

This article discusses the diffusion and solubility behavior of methanol/methyl tert‐butyl ether (MTBE) in glassy 6FDA–ODA polyimide prepared from hexafluoroisopropylidene 2,2‐bis(phthalic anhydride) (6FDA) and oxydianiline (ODA). The diffusion coefficients and sorption isotherm of methanol vapor in 6FDA–ODA polyimide at various pressures and film thicknesses were obtained with a McBain‐type vapor sorption apparatus. Methanol/MTBE mixed‐liquid sorption isotherms were obtained by head‐space chromatography and compared with a pure methanol sorption isotherm obtained with a quartz spring balance. Methanol sorption isotherms obtained with the two methods were almost identical. Both methanol sorption isotherms obeyed the dual‐mode model at a lower activity, which is typical for glassy polymer behavior. The MTBE was readily sorbed into the polymer in the presence of methanol, but the MTBE sorption isotherm exhibited a highly nonideal behavior. The MTBE sorption levels were a strong function of the methanol sorption level. Methanol diffusion in the polymer was analyzed in terms of the partial immobilization model with model parameters obtained from average diffusion coefficients and the dual‐mode sorption parameters. Simple average diffusion coefficients were obtained from sorption kinetics experiments, whereas the dual‐mode sorption parameters were obtained from equilibrium methanol sorption experiments. An analysis of the mobility and solubility data for methanol indicated that methanol tends to form clusters at higher sorption levels. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2254–2267, 2000     

Sorption and diffusion of water in poly(vinylpyrrolidone)

The effect of molecular mass, thermal prehistory, physical state, and three-dimensional chemical crosslinked structure of a polymer on dissolution and diffusion in the PVP-water system has been studied. The kinetic dependences of sorption that correspond to the Fickian or pseudonormal type have been measured. In a certain concentration range, sorption is accompanied by transition of the system to the rubbery state. In the glassy state, the negative concentration dependence of the diffusion coefficient related to the nonequilibrium state of the polymer sorbent is observed. Sorption isotherms are described by S-shaped curves. It has been shown that the thermal prehistory of the polymer sorbent has the most pronounced effect on its sorption behavior. The effect of molecular mass is insignificant, while three-dimensional chemical crosslinks in PVP manifest themselves only in the region of the rubbery state. In accordance with the double sorption model, the experimental isotherms are represented as the superposition of two isotherms described by the Langmuir and Flory-Huggins equations. For the glassy state of the polymer sorbent, the degree of the nonequilibrium state has been estimated. With due regard for the excess free volume, the detailed thermodynamic analysis of isotherms has been performed; namely, the pair interaction parameters and the free energy of mixing have been calculated. The state of water in the polymer has been examined within the framework of hydrate contributions and clusterization theory.     

Sorption of light gases in glassy poly(ethyl methacrylate)

Although gas sorption in glassy polymers is a well‐studied phenomenon, no general microscopical model is developed which is able to describe the gas sorption in a wide temperature range using only characteristics of polymer and gas molecule. In this work, sorption isotherms and desorption kinetics of O₂, Ar, and N₂ for glassy poly(ethyl methacrylate) have been measured in the temperature range from 160 to 308 K. To describe both the phenomena, the model is developed which postulates that, in the frozen structure of glassy polymer, any cavities between macromolecules are the sorption sites for small molecules. The cavities of small size can expand elastically to accommodate a gas molecule. The sorption sites are considered to be the potential wells and their depths are distributed according to Gaussian law. The concentration of sorption sites, their mean depth and depths dispersion, and the frequency of molecules oscillations in the sorption sites are the only parameters which determine both the gas transport and sorption. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2018 , 56, 288–296     

Sorption and permeation in elastic solids: applicability to gas transport in glassy polymeric materials

A thermodynamic analysis of sorption and transport in elastic solids is presented. The penetrant's chemical potential in the solid is calculated from the free energy changes that accompany deformation of the solid to accommodate the penetrant and mixing of the deformed solid and penetrant. Sorption isotherms are obtained by equating this chemical potential to that of that of the gas phase. The penetrant diffusivity is determined from a statistical mechanical analysis described in the literature. Both pure and mixed gas solubility, diffusivity and permeability are predicted. Mixed gas behavior is predicted using material parameters obtained from pure gas data only. The results suggest certain relationships between the dual mode model parameters that are found experimentally. Comparison with several experimental gas–polymer systems indicate the elastic solid analysis can reproduce pure gas data and predict mixed gas data well with physically realistic material parameters. The results provide a basis for investigating the relationships between the mechanical, volumetric, gravimetric and thermodynamic processes that give rise to transport.     

Sorption of CO2, C2H4, N2O and their binary mixtures in poly(methyl methacrylate)

Sorption of carbon dioxide, ethylene, and nitrous oxide in poly(methyl methacrylate) (PMMA) at 35°C has been characterized for each gas as a pure component and for mixtures of carbon dioxide/ethylene and carbon dioxide/nitrous oxide. Pressures up to 20 atm were examined. Pure-component sorption isotherms are concave to the pressure axis for each of the gases. This behavior is accurately described by the dual-mode sorption model. Using only the purecomponent dual-mode parameters and the generalization of the model for gas mixtures, one can predict the total concentration of gas sorbed in the polymer to within an average deviation of ±2.01% for the CO₂/C₂H₄/PMMA system and ±0.98% for the CO₂/N₂O/PMMA system. In both systems, for each component of the mixture, sorption levels were lower than corresponding pure-component sorption levels at pressures equal to the partial pressure of the respective components in the mixture. Depression of the sorbed concentration in mixture situations appears to be a general feature of the above systems and can be substantial in some situations. For the CO₂/C₂H₄/PMMA system, use of pure-component sorption data to estimate the total sorbed concentration in the mixture would be in error by as much as 40% if one failed to account for competition phenomena responsible for depression in mixed-gas situations. Mixture pressures as high as 20 atm were studied for both systems and in the CO₂/N₂O/PMMA system sorbed concentrations reach 33.90 [cm³(STP)/cm³ polymer] without any significant deviation from model predictions.     

Competitive permeation of gas and water vapour in high free volume polymeric membranes

Highly permeable glassy polymeric membranes based on poly (1‐trimethylsilyl‐1‐propyne) (PTMSP) and a polymer of intrinsic porosity (PIM‐1) were investigated for water sorption, water permeability and the separation of CO₂ from N₂ under humid mixed gas conditions. The water sorption isotherms for both materials followed behavior indicative of multilayer adsorption within the microvoids, with PIM‐1 registering a significant water uptake at very high water activities. Analysis of the sorption isotherms using a modified dual sorption model which accounts for such multilayer effects gave Langmuir affinity constants more consistent with lighter gases than the use of the standard dual mode approach. The water permeability through PTMSP and PIM‐1 was comparable over the water activities studied, and could be successfully model ed through a dual mode sorption model with a concentration dependent diffusivity. The water permeability through both membranes as a function of temperature was also measured, and found to be at a minimum at 80 ° C for PTMSP and 70 °C for PIM‐1. This temperature dependence is a function of reducing water solubility in both membranes with increasing temperature countered by increasing water diffusivity. The CO₂ ‐ N₂ mixed gas permeabilities through PTMSP and PIM‐1 were also measured and model ed through dual mode sorption theory. Introducing water vapour further reduced both the CO₂ and N₂ permeabilities. The plasticization potential of water in PTMSP was determined and indicated water swelled the membrane increasing CO₂ and N₂ diffusivity, while for PIM‐1 a negative potential implied that water filling of the microvoids hampered CO₂ and N₂ diffusion through the membrane. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 719–728     

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.     

Sorptive dilation of poly(vinyl benzoate) and poly(vinyl butyral) by carbon dioxide

Dilation of poly(vinyl benzoate) and poly(vinyl butyral) accompanying sorption of carbon dioxide is measured with a cathetometer under pressures up to 50 atm at 25°C. Sorption isotherms for carbon dioxide in these polymers were also determined gravimetrically. Each dilation isotherm plotted versus pressure, as well as the sorption isotherm, showed an inflection point corresponding to the glass transition of the polymer-gas system. The dilation isotherms changed their form at that point from concave to convex to the pressure axis or to a straight line. Dilation and sorption isotherms exhibited time-dependent hysteresis below the inflection point but not above the point. Partial molar volumes of carbon dioxide in polymers, which were determined from dilation and sorption data above the point, were found to be independent of concentration and larger than those below the point. The latter volumes depended on concentration. Based upon the extended dual-mode sorption concept, which takes account of plasticization of polymer by sorbed gas, a dilation model was developed. Dilation data were described well by the model.     

Swelling and sorption in polymer–CO2 mixtures at elevated pressures

Experimental data on gas sorption and polymer swelling in glassy polymer—gas systems at elevated pressures are presented for CO₂ with polycarbonate, poly(methyl methacrylate), and polystyrene over a range of temperatures from 33 to 65°C and pressures up to 100 atm. The swelling and sorption behavior were found to depend on the occurrence of a glass transition for the polymer induced by the sorption of CO₂. Two distinct types of swelling and sorption isotherms were measured. One isotherm is characterized by swelling and sorption that reach limiting values at elevated pressures. The other isotherm is characterized by swelling and sorption that continue to increase with pressure and a pressure effect on swelling that is somewhat greater than the effect of pressure on sorption. Glass transition pressures estimated from the experimental results for polystyrene with CO₂ are used to obtain the relationship between CO₂ solubility and the glass transition temperature for the polymer. This relationship is in very good agreement with a theoretical corresponding-states correlation for glass transition temperatures of polystyrene-liquid diluent mixtures.     

Sorption Isotherms of Aromatic Compounds in Organophilic Polymer Membranes Used in Pervaporation

The sorption isotherms of seven aromatic compounds and some mixtures of them in soft-segment elastomeric polyether-polyamide block-copolymer (PEBA) have been measured at 25 and 50°C using UV-VIS spectroscopic and gas chromatographic analysis. The solubility in PEBA depends significantly on the molecular size of the aromatic compound. Sorption coefficients have been determined. The temperature dependence of the sorption coefficient has been used to determine also the sorption enthalpy of the aromatic compounds from the aqueous solution to the polymer phase. Ternary mixtures consisting of two aromatics in dilute aqueous solution have been investigated in order to study possible synergistic solubility effects. The sorption coefficients can be used for the prediction of the permeation rate across the nonporous PEBA membrane in the so-called pervaporation process. An example is presented.     

Dual-mode sorption kinetics of gases in glassy polymers

The transport of gases in many glassy polymers can be described satisfactorily by means of a “dual-mode sorption” model. The transport behavior observed with a given gas/polymer system can be characterized by the model parameters, which are obtained from solubility measurements in conjunction with absorption/desorption or permeability measurements. The present study discusses the inverse problem, namely, the prediction of the absorption/desorption behavior of a gas in a glassy polymer from a specified set of dual-mode sorption parameters. Satisfactory agreement is obtained between reported absorption rates of sulfur dioxide in glassy polycarbonate and of water vapor in Kapton® ?

1 ?Trademark of E. I. du Pont de Nemours & Co.

and the rates predicted by the dual-mode sorption model. This study also confirms the consistency of the model.     

Misleading information on homogeneity and heterogeneity obtained from sorption isotherms

In this paper, the applications and the differences among the widely applied sorption isotherms (Langmuir isotherm for adsorption, competitive adsorption, ion exchange, Freundlich isotherm) are shown. The misleading information obtained by the formal applications of the isotherms is demonstrated using a simple mathematical model of homovalent ion exchange on homogeneous surface. This model calculation clearly reveals that when studying surface accumulation processes, the mechanism of the sorption process has to be determined. The correct thermodynamic interpretation of the data of sorption experiments is possible only if the isotherm is adapted to the sorption mechanism. It is emphasized that the regression values of the applied models themselves provide correct information neither on the surface heterogeneity nor the interactions among the sorbed species. The curved shape of an inadequate applied isotherm gives no information about the heterogeneity. In order to study the energy distribution of surface sites, the thermodynamic equations, including isotherms, has to be selected on the basis of sorption mechanism. All variable quantities involved in the given model (concentrations of the competing substances in all phases) have to be measured experimentally and included into the isotherm.     

Argon sorption and partial molar volume in poly(ethyl methacrylate) above and below the glass transition temperature

Sorption and dilation isotherms for argon in poly(ethyl methacrylate) (PEMA) are reported for pressures up to 50 atm over the temperature range 5–85°C. At temperatures below the glass transition (T_g=61°C), sorption isotherms are well described by the dual-mode sorption model; and isotherms above T_g follow Henry's law. However, isotherms for dilation due to sorption are linear in pressure at all temperatures over the range investigated. Partial molar volumes of Ar in PEMA are obtained from these isotherms. The volumes are approximately constant above T_g (about 40 cm³/mol), whereas the volumes below T_g are smaller and dependent on both temperature and concentration (19–26 cm³/mol). By analyzing the experimental data according to the dual-mode sorption and dilation model, the volume occupied by a dissolved Ar molecule and the mean size of microvoid in the glass are estimated to be 67 129 ų, respectively. The cohesive energy density of the polymer is also estimated as 61 cal/cm³ from the temperature dependence of the dual-mode parameters.     

Equilibrium sorption of heavy metals and phosphate from single- and binary-sorbate solutions on goethite

The amounts of Cu(II), Zn(II), and phosphate sorbed from single- and binary-sorbate systems on goethite (alpha-FeOOH) were measured. Experiments were carried out as a function of equilibrium pH (2-7), sorbate concentration (0.21-1.57 mM), and temperature (15-35 degrees C). The aqueous phase contained 0.1 M NaNO3 to maintain ionic strength constant. A convenient method was used to obtain sorption isotherms of single Cu(II), Zn(II), and phosphate at a fixed equilibrium pH, which could be well described by the Langmuir equation. Thermodynamic parameters for the sorption of single Cu(II) and phosphate including the free energies, isosteric enthalpies, and entropies were determined. In contrast to the single-sorbate systems, the sorption of metals was inhibited in the binary Cu(II)-Zn(II) system, whereas the sorption of both sorbates was enhanced in the binary Cu(II)-phosphate system under the conditions studied. The validity of the Langmuir competitive model for the prediction of the sorption isotherms in a binary Cu(II)-Zn(II) system was also discussed.     

Sorption of ammonia and nitrogen on cellulose acetate

Adsorption isotherms of ammonia and nitrogen on cellulose acetate in the temperature range 303–333 K are measured by reversed-phase chromatography. It is established that ammonia specifically interacts with the polymer matrix of the sorbent. The experimental data are described in terms of the Laatikainen-Lindstrom quasi-chemical model of vapor sorption. The enthalpies of sorption are determined, and their dependences on the ammonia and nitrogen concentrations in the sorbent are established.