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Since its inception, the non-equilibrium lattice fluid (NELF) model has become a vital tool in correlating and predicting the gas solubility behaviour in glassy polymeric membranes. But like its equilibrium variant, the NELF model is highly constrained by the availability of the pure polymer characteristic parameters, which are not always convenient to obtain as the need arises. In this study, we provide a proof-of-concept for building a machine learning-based group contribution method (ML-GC) for the Sanchez–Lacombe equation of state (EoS) pure polymer parameters. The ML-GC model was built using a modified version of the Marrero and Gani's method, which incorporates machine learning regression into the GC parameterisation process. The final model was capable of reproducing the parameters of a randomly selected test set of polymers, with a diverse range of chemical structures. The resultant average AARD % of the predicted densities in this set is 5.59%, with no polymer exceeding 15%. Moreover, to test the model's capabilities in estimating the parameters of high glass transition temperature polymers, we predicted a priori the characteristic parameters of 6 polyimides from the knowledge of their molecular structure. The ML-GC parameters were also incorporated into the NELF model to predict the infinite dilution solubility coefficients (S 0) of some of these polymers and the results were validated against experimental data. Furthermore, the ML-GC-NELF model was also used for the first time to represent effectively the gas solubility isotherms in PIM-PI-SBI and PIM-PI-EA with relatively small magnitudes of the binary interaction parameters (k i j). Despite the small data-set used herein, the model performance was satisfactory, however, as more data are being published in literature, the proposed ML-GC model has the potential of providing even more accurate predictions for a wider range of polymers, ultimately leading to lesser reliance on experimental data for modelling the gas sorption. [Display omitted] • A machine-learning group contribution (ML-GC) model for the Sanchez–Lacombe Equation of state (SL-EoS) was built. • The ML-GC model was capable of predicting the pure polymer parameters. • The ML-GC model reproduced the experimental pure polymer and solubility data satisfactorily. • The ML-GC-NELF model was used to calculate the gas solubility isotherms of PIMs. [ABSTRACT FROM AUTHOR]

The sorption of CO 2 , CH 4 and CO 2 /CH 4 mixtures in poly(trimethylsilyl propyne) (PTMSP) and PIM-1 (the first reported polymer of intrinsic microporosity) measured in previous works revealed that, in the multicomponent case, significant exclusion effects cause a decrease in solubility with respect to the pure-gas case, and this effect is more pronounced for the less soluble gas (CH 4). In this work we perform a similar experimental analysis on the tetrazole-modified PIM-1 (TZ-PIM), at three mixture compositions (10, 30, 50 mol.% CO 2) and three temperatures (25, 35, 50 °C), up to 35 bar. The obtained trends for solubility and solubility-selectivity of CO 2 , CH 4 , and their mixtures in TZ-PIM are found to be qualitatively and quantitatively comparable to those obtained for other high free volume polymers. Moreover, mixed-gas sorption data collected so far for three materials (TZ-PIM, PIM-1 and PTMSP) are modelled by means of the Non-Equilibrium Lattice Fluid (NELF) model and compared with experimental results. The model was found capable of representing the competitive effects in solubility and solubility-selectivity, in a wide range of temperature and pressure, with a better quantitative accuracy attained at higher temperatures. Finally, by coupling these results with mixed-gas permeation data, an estimate of the mixed-gas diffusivity was obtained, and it was revealed that, in multicomponent conditions, sorption provides a greater contribution to the selectivity than diffusivity, in agreement with experimental findings for similar materials. Image 1 • Mixed-gas CO 2 /CH 4 sorption in TZ-PIM displays strong competitive effects. • Competition enhances the solubility-selectivity in multicomponent conditions. • NELF model yields good predictions of mixed-gas sorption using only pure-gas data as input. • Diffusivity-selectivity in the multicomponent case estimated with the solution-diffusion model. • Solubility-selectivity is higher than diffusivity-selectivity in the mixed-gas case. [ABSTRACT FROM AUTHOR]

We studied in this work the properties of a new membrane (TR-PBO) obtained by solid state thermal rearrangement at 450 °C of a recently developed polyimide precursor, (mHAB-6FDA), which was synthesized by reaction of (3,3-diamino-4,4-dihydroxybiphenyl, mHAB) with 2,2-bis(3,4-dicarboxyphenyl) hexafluoropropane dianhydride (6FDA). The mHAB monomer is an isomer of the commercial 3,3′-dihydroxybenzidine (pHAB), used to form the more popular polyimide precursor pHAB-6FDA. TR-PBO membranes obtained from mHAB-6FDA showed excellent CO 2 permeability (720 Barrer) and good CO 2 /CH 4 ideal selectivity of 23. We found out that the thermal rearrangement enhances the solubility and diffusion coefficients of CO 2 at 10 bar by factors as high as 1.3 and 5, respectively. Larger enhancements, however, were observed in the case of CH 4 , causing the diffusivity selectivity to decrease by a factor of 2.6 and the solubility selectivity to decrease by a factor of ~1.5 upon rearrangement. The pure gas solubility was modeled with the Dual Sorption Mode model and the NELF model. The two models were then used to predict the mixed gas behavior in terms of solubility-selectivity, highlighting the effects of competition that are consistent with those observed in other glassy polymers. We also performed moisture sorption tests and gas permeability measurements in the presence of humidity. It was observed that the thermal rearrangement increases the membrane hydrophobicity and, consistently, the CO 2 and CH 4 permeability of mTR-PBO membranes is much more stable in the presence of humidity than that of the precursor polyimide membranes. Finally, the effect of aging on the membrane performance was analyzed. A 30% decrease in the CO 2 permeability of TR polymer membranes (around 50 µm thick) was observed after 6 months, while the selectivity increased by 20%. These results indicate that, even after 6 months, the performance of the TR polymer membrane was outstanding and close to Robeson's upper bound. [ABSTRACT FROM AUTHOR]

Sorption data of nitrogen, argon and krypton in different glassy polymers at different cryogenic temperatures are analyzed, including hydrogen in PIM-1; conventional and high free volume glassy polymers are included. A consistent interpretation of experimental data is obtained by considering penetrant dissolution in a uniform dense glassy polymer, undergoing volume swelling with all penetrants but hydrogen. All the data are properly described by using the NELF model which is appropriate for the solubility in glassy polymers. Remarkably, in each polymer the sorption isotherms of different penetrants are described well by using the same initial polymer density, and the same swelling coefficient value allows satisfactory description of the sorption behavior of one penetrant at different temperatures. For PIM-1, NELF model and molecular dynamics give exactly the same swelling, and desorption after sorption or subsequent sorption-desorption cycles show hysteresis effects clearly associated to irreversible volume changes; in addition, hydrogen sorption in PIM-1 is fully predicted by the NELF model, simply based on nitrogen sorption, as opposed to BET theory. On the bases of the above results the conclusion is drown that BET theory is not applicable to the cryogenic sorption isotherms in glassy polymers. [ABSTRACT FROM AUTHOR]

The solubility of various gases in several glassy polymers has been analyzed by the nonequilibrium thermodynamics for glassy polymers (NET-GP) framework, coupled with a lattice fluid equation of state model. Moreover, a simple rheological tool recently introduced for the a priori evaluation of the penetrant induced swelling is employed to predict the volumetric behavior of the solute/polymer mixture below T g . Such model accounts for an additional nonequilibrium parameter for the description of polymer phase that represents its pseudoequilibrium compressibility, and it can be retrieved from pure component pressure–volume–temperature data below T g . More in detail, the effect of polymer pretreatment and its prior history (e.g. annealing or conditioning by preswelling) on the resulting solubility behavior has been thoroughly analyzed and discussed, and the experimental data obtained for several penetrant/polymer couples have been well described by the present model. [ABSTRACT FROM AUTHOR]

The transport model proposed by Minelli and Sarti for the representation of gas and vapor permeability in glassy polymers has been extensively applied to various systems, and the model results are thoroughly analyzed. The approach is based on fundamental theory for the diffusion of low penetrant species in polymers, in which the diffusivity is considered as the product of the molecular mobility, and a thermodynamic coefficient, accounting for the concentration dependence of the chemical potential. The model relies on the thermodynamic description of the penetrant/polymer systems provided by the NonEquilibrium Thermodynamics for Glassy Polymers (NET-GP) approach. The penetrant mobility is assumed to depend exponentially on penetrant concentration, and the model contains two parameters only: mobility coefficient at infinite dilution and plasticization factor. The model parameters obtained from the analysis of the permeability behaviors of various systems have been examined and general correlations are derived. The mobility coefficient is indeed correlated to the properties of the pure penetrants (penetrant molecular size) and pure polymer (fractional free volume and characteristic energy). This allows the derivation of a simple and general expression for the prediction of the permeability of any penetrant species in glassy polymers in the range of low penetrant pressures, as well as the selectivity of any gas pair. Remarkably, the model predictions are able to represent quite accurately the experimental data available in the literature. Furthermore, the plasticization factor is correlated to the swelling produced by the penetrant into the glassy polymer matrix, obtaining thus a reliable tool for the estimation of the pressure dependence of gas permeability on upstream pressure. [ABSTRACT FROM AUTHOR]

The permeability of various low molecular weight species (both gases and vapors) in a series of glassy polymers has been extensively analyzed by means of a thermodynamic based approach for solubility and diffusivity, recently proposed and already applied to a few penetrant/polymer systems. The model relies on the thermodynamic description of the solubility behaviors of the solutes provided by the nonequilibrium thermodynamic model for glassy polymers (NET-GP), while the diffusivity is the product of the mobility coefficient, a purely kinetic quantity, and the thermodynamic factor, accounting for the dependence of the penetrant chemical potential on its concentration in the glassy polymer matrix. The model is applied to permeability data of many penetrant species from very light gases, such as hydrogen or helium, to hydrocarbons and fluorocarbons, in several different glasses, including very high free volume materials, polyimides and fluoropolymers. The model proved to be effective in the representation of all types of permeability behaviors with respect to penetrant upstream pressure, which may be either decreasing, increasing, or with a nonmonotonous trend showing a minimum value at the so-called plasticization pressure. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Molecular dynamics (MD) and grand canonical Monte Carlo (GCMC) simulations with Teflon AF2400 and Teflon AF1600 predicted diffusivities and solubilities of He and N 2 that corresponded reasonably with available experimental data. The results of the MD simulations further showed that NF 3 had a higher diffusivity than CF 4 in both polymers and that the NF 3 /CF 4 diffusion selectivity was 67% higher in Teflon AF1600 compared to Teflon AF200. However, because the GCMC simulations failed to offer a conclusive answer to the role of the solubility selectivity w.r.t. NF 3 and CF 4 , the non-equilibrium lattice fluid (NELF) model was used to further evaluate this aspect. Consequently, the combined NF 3 /CF 4 diffusion and solubility selectivity values predicted permeability selectivities that corresponded well with the previously reported experimental data, which confirmed that a lower FFV resulted in an increased NF 3 /CF 4 solubility and diffusion selectivity. [ABSTRACT FROM AUTHOR]

The solubility of several gases and vapors in glassy ethyl cellulose (EC) over wide ranges of temperature and pressure is described by means of a thermodynamic model, which specifically accounts for the nonequilibrium volumetric properties of EC. The behaviors of polymer/penetrant mixtures are represented by the well-established nonequilibrium lattice fluid model (NELF) in its latest development, suitable to be used in a pure predictive fashion for the representation of the volume swelling of the polymer matrix, as induced by penetrant sorption. The analysis reveals the ability of the model to describe all the experimental trends, and a sole binary parameter for each polymer/penetrant couple is considered. The NELF model allows the satisfactory prediction of the thermodynamic properties of all polymer/solute systems examined, in the entire range considered for temperature and vapor fugacity. [ABSTRACT FROM AUTHOR]

Abstract: The description of permeability and solubility of CO2 in poly(ethylene terephthalate) both in the glassy and in the rubbery state is analyzed by means of thermodynamic based models for solubility and diffusivity. The Non-Equilibrium Thermodynamic approach for Glassy Polymers (NET-GP), together with the Sanchez Lacombe equation of state, provides the description of the solubility isotherms above and below the glass transition temperature (NELF model). The penetrant diffusion coefficient is then taken as the product of a kinetic factor, the mobility, and a thermodynamic factor associated to the concentration dependence of the chemical potential of the diffusing species. The thermodynamic factor is predicted by the NELF model, while the mobility factor is considered to depend exponentially from penetrant concentration, following the usual trend commonly found experimentally, and its expression contains only two adjustable parameters. The model analysis shows that, in all the cases examined, the model used is able to describe carefully the experimental trends in a simple and effective way, accounting for all the different behaviors observed, for both solubility and permeability. [Copyright &y& Elsevier]

Abstract: The description of permeability and solubility of CO2 in different complex polymer matrices in the glassy state is analyzed by considering the diffusion coefficient as the product of a kinetic factor, mobility, and a thermodynamic factor associated to the concentration dependence of the chemical potential of the diffusing species, according to what recently presented for different pure polymers [Minelli and Sarti, J. Membr. Sci. 435 (2013) 176–185]. The thermodynamic factor is calculated in a predictive way by using the nonequilibrium lattice fluid model (NELF) or is obtained directly from experimental solubility isotherms, when pure component parameters for the NELF model are not available. The mobility factor is considered to depend exponentially from penetrant concentration, following the usual trend commonly found experimentally, and its expression contains only two adjustable parameters. The permeability model is used to describe steady state permeation of CO2 in a series of complex glassy phases, formed by polysulfone (PSf) and polyphenylene oxide (PPO) with different plasticizers, glassy polymer blends, glassy random copolymers and crosslinked polyimides. The analysis shows that in all the cases examined, the model used is able to describe the experimental trends in a simple and effective way, accounting for all the different behaviors observed, in which permeability is either decreasing or increasing with upstream pressure and even when permeability is non-monotonous and presents a minimum value due to the so-called plasticization effect. A general correlation is also found for both model parameters: the infinite dilution mobility correlates well with the reciprocal fractional free volume, according to the FFV theory, while the plasticization factor is associated to the swelling coefficient of the polymer matrix. [Copyright &y& Elsevier]

Abstract: The description of permeability and diffusivity in glassy polymers has been revisited by considering the diffusion coefficient as the product of a kinetic factor, mobility, and a thermodynamic factor associated to the concentration dependence of the chemical potential of the diffusing species. The latter term does not contain any adjustable parameter since it is obtained directly from solubility isotherm data or can be even calculated in a predictive way by using well established predictive procedures as the Non Equilibrium Thermodynamics for Glassy Polymers (NET-GP) models. The mobility factor considered is endowed with an exponential dependence on penetrant concentration, following the usual trend commonly found experimentally, thus containing only two adjustable parameters. The resulting expressions for diffusivity and for permeability describe rather carefully the pressure dependence observed in glassy polymers, both in steady state permeation and in transient mass uptake, in all the cases inspected, even in the presence of the so called plasticization effects. [Copyright &y& Elsevier]

Abstract: Sorption and swelling of alkane and alcohol vapors in addition-type poly(trimethyl silyl norbornene) (PTMSN) obtained in part I of this work were analyzed and compared to the predictions of the Non-Equilibrium Lattice Fluid (NELF) model. The polymer characteristic Lattice Fluid (LF) parameters were determined by best fitting the infinite dilution solubility coefficients, S 0, of a wide series of n-alkane penetrants, from ethane to n-dodecane. The solubility of alkanes, which is comparable quantitatively and qualitatively to that measured in PTMSP, is well represented by the model. The solubility isotherms of alcohols (methanol, ethanol and 1-propanol), which exhibit a peculiar sigmoidal behavior that finds no explanation within the dual mode model, are also in line with the NELF model prediction without any ad hoc correction. The model is also used to represent successfully the dependence of the solubility on the alkane size, in PTMSN as well as in PTMSP and Teflon AF2400. The three polymers exhibit different trends: in PTMSP the vapor-pressure normalized solubility coefficient (S 0 p vap ) increases with the penetrant size, while in Teflon AF2400 the behavior is opposite and in PTMSN the size plays a very little role on the activity-based solubility. Such a behavior, that is useful to determine the selective performance of the polymer, was analyzed and discussed taking advantage of the support offered by the NELF model: it was seen that the three main factors affecting the extent of solubility dependence on the alkane size are the polymer fractional free volume and cohesive energy density, as well as its energetic interaction with the penetrant; an approximate formula is used to estimate a priori the behavior of the solubility based on the knowledge of the polymer properties. [Copyright &y& Elsevier]

[Display omitted] • Ternary mixed-gas sorption (C 2 H 6 /CO 2 /CH 4) was measured using a pressure decay apparatus. • Ethane has a strong exclusion power over CH 4 and CO 2 limiting their solubility in the mixed-gas case. • Binary and ternary sorption tests were systematically compared to analyze 3rd component effects. • The NELF model predicts binary and ternary mixed-gas sorption using only pure-gas data as input. • Leveraging high mixed-gas solubility-selectivity strongly enhances separation performance. In this work, a new experimental protocol was designed to measure sorption of ternary gas mixtures in polymer membranes for the first time. We measured the sorption isotherms of binary and ternary mixtures of C 2 H 6 , CO 2 , and CH 4 in PIM-1, using a pressure decay apparatus, working at constant composition and variable pressure of the gas phase. The results show competitive sorption effects in all gas mixtures analyzed, affecting each species in proportion to the total sorbed concentration of the other components. It was found that the presence of C 2 H 6 in the mixture reduces the sorption of both CH 4 and CO 2 to a similar extent, so that the CO 2 /CH 4 solubility-selectivity is not markedly affected. The experimental data were used to validate predictions of the Non-Equilibrium Lattice Fluid (NELF) model of binary and ternary sorption isotherms. The NELF model is generally in close agreement, both in the prediction of pressure and mixture composition effects, proving to be a reliable tool to assess mixed-gas sorption, even in complex scenarios. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

This study presents an extended thermodynamic and phenomenological combined model to mitigate the environmental hazardous acid gas over composite membranes. The model has been applied to an acid gas such as carbon dioxide (CO 2) for its permeation through polyetherimide incorporated montmorillonite (Mt) nanoparticles hollow fiber asymmetric composite membranes. The well-established non-equilibrium lattice fluid (NELF) model for penetrating low molecular weight penetrant in a glassy polyetherimide (PEI) was extended to incorporate the other important polymer/filler system features such as tortuosity in acid gas diffusion pathways resulted from layered filler aspect ratio and concentration. The model mentioned above predicts the behavior of acid gas in PEI-Mt composite membranes based on thermodynamic characteristics of CO 2 and PEI and tortuosity due to Mt. The calculated results are compared to experimentally determined values of CO 2 permeability through PEI-Mt composite asymmetric hollow fiber membranes at varying transmembrane pressures and Mt concentrations. A reasonable agreement was found between the model predicted behavior and experimentally determined data in terms of CO 2 solubility, Mt concentration and aspect ratio were calculated based on average absolute relative error (%AARE). The proposed modified model efficiently predicts the CO 2 permeance across MMMs up to 3 wt% Mt loadings and 6 bar pressure with ± 10%AARE. [Display omitted] • Thermodynamic and phenomenological models over membrane have been developed. • The developed model predicts acid gas behavior in the composite membranes. • Predicted and experimentally obtained acid gas permeability are compared over the membranes. • AARE% was lower than 5% over the composite membranes. • Predicted and experimentally determined data are found to be comparable. [ABSTRACT FROM AUTHOR]

This work aims at obtaining a systematic description and prediction of the multicomponent gas separation behavior of membranes, using the Non-Equilibrium Lattice Fluid (NELF) model for gas sorption and the Standard Transport (ST) model for gas permeation. The scheme is applied to a comprehensive analysis of CO 2 /CH 4 separation with cellulose acetate membranes. A dedicated experimental campaign (pressure-volume-temperature (P V T), differential scanning calorimetry (DSC) and pure gas sorption tests) was performed to obtain reliable model parameters, accounting also for crystallinity. The approach was validated against a complete set of literature data, including mixed gas sorption and permeation. The parameters obtained were used to perform predictive simulations of mixed CO 2 /CH 4 sorption and permeation in a wide mixture composition range. The NELF model accurately predicts the effect of temperature on sorption, as well as the strong competitive exclusion of CH 4 when CO 2 is present, that enhances the solubility-selectivity. The ST model correctly estimates the experimentally observed lower-than-ideal CO 2 /CH 4 perm-selectivity. The model shows that low perm-selectivity is due to mixed gas diffusivity-selectivity (which is not available experimentally): even though solubility-selectivity increases in the mixed gas case, diffusivity-selectivity suffers a larger departure from pure gas conditions, ultimately leading to a lower perm-selectivity under multicomponent conditions. [Display omitted] • A framework to model gas transport through semicrystalline glasses was presented. • Sanchez-Lacombe equation of state parameters were obtained for cellulose acetates. • The models gave good predictions of CO 2 /CH 4 mixed-gas solubility and permeability. • A comprehensive analysis of mixed-gas sorption and diffusion effects was carried out. [ABSTRACT FROM AUTHOR]

CO 2 and CH 4 mixed-gas solubility was measured in 3,3′-dihydroxy-4,4′-diamino-biphenyl (HAB) 2,2′-bis-(3,4-dicarboxyphenyl) hexafluoropropane dianhydride (6FDA) polyimide and in its thermally rearranged (TR) derivative, TR450. Due to competitive sorption effects, the solubility of both species in mixed-gas conditions is lower than the corresponding pure-gas solubility at the same fugacity. CH 4 sorption is significantly affected by the presence of the second gas, while CO 2 behavior is hardly altered. Therefore, the multicomponent solubility-selectivity is higher than the ideal value calculated from pure-gas sorption data, and this has a positive impact on CO 2 /CH 4 separation properties. The multicomponent solubility data were modelled with the Non-Equilibrium Lattice Fluid (NELF) model, using only pure component parameters and binary interaction parameters obtained from pure-gas sorption data available in the literature, with no parameters determined from the mixed-gas sorption data. Although it is easier to use, the multicomponent Dual Mode Sorption (DMS) model yielded less accurate predictions of mixed-gas sorption. Mixed-gas sorption experiments and modelling, coupled with mixed-gas permeation data, enabled a better fundamental understanding of the separation properties of these materials. Unlike the case of pure-gas experiments, where diffusivity contributes more to the overall ideal selectivity, competitive sorption is the main effect governing the permselectivity of these membrane materials at multicomponent conditions. A systematic comparison with literature data on mixed-gas CO 2 /CH 4 sorption and permeability revealed these to be generalized trends, obeyed by materials of very different chemical constitution. This finding can inform on the criteria that make polymers performing better in multicomponent scenarios, with potential impact on the design and synthesis strategies of new materials. Image 1 • Competitive sorption enhances CO 2 /CH 4 solubility-selectivity in HAB-6FDA and TR450. • The NELF model predicts multicomponent sorption using only pure-gas data as input. • Solution-diffusion analysis was performed to extract mixed-gas diffusivities. • The multicomponent permselectivity of glassy polymers is driven by solubility. [ABSTRACT FROM AUTHOR]