Flory-Huggins Expression Research Articles

Integrating theory with machine learning for predicting polymer solution phase behavior

Flory-Huggins (FH) theory is foundational to understanding macro-phase separation in polymer solutions; however, its predictions often quantitatively disagree with experiment. Recent machine-learning (ML) methods have generated predictive models of phase behavior across a broad range of chemistries and state variables with uncertainty comparable to experiment, but they lack interpretability. In this work, we develop several hybrid frameworks that combine Flory-Huggins theory with ML to (i) further improve interpolation and extrapolation with less experimental data, as well as (ii) provide interpretability of the ML model. Using the well-studied binodal of polystyrene-cyclohexane as a case study, we compare data-derived ML models to hybrid models where the prediction is confined by a theoretical expression (theory-constrained model), or the feature vector input incorporates theoretical expressions (theory-informed model). Even though Flory-Huggins theory is imperfect, its incorporation improves performance under data sparse situations, such as when only 2 or 3 molecular masses are in the training set. However, neither approach to integrate theory provides advantages in accuracy or computational efficiency when greater coverage of the parameter space or quantities of experimental data are available, indicating that the greatest impact on predictability occurs in data sparse situations. More important than impacting the accuracy of predictions though, these hybrid models provide physical relationships, such as the molecular mass dependence of the critical point or the coefficients within a FH expression. They also afford determination of scaling behavior of theoretical parameters from cloud point experiments instead of more complicated methods. This aspect of physics-incorporated ML models enhances trust in predictions, as well as providing a systematic means to identify anomalous behavior, assess experimental data quality, and reveal unanticipated correlations among factors.

Chain-Increment Approach to the Mutual Miscibility of Polymers with All-Atom Molecular Simulation

The mutual miscibility of polymers was addressed with all-atom molecular dynamics (MD) simulation and the energy-representation theory of solvation. The chain-increment method was extended to polymer mixtures, and the mixing free energy was related to the incremental free energy for the constituent polymer species that is the free-energy change of turning on the interaction of a monomer in the tagged polymer chain with the surrounding molecules. The free energy was then computed for the mixing of polyethylene (PE) and poly(vinylidene difluoride) (PVDF) and of poly(vinyl alcohol) (PVA) and polyvinylpyrrolidone (PVP). It was seen that the incremental free energy is insensitive to the location of the monomer in the inner part of the polymer chain and depends on the mixing ratio of the polymer system in parallel with the average interaction energy of the incremented monomer with the surrounding polymers. The intermolecular interactions of each polymer species were further analyzed at atomic resolution. The PVDF–PVDF interaction was found to become less favorable (more positive) upon addition of PE in the mixture of PE and PVDF, making them immiscible with each other. PVP favors PVA more than PVP, on the other hand, leading to the miscibility of PVA and PVP. The connection of our formulation of the mixing free energy with the Flory–Huggins expression is also described.

Predicting wax appearance temperature and precipitation profile of normal alkane systems: An explicit co-crystal model

Wax precipitation is a major flow assurance issue for petroleum production, oil blending, downstream processing, and oil products usage. While numerous thermodynamic models have been proposed to calculate wax appearance temperature (WAT) and precipitation profile at temperatures below WAT, the existing models are correlative in nature and fail to comply with phase behavior experimentally observed for wax precipitation. This work presents a predictive thermodynamic model for wax precipitation based on an explicit co-crystal formation assumption for precipitated wax, a treatment consistent with experimental findings. With a simple Flory-Huggins expression accounting for the liquid phase nonideality and the treatment of lamella structure for wax precipitation below WAT, the model satisfactorily predicts WAT at low pressures for multicomponent paraffin mixtures and wax precipitation amounts and contents as the temperature drops below WAT.

50th Anniversary Perspective: Phase Behavior of Polymer Solutions and Blends

We summarize our knowledge of the phase behavior of polymer solutions and blends using a unified approach. We begin with a derivation of the Flory–Huggins expression for the Gibbs free energy of mixing two chemically dissimilar polymers. The Gibbs free energy of mixing of polymer solutions is obtained as a special case. These expressions are used to interpret observed phase behavior of polymer solutions and blends. Temperature- and pressure-dependent phase diagrams are used to determine the Flory–Huggins interaction parameter, χ. We also discuss an alternative approach for measuring χ due to de Gennes, who showed that neutron scattering from concentration fluctuations in one-phase systems was a sensitive function of χ. In most cases, the agreement between experimental data and the standard Flory–Huggins–de Gennes approach is qualitative. We conclude by summarizing advanced theories that have been proposed to address the limitations of the standard approach. In spite of considerable effort, there is no con...

A new thermodynamic model of volume changes in temperature-sensitive polymer gels

In the classical swelling theory a polymer gel is a very (infinitely) large cross-linked polymer molecule that encages the solvent molecules. Here, a new swelling model is developed. Firstly, this model permits many cross-linked polymer molecules with finite molar mass instead of a single one. The gel formation is described by a huge number of consecutive swelling equilibria incorporating a solvent molecule by solvent molecule. Thus, the gel is a polydisperse mixture of cross-linked polymer molecules encaging different amounts of solvent. The polydispersity is expressed by a gel-distribution function, the width of which decreases with increasing molar mass of the cross-linked polymer. In the limit of infinite molar mass the new swelling theory is equivalent to the classical one. Secondly, different to the classical treatment, the polymer network is considered to be finitely extensible. Both theories are applied to hydrogels of poly(N-isopropylacrylamide). The mixing contribution of the Helmholtz energy is described by a modified Flory–Huggins expression, the parameters of which are fitted to cloud-point data of a solution of a linear poly(N-isopropylacrylamide). For the elastic contribution the phantom network theory is applied. The new model calculates the swelling degree as a function of temperature nearly perfect in comparison with the experimental data. The classical theory fails in the range of high swelling degrees. Furthermore, the classical treatment predicts a first-order transition between a swollen and a shrunken gel. The new model predicts also a sharp but continuous transition.

Representation of vapor–liquid equilibria of aqueous polymer–salt solutions by a new modified segment-based Wilson model

A new model for representation of the excess Gibbs energy of polymer–electrolyte solutions is proposed. The excess Gibbs energy of a polymer–electrolyte solution is expressed as a sum of contributions of a combinatorial, a long-range and a short-range excess Gibbs energy term. The Flory–Huggins expression and the Pitzer’s extension of the Debye–Hückel function are used respectively as a combinatorial and a long-range contribution to the excess Gibbs energy. A new expression based on the local composition concept, which is the modified Wilson model, is developed to account for the short-range contribution to the excess Gibbs energy. The model provides a versatile and flexible thermodynamic framework for both correlating and predicting the phase equilibrium of electrolyte solutions, polymer solutions and complex systems containing both electrolytes and polymers. The utility of the model is demonstrated with successful representation of vapor–liquid equilibrium of several PEG or PPG-salt–H 2O systems at different polymer molar masses. It was found that, the absolute relative percentage deviations of water activity for polymer solutions, salt solutions and polymer + salt solutions are about 0.1%, 0.04% and 0.1%, respectively. Results are compared with those obtained from the Wilson model.

Water activities of ternary mixtures of PPG425 + K 2CO 3 + H 2O and PPG425 + Na 2CO 3 + H 2O at 298.15 K: Experiments and correlation

Water activities of polypropylene glycol (PPG425) + K 2CO 3 + H 2O and PPG425 + Na 2CO 3 + H 2O systems have been measured using isopiestic method at 298.15 K. The water activities were correlated successfully with segment-based local composition Wilson model and extended form of Chen-NRTL models. The Pitzer–Debye–Hückel (PDH) equation and the Flory–Huggins expression were, respectively, used for the long-range interactions and the combinatorial contribution in the models. The parameters of the models for these systems were calculated. Good agreement was obtained with the experimental water activity data with both models.

A modified segment-based nonrandom two-liquid model for the calculation of vapor–liquid equilibrium of aqueous polymer–salt solutions

A new model for representation of the excess Gibbs energy of polymer–electrolyte solutions is proposed. The excess Gibbs energy of a polymer–electrolyte solution is expressed as a sum of contributions of a combinatorial, a long-range and a short-range excess Gibbs energy term. The Flory–Huggins expression and the Pitzer's extension of Debye–Hückel function are used, respectively, as a combinatorial and a long-range contribution to the excess Gibbs energy. A new expression based on the local composition concept, which is the modified nonrandom two-liquid (NRTL) model, is developed to account for the short-range contribution to the excess Gibbs energy. The model provides a versatile and flexible thermodynamic framework for both correlating and predicting the phase equilibrium of electrolyte solutions, polymer solutions and complex systems containing both electrolytes and polymers. The utility of the model is demonstrated with successful representation of vapor–liquid equilibrium of several PEG or PPG–salt– H 2 O systems at different polymer molar masses. Results are compared with those obtained from the NRTL model.

New local composition model for polymer solutions

A new segment-based local composition model is presented for the excess Gibbs energy of polymer solutions. The excess Gibbs energy of a polymer solution is expressed as a sum of contributions of a combinatorial and a residual excess Gibbs energy term. The truncated Freed correction to Flory–Huggins expression as first correction for the configurational entropy of mixing is used as a combinatorial contribution to the excess Gibbs energy. A new expression based on the local composition concept, which is the NRF-Wilson model, is developed to account for the contribution of the residual excess Gibbs energy. The main difference between this model and the segment-based NRTL model available in the literature is in the nature of the short-range energy parameter and their references states. The utility of the model is demonstrated with the successful representation of the vapor–liquid equilibria, density and viscosity of several polymer solutions. The results show that, the model is valid for the whole range of polymer concentration, from dilute solution up to saturation. The results are compared with those obtained from the segment-based NRTL and segment-based Wilson models. The model presented in this work produces better results.

Extension of the Wilson model to multicomponent polymer solutions: applications to polymer–polymer aqueous two-phase systems

The Wilson model for non-electrolytes has been extended for the representation of the excess Gibbs energy of multicomponent polymer solutions. The excess Gibbs energy of a polymer solution is expressed as a sum of contributions of a combinatorial and a residual excess Gibbs energy term. The contribution of the combinatorial excess Gibbs energy is represented by the Flory–Huggins expression. A new expression based on the local composition concept, which is a modified Wilson model, has been developed to account for the contribution of the residual excess Gibbs energy. The model has been applied to several polymer–polymer aqueous two-phase systems. The results show that the model is reliable both for correlation and prediction of the liquid–liquid equilibrium phase diagrams of aqueous two-phase systems at different polymer molecular weights.

Liquid–liquid demixing in a binary polymer blend driven solely by the component-selective crystallizability

On the basis of an extended Flory–Huggins expression for the mixing free energy of the long-chain binary polymer blend, we showed that liquid–liquid demixing can be driven solely by the propensities of crystallization differing with respect to components. Demixing prior to crystallization on cooling was demonstrated by dynamic Monte Carlo simulations of the lattice model for a symmetric polymer blend where only one component is crystallizable. We discussed the implications of this principle to several experimental observations including the (in)compatibility of polymer chains with different stereochemical compositions, the memory effect of polymer crystallization, and the spinodal decomposition at an early stage of polymer crystallization.

UCST behavior in blend systems of isotactic polystyrene/poly(4-methyl styrene) in comparison with atactic polystyrene/poly(4-methyl styrene)

Blends of poly(4-methylstyrene) (P4MS) with polystyrene (iPS) exhibit an upper critical solution temperature (UCST) at ca. 270 °C. The overall phase behavior and trend of variation in the phase diagrams for the iPS/P4MS blend system with respect to molecular weights of iPS is similar to an earlier studied blend system of atactic PS with P4MS. This suggests that the crystal phase-related tacticity and crystallinity in iPS does not influence the amorphous phase behavior and UCST behavior of the polymer mixtures. A model based on a modified Flory-Huggins expression for binary interactions was constructed to describe the UCST-type behavior of the iPS/P4MS blend and to compare the qualitative effects of molecular weights on iPS/P4MS blend vs. atactic PS/P4MS systems.

Binary versus ternary interactions in a completely miscible three‐polymer blend system: Poly(ϵ‐caprolactone) with two different methacrylic polymers

AbstractA truly miscible ternary miscible blend consisting of poly(ϵ‐caprolactone) (PCL), poly(phenyl methacrylate), and poly(benzyl methacrylate) (PBzMA) was discovered. The three‐polymer blend system was completely miscible within the entire composition range at ambient temperature up to about 150 °C, and ternary phase diagrams at increasing temperatures were characterized and interpreted. A ternary‐interaction model based on the modified Flory–Huggins expression was used to describe the phase diagrams with the individual binary interaction strengths. The model fitted well with the experimental‐phase diagram for the ternary blend system at T = 250 °C, where the binary PCL‐PBzMA blend system is on the critical points of phase separation. Interpretation of discrepancy between the model and experimental at other temperatures was handled with an empirical approach. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 747–754, 2002

Non-equilibrium chemical potential and stress-induced migration of polymers in tubes

A non-equilibrium chemical potential depending on the viscous pressure tensor is used to describe shear-induced diffusion in polymer solutions flowing along cylindrical tubes. Our results generalize previous ones in three main aspects: a Flory–Huggins expression for the equilibrium contribution to the chemical potential is used instead of the ideal-gas like expression, the full expression for the steady-state compliance of the solution is taken into account instead of only the polymer contribution, and the influence of the solute molecular mass is explicitly considered. As a qualitatively new result of considerable practical interest, it stands the prediction that in some circumstances, a dynamical instability may appear, which accelerates and enhances the separation process.

Morphology, Phase Diagrams, and UCST Behavior in Blends of Polystyrene with Poly(4-methylstyrene)

The upper-critical-solution-temperature (UCST) phase behavior of blends composed of two styrenic polymers, poly(4-methylstyrene) (P4MS) and polystyrene (PS), and the morphology evolution during the de-mixing/mixing phase transition were examined. The blend system is immiscible at ambient temperature and remains so until it becomes miscible upon heating at high temperatures, with an upper critical solution temperature (UCST) at ≈ 235°C. The UCST-phase diagrams were constructed by experimental observations of blend morphology using both optical and scanning electron microscopy. Interesting phase-in-phase micro-domains existed in the immiscible PS/P4MS blend although it exhibited a single Tg. This apparent contradiction was resolved. For expanding the interpretation of the effects of differing molecular weights, a model based on a modified Flory-Huggins expression for binary interactions was constructed to describe the UCST-type phase behavior of the PS/P4MS blend. The model predicts that the UCST is below the ambient temperature for a PS/P4MS blend system if PS possesses Mw = 20 000 g/mol or lower, which essentially means that the blends of lower PS Mw are miscible at ambient or any temperatures above. This model was verified experimentally.

A THERMODYNAMIC MIXED-SOLID ASPHALTENE PRECIPITATION MODEL

ABSTRACT A simple model for the prediction of asphaltene precipitation is proposed. The model is based on an equation of state and uses standard thermodynamics, thus assuming that the precipitation phenomenon is a reversible process. The solid phase is treated as an ideal multicomponent mixture. An activity coefficient model based on the athermal Flory-Huggins expression was included to account for solid phase non-ideality, but was found to have little effect on the results. Only a fraction of each of the heaviest components are considered potentially solid forming. This approach is based on the assumption, that out of the total PNA composition of a crude, the asphaltenes and resins are mainly aromatic-naphtenic in their nature. The predictions shows the correct qualitative behavior with respect to temperature and pressure, but due to lack of data the model was not evaluated quantitatively with respect to these effects

Kinetic theory and the Flory–Huggins approximation

A kinetic approach is used to examine the structure of the approximations that lead to the Flory–Huggins expression for the free energy of mixing of a polymer solution. The kinetic model that we study can be solved exactly in one dimension, and also on an infinite-dimensional lattice, in which case it reduces to the Flory–Huggins formula. An approximate solution that interpolates between these limits gives results for a three-dimensional lattice that differ rather little from those of the original Flory–Huggins theory.

Thermodynamics of complex formation in chloroform-oxygenated solvent mixtures

Complex formation equilibria in binary mixtures of chloroform with dipropyl ether (PE), diisopropyl ether (IPE), methyl tert-butyl ether (MBE), tetrahydrofuran (THF). 1,4-dioxane (DOX), acetone (AC), and methyl acetate (MA) have been analyzed in detail using several association models. Vapor-liquid equilibria, excess enthalpy and excess heat capacity data for these mixtures have been correlated using a multiproperty global fitting procedure. The thermodynamic properties for chloroform +PE, +IPE, +MBE, +AC, and +MA are best correlated using the ideal association model while for chloroform +THF and +DOX the best model is an athermal solvation model where the Flory-Huggins expression for the species activity coefficients is considered. The model parameters, i.e., the equilibrium constant, enthalpies and heat capacities of complexation, were found to be reliable, well representing the chloroform-oxygenated solvent H-bonded complexes. A detailed discussion is given on the test proposed by McGlashan and Rastogi to decide whether the solution contains only 1∶1 complexes or 2∶1 complexes as well. The complex formation equilibria in chloroform mixtures is compared to those previously examined for halothane (2-bromo-2-chloro-1,1,1-trifluoroethane) mixed with the same oxygenated solvents. It was found that the H-bonds formed by halothane are stronger than those formed by chloroform.

Thermodynamic study of complex formation by hydrogen bonding in halogenoalkane–oxygenated solvent mixtures. Halothane with propyl ether, isopropyl ether, 1,4-dioxane and 2,5-dioxahexane

Vapour–liquid equilibria (VLE) for binary mixtures of halothane with propyl ether (PE), isopropyl ether (IPE), 1,4-dioxane (DOX) and 2,5-dioxahexane (monoglyme)(G1) have been measured at 293.15 K using a gas chromatographic headspace analysis method. Excess molar heat capacities and excess molar volumes at 298.15 K were also measured for halothane–G1. Complex-formation equilibria for these four halothane–oxygenated solvent mixture have been analysed in detail using several association models. The data reported here, together with previously published excess enthalpies and excess heat capacities, have been correlated using a multiproperty global fitting procedure. The thermodynamic properties for halothane–PE and –IPE are best correlated using the ideal association model while for halothane–DOX and –G1 the best model is the athermal solvation model where the Flory–Huggins expression for the activity coefficients is considered. In halothane–PE and –IPE mixtures there are only 1:1 H-bonded complexes while for halothane–DOX and –G1 the 2:1 complexes are also present. The model parameters, i.e. the equilibrium constants, enthalpies and heat capacities of complexation were found to be reliable, truly representing the halothane–oxygenated solvent H-bonded complexes. Enthalpies of complexation were also calculated using quantum mechanical methods and found to be in good agreement with those obtained from thermodynamics. Previously, we have presented an analogous thermodynamic study for mixtures of halothane with four other oxygenated solvents viz. acetone, methyl acetate, tetrahydrofuran and methyl tert-butyl ether. Here, a comparative analysis of the thermodynamic behaviour of the eight halothane–oxygenated solvent mixtures is presented. It was found that the enthalpy for the complexation reaction A + B ⇔ AB correlated with the oxygen surface area per H bond as given by PC MODEL calculations. Finally, a clear correlation between the solvent Kamlet–Taft β parameter (H-bonding acceptor ability) and equimolar GE, HE, TSE and TCEp was found: as β increases GE, HE and TSE are seen to become more negative and TCEp more positive, i.e. as the basicity of the solvent increases the halothane–oxygenated solvent H bonds are stronger and the degree of organization or order in solution increases. This correlation might be used to make rough estimates of the excess functions for other halothane–oxygenated solvent mixtures.

Modeling of thermodynamic properties of associated solutions with equilibrium constants defined on activities

Association constants (for example demerisation constants of carboxylic acids) defined in concentration units are strongly dependent on concentration and solvent. Depending on the solvent, these equilibrium constants vary two orders of magnitude at infinite solution. This is due to the differences in size and the physical interaction between monomer, associated and solvent molecules. This work presents a thermodynamic model to describe the properties ( VLE, H E, K x = x A i x A i ) of associated mixtures (carboxylic acids-inert solvent) where the equilibrium constants are based on activities and not on concentration units. The physical part of the activity coefficients is expressed by a group contribution method (distinguishing between monomers, chemical bonded and solvent molecules) based on the Bragg-Williams approximation. For the combinatorial part, the Flory-Huggins expression is used. In addition to being able to describe VLE-and H E-data as good as modified UNIFAC by Gmehling et al. (1993), the model yields values for K x,∞ ( K x at infinite dilution) of the correct order.