Lattice Fluid Theory Research Articles

Multi-scale modeling of gas solubility in semi-crystalline polymers: bridging Molecular Dynamics with Lattice Fluid Theory

Multi-scale modeling of gas solubility in semi-crystalline polymers: bridging Molecular Dynamics with Lattice Fluid Theory

Modelling changes in glass transition temperature in polymer matrices exposed tolow molecular weight penetrants.

Polymer matrices, when placed in contact with a fluid phase made of low molecular weight compounds, undergo a depression of their glass transition temperature (Tg) determined by the absorption of these compounds and the associated plasticization phenomena. Frequently, this effect is coupled with the mechanical action of the compressive stress exerted by the pressure of the fluid phase that, in contrast, promotes an increase in the Tg. This issue is relevant for technological and structural applications of composites with high-performance glassy polymer matrices, due to their significant impact on mechanical properties. We propose an approach to model and predict rubbery-glassy states maps of polymer-penetrant mixtures as a function of pressure and temperature based on the Gibbs-Di Marzio criterion. This criterion establishes that a 'thermodynamic' glass transition does occur when the configurational entropy of the system vanishes. Although questioned and criticized, this criterion constitutes a good practical approach to analyse changes of Tg and, in some way, reflects the idea of an 'entropy catastrophe' occurring at the glass transition. Several polymer-penetrant systems have been analysed modelling configurational entropy by means of the Non-Random Hydrogen Bond lattice fluid theory, able to cope with possible non-random mixing and occurrence of strong interactions. This article is part of the theme issue 'Ageing and durability of composite materials'.

Modelling solubility in semi-crystalline polymers: a critical comparative review

The prediction of gases and vapors solubility isotherms in semi-crystalline polymers is still an unsolved problem of great technological relevance. This work provides an overview of the existing models developed for the assessment of solubility in semi-crystalline materials and presents a comparative discussion of their strengths and limitations. Experimental evidence suggests that the presence of impermeable crystalline domains is responsible for a reduced sorption capacity of the amorphous network compared to the unconstrained amorphous state. Three different modeling categories are distinguished. A set of models ascribes the reduced sorption capacity exhibited by the amorphous phase to an increased density state induced by the confinement provided by the crystals, which is represented with the application of a hydrostatic constraint pressure term. Using a different perspective, other modeling strategies assume the fraction of elastically effective chains, or ‘tie-chains’ linking crystal domains, to be responsible for increased activity of the penetrating molecules in the matrix and consequently for their lower solubility. Finally, the last category of models proposes the application of the Non-Equilibrium Lattice Fluid theory to the amorphous phase, which is regarded as in a pseudo glassy state, due to the presence of the surrounding crystallites, which are responsible for a hindered structure mobility. The theoretical foundations of the listed approaches are presented, and the relative results are collected and discussed. In conclusion, some prospective work guidelines for modeling strategies improvement are provided.

A modified non-equilibrium lattice fluid model based on corrected fractional free volume of polymers for gas solubility prediction

We propose a model based on non-equilibrium lattice fluid (NELF) theory and corrected fractional free volume of polymers to effectively and accurately predict the solubility of gases in different polymers. The method to achieve this purpose is based on the utilization of NELF model infinite dilution solubility coefficient (S0) as the base of predictive calculations. To account for the isolated pore in the polymer matrix in density estimation, a fractional free volume correction factor (β) was introduced in NELF model. The modified NELF model was successfully applied for prediction of solubility of C3H8 and CO2 in polyethylene oxide (PEO) and CO2 in polyethylene terephthalate (PET), isotactic polypropylene (i-PP), polyetherimide (PEI), polymethyl methacrylate (PMMA) and polyethyl methacrylate (PEMA) with adjustments in β value and depth of diffusion of gases in polymer matrix (ζ) at different pressures and temperatures. This work involves multi-objective optimization using genetic algorithm of MATLAB toolbox with adjusted settings. It applies to find the optimum temperature at which the minimum standard deviation of β for different gas-polymer systems is obtained. β showed the same trend of change with temperature as the constrained pressure imposed on the amorphous phase in semi-crystalline polymers. A cubic correlation for standard deviation for β versus temperature was obtained which was able to anticipate the changing trend of β at different temperatures. The chi-square test results verified that compared with original NELF model, a more accurate model for prediction of gas solubilities in polymers has been proposed.

Volumetric Properties and Sorption Behavior of Perfluoropolymers with Dioxolane Pendant Rings

Lattice fluid theory is used to develop transport property–structure correlations for glassy perfluoropolymers with dioxolane pendant rings, a new class of membrane materials for gas separation. Po...

Improved prediction of solubility of gases in polymers using an innovative non-equilibrium lattice fluid/Flory-Huggins model

A combination of Flory-Huggins and non-equilibrium lattice fluid models are utilized for predicting the solubility coefficients of CO2, CH4, N2, n-C4H10 and i-C4H10 in low and high-density polyethylene, polysulfone and polycarbonate. The genetic programming has been used to acquire the appropriate function for this model. The solubility coefficients at infinite dilution are calculated based on non-equilibrium lattice fluid theory and the gas-polymer interaction is expressed by the Flory-Huggins interaction parameter. The solubility coefficients at infinite dilution, Flory-Huggins interaction parameter and pressure were selected as terminal sets and some simple mathematical functions and operators such as multiplication, division, summation, power and absolute value were regarded as mathematical function sets. The adjustable parameters of the proposed function were determined for each gas-polymer system based on nonlinear data fitting. The first adjustable model parameter was in the form of constant power and the second was obtained as a variable coefficient in the form of quadratics function of temperature. The results of the presented model demonstrate improved ability to predict the solubility of investigated gases in the considered polymers at high pressures in comparison to non-equilibrium lattice fluid and Sanchez-Lacombe equation of state models. In some cases, the absolute errors between experimental and predicted values of solubility coefficients were below 1% at the considered conditions.

Sorption Thermodynamics of CO₂, H₂O, and CH₃OH in a Glassy Polyetherimide: A Molecular Perspective.

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.

A predictive model for the permeability of gas mixtures in glassy polymers

The transport of gaseous mixtures in glassy polymers is analyzed by means of a thermodynamic based model, which is applied to describe the permeability of CO2/CH4 50/50 binary mixtures in various glassy polymeric membranes. The approach relies on the description of the solubility behavior of penetrant/polymer mixtures provided by the nonequilibrium thermodynamics for glassy polymers (NET-GP), and considers the gradient in penetrant chemical potential of each species as the actual driving force of the diffusive mass fluxes. Such an approach is specialized to dilute solutions conditions, as it is typically of interest for the transport of light gas (e.g. CO2, N2, CH4, O2) in glassy polymeric membranes; that allows for the simple and successful prediction of the gas permeability of gas mixtures based on single component transport data, with no additional parameters required.The NET-GP model is used in combination with an equation of state (lattice fluid theory by Sanchez and Lacombe) to obtain the solubility of pure and mixed gases at various pressures and compositions, as well as the thermodynamic factors accounting for the dependence of chemical potentials of the solutes on the concentrations of both penetrants. An exponential dependence on penetrant concentration is used to describe the mobility coefficient behavior, so that only two adjustable parameters are required for the pure penetrant case (infinite dilution mobility and plasticization factor). A simple but effective linear mixing rule is considered to describe transport in the binary mixture case, which does not introduce any additional adjustable parameter due to the presence of a second penetrating species. The comparison with permeation data of gas mixtures in different polymers shows the good predictive ability of the model.

Quasi-Chemical PC-SAFT: An Extended Perturbed Chain-Statistical Associating Fluid Theory for Lattice-Fluid Mixtures

In this study, a new modification of the perturbed chain-statistical associating fluid theory (PC-SAFT) has been proposed by incorporating the lattice fluid theory of Guggenheim as an additional term to the original PC-SAFT terms. As the proposed model has one more term than the PC-SAFT, a new mixing rule has been developed especially for the new additional term, while for the conventional terms of the PC-SAFT, the one-fluid mixing rule is used. In order to evaluate the proposed model, the vapor-liquid equilibria were estimated for binary CO2 mixtures with 16 different ionic liquids (ILs) of the 1-alkyl-3-methylimidazolium family with various anions consisting of bis(trifluoromethylsulfonyl) imide, hexafluorophosphate, tetrafluoroborate, and trifluoromethanesulfonate. For a comprehensive comparison, three different modes (different adjustable parameters) of the proposed model were compared with the conventional PC-SAFT. Results indicate that the proposed modification of the PC-SAFT EoS is generally more reliable with respect to the conventional PC-SAFT in all the three proposed modes of vapor-liquid equilibria, giving good agreement with literature data.

Modeling gas permeability and diffusivity in HAB-6FDA polyimide and its thermally rearranged analogs

Gas permeability in HAB-6FDA polyimide and its thermally rearranged analogs was described using a thermodynamic model based on the non-equilibrium lattice fluid (NELF) theory. This study is part of an ongoing effort to describe gas sorption and transport behavior of TR polymers theoretically. Hydrogen, nitrogen and methane permeability over a broad range of pressures (up to 32atm) and temperatures (−10 to 50°C) was calculated with one adjustable parameter at each temperature, i.e., the infinite dilution mobility coefficient. For highly soluble, swelling gases, such as CO2, matrix plasticization was accounted for by a second adjustable parameter, the plasticization factor, which describes the dependence of penetrant mobility on concentration. Model parameters correlate with membrane structure and gas properties. At fixed temperature, the infinite dilution mobility correlates with penetrant critical volume and polymer fractional free volume. For each penetrant, the temperature dependence of infinite dilution mobility is described by the Arrhenius law. Based on the modeling results, unique separation performance of TR polymers is a manifestation of their strong size-sieving ability. Finally, diffusion coefficients and ideal selectivities were predicted with no adjustable parameters.

Modeling of gas solubility and permeability in glassy and rubbery membranes using lattice fluid theory

In this study, the lattice fluid (LF) and non-equilibrium lattice fluid (NELF) theories combined with the modified Fick's law and free volume theory were employed to develop mass transfer models for the prediction of gas sorption and permeation in glassy and rubbery membranes in a wide range of temperature and pressure. The finite element method using COMSOL multi-physics software was used to solve the governing transport equations. The gas sorption into the glassy membranes shows a non-equilibrium behavior in spite of the rubbery membranes in which the gas sorption is in the equilibrium state. The results indicated that the membranes with higher fractional free volume show higher solubility and diffusion coefficients. The proposed model can predict satisfactory the gas solubility and permeability in the glassy and rubbery membranes and determine the influence of operating pressure and temperature on the transport properties of the membranes without need of any adjustable parameters.

Raman Line Imaging of Poly(ε-caprolactone)/Carbon Dioxide Solutions at High Pressures: A Combined Experimental and Computational Study for Interpreting Intermolecular Interactions and Free-Volume Effects.

In the present study, a Raman line-imaging setup was employed to monitor in situ the CO2 sorption at elevated pressures (from 0.62 to 7.10 MPa) in molten PCL. The method allowed the quantitative measurement of gas concentration in both the time-resolved and the space-resolved modes. The combined experimental and theoretical approach allowed a molecular level characterization of the system. The dissolved CO2 was found to occupy a volume essentially coincident with its van der Waals volume and the estimated partial molar volume of the probe did not change with pressure. Lewis acid-Lewis base interactions with the PCL carbonyls was confirmed to be the main interaction mechanism. The geometry of the supramolecular complex and the preferential interaction site were controlled more by steric than electronic effects. On the basis of the indications emerging from Raman spectroscopy, an equation of state thermodynamic model for the PCL-CO2 system, based upon a compressible lattice fluid theory endowed with specific interactions, has been tailored to account for the interaction types detected spectroscopically. The predictions of the thermodynamic model in terms of molar volume of solution have been compared with available volumetric measurements while predictions for CO2 partial molar volume have been compared with the values estimated on the basis of Raman spectroscopy.

Lower and upper critical solution temperatures of binary polymeric solutions

The phase behavior of binary polymeric solutions such as lower and upper critical solution temperatures has an important role in many polymeric processes. For theoretical investigation on the prediction of these temperatures, a substantial number of data points on binary polymeric solutions were collected from literature and used to present a reliable calculation routine through chemical engineering thermodynamic modeling approach. The thermodynamic model of Compressible Regular Solution was used. The minimization of errors and predefined objective function was done by applying Particle Swarm Optimization technique. An efficient and accurate empirical correlation employing some quantitative structure–property relationship concept through statistical modeling was developed. To develop the statistical model, the connectivity indices of polymer and solvent were used as the independent variables of the model. Four statistical parameters were defined as auxiliary criteria to evaluate the models and convergence of calculations. In addition, attempts were made to develop and correlate the connectivity indices (topological descriptors) of polymer and solvent to the lattice fluid theory parameters of Sanchez-Lacombe Equation of State. The reliability and accuracy of proposed approaches were discussed in-details and the results were compared to the available experimental data. Desirable agreements between calculated and experimental data were found in thermodynamic model as demonstrated by a maximum Individual Absolute Relative Deviation of 5%. An averaged IARD of 4.3% was obtained for the empirical model. The new correlation predicts connectivity indices of components with acceptable accuracy.

Water sorption thermodynamics in poly(propylene sebacate)

Water sorption thermodynamics in poly-propylene sebacate, PPSeb, a biodegradable semi-crystalline polyester, has been studied above its glass transition temperature. Experimental results have been obtained by using gravimetric methods to determine water sorption isotherms at several temperatures, as well as by means of in-situ FTIR transmission spectroscopy, to obtain qualitative information and quantitative estimates on the self- and cross-Hydrogen Bonds occurring within the water-polymer mixture. In fact, the water-polymer system has been found to be populated by several water ‘species’, each one characterized by a particular configuration of HB interactions. Different moieties on the polymer backbone are involved in the formation of specific interactions. The experimental results have been interpreted in the light of the Non Random Hydrogen Bonding (NRHB) lattice fluid theory for mixtures, which accounts both for the presence of Hydrogen Bonding (HB) interactions and for non randomness of the lattice contacts. An excellent fitting of experimental water sorption isotherms has been obtained, from which relevant mean-field and HB interactional parameters of the model have been calculated. On that basis, NRHB model has been then used to obtain qualitative and quantitative predictions, at equilibrium, of self and cross HB interactions involving both water molecules and groups present on the macromolecules within the polymer-water mixture, which compare well with the experimental results of FTIR spectroscopy.

Experimental Density of Ionic Liquids and Thermodynamic Modeling with Group Contribution Equation of State Based on the Lattice Fluid Theory

In this work are reported densities of protic and aprotic ionic liquids in a wide temperature range, at ambient pressure, and a thermodynamic modeling is performed through Mattedi–Tavares–Castier, the EOS is based on lattice gas theory. The results are quite satisfactory with deviations smaller than 0.05%. The densities of aprotic ionic liquids are presented with greater magnitudes compared to protic ILs, revealing significant differences between levels of cohesion and electrostatic contributions among the protic and aprotic ionic liquids.

Modeling gas hydrate-containing phase equilibria for carbon dioxide-rich mixtures using an equation of state

Thermodynamic modeling of phase behaviors for CO2-rich mixtures in gas hydrate forming conditions are required for the process design in the field of carbon dioxide sequestration and enhanced oil recovery. With recent experimental data published for solubility of water in CO2–rich mixtures that are significantly different from those previously published, improved modeling studies become necessary for phase equilibria containing gas hydrates. In the present study, an equation of state based on hydrogen-bonding nonrandom lattice fluid theory was applied for both vapor and liquid phases. The model for hydrogen-bonding contribution is simplified and a weak hydrogen bonding between water and carbon dioxide was included for improved calculation of mutual solubility. Hydrate phase was modeled by van der Waals and Platteeuw method but without guest specific parameters other than Kihara potential parameters. The method was applied to single and binary CO2-rich guest mixtures containing methane, ethane, propane, isobutene, nitrogen, hydrogen sulfide and methanol for temperatures above 180 K and pressures below 100 MPa. Results of two- and three-phase equilibrium calculations containing gas hydrates were found to be comparable with those of CSMGem (Sloan and Koh, Clathrate and Hydrates of Natural Gases, 3rd ed., CRC Press, Boca Raton, FL, 2008) in general and better for water contents in liquid carbon dioxide in equilibrium with gas hydrates.

Thermodynamic modeling of refrigerants solubility in ionic liquids using original and ϵ*-Modified Sanchez–Lacombe equations of state

Ionic liquids can dramatically improve the efficiency of absorption refrigeration processes due to their negligibly small vapor pressure; where the IL absorbs the refrigerant gas in one stage and subsequently releases the high-pressure gas upon addition of heat. In the current study, phase equilibrium data (P-T-x) of a diverse set of refrigerants in different types of ionic liquids are modeled with the aid of two equations of state based on lattice fluid theory, i.e. Sanchez–Lacombe and ϵ*-Modified Sanchez–Lacombe EOSs. The characteristic parameters of the pure components were determined from a database of 3086 experimental liquid density datapoints for 11 refrigerants and 12 ILs. According to the results obtained by considering the experimental data of 25 binary IL/refrigerant mixtures comprised of 88 isotherms with a total of 658 datapoints, both EOSs can successfully correlate the solubility data with AARDs of %5.773 and %5.771 for SL and ϵ*-Modified SL EOSs, respectively.

Predictive calculation of hydrogen and helium solubility in glassy and rubbery polymers

Hydrogen sorption isotherms in selected glassy and rubbery polymers, available over a wide range of temperatures (−20 to 70°C) and pressures (0–60atm) have been modeled and correlated using equilibrium and non-equilibrium thermodynamic models based on the lattice fluid theory. A good representation of the experimental data can be obtained for the systems considered over the whole range of pressures and temperatures inspected by using just one fitting parameter, under the fundamental assumption that hydrogen behaves as a non-swelling penetrant. The theoretical estimates of infinite dilution solubility coefficients are in excellent agreement with the experimental data. Remarkably, the model analysis allows a reliable estimate of the isosteric heat of sorption and its dependence on the hydrogen concentration over the whole range of pressures considered. A similar theoretical analysis has been performed by considering the helium sorption data available at 35°C for a series of polymers considered for membrane-based gas separations. Finally, He/H2 solubility–selectivity at 35°C has been correctly predicted: as expected, the glassy Teflon® AF-series perfluorinated copolymers display a higher He/H2 solubility selectivity compared to the hydrocarbon-based polymers.

Trends in the Athermal Entropy of Mixing of Polymer Solutions

Polymeric mixtures of hydrocarbons and alcohols have been simulated with discontinuous potential models to characterize the Helmholtz energy of the repulsive reference fluids. This quantity is equivalent to the athermal mixture entropy. The reference compressibility factor and Helmholtz free energy have been correlated for various molecular structures from single to infinite chain lengths. The mixtures included small n-alkanes, branched alkanes, aromatics, and alcohols, with polymeric molecules of: n-alkanes, ethyl-styrenes, ethyl-propylenes, and isoprenes. We find that the athermal entropy of mixing at constant packing fraction deviates significantly from ideality as the volume ratio increases, but the nonideality is fairly insensitive to structural details like branching and rings. Volume ratio alone does not provide a complete characterization, however. For example, a mixture of C40 and C80 would yield a small deviation whereas a mixture of C2 and C4 would provide a relatively large deviation. This observation leads to the introduction of a characteristic parameter in terms of entropy density, designated as an entropic solubility parameter. In both ideal and nonideal solutions, the trends still follow van der Waals (vdW) mixing. This leads to an accurate characterization of the entropic contribution to the χ parameter (χS) of Flory–Huggins theory for mixtures of all sizes, shapes, and compositions of molecular structures. A general rule is developed for predicting the athermal entropy of mixing based on knowledge of the volume ratios and entropic solubility parameter of the constituent molecules. The simulations are compared to Flory–Huggins (FH), group contribution lattice fluid theory (GCLF), statistical associating fluid theory (SAFT), Sanchez–Lacombe (SL), and Guggenheim-Staverman (GS) theories of polymer chains.

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.