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The prediction of the solubility of gasses in semi-crystalline polymers is still a challenging task due to the difficulty in providing a comprehensive description of the morphological and mechanical perturbation felt by the amorphous phase intercalated with the impermeable crystal domains. Among the different modeling techniques, a frequently adopted strategy models the reduced solubility experienced by the confined amorphous phase via an additional pressure to the external gas pressure acting on the latter, the so-called constraint pressure ‘pc’. The work presented here is dedicated to a newly developed multi-scale modeling strategy, belonging to the aforementioned category, that innovatively couples Molecular Dynamics simulations with Lattice Fluid theory. The model was applied to carbon dioxide, ethylene, and propane solubility isotherms in High-Density Polyethylene, and validated against experimental literature data, confirming its ability to model the solubility in semi-crystalline polymers. In addition, it showed good accordance with a fully macroscopic model already present in the literature. The successful multi-scale coupling presented here paves the way for the development of a fully predictive modeling strategy.

The sorption and transport of CO2 in two polymers, Matrimid and PDMS, were modelled using data available across the critical region, at various temperatures and up to 18 MPa. The experimental trends show a complex behavior that is affected by the transition from gas-like to liquid-like density of CO2, as well as by the sorption induced glass transition of the polymer. The Non Equilibrium Thermodynamics (NET-GP) approach for the solubility, coupled to its complementary tool for the permeability, the Standard Transport Model (STM), allows to represent thoroughly the complexity of CO2 sorption and permeation in this operative range with a selfconsistent set of parameters. Furthermore, the model offers a deep insight in the swelling induced by CO2 in the different states of the polymers, and allows to decouple the kinetic and thermodynamic contributions to the transport phenomena in a meaningful way. This work takes a step forward in the understanding and simulation of the complex interactions between high pressure, supercritical CO2 and industrially relevant polymeric materials.

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 C2H6, CO2, and CH4 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 C2H6 in the mixture reduces the sorption of both CH4 and CO2 to a similar extent, so that the CO2/CH4 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.

The gas solubility in polymeric membranes affects the separation performance, particularly in the case of CO2 capture processes. Solubility and solubility-selectivity in membranes of multicomponent mixtures can deviate rather markedly from the corresponding pure gas values, due to swelling and competition phenomena, and require dedicated time-consuming measurements. Many experiments can be avoided by using a suitable thermodynamic tool, such as an Equation of State (EoS) model, to represent the gases sorption in the membrane. Such models require, for parameterization, knowledge of the polymer behavior above the glass transition Tg, which is a limit for membrane modeling, because the most attractive polymers for gas separation are rigid matrices characterized by very high Tg values, difficult to reach experimentally. In this work, we study the sorption of CO2/CH4 mixtures in a high-Tg polyimide membrane (Matrimid®) using a bottom-up approach. Pressure-volume-temperature data for Matrimid® above Tg are generated using NPT Molecular Dynamics simulations: the results are regressed to find Matrimid® parameters for the PC-SAFT Equation of State. Finally, the Non Equilibrium PC-SAFT macroscopic model (NE-PC-SAFT) is used to calculate CO2 and CH4 solubility and solubility-selectivity as a function of gas mixture pressure, composition and temperature. The approach is tested successfully over many experimental pure gas and vapor sorption data in Matrimid®. Mixed gas calculations predict a marked competition, which affects more methane than CO2 sorption, and results in a higher-than-ideal value of solubility-selectivity. Combined with the fact that experimental mixed gas permeability-selectivity is lower than the ideal value, such results indicate that the diffusivity of CH4 in Matrimid is significantly enhanced in presence of CO2, causing a decrease of diffusivity-selectivity. The gas solubility in polymeric membranes affects the separation performance, particularly in the case of CO2 capture processes. Solubility and solubility-selectivity in membranes of multicomponent mixtures can deviate rather markedly from the corresponding pure gas values, due to swelling and competition phenomena, and require dedicated time-consuming measurements. Many experiments can be avoided by using a suitable thermodynamic tool, such as an Equation of State (EoS) model, to represent the gases sorption in the membrane. Such models require, for parameterization, knowledge of the polymer behavior above the glass transition Tg which is a limit for membrane modeling, because the most attractive polymers for gas separation are rigid matrices characterized by very high Tg values, difficult to reach experimentally. In this work, we study the sorption of CO2/CH4 mixtures in a high-Tg polyimide membrane (Matrimid®) using a bottom-up approach. Pressure-volume-temperature data for Matrimid® above Tg are generated using NPT Molecular Dynamics simulations: the results are regressed to find Matrimid® parameters for the PC-SAFT Equation of State. Finally, the Non Equilibrium PC-SAFT macroscopic model (NE-PC-SAFT) is used to calculate CO2 and CH4 solubility and solubility-selectivity as a function of gas mixture pressure, composition and temperature. The approach is tested successfully over many experimental pure gas and vapor sorption data in Matrimid®. Mixed gas calculations predict a marked competition, which affects more methane than CO2 sorption, and results in a higher-than-ideal value of solubility-selectivity. Combined with the fact that experimental mixed gas permeability-selectivity is lower than the ideal value, such results indicate that the diffusivity of CH4 in Matrimid® is significantly enhanced in presence of CO2, causing a decrease of diffusivity-selectivity.

We have explored the effect of temperature, in the range between 25 and 50 °C, on the mixed gas CO2/CH4 sorption in a polymer of intrinsic microporosity (PIM-1), that has a solubility-selectivity higher than diffusion-selectivity for this mixture. The new data obtained in this work at 25 and 50 °c were combined with previously published data obtained at 35 °C to determine the temperature-dependence of mixed gas solubility and solubility-selectivity, a type of information that has not been obtained before in the literature. The data were collected at different total pressures and gas mixture compositions using a pressure decay − gas chromatographic device. The data collected indicate that the sorption of such mixture in PIM-1 is dominated by competition, whose major effect is to reduce the solubility of gases with respect to the pure gas value at same fugacity. The competitive phenomena follow a generalized trend that is not dependent on total gas pressure, composition and gas type but is only a function of the second gas concentration in the polymer and temperature. In particular the competition, expressed as reduction of gas solubility with respect to pure gas value, decreases with the concentration of the second gas, and increases with increasing temperature. Such effects are however generally favourable to separation, with positive deviations of the CO2/CH4 solubility-selectivity from ideal values calculated from pure gas solubility, by factors as high as 4. As a rule of thumb, observed at all temperatures and also in other glassy polymers, the real solubility-selectivity deviates positively from ideal value calculated from pure gas behaviour if the molar content of CO2 in the membrane is higher than that of methane, which is usually the case, unless low CO2 gas fractions are considered. A new type of generalized plot, reporting departures of multicomponent properties from the corresponding pure gas values, has been traced for this system and indicates that solubility selectivity departure is univocally correlated to CH4 solubility departure, independent of the operative conditions (pressure, composition, temperature) explored during the experiments.

In this work, we studied CO 2 sorption and transport in two series of aliphatic-aromatic copolyimide membranes, as a function of chemical formulation, pressure and temperature. Such materials are formed by two distinct phases with rather different properties: the first one is formed of rubbery polyether segments (poly(propylene oxide) (PPO) or poly(ethylene oxide) (PEO)), characterized by favorable energetic interactions with CO 2 and high flexibility, which endows the copolymers with high CO 2 permeability, suitable for capture processes. The second phase is formed by hard glassy polyimide blocks, randomly distributed at the microscopic level, which provide the necessary thermal, chemical and mechanical stability. Previous studies indicate that CO 2 permeability increases with increasing the amount of polyether phase in the copolymers; in the present work we investigate more deeply the interactions and synergies occurring between the two phases, by focusing separately on the CO 2 solubility and diffusivity terms that contribute to permeability. In particular, by studying the shape of the solubility isotherm, as well as the values of diffusivity, sorption enthalpy and activation energy, we were able to monitor the transition from a glassy-like to a rubbery-like behavior as the fraction of rubbery component in the copolymer increases. The data indicate that polyether enhances CO 2 permeability by acting mostly on diffusivity, while the solubility contribution is less affected on a quantitative basis. However, the qualitative behavior of solubility allows understanding the nature of interactions between the two phases. In particular, by using a simple additive approach to estimate the CO 2 solubility of the copolymer, and the Non-Equilibrium Lattice Fluid (NELF) to evaluate the CO 2 solubility in the pure homopolymers, one concludes that the copolymers sorption behavior is “ideal”, i.e. purely additive, indicating a good combination of the two phases. The copolymer volume, on the other hand, shows a contraction upon combination of the two phases. The NELF modeling of solubility data allows attributing such a contraction only to the glassy phase, whose excess free volume is reduced in the presence of the rubbery portion in the copolymer, which possibly partly occupies such excess volume, indicating a strong interpenetration of the two phases.

Semicrystalline polymers are employed in a variety of applications in view of their good stability, mechanical and barrier properties. In spite of their widespread use, the fundamental knowledge about their fluid transport and sorption behavior is somehow poor and limited to approximate models. As far as their thermodynamic properties of their mixtures with gases and vapors are concerned, it is generally accepted that semicrystalline polymers absorb lower amounts of fluid than the corresponding, wholly amorphous polymers. This behavior is due to the fact that the crystalline domains are practically impermeable to fluids and can hardly accommodate any host molecule, and furthermore, to a reduced sorptive capacity of the amorphous phase with respect to the pure amorphous polymer. The equation of state models, which accurately represent the experimental solubility of fluids in molten polymers, often overestimate the same property below Tm, even accounting for the negligible penetrant uptake in the crystalline phase. In this work we attribute the significant overestimation of solubility to the constraining effect that the rigid crystallites impose on the amorphous phase [1,2]. The present approach follows a similar one applied by Memari et al. to the same case, which made us of a Montecarlo technique to calculate the polymer amorphous phase solubility [3]. The constraining effect is accounted by considering that a constraint pressure, pc, acts on the amorphous solid phase in addition to the one prevailing in the fluid phase, p [4]. In particular we use the Sanchez Lacombe Equation of State (SL EoS) [5], as it describes accurately the behavior of amorphous polymer phases. The binary parameter for the fluid-polymer energetic interactions, kij, and the constraint pressure pc are adjusted on the experimental solubility data above and below the polymer melting point Tm, respectively. The approach is applied to the description of the solubility of different solutes in a variety of polymers above and below their melting points. In spite of its simplicity, the model is effective and accurate in representing the thermodynamic behavior of several fluid/polymer systems: n-C4, i-C4, N2 and CO2 in LDPE, CO2 in HDPE, 1-hexene in LLDPE, CO2 in i-PP, CO2 and C3H8 in PEO in wide ranges of temperature. In all those cases, the binary energetic parameter of the model, kij, is adjusted on the solubility data above Tm, where pc is zero by default, as no crystallites are present in the polymer phase, and then extrapolated below the melting point, where the only adjusted parameter is pc. Interestingly, the values of pc do not depend on the penetrant type but only on the polymer type, degree of crystallinity, and on the temperature, as it is consistent with the physical meaning of such parameter. In particular, the values of pc for polyolefins increase exponentially with the crystalline mass fraction, up to about 80 MPa at a crystalline fraction of 0.73, and decrease exponentially with increasing temperature, vanishing across the melting point.

Reactive absorption of CO2 and H2S in aqueous methyldiethanolamine (MDEA) solutions is considered within the ePC-SAFT equation of state. We demonstrate that ePC-SAFT can be employed in a predictive manner without regression of additional temperature-correlated terms. Mixed system predictions are tested using a consistent set experimental data covering a wide range of temperatures (313 K–413 K), partial pressures (0.001 kPa–1000 kPa), and MDEA mass fractions (0.05–wMDEA 0.75 wMDEA). Predicted partial pressures for acid gas absorption show good agreement for low MDEA fractions (wMDEA < 0.5). Absorption selectivity in binary H2S + CO2 absorption is correctly predicted, with absolute average deviations of 57.18% and 79.32% for partial pressures of CO2 and H2S. We identify a significant deterioration in ePC-SAFT predictive power for the high-MDEA regime (wMDEA > 0.5), likely originating from underlying assumptions in the Debye-Hückel electrolyte free energy treatment and representation of ionic species.

The solubility and diffusivity of several liquid species in crosslinked polydimethylsiloxane (PDMS) films were determined with a gravimetric method at 35 °C. The series of liquids considered includes alkanes (from n-C 5 to n-C 18 ; cyclo-C 6 ), water, acetone, alcohols (ethanol, 1-propanol, 1-butanol, iso-butanol, tert-butanol, 1-pentanol, 1-hexanol), terpenes (squalene, limonene, linalool, geraniol) and edible oils with different oleic acid contents. The effect of size, structure and solubility parameter of the different molecules on the sorption and transport properties in PDMS was discussed. The diffusivity values of the different penetrants inspected span over one decade while the molar solubility spans over four decades, indicating that the membrane selective behavior is a strong function of liquid solubility. In particular, the solubility of hydrocarbons (alkanes, hydrocarbon terpenes) that have favorable energetic interactions with the polymer is entropy-driven, i.e. decreases with molecule size. The solubility of substances bearing one alcoholic group is also strongly affected by the energetic interactions with the polymer, especially if their alkyl chain length is small. The trends are due to the dual entropic/enthalpic nature of solubility, which obeys a nice exponential decreasing trend when reported versus the product M W 0.75 ( δ − δ PDMS ) 2 which accounts for both entropic and energetic nonideality of the polymer–penetrant mixture. The diffusivity decreases roughly monotonically with the molecular size of the penetrant within a series of homologous penetrants, unless the penetrants have very high solubility differences and induce different swelling in the polymer matrix. Solubility data are well represented by the Flory−Rehner model, that also allows to explain differences between experimental sources based on the crosslinking degree.

The representation of liquid–liquid equilibria (LLE) in ternary systems composed by water, 1,4-dioxane and different grades of poly(lactic acid) (PDLLA and PLLA), has been addressed through the PC-SAFT equation of state (EoS), in which the scheme of induced association is used to represent the interaction between solvent (dioxane) and non-solvent (water). The model parameters devoted to the description of pure component properties, as well as those pertinent to the representation of thermodynamic behaviour of solvent/non-solvent mixtures, were tuned on the basis of specific pressure–volume–temperature (PVT) data for the corresponding systems. Only the binary parameters for polymer–solvent and polymer/non-solvent pairs were adjusted to obtain a useful representation of experimental LLE data for the ternary systems. A suitable description of the thermodynamic properties of ternary mixtures was obtained using temperature-independent binary interaction parameters in the range 25–80 °C, and the consistency of the approach in the entire composition range was verified against experimental solubility data specifically measured for the polymer/non-solvent pair. The model shows good ability in the description of the thermodynamic properties of the system and it represents a reliable tool for the prediction of LLE also at conditions different from those considered for its set-up. This approach thus represents a useful designing tool for processes, such as thermally induced phase separation (TIPS), used in the preparation of microporous polymeric scaffolds.

The individual solubility of CH 4 and CO 2 from binary gas mixtures was measured at 35 °C and up to 35 bar in a polymer of intrinsic microporosity (PIM-1), at different compositions of the gas phase (from 0 to 50 mol% of CO 2 ). The experiments were conducted on a pressure-decay apparatus equipped with a gas chromatograph, allowing a highly flexible measuring procedure. The gas solubility was plotted versus gas phase composition, total pressure, gas fugacity and second gas concentration. The mixed gas solubility of both species, CH 4 and CO 2 , is lower than the pure gas value at the same fugacity, but the reduction of methane solubility due to the presence of CO 2 is generally more significant. Such behavior is due to the fact that CO 2 has normally higher solubility than methane: indeed the depression of the solubility coefficient with respect to the pure gas value is similar for both gases, when reported at the same concentration of the second gas. The real, mixed gas solubility selectivity is in general higher than the ideal value calculated from pure gas behavior. The ratio between real and ideal solubility selectivity increases with CO 2 concentration in the membrane, according to a single mastercurve, reaching a maximum value of 4, and it also increases with the ratio between CO 2 and CH 4 concentration in the membrane. In particular, as in the case of other glassy polymers, the real solubility selectivity of CO 2 over CH 4 is higher than the ideal value if c (CO 2 )> c (CH 4 ), and it is lower than the ideal value if the opposite condition holds true. Such behavior occurs because the competition for sorption is normally less effective on the more abundant penetrant in the polymer. A selectivity–solubility performance plot can be drawn for this system.

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 .

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, S0, 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 (S0 pvap) 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.

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 sorption and transport properties of n -alkane and alkyl-alcohol vapors in addition-type poly(trimethyl silyl norbornene) (PTMSN) were studied at 35 °C and at various activity values. PTMSN is a high free volume polymer that shows very high solubility to the condensable vapors and can be considered a promising material for several applications, such as gas/vapor membrane separations. Sorption and diffusion experiments with n -alkanes ( n -C 4 , n -C 5 , n -C 6 ) and alkyl alcohols (CH 3 OH, C 2 H 5 OH, n -C 3 H 7 OH) were performed as well as dilation measurements with n -C 5 and n -C 6 , at 35 °C. The behavior of the alcohol vapors differs from that of alkanes, mainly for the shape of the solubility, diffusivity and mobility isotherms: this phenomenon, as well as the diffusion trend of the different vapors, was explained by the different interactions between the penetrants and the hydrocarbon-based polymer. The permeability isotherms were also estimated from solubility and diffusivity data, showing that the permeability of PTMSN lies in between those of AF2400 and PTMSP. Finally, the dependence of the transport parameters of n -alkanes in PTMSN on penetrant pressure and molecular weight was analyzed, and compared to that observed in other high free volume glassy polymers. The activity-based solubility coefficients as well as permeabilities of alkanes in PTMSN show a weak dependence on molecular weight, while in PTMSP permeability increases and in AF2400 decreases with increasing the number of carbon atoms.

The gas and vapor transport into films of Hyflon® Ion H, a short-side-chain perfluorosulfonic acid ionomeric (PFSI) membrane, suitable for use in proton exchange membrane fuel cells (PEMFC), has been studied at various temperatures (35°C, 50°C, and 65°C). The permeability and diffusivity values of He, N2, and O2 show an Arrhenius type dependence on temperature in the range inspected. Pure water vapor permeation was studied at 65°C, at low/medium activity values. The determination of water transport parameters has been performed by solving numerically the water mass balance with a variable diffusion coefficient and accounting for a water immobilization reaction onto the hydrophilic sites of the matrix. The boundary conditions vary in time according to the mass balance on the penetrant volume. The solution allows to represent closely the experimental permeation behavior in all its stages.

A water vapor soprtion study has been performed on Hyflon Ion membranes obtained by casting and the results have been compared to those obtained in an extruded samples, obtaining lower water uptake.

The knowledge of the factors that can influence directly or indirectly the productivity of a credit institution has a considerable importance for the decisions' makers especially in order to decide the optimal solution of management problems. This paper describes a methodology for a systematic approach to the problem of the business control. Particularly it is shown an operational model representing the business behaviour of a credit institution. This model allows an immediate evaluation of the productivity of a credit institution by some indexes drawn from the normal system of banking book-keeping.