The QCHB model of fluids and their mixtures

The essentials of the QCHB (quasi-chemical hydrogen-bonding) equation-of-state model are presented along with some applications for calculations of phase equilibria and interfacial properties of fluids and their mixtures. This is a model applicable to non-polar systems as well as to highly non-ideal systems with strong specific interactions, to systems of small molecules as well as to macromolecules, including polydisperse polymers, glasses, and gels, to liquids as well as to vapours including supercritical systems, to homogeneous as well as to inhomogeneous systems. A quasi-thermodynamic approach of inhomogeneous systems is used for modeling the fluid–fluid interface. Consistent expressions for the interfacial tension and interfacial profiles for various properties are presented. A satisfactory agreement is obtained between experimental and calculated surface tensions. Extension of the approach to mixtures is examined along with the associated problems for the numerical calculations of the interfacial profiles. A new equation is derived for the chemical potentials in the interfacial region, which facilitates very much the calculation of the composition profiles across the interface. The relation of the model with the COSMO-RS approach is also discussed.     

Nematic–isotropic phase behaviors of polydisperse polymer/liquid‐crystal systems: Chain‐length‐dependent interaction parameter

Phase behaviors of polydisperse polystyrene (PS)/nematic liquid‐crystal systems [P‐ethoxy ‐ benzylidene ‐ p ‐ n‐butylaniline (EBBA)] are investigated with a thermo‐optical analysis technique. We also develop a thermodynamic framework to describe the phase behaviors of polydisperse PS/EBBA systems. The proposed model is based on a modified double‐lattice model to describe isotropic mixing and Maier–Saupe theory for anisotropic ordering. To correlate the polymer chain length and energy parameters in a nematic–isotropic biphasic region and to apply the primary interaction parameter in an isotropic–isotropic phase‐transition behaviors of polydisperse PS/EBBA systems. The proposed model shows remarkable agreement with experimental data for the model systems in comparison with an existing model. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1031–1039, 2006     

The prediction of isobaric phase equilibria for non-ideal multicomponent mixtures from heat of mixing data

A method for predicting isobaric binary and ternary vapor—liquid equilibrium data using only isothermal binary heat of mixing data and pure component vapor pressure data is presented. Three binary and two ternary hydrocarbon liquid mixtures were studied. The method consists of evaluating the parameters of the NRTL equation from isothermal heat of mixing data for the constituent binary pairs. These parameters are then used in the multicomponent NRTL equation to compute isobaric vapor—liquid equilibrium data for the ternary mixture. No ternary or higher order interaction terms are needed in the ternary calculations because of the nature of the NRTL equation. NRTL parameters derived from heat of mixing data at one temperature can be used to predict vapor—liquid equilibrium data at other temperatures up to the boiling temperature of the liquid mixture.For the systems studied this method predicted the composition of the vapor phase with a standard deviation ranging from 1–8% for the binary systems and from 4–12% for the ternary systems.     

Liquid–liquid equilibria of polydisperse polymer systems: applicability of continuous thermodynamics

Continuous thermodynamics is a suitable tool for describing the thermodynamic properties of solutions of polydisperse polymers. To represent liquid–liquid equilibria of polydisperse polymer/solvent systems, a continuous distribution function to represent the composition of polydisperse polymers has to be considered. In this study, we calculate the molar mass distributions of both principal phases and conjugate phases, using the extended Flory–Huggins model. The results for various polydisperse polymer systems are compared with experimental data.     

Self-consistent field theory and its applications in polymer systems

This review article addresses the widely used self-consistent field theory (SCFT) in interacting polymer systems. The theoretical framework and numerical method of solving the self-consistent equations are presented. In this paper, different structures of polymer can be considered, such as homopolymer, block copolymer, polydisperse polymer and charged polymer. Several systems, micro/macro phase separation, interface, self-assembly, are presented as examples to demonstrate its applications in details. Besides, the fluctuation effects are considered. The first order is Gaussian fluctuation theory, which can be used to determine the stability of the mean-field solution and predict the kinetics of unstable structure. The derivation and applications of Gaussian fluctuation theory are presented as well.     

Effects of colloid polydispersity on the phase behavior of colloid-polymer mixtures

We study theoretically the equilibrium phase behavior of a mixture of polydisperse hard-sphere colloids and monodisperse polymers, modeled using the Asakura-Oosawa model [S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255 (1954)] within the free volume approximation of H. N. W. Lekkerkerker, W. C. K. Poon, P. N. Pusey, A. Stroobants, and P. B. Warren [Europhys. Lett. 20, 559 (1992)]. We compute full phase diagrams in the plane of colloid and polymer volume fractions, using the moment free energy method. The intricate features of phase separation in pure polydisperse colloids combine with the appearance of polymer-induced gas-liquid coexistence to give a rich variety of phase diagram topologies as the polymer-colloid size ratio xi and the colloid polydispersity delta are varied. Quantitatively, we find that polydispersity disfavors fluid-solid against gas-liquid separation, causing a substantial lowering of the threshold value xi(c) above which stable two-phase gas-liquid coexistence appears. Phase splits involving two or more solids can occur already at low colloid concentration, where they may be kinetically accessible. We also analyze the strength of colloidal size fractionation. When a solid phase separates from a fluid, its polydispersity is reduced most strongly if the phase separation takes place at low colloid concentration and high polymer concentration, in agreement with experimental observations. For fractionation in gas-liquid coexistence we likewise find good agreement with experiment, as well as with perturbative theories for near-monodisperse systems.     

Phase coexistence in a polydisperse charged hard-sphere fluid: polymer mean spherical approximation

We have reconsidered the phase behavior of a polydisperse mixture of charged hard spheres (CHSs) introducing the concept of minimal size neutral clusters. We thus take into account ionic association effects observed in charged systems close to the phase boundary where the properties of the system are dominated by the presence of neutral clusters while the amount of free ions or charged clusters is negligible. With this concept we clearly pass beyond the simple level of the mean spherical approximation (MSA) that we have presented in our recent study of a polydisperse mixture of CHS [Yu. V. Kalyuzhnyi, G. Kahl, and P. T. Cummings, J. Chem. Phys. 120, 10133 (2004)]. Restricting ourselves to a 1:1 and possibly size-asymmetric model we treat the resulting polydisperse mixture of neutral, polar dimers within the framework of the polymer MSA, i.e., a concept that--similar as the MSA--readily can be generalized from the case of a mixture with a finite number of components to the polydisperse case: again, the model belongs to the class of truncatable free-energy models so that we can map the formally infinitely many coexistence equations onto a finite set of coupled, nonlinear equations in the generalized moments of the distribution function that characterizes the system. This allows us to determine the full phase diagram (in terms of binodals as well as cloud and shadow curves), we can study fractionation effects on the level of the distribution functions of the coexisting daughter phases, and we propose estimates on how the location of the critical point might vary in a polydisperse mixture with an increasing size asymmetry and polydispersity.     

The morphology of the monomer–polymer particle in styrene emulsion polymerization

A heterogeneous model for the monomer–polymer particle in styrene emulsion polymerization is presented. In this model, the growing particle consists of an expanding polymer-rich core surrounded by a monomer-rich shell which serves as the major locus of polymerization. This core-shell model was suggested by kinetic studies with continuously uniform latices which showed that the systems of interest were of the Smith-Ewart case II type but that the dynamic—as opposed to equilibrium swelling—particle monomer concentrations were continuously variable. Supporting evidence for the suggested morphology was obtained by electron microscope observation of ultrathin sections of latex particles.     

Examining the effect of chain length polydispersity on the phase behavior of polymer solutions with the statistical associating fluid theory (Wertheim TPT1) using discrete and continuous distributions

Polymers are naturally polydisperse. Polydispersity may have a large effect on the phase behavior of polymer solutions, in particular, on the liquid-liquid phase equilibria. In this paper, we determine the cloud and shadow curves bounded by lower critical solution temperatures for a number of polymer+solvent systems where the polymer is polydisperse in terms of molecular weight (chain length). The moment method [P. Sollich, P. B. Warren, and M. E. Cates, Adv. Chem. Phys. 116, 265 (2001)] is applied with the SAFT approach to determine cloud and shadow curves with continuous Schulz-Flory distributions. It is seen that chain length polydispersity always enhances the extent of liquid-liquid phase equilibria. The predicted cloud curves obtained for continuous distributions are very similar to those obtained for simple ternary mixtures with the same polydispersity index, while the corresponding shadow curves can be very different depending on the composition of the parent distribution. The ternary phase behavior can be used to provide an understanding of the shape of the cloud and shadow curves. Regions of phase equilibria between three liquid phases are found for ternary systems when the chain length distribution is very asymmetrical; such regions are not observed for Schulz-Flory distributions even in the case of a large degree of polydispersity.     

Calculation of the stability and of the phase equilibrium in the system polystyrene+cyclohexane+carbon dioxide based on equations of state

In order to calculate spinodals for polymer systems with an equation of state (EOS), we developed a stability theory using continuous thermodynamics. Here, the mixture considered consists of a polydisperse polymer and two monodisperse components as for example a solvent and a gas. We derived the spinodal equation on the base of the segment-molar Helmholtz energy of the mixture. As a result, a determinant similar to that of the monodisperse case is obtained, but the polydisperse polymer is identified by its weight average of the molecular weight. Furthermore, our paper provides the equations for the cloud-point curve derived with the aid of continuous thermodynamics. The final equations are applied to the system polystyrene+cyclohexane+carbon dioxide using the EOS of Sako, Wu and Prausnitz (SWP-EOS). For parameter fit and to prove the accuracy of the treatment, experimental data of the high-pressure equilibrium of the binary subsystems and of the ternary system were taken from literature.     

Effect of small amounts of methanol on the vapour—liquid equilibrium for the water-formaldehyde system

The total vapour pressures of three mixtures of the water-methanol-formaldehyde system have been measured over the range of temperatures 55–95°C. The formaldehyde composition was in the range 34–44 wt.%, while that of methanol was <1 wt.%. These measurements were carried out with the aim of demonstrating that, for studying the vapour—liquid equilibrium for the water—formaldehyde system, it is necessary to eliminate methanol to a greater extent than is usual.The experimental results compare favourably with those calculated by a predictive thermodynamic model which explains nonideality in the liquid phase in terms of chemical forces for the binary systems water—formaldehyde and methanol—formaldehyde and in terms of physical forces (Wilson equation) for the binary system water—methanol.The proposed model was also utilized in bubble-point calculations for the binary system methanol-formaldehyde and for the ternary system over a large range of compositions. The results were compared with literature data.     

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     

Prediction of polymer composition in batch emulsion copolymerization

Monomer partitioning in emulsion copolymerization plays a key role in determining composition drift and polymerization rates. The combination of recently developed thermodynamically based monomer partitioning relationships with mass balance equations, makes predictions of monomer partitioning in emulsion copolymerizations possible in terms of monomer mole fractions and monomer concentrations in the particle and aqueous phases. Using this approach, the effects of monomer to water ratios and polymer volumes on the monomer mole fraction within the polymer particle phase in a nonpolymerizing system at thermodynamic equilibrium can be determined. Comparison of these monomer partitioning predictions with experiments for the monomer system methyl acrylate—vinyl acetate shows good agreement. Furthermore, composition drift occurring in a polymerizing system as a function of conversion can be predicted if the assumption is made that equilibrium is maintained during reaction. Comparison of predictions with experimental results for emulsion copolymerizations of the monomer systems methyl acrylate—vinyl acetate and methyl acrylate—indene shows good agreement. © 1994 John Wiley & Sons, Inc.     

Isothermal vapor—liquid equilibrium of ternary mixtures formed by 1-propanol,acetonitrile and benzene

Vapor—liquid equilibrium data are presented for the ternary system 1-propanol-acetonitrile-benzene, at 45°C. The experimental vapor—liquid equilibrium results of the three constituent binary systems are well reproduced with the UNIQUAC associated-solution model and the ternary results are compared with those calculated from the model with binary parameters alone. Ternary prediction of liquid—liquid equilibria is given for the 1-propanol-acetonitrile-n-hexane and 1-propanol—acetonitrile-n-heptane systems at 25°C.     

pePC-SAFT: Modeling of polyelectrolyte systems 2. Aqueous two-phase systems

This work considers aqueous two-phase systems (ATPS) containing one polymer-polyelectrolyte as well as one salt. To model the liquid-liquid equilibria (LLE) of these systems, the recently presented model pePC-SAFT has been employed. ATPS containing poly(acrylic acid) of different degrees of neutralization or poly(vinyl pyrrolidone), respectively, were considered. The binary interaction parameters used between water-poly(acrylic acid) and water-poly(vinyl pyrrolidone) were adjusted to vapor-liquid equilibrium (VLE) data of these systems. ATPS consisting of poly(vinyl pyrrolidone)-water-sodium sulfate were predicted as function of temperature as well as of molar mass of the polymer. For poly(acrylic acid) systems, ATPS were predicted as function of charge density (degree of neutralization) for different types of salt. For these calculations, the polyelectrolyte model parameters were determined from the non-charged polymer whereas the effect of increasing charge density has been purely predicted by the model. Using this approach, it is possible to predict the shrinking of the liquid-liquid equilibrium region with increasing charging of the polyelectrolyte.     

Thermodynamic equilibrium of adsorbed phases

The types of phase equilibrium behavior for adsorbed binary mixtures that can be predicted by an equation of state (EOS) based on the lattice gas theory are investigated. The equilibrium conditions were obtained by solving the isofugacity equations between adsorbed phases. It is observed that the investigated EOS can predict complex behavior for adsorbed phases such as the existence of azeotropes, and retrograde and double retrograde phase transition phenomena, that are analogous to those found in bulk phase equilibrium. Furthermore, it was possible to find systems that presented phase equilibrium between two dense adsorbed phases, analogously to liquid–liquid equilibrium for bulk phases. Experimental data would be necessary to confirm the types of adsorbed phase behavior predicted by the calculations presented.     

Effects of Polydispersity on the Phase Behavior of Additive Hard Spheres in Solution

The ability to separate enzymes, nucleic acids, cells, and viruses is an important asset in life sciences. This can be realised by using their spontaneous asymmetric partitioning over two macromolecular aqueous phases in equilibrium with one another. Such phases can already form while mixing two different types of macromolecules in water. We investigate the effect of polydispersity of the macromolecules on the two-phase formation. We study theoretically the phase behavior of a model polydisperse system: an asymmetric binary mixture of hard spheres, of which the smaller component is monodisperse and the larger component is polydisperse. The interactions are modelled in terms of the second virial coefficient and are assumed to be additive hard sphere interactions. The polydisperse component is subdivided into sub-components and has an average size ten times the size of the monodisperse component. We calculate the theoretical liquid–liquid phase separation boundary (the binodal), the critical point, and the spinodal. We vary the distribution of the polydisperse component in terms of skewness, modality, polydispersity, and number of sub-components. We compare the phase behavior of the polydisperse mixtures with their concomittant monodisperse mixtures. We find that the largest species in the larger (polydisperse) component causes the largest shift in the position of the phase boundary, critical point, and spinodal compared to the binary monodisperse binary mixtures. The polydisperse component also shows fractionation. The smaller species of the polydisperse component favor the phase enriched in the smaller component. This phase also has a higher-volume fraction compared to the monodisperse mixture.     

Application of the sCPA equation of state for polymer solutions

Specific interactions, for example hydrogen bonding, dominate in numerous industrially important polymeric systems, both polymer solutions and blends. Typical cases are water-soluble polymers including biopolymers of special interest to biotechnology (e.g. the system polyethyleneglycol/dextran/water). Furthermore, most polymer blends are non-compatible and the requirement for compatible polymer pairs is often the presence of hydrogen-bonding interactions (e.g. polyvinylchloride/chlorinated polyethylene). In this work we give at first a short, comparative evaluation of existing thermodynamic models suitable for polymeric systems that take into account, explicitly, specific interactions like HB. The range of application of the models in terms of phase equilibria and their specific characteristics (accuracy of calculation, degree of complexity) are discussed. Finally, vapor–liquid equilibria (VLE) calculations for a number of polymer+solvent systems (including five different polymers) with a novel and very promising model are presented. This model is in the form of an equation of state that is (in its general formulation) non-cubic with respect to volume and has separate terms for physical and chemical interactions. The model has recently been proposed and has already been successfully applied to non-polymeric hydrogen-bonding systems (alcohol/water/hydrocarbons). This is the first time that it is extended to polymer solutions.     

Reaction-induced phase separation of pseudo-interpenetrating polymer networks in polydisperse polymer blends: a simulation study

We develop a minimal model for the process of reaction-induced phase separation in a polydisperse polymer blend. During the reaction, one component undergoes polymerization, leading to phase separation via spinodal decomposition. The effect that changing the final degree of polymerization has on the phase-separation process is studied. Finally an elastic energy term is included mimicking the cross-linking process and the generation of a semi-interpenetrating polymer network. We show that the scaling of the dominant lengthscale with time varies according to the reaction conditions.     

Optical absorption of the β‐phase of poly(9,9‐dioctylfluorene)

We outline a theory for the optical absorption of the β‐phase of poly(9,9‐dioctylfluorene) (PFO) that is based on the Frenkel exciton model. The absorption peak at 435 nm is attributed to polymer segments having torsion angles equal to π that are weakly perturbed by the presence of random monomer junctions with torsion angles equal to 0. The broad band below 435 nm is associated with disordered segments having a broad distribution of random torsion angles. The effects of small random deviations from π in the torsion angles are discussed. The calculations support the interpretation that the β‐phase is characterized by alternating segments of highly ordered and strongly disordered regions. PFO is a widely studied fluorene‐based polymer with interesting and potentially useful photophysical properties. In this work, Frenkel exciton states in the β‐phase of PFO are studied, and a two‐region model—weak torsional disorder and strong torsional disorder—is presented. The peak in the optical absorption at 435 nm is associated with the weakly disordered regions. The broad background in the absorption is attributed to the strongly disordered regions. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016 , 54, 1109–1111