Nonequilibrium Lattice Fluid Research Articles

Nonequilibrium Lattice Fluid Modeling of Gas Solubility in HAB-6FDA Polyimide and Its Thermally Rearranged Analogues

For the first time, a theoretical analysis of gas sorption, based on the nonequilibrium lattice fluid (NELF) model, in chemically imidized HAB-6FDA polyimide and its thermally rearranged analogues is presented. Because of the inaccessibility of pVT data in the rubbery region, the characteristic lattice fluid parameters of the polymers considered in this study were obtained from a collection of infinite dilution solubility data at multiple temperatures. Hydrogen, nitrogen, and methane sorption isotherms at 35 °C were fit to the NELF model using one adjustable parameter, i.e., the polymer–penetrant binary interaction parameter, k12. The optimal value of k12 for each polymer–penetrant pair was used to predict hydrogen, nitrogen, and methane sorption isotherms at other temperatures and at pressures up to 6 MPa. For carbon dioxide, a second adjustable parameter, the swelling coefficient, was introduced to account for sorption-induced matrix dilation. The ideal solubility–selectivity is also predicted for sever...

Sorption and transport of CO2 in copolymers containing soft (PEO, PPO) and hard (BKDA-ODA and BPDA-ODA) segments at different temperatures: Experimental data and modeling

In this work, we studied CO2 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 CO2 and high flexibility, which endows the copolymers with high CO2 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 CO2 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 CO2 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 CO2 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 CO2 solubility of the copolymer, and the Non-Equilibrium Lattice Fluid (NELF) to evaluate the CO2 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.

Thermodynamic model for the permeability of light gases in glassy polymers

The permeability of light gases in a series of different glassy polymers is analyzed through a thermodynamic‐based approach for solubility and diffusivity. The nonequilibrium thermodynamic model for glassy polymers describes the solubility of the different penetrants; diffusivity is given as the product of a mobility factor and a thermodynamic factor. The latter is predicted by the nonequilibrium lattice fluid thermodynamic model, while the mobility coefficient is determined using the experimental permeability data. For rather soluble penetrants (e.g., CO2), a plasticization factor is also accounted for, considering the mobility to depend exponentially on penetrant concentration, as often observed experimentally. The model is able to describe accurately the experimental behavior in a simple and effective way, considering only two adjustable parameters. The mobility coefficient is found to depend on the penetrant size (critical volume) and on the fractional free volume of the polymer matrix, following rather general and reasonable correlations. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2776–2788, 2015

Thermodynamic basis for vapor permeability in Ethyl Cellulose

The vapor permeability in glassy Ethyl Cellulose may show rather different behaviors versus upstream pressure, depending on the penetrant considered, including a monotonous decrease as well as a monotonous increase with increasing pressure. Such a spectrum of behaviors, experimentally well known, is still needing a unifying interpretation and a mathematical description, which we are now proposing in the present work. For each solute, at all different temperatures, the permeability in Ethyl Cellulose is described by means of a simple model with a solid thermodynamic basis. The thermodynamic behavior of the polymer/penetrant mixture is given by the nonequilibrium lattice fluid model (NELF), which provides the vapor solubility at different values of T and p. The penetrant permeability is described considering the diffusion coefficient as the product of a purely kinetic factor, the mobility, and a thermodynamic factor related to the dependence of the chemical potential of the diffusing species on its concentration in the polymer. The thermodynamic factor is readily calculated from the NELF model, while the mobility factor is endowed with an exponential dependence on penetrant concentration, as often suggested by experimental data; its expression utilizes only two adjustable parameters: the infinite dilution mobility coefficient and the plasticization factor. The analysis indicates that this simple mathematical theory is able to describe accurately all the different behaviors observed experimentally, for all the penetrants inspected, at all temperatures. In addition, the two model parameters for mobility follow clear and simple correlations: the infinite dilution mobility coefficient scales with the penetrant size, and the plasticization factor is related to the ability of the penetrant to swell Ethyl Cellulose, as quantified by the volume swelling coefficient, which is available from independent information.

Predictive model for gas and vapor solubility and swelling in glassy polymers I: Application to different polymer/penetrant systems

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 Tg. 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.

Modelling sorption and diffusion of NF3 and CF4 in Teflon AF perfluoropolymer membranes

Molecular dynamics (MD) and grand canonical Monte Carlo (GCMC) simulations with Teflon AF2400 and Teflon AF1600 predicted diffusivities and solubilities of He and N2 that corresponded reasonably with available experimental data. The results of the MD simulations further showed that NF3 had a higher diffusivity than CF4 in both polymers and that the NF3/CF4 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. NF3 and CF4, the non-equilibrium lattice fluid (NELF) model was used to further evaluate this aspect. Consequently, the combined NF3/CF4 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 NF3/CF4 solubility and diffusion selectivity.

Thermodynamic basis for vapor solubility in ethyl cellulose

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.

NELF predictions of a solubility–solubility selectivity upper bound

Non-equilibrium lattice fluid theory (NELF) is used to evaluate the relationship between solubility and solubility selectivity for glassy polymeric materials. The NELF model provides a priori predictions of gas solubility and solubility selectivity for gas penetrants in glassy polymers without adjustable parameters. The results suggest an upper bound exists on solubility and solubility selectivity. The solubility upper bound depends on gas and polymer material properties through NELF model parameters. The polymer–polymer interaction energy determines the location of the upper bound. The upper bound moves in a favorable direction as polymer cohesive energy density increases and is limited by the highest value achievable in real materials. Furthermore the temperature dependence of solubility and solubility selectivity is predicted by the analysis.

Prediction of Water Solubility in Glassy Polymers Using Nonequilibrium Thermodynamics

In this study, the sorption of water in poly(methyl methacrylate) (PMMA) was measured at various water vapor activities (0–0.85) at 25, 35, and 45 °C using a quartz spring microbalance. Furthermore, the water sorption isotherms in PMMA were predicted using two nonequilibrium thermodynamic models: the nonequilibrium lattice fluid (NELF) model and nonequilibrium statistical associating fluid theory (NE-SAFT), where excellent agreement between the NE-SAFT prediction and experimental data was observed. In contrast, deviation between the NELF model prediction and water sorption isotherms in PMMA was observed above a water activity of ca. 0.50. In situ time-resolved Fourier transform infrared attenuated total reflectance spectroscopy confirmed the presence of self-associated water (i.e., water clusters) at elevated water activities, providing a rationale for deviation between the NELF model and experimental data, where unlike NE-SAFT, the NELF model does not account for these self-association interactions. The ...

Gas Solubility, Diffusivity, Permeability, and Selectivity in Mixed Matrix Membranes Based on PIM-1 and Fumed Silica

The effects that the addition of fumed silica (FS) nanoparticles has on the gas permeability, solubility, diffusivity, and selectivity of a polymer of intrinsic microporosity (PIM-1) are modeled considering the density of the composite matrix as the key input information. PIM-1 is treated as a homogeneous glassy polymer endowed with a specific free volume that increases with the amount of nanoparticles loaded, as indicated by the experimental values of mixed matrix density. The solubility isotherms of H2, He, O2, N2, CH4, and CO2 in matrices of PIM-1 with different FS loadings are calculated with the nonequilibrium lattice fluid (NELF) model. The gas diffusivity and permeability variation due to FS addition are related to the fractional free volume of the polymer phase, according to the semiempirical free volume theory equation. Remarkably, the coefficient amplifying the free volume effect increases with the molecular size of the gas, expressed by the van der Waals volume, thus allowing an estimation of t...

Permeability and solubility of carbon dioxide in different glassy polymer systems with and without plasticization

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.

Modeling gas and vapor sorption in a polymer of intrinsic microporosity (PIM-1)

The sorption of gases (CO2, N2, CH4, O2) and vapors of different nature (n-butane, n-pentane, i-pentane, n-hexane, n-heptane, methanol, ethanol, 1-propanol, 1-butanol, water, chloroform, toluene, 1,4-dioxane, tetrahydrofurane) in films of a polymer of intrinsic microporosity (PIM-1) was studied at room temperature and various pressures. For the first time, the solubility properties of PIM are studied with a rigorous macroscopic model based on the Lattice Fluid (LF) theory; the LF parameters, not available for PIM-1, were obtained from independent gas sorption data at low pressure. The solubility of gases and water vapor follows the typical concave trend of dense glassy polymers and is described accurately by the Non Equilibrium Lattice Fluid (NELF) model, given the dry polymer density value and a slight adjustment of the binary parameter kij. The organic vapor solubility isotherm in PIM-1 can reach very high values and even exhibit a positive concavity at high penetrant contents, that is typical of sorption in rubbery phases. Such portion of the solubility isotherm was thus modeled with the equilibrium LF Equation of State (EoS) and one adjustable parameter kij, while at lower activities the sorption was described with the same value of kij and the NELF model. The value of PIM-1 density in the swollen glassy state required by the NELF model is estimated with an a priori scheme, based on pure penetrant liquid volume.The present work shows that, despite its peculiar microscopic structure, the solubility of PIM-1 can be estimated successfully with the same thermodynamic tools used for dense homogeneous phases and a limited number of parameters.

Vapor and Liquid Sorption in Matrimid Polyimide: Experimental Characterization and Modeling

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.

A Predictive Model for Vapor Solubility and Volume Dilation in Glassy Polymers

The nonequilibrium lattice fluid (NELF) model is here implemented with a simplified relation for bulk rheological behavior of glassy polymers, to obtain a predictive model for pseudoequilibrium solute content and volume dilation induced by swelling agents. The rheological model discussed in this work is ultimately used to derive a relation for the pseudoequilibrium volume of the polymeric system as a function of temperature and solute fugacity. The model makes use of two nonequilibrium parameters which, in turn, can be determined through the analysis of nonequilibrium pVT properties of the polymeric species. The predictive ability of the new model for solubility and volume swelling was tested by means of the comparison of model predictions and correlations with experimental data, available in the literature, for different polymer/penetrant pairs. The results obtained from the comparison allow one to conclude that the model is not just useful to describe the swelling effect of penetrants but also to recogn...

Nonequilibrium Sorption of Water in Polylactide

The sorption of water in polylactide (PLA) was measured at various vapor activities (0–0.85) and temperatures using a number of experimental techniques, including quartz spring microbalance (QSM), quartz crystal microbalance (QCM), and in situ time-resolved Fourier transform infrared-attenuated total reflectance (FTIR-ATR) spectroscopy. Additionally, a prediction of the water sorption isotherm in PLA was obtained with the use of the nonequilibrium lattice fluid (NELF) model, where an excellent agreement between the model prediction and experimental sorption data was observed at different experimental temperatures (all below the glass transition temperature of PLA) for water vapor activities less than 0.65. Results from in situ time-resolved FTIR-ATR spectroscopy revealed that water is present predominately as dimers in PLA at water vapor activities less than 0.65, and the presence of larger hydrogen-bound water clusters was observed at water vapor activities >0.65. This provides a rationale for the deviat...

Predicting Solubility and Diffusivity of Gases in Polymers under High Pressure: N2 in Polycarbonate and Poly(ether-ether-ketone)

The aim of this study was to develop a model that predicts the gas solubility and the sorption and desorption kinetics in polymer granulates over large temperature and pressure intervals. Besides the part predicting the solubility and diffusivity, the model involves the simultaneous solution of the diffusion equation and the heat equation in three dimensions using a finite element method (FEM). When the temperature- and pressure-dependent solubility of a specific polymer/gas combination is not known, an improved version of the non-equilibrium lattice fluid model (NELF) is used to predict the solubility. The improvement of the NELF model includes the use of Hansen’s solubility parameters, and it uses pressure–volume–temperature (PVT) data from two new empirical models, which accurately estimate polymer densities over a wide range of temperatures and pressures. The new solubility model predicted the solubility–pressure data of N2 in poly(ethyl methacrylate) and N2 and CH4 in polycarbonate (PC) at pressures ...

Sorption of hydrocarbons and alcohols in addition-type poly(trimethyl silyl norbornene) and other high free volume glassy polymers. II: NELF model predictions

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, 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.

Chloroform sorption in nanoporous crystalline and amorphous phases of syndiotactic polystyrene

AbstractThe transport of chloroform in films of atactic polystyrene and of semicrystalline syndiotactic polystyrene in its nanoporous form (δ‐form) has been investigated by gravimetric analysis. Experimental tests have been performed at 35, 49, and 56 °C and at several vapor pressures ranging from 0.5 to 100 Torr. Nonequilibrium lattice fluid prediction of the amorphous sorption behavior was used to enucleate the sorption contribution of the crystalline nanoporous phase from semicrystalline samples. Sorption behavior of the crystalline phase has been interpreted on the basis of Langmuir equation. Moreover, the chloroform sorption at low activities in the crystalline nanoporous phase has been predicted by using Grand Canonical Monte Carlo molecular simulations. Isosteric heats of sorption were also experimentally evaluated for the crystalline phase, and compared with the corresponding prediction of molecular simulation. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 8–15, 2008

Correlations between Penetrant Properties and Infinite Dilution Gas Solubility in Glassy Polymers: NELF Model Derivation

An analysis of the empirical correlations that relate the solubility coefficient of gases at infinite dilution in glassy polymers to penetrant properties has been performed, by considering gas sorption as a succession of three steps: condensation of the penetrant, dissolution in the equilibrium polymeric phase, and departure of the system from equilibrium. The infinite dilution solubility coefficient is obtained from the contributions of the three steps, the first two of which involve standard equilibrium thermodynamics, while the latter requires nonequilibrium considerations and is described by using the nonequilibrium lattice fluid (NELF) model. The various terms have been evaluated for several gases in polycarbonate, polysulfone, poly(phenylene oxide), and poly(methyl methacrylate). It has been found that the largest term in the expression of solubility is due to condensation, which explains the observed correlation between solubility and critical temperature, while the out of equilibrium contribution to solubility, which generally gives numerical values lower than the condensation term, is in turn more strongly correlated to the penetrant critical volume.

Solubility of gases and vapors in glassy polymers modelled through non-equilibrium PHSC theory

Well established equation-of-state (EoS) models from Perturbed Hard Sphere Chain Theory have been adopted within the framework of non-equilibrium thermodynamics for glassy polymers (NET-GP) to describe the solubility of small penetrants in glassy polymers. The procedure presented here parallels the one already applied to the Lattice Fluid EoS, obtaining the Non-Equilibrium Lattice Fluid (NELF) model. The solubility is calculated as pseudo-equilibrium solute content in the glassy polymeric phase at fixed temperature and solute fugacity, and is compared to the experimental data. The NE-PHSC model predictions require pure component and binary EoS parameters, together with the value of the glassy polymer density during sorption. Several common glassy polymers, as well as a high free volume fluoropolymer, have been considered to test the model ability to represent the thermodynamic properties of glassy gas–polymer mixtures. A comparison between the performance of two NE-PHSC models, endowed with different expressions for the interaction potential, is also presented, offering interesting conclusions on their accuracy in the representation of the thermodynamic properties of relatively high-density systems. The results obtained in this work show that the NET-GP approach provides a reliable description of the properties of homogeneous glassy mixtures.