A procedure for predicting sorption equilibrium in ternary polymer systems from Flory–Huggins binary interaction parameters and the inversion point of preferential solvation

A procedure has been developed, based on the Flory–Huggins theory as generalized by Pouchlý, which permits the calculation of preferential (λ) and total (Y) sorption coefficients from previous information on the binary interaction parameters, χ, χ, and g₁₂(?₁₀) and on the mixture composition at which the sign of λ inverts. The expressions obtained were applied to 10 cosolvent polymer systems for which experimental values of λ and Y are known. Practically in all the studied systems, the theoretical predictions are in fair accordance with the experimental data.     

A thermodynamic analysis on the coincidence of extrema conditions in the sorption equilibrium for ternary polymer systems

Flory-Huggins theory of polymer solutions has been used to express the condition of extrema values in the total sorption, as well as the inversion point in the preferential adsorption parameters for termary polymer systems. Two approaches have been followed, the first considers the binary and ternary interaction parameters independent of polymer concentration and solvent composition. In the second one, this dependence has been introduced. Our attention is focused on the volume fraction of solvent mixture dependence of the above parameters, in order to confirm or not the coincidence between the extrema values and the inversion point. Several cosolvent and cononsolvent ternary polymer systems, have been used to test the validity of the equations obtained. Also, it has been verified, from an experimental point of view, that in cosolvent ternary polymer systems there is coincidence in both compositions while in cononsolvent ternary polymer systems, such coincidence does not appear.     

The use of Flory–Huggins parameters as a measure of interactions in polymer‐filler systems

Any quantitative information on the strength of interactions between inorganic filler and polymer is substantial for the future application of the composite. The magnitude of adhesion of two phases may be deduced from results collected by various experimental techniques. Polyether‐urethane/modified carbonate‐silicate fillers systems containing different amount of filler (5, 10, and 20 wt %) were the materials investigated. We propose to express the magnitude of modified filler/polymer interactions by Flory–Huggins χ parameter. It may be deduced from the results collected by inverse gas chromatographic (IGC) experiment. We have also tried to explain the influence of the solvent on values of the evaluated parameters and to check the usefulness of some of presented methods to minimize Δχ effect. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1853–1862, 2006     

A thermodynamic approach to study hydrogen-bonding interactions in solvent/solvent/polymer ternary systems

A thermodynamic approach based on both the classical Flory-Huggins (FH) formalism and the association equilibria (AE) theory has been developed to study the solubility properties of a system formed by a proton-donor solvent (A), a proton-acceptor solvent (B) and a proton-acceptor polymer (C). The miscibility of this ternary system is attained by competitive specific interactions via hydrogen-bonding established between the hydroxyl and carbonyl interacting groups of either solvent-solvent (AB) or solvent-polymer (AC) system components. The binary AB and AC specific interactions and their dependence with the system composition as well as with the extent of the association equilibrium have been quantified by means of two new parameters, Δg_AB and Δg_AC. These excess functions have appeared to be equivalent to the combinatorial or entropic term of the Gibbs free energy of the complex formation process, which accounts for the entropy of mixing plus the intermolecular specific interactions. The theoretical predictions have reasonablely agreed with experimental data on preferential solvation of two systems taken from literature: methanol(A)/1,4-dioxane(B)/poly(alkyl methacrylate)(C) and n-alcohol(A)/heptan-3-one(B)/poly(vinyl pyrrolidone)(C).     

Phase behaviour of nematic liquid crystal/linear polymer systems: Applicability of the extended Flory‐Huggins theory

Phase behaviour of nematic liquid crystals (N‐(p‐ethoxybenzylidene)‐p‐butylaniline and N‐(p‐methoxybenzylidene)‐p‐butylaniline) in linear polystyrene are investigated by thermo‐optical analysis. Results are compared to the model based on the extended Flory‐Huggins equation. The model used in this work includes two adjustable model parameters (d₀, d₁). The proposed semi‐empirical model gives a very good agreement with new data obtained for liquid crystal and polystyrene systems.     

Solvent quality as reflected in concentration‐ and temperature‐dependent Flory–Huggins interaction parameters

Flory–Huggins interaction parameters (χ) between poly(dimethylsiloxane) (weight‐average molecular weight = 152 kg/mol) and various solvents (methyl ethyl ketone, toluene and n‐octane) were determined as a function of composition and temperature with vapor‐pressure measurements. These data, complemented by independent information for dilute and very concentrated solutions, serve as the basis for a discussion of solvent quality via different theoretical relations. Regardless of polymer concentration, the χ values fall from methyl ethyl ketone via toluene to n‐octane, the ketone being the worst solvent and the hydrocarbon being the best solvent. The variation of χ with composition and temperature is complex. Within the range of moderate polymer concentrations, the influences of composition decrease with increasing solvent quality. Additional effects become noticeable at the ends of the composition scale. The enthalpy parts (χ_H) and entropy parts (χ_S) of the Flory–Huggins interaction parameter, obtained from χ(T), vary considerably with composition and change their sign in some cases; these constituents of the Flory–Huggins interaction parameter do not permit a direct assessment of solvent quality. A clear‐cut picture is, however, regained with a comparison of the interdependence of χ_S and χ_H. The elimination of explicit concentration influences re‐establishes the order in the solvent quality setup via χ. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 651–662, 2001     

Determination of thermodynamic parameters of polymer–solvent systems from sedimentation–diffusion equilibrium in the ultracentrifuge

Sedimentation equilibrium in the ultracentrifuge means that there is such a distribution of molecular species throughout the cell, that the centrifugal forces are balanced by differences in the activities. This provides a method for determination of the activities and the chemical potentials in polymer solutions which, in principle, is very simple and reliable. A complication is caused by polydispersity of the dissolved polymer. If one assumes that the interaction parameter depends on concentration and temperature, but not on molecular weight, it is possible to determine the chemical potential of polymer and solvent from the ultracentrifugal data. Experiments have been carried out on the systems polystyrene–toluene and polystyrene–cyclohexane at different temperatures and in the concentration range 0–80 wt-%. The results are expressed in the data for the chemical potential of the solvent, the number average chemical potential of the polymer and the interaction parameter χ.     

On linear programming approach for the calculation of chemical equilibrium in complex thermodynamic systems

The advantages of linear programming approach for the calculation of chemical equilibrium in complex thermodynamic systems are demonstrated. The algorithm presented is used for the development of software package for thermodynamic modeling.     

Liquid–liquid equilibria of polymer solutions: Flory‐huggins with specific interaction

We have developed a new Flory‐Huggins model by adding a specific interaction parameter derived from a modified double‐lattice model for the Helmholtz energy of mixing for binary liquid mixtures. This model is very simple and could be easily integrated into engineering applications. Using this revised model, we can successfully describe the phase behavior of polymer solutions with an upper critical solution temperature (UCST), a lower critical solution temperature (LCST), both UCST and LCST, and a closed miscibility loop. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 162–167, 2010     

A new model for predicting thermodynamic properties of ternary metallic solution from binary components

A model has been derived to predict thermodynamic properties of ternary metallic systems from those of its three binaries. In the model, the excess Gibbs free energies and the interaction parameter ω₁₂₃ for three components of a ternary are expressed as a simple sum of those of the three sub-binaries, and the mole fractions of the components of the ternary are identical with the sub-binaries. This model is greatly simplified compared with the current symmetrical and asymmetrical models. It is able to overcome some shortcomings of the current models, such as the arrangement of the components in the Gibbs triangle, the conversion of mole fractions between ternary and corresponding binaries, and some necessary processes for optimizing the various parameters of these models. Two ternary systems, Mg–Cu–Ni and Cd–Bi–Pb are recalculated to demonstrate the validity and precision of the present model. The calculated results on the Mg–Cu–Ni system are better than those in the literature. New parameters in the Margules equations expressing the excess Gibbs free energies of three binary systems of the Cd–Bi–Pb ternary system are also given.     

Phase equilibrium in polymer systems

The topological analysis of phase equilibria in polymer systems, which was developed by S.P. Papkov; the classification of the types of phase equilibrium; the principle of mutual independence of the two types of equilibrium; the concept of the generality of the phase equilibrium in polymer-solvent, polymer-plasticizer, polymer-oligomer, and polymer-polymer systems; and the processes of water sorption by polymers are considered. It is shown that the problems that Papkov dealt with remain topical and the concepts that he elaborated can underlie new studies in the field of phase equilibria in polymer systems.     

A continuous polydisperse thermodynamic algorithm for a modified flory–Huggins model: The (polystyrene + nitroethane) example

A modified Flory–Huggins model is presented, considering a concentration‐ and temperature‐dependent interaction parameter, and using the methodology of Continuous Thermodynamics to take into account both polydispersity and its effect on phase equilibrium of polymeric systems. This model describes all commonly found, as well as other unusual polymer + solvent and polymer + polymer, liquid–liquid phase diagrams and is easily extended to take all possible pressure effects into consideration. Modeling and least‐squares fit of polystyrene + nitroethane liquid–liquid cloud‐point data have produced results in good accord with the experimental ones by using meaningfully physical parameters. These results have been used to discuss polystyrene molecular weight, pressure, and isotopic substitution effects on polystyrene + nitroethane systems. A first‐order interpretation of phase equilibrium isotopic substitution effect has also been applied. It combines the simplest form of the Flory–Huggins model with the statistical theory of condensed phase isotope effects. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 632–651, 2000     

Correlation of thermodynamic modeling and molecular simulations for liquid-liquid equilibrium of ternary polymer mixtures based on a phenomenological scaling method

In our previous study [S.Y. Oh, Y.C. Bae, J. Phys. Chem. B 114 (2010) 8948-8953], we presented a new method to predict liquid-liquid equilibria in ternary simple liquid mixtures by using a combination of a thermodynamic model and molecular simulations. As a continuation of that effort, we extend our previously developed method to ternary polymer systems. In the simulations, we used the dummy atoms to calculate the pair interaction energy values between the polymer segments and the solvent molecules. Furthermore, a thermodynamic model scaling concept is introduced to consider the chain length dependence of the energy parameters. This method was applied to ternary mixtures incorporating low to high molecular weight polymers. The method presented here well described the experimental observations using one or no adjustable parameters.     

Prediction of maxima and minima in the curve of total sorption parameter in ternary polymer systems. Influence of ternary interaction parameter

Flory-Huggins theory modified by Pouchly has been applied to predict maxima and minima in the curve of total sorption in ternary polymer systems formed by a polymer and two liquids. In this work, different diagrams based on experimental magnitudes easily obtained such as the difference in affinities of liquids, solvents and non solvents, and the solvent molar volume ratio. Total sorption parameter has been considered to be the decisive magnitude to define extrema conditions in both cosolvent and cononsolvent ternary polymer systems. The theoretical prediction is not altered by the inclusion of ternary interactions. Different examples of ternary systems dealing with vinyl polymers and polydimethyl siloxane have been used to test the above formalism.     

A differential approach to calculating osmotic equilibrium for multisolvent systems

A differential approach to the calculation of osmotic pressure of multisolvent systems within the Lewis–Randall framework is presented in this paper. Exact differential equations relating the osmotic pressure and the system composition along paths of constant solvent chemical potential are obtained and numerically solved. Although even for the simple case of an ideal solution no analytic expression for the osmotic pressure can be obtained, the system of differential equations does not pose numerical difficulties to be solved. Examples of the use of the proposed methodology are presented using the two-suffix Margules and Flory–Huggins equations, allowing an assessment of the influence of liquid-phase non-ideality on the performance of the method, and showing that it can be applied even for systems wherein liquid–liquid phase equilibrium occurs.     

Polymer-polymer interaction parameters in solvent/polymer/polymer ternary systems

Empirical equations defining the relationships between b₂₃ and χ₂₃ interaction parameters for the ternary systems toluene (1)/poly(dimethylsiloxane), PDMS, (2)/polystyrene, PS, (3) and benzene (1)/PDMS(2)/PS(3) have been determined. The dependences of both interaction parameters on the mean molecular weights of the macromolecular components have also been studied. A single dependence of b₂₃ (or χ₂₃) upon the mean molecular weight is found when both polymeric components have intermediate to high molecular weights. However, that single dependence is not found when the molecular size of polymer (2) is low.     

计算离子液体溶液汽液相平衡的分子热力学模型

用平均球近似理论、微扰理论和UNIFAC基团贡献方法分别考虑离子之间的长程静电作用、离子与溶剂之间的中程静电作用以及所有粒子之间的短程作用,本文提出了一种新的分子热力学模型,可用于离子液体溶液中溶剂活度系数的计算.通过对含烷基咪唑磷酸酯类离子液体与水、甲醇或乙醇组成的9个二元体系的饱和蒸汽压数据进行关联,获得了相关的模型参数,即溶剂的分子直径和基团之间的交互作用能参数.溶剂活度系数及饱和蒸汽压的计算结果与实验值的平均偏差为1.40%,符合良好,因此本模型可望用于含离子液体体系汽液相平衡的预测.     

A thermodynamic surface model for caesium sorption on bentonite

Caesium sorption on Wyoming bentonite MX-80 has been studied in solutions of NaCl, KCl, MgCl(2), CaCl(2), NaNO(3) and Ca (NO(3))(2) of concentrations varying between 0.025 and 1 mol/L, as well as in a weakly saline (I=0.004 ml/L) and a strongly saline (I=0.46 mol/L) natural groundwater. These experiments have been used to derive a thermodynamic model for the interaction of caesium with the bentonite surface in accordance with a surface chemical model, including acid/base reactions developed recently for montmorillonite. The sorption behaviour of caesium on bentonite can be described, within the experimental and model uncertainties, in terms of a one-site ion exchange model. The ion exchange constant obtained for the reaction NaX+Cs(+) left arrow over right arrow CsX+Na(+) (where X represents the ion exchange sites on montmorillonite) is log(10) K(0)(ex)=1.6. Impurities in the bentonite, influencing the concentrations of competing cations, such as Na(+), K(+), Mg(2+) and Ca(2+), have a crucial impact on the sorption of caesium. This impact can be adequately quantified with the present model. The model predictions compare well with sorption data published in the open literature on both Wyoming bentonite MX-80 and other types of bentonite. Distribution coefficients from the literature obtained from both batch and diffusion experiments and varying over four orders of magnitude are reproduced and explained successfully by the model.