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A new fitting model for sorption by glassy polymers is suggested based on the Flory–Huggins (FH) equation with a composite formula for the FH interaction parameter, χ, which is applicable if sorption experimental data shows a single-maximum variation of the FH parameter. Namely, a power-like and a linear approximation is assumed for χ ( φ 1 ) , as a function of solvent volume fraction φ 1 , before and after the point of its maximum. After determining the maximum point from a direct inspection of the sorption data, the three fitting parameters are evaluated by solving two independent least-square minimization problems. Several sorption studies of biopolymers taken from the literature show that the endset of the glass transition region is correlated with the position of the maximum of the FH interaction parameter. Based on this hypothesis and the Vrentas–Vrentas model for sorption of glassy polymers, a theoretical framework for the glass transition analysis is developed. In particular, the solvent-induced glass transition temperature variation can be estimated from the sorption isotherm as a function of the solvent content corresponding to temperatures above the temperature of sorption.

Understanding and predicting sorption thermodynamics of low molecular weight compounds in rubbery and glassy polymers is of great relevance to elucidate important phenomena in areas at the interface of various scientific branches, such as the colloid and interface science, membrane science, polymer foaming, tissue engineering, scaffolding, microcellular materials, aerogels, and for the implementation of technological applications. The development of thermodynamic models for polymer-based mixtures, applicable over a wide range of conditions, remains an active and fascinating research area. Recent advances in statistical thermodynamics and a better understanding of intra- and inter-molecular interactions, thanks to accurate experimental measurements and molecular simulations using realistic force fields, have contributed significantly to this end. In fact, sorption thermodynamics in polymers plays a relevant role in describing phase equilibria of polymer mixtures, (hydro)gel swelling, intramolecular association, hydrogen-bonding cooperativity and polymer degradation and stability, in assessing durability of polymers exposed to aggressive environments, in predicting penetrant induced crystallization and plasticization phenomena in polymers, in designing polymer-based separation processes, in tailoring polymer foaming processes, in improving gas and vapor barrier properties of polymer packaging, in modelling devolatilization of polymer solutions and migration phenomena of additives, in designing drug delivery systems, to mention a few. In the last decades, models have been introduced rooted on Equation of State theories, some of them based on compressible lattice frameworks. Notably, these models have been structured to specifically account for non-random distribution of molecular species and for dealing with several kinds of self-interactions that establish between polymer molecules and between penetrant molecules as well as cross-interactions that establish between moieties present on polymer backbone and penetrants. These models have been built to describe the behaviour of both rubbery polymers and out-of-equilibrium glassy polymers. Towards the further development of these approaches to gain an increased predictive capability of this thermodynamic description, recently have been also introduced approaches aimed at the estimation of relevant parameters based on molecular descriptors for calculations of properties of pure-components bulk phases and solutions. Such a quantitative description of the sorption process by use of advanced thermodynamic theories invariably relies on a molecular-level characterization of the system under scrutiny to validate and support the theoretical framework. Information is required on the molecular aggregates formed in the system, their structure, stoichiometry and, whenever possible, their population. In this respect, vibrational spectroscopy (FTIR, Raman) has demonstrated to be among the most powerful techniques, due to its sensitivity towards H-bonding detection and to its sampling flexibility, which allows the development of in-situ, time-resolved measurements. In the last ten years, significant advancements have occurred in terms of both experimental approaches and data analysis techniques, which considerably contributed to deepening the interpretation of the molecular interactions scenario. In particular, Two-dimensional correlation spectroscopy (2D-COS), Difference spectroscopy (DS) and first-principles quantum chemistry calculations have made a strong impact on the amount and quality of the acquired information. In view of the progress in this rapidly advancing and technologically relevant subject, this review article summarizes the state of the art on sorption thermodynamics modelling and on synergic combination with the wealth of information recently made available thanks to advanced vibrational spectroscopy techniques.

For the past few decades, researchers have been intrigued by glassy polymers, which have applications ranging from gas separations to corrosion protection to drug delivery systems. The techniques employed to examine the sorption and diffusion of small molecules in glassy polymers are the subject of this review. Diffusion models in glassy polymers are regulated by Fickian and non-Fickian diffusion, with non-Fickian diffusion being more prevalent. The characteristics of glassy polymers are determined by sorption isotherms, and different models have been proposed in the literature to explain sorption in glassy polymers over the last few years. This review also includes the applications of glassy polymers. Despite having many applications, current researchers still have difficulty in implementing coating challenges due to issues such as physical ageing, brittleness, etc., which are briefly discussed in the review.

In this paper, the sorption thermodynamics of low-molecular-weight penetrants in a glassy polyetherimide, endowed with specific interactions, is addressed by combining an experimental approach based on vibrational spectroscopy with thermodynamics modeling. This modeling approach is based on the extension of equilibrium theories to the out-of-equilibrium glassy state. Specific interactions are accounted for in the framework of a compressible lattice fluid theory. In particular, the sorption of carbon dioxide, water, and methanol is illustrated, exploiting the wealth of information gathered at a molecular level from Fourier-transform infrared (FTIR) spectroscopy to tailor thermodynamics modeling. The investigated penetrants display a different interacting characteristic with respect to the polymer substrate, which reflects itself in the sorption thermodynamics. For the specific case of water, the outcomes from molecular dynamics simulations are compared with the results of the present analysis.

The transport of CO2, CH4, N2 and He in films of polyimide 6FDA–6FpDA was studied at different temperatures, from 25 to 45 °C, and Relative Humidity (RH) from 0% to 90%. The permeability of all gases decreases with increasing RH, by factors as high as 44% for CH4 and 35% for the other gases; moreover, the relative decrease of gas permeability with respect to the dry polymer value is constant with temperature and just weakly dependent on the gas type. Pure water vapour sorption experiments were also conducted on the polymeric films in the same RH and temperature range, showing that water solubility in 6FDA–6FpDA is almost temperature-independent. A strong correlation therefore seems to exist between gas permeability decrease and water content in the membrane, due to the fact that absorbed water molecules occupy a portion of the polymer free volume, that is no longer available for gas permeation. Based on these considerations, the humid permeation results were analysed quantitatively and modelled with a simple approach based on free volume theory.

The influence of water vapor on the gas permeability of a commercial polyimide, Matrimid ® 5218, has been extensively investigated at three different temperatures (25, 35 and 45 °C), and with four different penetrant gases (CH 4 , N 2 , CO 2 and He), varying the relative humidity in the range 0–75%. In all tests performed, the permeability coefficient decreases as the concentration of water vapor in the membrane increases. In particular, the influence of the presence of water on gas permeability is very similar for all penetrants, as the same permeability decrease is found, at a given relative humidity, despite the different thermodynamic and kinetic characteristics of the probe gases considered. As temperature is raised, the gas permeability is enhanced, as expected. On the other hand, its decrease with respect to the dry polymer values, as relative humidity increases, is not affected by temperature, and it remains substantially unaltered from 25 to 45 °C, suggesting that such phenomenon can be directly related to the amount of water dissolved in the membrane, which is also unaffected by temperature. Based on the experimental evidence, a simple model is proposed to describe the permeation process under humid conditions, in the framework of the free volume theory. In particular, it has been considered that absorbed water molecules influence gas permeability by occupying polymer free volume, reducing its availability to other penetrants with lower condensability. The model describes accurately the experimental data using only two adjustable parameters for the polymer-water-penetrant system, once the water solubility is estimated from sorption measurements.

An extensive analysis is performed for the use of a model recently introduced for sorption induced volume dilation in glassy polymers, which can be combined with the nonequilibrium lattice fluid model (NELF) for the description of gas and vapor solubility in glassy polymers in a wide temperature range below the glass transition, from dry to fully plasticized conditions. The procedure extends the capability of previous versions of NELF model, as it refers to correlation and prediction of solubility of vapors and swelling agents. To this aim, the model counts on an additional out-of-equilibrium parameter for the polymeric species that interprets its pseudoequilibrium compressibility, and it can be retrieved from the analysis of pure component pressure–volume–temperature data below T g . The examples discussed in this work show that the overall procedure is highly reliable and addresses the key properties of the system for the direct interpretation of the effect of temperature, solute fugacity and state of glassy polymers on pseudoequilibrium volume swelling and solute content.

The aim of this work is to study in detail the sorption of binary gas mixtures containing methane into polymers suitable for membrane separations. A novel measuring procedure has been developed and validated by performing mixed gas sorption tests on the n -C 4 /CH 4 mixture in films of poly(1-trimethylsilyl-1-propyne) (PTMSP), for which literature data are available. Then, individual uptakes of CH 4 and CO 2 from binary gas mixtures were measured at 35 °C and up to 35 bar in PTMSP, in the whole gas composition range. The experiments were conducted on a pressure-decay apparatus equipped with a gas chromatographic device. The novel experimental procedure was set up in order to obtain data either at constant partial pressure of one gas, as done by previous authors, or at constant gas composition and variable total pressure. The latter protocol allows to mimic more closely the real membrane separation processes, where only the total pressure of the mixture can be varied arbitrarily. It was observed that the presence of CH 4 does not alter significantly the sorption of CO 2 and of n -C 4 in PTMSP, while the mixed gas solubility of CH 4 is lower than the pure gas value at the same CH 4 fugacity. In particular, the CH 4 solubility coefficient, as well as the mixed gas solubility selectivity are univocal functions of CO 2 fugacity (or concentration) and are otherwise independent of the gas phase composition. The real CO 2 /CH 4 solubility-selectivity of PTMSP is similar to the ideal value at low CO 2 fugacity but it becomes significantly higher, up to 4.5 times, at 25 bar of CO 2 fugacity. A quantitative rule can be drawn using data of several binary gas mixtures in glassy polymers, based on which the ratio between actual mixed gas and pure ideal solubility selectivity of CO 2 over CH 4 is a single, monotonously increasing function of the ratio between the concentration of the two components, c (CO 2 )/ c (CH 4 ), which becomes higher than unity as c (CO 2 )> c (CH 4 ). In other words, the competition effects depress more significantly the less abundant penetrant in the polymer, that is usually CH 4 .

The sorption of organic vapors of different chemical nature, molecular weight and polarity in glassy Matrimid 5218 polyimide films is measured and modeled in the entire activity range up to saturated vapor conditions. Experimental isotherms show several different peculiarities according to the nature of the solute component, consistent with the glassy character of the polymeric matrix. These features range from high vapor solubility coefficient at infinite dilution to swelling and plasticization of the polymer matrix depending on temperature and solute affinity. The various behaviors are then discussed by means of a thermodynamic model for properties of glassy polymeric phases. Indeed, the Non-Equilibrium Lattice Fluid (NELF) model is used to interpret the Vapor-Liquid-Equilibrium below Tg, while the corresponding equilibrium Lattice Fluid model was considered to the same purpose for the case of plasticization conditions in the system. This representation is able to describe consistently all different observed features and allows for a predictive procedure to reproduce solubility isotherms over a wide activity range.

The improvement of the separation performance of polymeric membranes for gas separations has been pursued by the addition of specific fillers, which produced diverse and even unexpected behaviors, hard to rationalize and to predict, both quantitatively and qualitatively. In particular, the addition of fumed silica nanoparticles in glassy polymeric membranes has shown unusual properties that apparently lead to the indication that only a specific experimental analysis could provide the information required for the permeability and separation of the gases of interest. We review transport properties, solubility and permeability in mixed matrix membranes obtained by loading fumed silica nanoparticles in different glassy polymers, interesting for membrane separations, and revise the main modeling procedure suitable to calculate and predict the relevant transport properties in such mixed matrix membranes.

The transport of gases in glassy polymeric membranes has been analyzed by means of a fundamental approach based on the nonequilibrium thermodynamic model for glassy polymers (NET-GP) that considers the penetrant chemical potential gradient as the actual driving force of the diffusional process. The diffusivity of a penetrant is thus described as the product of a purely kinetic quantity, the penetrant mobility, and a thermodynamic factor, accounting for the chemical potential dependence on its concentration in the polymer. The NET-GP approach, and the nonequilibrium lattice fluid (NELF) model in particular, describes the thermodynamic behavior of penetrant/polymer mixtures in the glassy state, at each pressure or composition. Moreover, the mobility is considered to follow a simple exponential dependence on penetrant concentration, as typically observed experimentally, using only two adjustable parameters, the infinite dilution penetrant mobility L10 and the plasticization factor β, both determined from the analysis of the dependence of steady state permeability on upstream pressure. The available literature data of diffusional time lag as a function of penetrant upstream pressure has been reviewed and compared with model predictions, obtained after the values of the two model parameters (L10 and β), have been conveniently determined from steady state permeability data. The model is shown to be able to describe very accurately the experimental time lag behaviors for all penetrant/polymer pairs inspected, including those presenting an increasing permeability with increasing upstream pressure. The model is thus more appropriate than the one based on Dual Mode Sorption, which usually provides an unsatisfactory description of time lag and required an ad hoc modification.

A model is introduced here for volume relaxation in glassy polymers, induced by sorption of low molecular weight species. The model relies on description of thermodynamic properties of glassy mixtures obtained through NET-GP model and it considers simple linear viscoelastic laws to interpret the stress–strain relation for the polymeric material. Results for the case of sorption kinetics in constant fugacity processes, obtained from the model, are compared with experimental values for vapour sorption in glassy polymeric microspheres. Satisfactory representations were obtained both for instant solute content measured in experimental run and for mass uptake registered in ‘relaxation stage’ of sorption processes. Viscosity data obtained from fitting of sorption experimental results are finally discussed in terms of temperature dependence and compared with values obtained form purely mechanical volume relaxation experiments.

Hybrid silica/poly(trimethylsilyl propyne) (PTMSP) membranes, in which the silica particles are generated in the polymer solution with a sol-gel reaction from a tetraethoxysilane (TEOS) precursor, were fabricated with silica loadings up to 30 wt%. The membranes are characterized with microscopic, thermogravimetric and pycnometric analysis; their sorption and diffusion properties with respect to n-C4 and n-C5 vapors are also measured at 25°C. A significant reduction of the polymer specific volume, organic vapour solubility and diffusivity is observed after incorporation of silica, according to a combination of free volume filling and compressive constraints. The behaviour of the hybrid membranes (HM) is opposite to that of mixed matrix membranes (MMM) obtained by physical mixing of preformed silica nanoparticles into PTMSP solutions, which show, on the contrary, free volume, vapour solubility and diffusivity values higher than pure PTMSP. The hybrid membranes are characterized by larger silica domains (around 1 m) than mixed matrix ones. Remarkably, however, the transport behaviour of both type of membranes can be interpreted, by the same theoretical background, on the basis of a single structural parameter, namely the density of the polymer phase. This parameter allows a direct estimation of the variation of free volume and vapour solubility and diffusivity in both type of membranes, through a NELF/free volume approach.

Giulio C. Sarti and Maria Grazia De AngelisDipartimento di Ingegneria Chimica, Mineraria e delle Tecnologie Ambientali (DICMA), Alma MaterStudiorum-Universita` di Bologna, via Terracini 28, 40131 Bologna, ItalyDOI 10.1002/aic.12571Published online March 2, 2011 in Wiley Online Library (wileyonlinelibrary.com).The solubility of liquid molecules in glassy polymers has been considered by using thegeneral results of the Non-Equilibrium Thermodynamics of Glassy Polymers (NET-GP),which proved successful to calculate solubility isotherms of gases in glassy polymers forrather different situations, including polymer blends, mixed gases, and mixed matrices. Itis shown that the existing model is suitable also for liquid penetrants in a glassy phase:water and ethanol sorption in polycarbonate (PC) and water sorption in polysulfone(PSf) have been examined as examples. The model’s ability to predict the solubilityfrom both liquid and vapor phases was tested successfully, using the same values of theparameters for both phases. In the case of PC, the model was also applied to calculatesuccessfully the solubility of liquid water at different temperatures from 25 to 130 C,with a single value of the energetic binary parameter.

The enhancement of gas and vapor transport rates induced by the addition of fumed silica nanoparticles to fluorinated glassy polymers is interpreted and quantitatively modeled considering the additional free volume created by incorporation of filler. That effect can be evaluated accurately from gas solubility data, using the NELF model. The solubility of CH4 and CO2 in matrices of Teflon AF1600 and AF2400, filled with variable amounts of fumed silica nanoparticles, was measured at 35 °C; the solubility of n-C4 and n-C5 vapors, as well as their diffusivity and the dilation induced in the same polymer matrices, was also measured at 25 °C. The fractional free volume (FFV) values, estimated on the basis of CH4 solubility data, were used to predict the solubility of the other penetrants inspected, with excellent agreement with experimental data. In addition, a single empirical correlation can be drawn, for both AF1600 and AF2400-based mixed matrices, between the infinite dilution diffusivity of vapors and the FFV value calculated from solubility data. Similarly, a simple correlation valid for both matrices is obtained as well for the dependence of diffusivity on penetrant concentration. Finally, use of the NELF model also allows an estimate of the swelling induced by the sorption process on the basis of virtually one simple data point of gas solubility.

Based on NonEquilibrium Thermodynamics (NET-GP) approach, a model is here presented and discussed, aimed to the prediction of anomalous characteristics in kinetics for differential sorption of swelling agents in amorphous glassy polymer films. For mentioned conditions, the “two-stage sorption” kinetics is typically observed, in which a first stage of Fickian character is followed by a second stage with kinetic order larger than 0.5 and characteristic time independent of film thickness. NET-GP model, which interprets the out of equilibrium conditions of polymer mixtures below Tg through the use of polymer mass density as order parameter, was successfully used since its introduction [Doghieri and Sarti, Macromolecules 1996, 29, 7885] to describe several non-equilibrium features in solubility isotherms for a number of glassy polymeric systems. By coupling the description of thermodynamic properties from NET-GP with simplified model for the bulk rheology of the system and generalized Fickian law for mass transport, a modeling tool is derived in this work which allows to satisfactorily describe results from experimental analysis of two stage sorption kinetics.

The solubility of C2H6 and C2F6 in two amorphous glassy fluoropolymers, Teflon® AF1600 and AF2400, has been modeled with the non equilibrium statistical associating fluid (NE-SAFT) and non equilibrium perturbed hard sphere chain (NE-PHSC) theories. Experimental swelling data during sorption are available to obtain a complete prediction of the solubility isotherms at 35°C. The solubility of C2F6 in both fluoropolymers is rather satisfactorily predicted by the models with no adjustable parameters, while that of C2H6 is overestimated, and a correction factor equal to about 0.90 must be applied on the binary energetic interactions estimated with the first approximation mixing rule. This result, combined with what previously obtained with the NE-lattice fluid (NE-LF) model, reveals inadequate representation of the hydrocarbon solubility in fluoropolymers by the geometric mean rule used to estimate the binary energetic interactions.

A new procedure is introduced for the calculation of solubility isotherms of plasticizing agents in glassy polymer matrices with particular application to the case of absorption of supercritical gases in bulk glassy polymer films. The model presented is an extension of the nonequilibrium thermodynamics for glassy polymers (NET-GP) approach, modified to allow for the calculation of the effects of pressure, temperature, and gas concentration on the glass transition. Mass sorption and one- dimensional swelling behavior are analyzed for the carbon dioxide (CO2)-poly(methyl methacrylate) (PMMA) system at high pressure. A quantitative comparison is presented between the model performance and experimental data measured using quartz crystal microbalance (QCM) and high-pressure ellipsometry (HPE).

A thermodynamic model is discussed for the representation of volume relaxation phenomenon induced by sorption of gases or vapours in glassy polymer matrixes. The model is based on results from NELF approach to the description of thermodynamic properties of glassy polymeric mixtures and it assumes a general linear viscoelastic law holds for the bulk rheology of the glassy polymer. Representation of volume relaxation properties of the mixture allows for the prediction of sorption kinetics in glassy polymer samples for those conditions in which diffusive resistance is negligible. Results from the use of the model are compared with experimental data for n-hexane sorption kinetics in glassy polystyrene microspheres, for relatively low vapour activity, at different temperatures. Satisfactory representation of experimental data is obtained by means of temperature dependent bulk viscosity which is consistent with results obtained form purely mechanical volume relaxation experiments.