Calculation of phase diagrams not requiring the derivatives of the Gibbs energy demonstrated for a mixture of two homopolymers with the corresponding copolymer

A method is presented which allows the calculation of phase diagrams (spinodal, binodal and tie lines) on the basis of the Gibbs energy of mixing ΔG. No derivatives of ΔG with respect to the composition variables are required. This method is particularly useful in cases where the composition dependence of ΔG is very complex and no analytical representation of the derivatives can be given. The method is applied to a ternary mixture of two homopolymers with a copolymer consisting of the same monomers. The sequence distribution of the copolymer is kept constant between random and purely alternating, and phase diagrams are calculated for different chemical compositions of the copolymer. The complex phase separation behavior resulting for a 1 : 1 copolymer becomes much simpler as one monomeric unit starts to predominate in the copolymer.     

The Fractionation of Copolymers by Chemical Composition

Abstract

Phase diagrams for the system of methyl ethyl ketone, cyclo-hexane, and styrene-acrylonitrile copolymer were determined. The phase diagrams indicate that the copolymer may be fractionated by chemical composition in this system. Discussions of the thermodynamics are also presented, to show that copolymers can effectively be fractionated into fractions of different compositions if a system can be found in which the difference between the Flory interaction parameters (x parameters) of two constituents of the copolymer with solvent is sufficiently large. Theoretically, the fractionation of copolymer must always occur to a certain extent, depending both on chemical composition and molecular weight. The composition fractionation results of styrene-acrylonitrile copolymers are given to confirm the discussions.     

Nonlinear depression of the lower critical solution temperatures in aqueous solutions of thermo-sensitive random copolymers

We develop a theoretical model of cooperative hydration to clarify the molecular origin of the observed nonlinear depression of the lower critical solution temperature (LCST) in the aqueous solutions of thermosensitive random copolymers and find the monomer composition at which LCST shows a minimum. Phase diagrams of poly(N-isopropylacrylamide-co-N,N-diethylacrylamide) copolymer solutions are theoretically derived on the basis of the theory of cooperative hydration by introducing the microscopic structure parameter η which characterizes the distribution of the monomer sequences along the chains. We compared them with the experimental data of LCST of random copolymers with various monomer compositions and also of the diblock copolymers with equimolar monomer composition. The transition temperature shifts to lower than those of homopolymer counterparts when the monomer sequence of the chains has an alternative tendency. On the contrary, for the blocky polymers such as diblock copolymers, the transition temperature remains almost the same as those of the homopolymers. Thus, the nonlinear effect in phase separation appears when the average block length of the copolymers is shorter than the average sequence length of the cooperative hydration. The degree of hydration is calculated as a function of the temperature and polymer concentration for varied distribution of the copolymer compositions. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2013, 51, 1112–1123     

Phase equilibria of poly(ɛ-caprolactone)-<Emphasis Type="Italic">co</Emphasis>-poly(acryl methacrylate)-methylene chloride systems and the phase structure of polymer binary mixtures

Isothermal cross sections of amorphous separation diagrams of the ternary systems poly(ɛ-caprolactone)-co-poly(acryl methacrylate) Eudragite RS-methylene chloride and poly(ɛ-caprolactone)-co-poly(acryl methacrylate) Eudragite E-methylene chloride are obtained via the experimental methods of turbidity points and analytical transmission electron microscopy and the calculation semi-empirical method of constructing boundary lines of phase diagrams. It is shown that the composition of copoly(acryl methacrylate) affects the length of the solubility region in the ternary phase diagram and the character of the anisotropy of films prepared via solvent evaporation. The determination of coordinates of figurative points corresponding to the completion of phase separation shows that the mechanism of phase separation for both systems is of the spinodal type. The specific orientation of polycaprolactone crystallites during film preparation from the poly(ɛ-caprolactone)-Eudragite RS mixture is determined by the morphology of the film formed during solvent evaporation and the phase composition.     

Free‐energy model of asymmetry in side‐chain liquid‐crystalline diblock copolymers

Side‐chain liquid‐crystalline‐b‐amorphous copolymers combine the thermotropic ordering of liquid crystals (LCs) with the physics of block copolymer phase segregation. In our earlier experiments, we observed that block copolymer order–order and order–disorder transitions could be induced by LC transitions. Here we report the development of a free‐energy model to understand the interplay between LC ordering and block copolymer morphology in an incompressible melt. The model considers the interaction between LC moieties, the stretching of amorphous chains from curved interfaces, interfacial surface contributions, and elastic deformation of the nematic phase. The LC block is modeled with Wang and Warner's theory, in which nematogens interact through mean‐field potentials, and the LC backbone is modeled as a wormlike chain. Free energy is estimated for various morphologies: homogeneous, lamellar, cylinder micelle, and spherical micelle. Phase diagrams were constructed by iteration over temperature and composition ranges. The resulting composition diagrams are highly asymmetric, and a variety of first‐order transitions are predicted to occur at the LC clearing temperature. Qualitatively, nematic deformation energies destabilize curved morphologies, especially when the LC block is in the center of the block copolymer micelle. The thermodynamics of diblocks with laterally attached, side‐on mesogens are also explored. Discussion focuses on how well the model captures experimental phenomena and how the predicted phase boundaries are affected by changes in polymer architecture. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2671–2691, 2001     

Phase-Separation Phenomena of Copolymer Solutions and Fractionation of Copolymers by Chemical Composition

Precipitation temperature-total polymer concentration diagrams for toluene solutions of two styrene-acrylonitrile copolymers different in chemical composition and their mixtures were determined and then triangular phase diagrams of this system were constructed from these diagrams. It is speculated from the triangular phase diagrams and experimentally shown that the copolymer may be effectively fractionated by chemical composition in this system.     

Systematic investigation of the global phase behavior of polymer-solvent systems in the density-density plane

The phase behavior of fluid mixtures is understood by the critical lines in fluid-gas diagrams. We investigated the critical lines of polymer-solvent systems at the mathematical double point, where two critical lines meet and exchange branches, and its environment within the framework of a model that combines the lattice gas model of Schouten, ten Seldam and Trappeniers with the Flory-Huggins theory. The critical lines are expressed as a function of x₁ and x₂, the density of type 1 polymer molecules and the density of type 2 polymer molecules, respectively; in this way global phase diagrams are presented and discussed in the density-density plane. Density-density plots are preferable when studying the differences in behavior of different classes, since they enable us to follow the connectivities in a systematic way. In this study the connectivity of critical lines at the mathematical double point and its around is investigated in detail. We also discuss the topology of the critical lines according to the Sadus classification scheme for ternary mixtures.     

PREPARATION OF EVOH MICROPOROUS MEMBRANES via THERMALLY INDUCED PHASE SEPARATION USING BINARY SOLVENTS

Microporous ethylene-vinyl alcohol copolymer （EVOH） flat membranes and hollow-fiber membranes with 38 mol% ethylene content were prepared via thermally induced phase separation （TIPS） using the mixture of 1,4-butanediol and poly（ethylene glycol）（PEG400） as diluents. Effects of the ratio of 1,4-butanediol to PEG400 on the phase diagrams, phase separation mechanism and membrane morphology were studied by small angle light scattering （SALS） measurements, differential scanning calorimetry （DSC）, and scanning ele~：tron microscopy （SEM）. It was found that by varying the composition of the binary solvent, the phase diagrams and membrane morphology can be controlled successfully. Moreover, the phase diagrams showed that broader regions of Liquid-Liquid （L-L） phase separation were obtained, as well as closer distances between L-L phase separation lines and Solid-Liquid （S-L） phase separation lines, Interconnected structures observed both in the flat membrane and hollow fiber membrane consist with the above results.     

Simultaneous interpenetrating networks of a polyurethane and poly([methyl methacrylate]-stat-[N,N-dimethylacrylamide])

In a previous study, tetrahedron metastable phase diagrams were presented for a model simultaneous interpenetrating network (SIN) system of cross-polyurethane-inter-cross-poly(methyl methacrylate) (PU-PMMA). One triangular face of the overall tetrahedron diagram represented the ternary system MMA-PMMA-“U”, wherein “U” denotes the monomer/prepolymer mixture for the PU. In this article, a comonomer, N,N-dimethylacrylamide (DMA), is incorporated into the PMMA network. Thus, the above-mentioned ternary system is altered to “A”-PA-“U,” where “A” denotes the acrylic monomer mixture [MMA + DMA] and PA denotes the resulting copolymer. Glass transitions of fully cured samples were determined by dynamic mechanical spectroscopy (DMS). Phase separation was determined by the onset of turbidity, and gelation of the first gelling polymer was determined by the sudden resistance of the system to flow. The critical point, representing simultaneous phase separation and PA gelation, divides the overall composition for the reaction mixture (and the final SIN) into two parts. For one, gelation of the acrylic network precedes phase separation, and vice versa for the other part. In the absence of DMA in the PA network, the gelation-first region is very narrow, but with increasing amounts of copolymerized DMA, the critical point moves along the triangular face to increase the working area of the gelation-first region. © 1996 John Wiley & Sons, Inc.     

Effects of poly(methyl methacrylate)-block-poly(vinyl acetate) copolymer on the spinodal decomposition of corresponding homopolymer blends

Effects of adding a small amount of poly(methyl methacrylate)-block-poly(vinyl acetate) (PMMA-b-PVAc) to poly(methyl methacrylate)/poly(vinyl acetate) (PMMA/PVAc) blends with a lower critical solution temperature (LCST) phase diagram on the kinetics of late-stage spinodal decomposition (SD) were investigated by time-resolved light scattering at 160°C. It is found that the coarsening process of the structure was slowed down or accelerated upon addition of PMMA-b-PVAc depending on the composition of the block copolymer and the blend. The effect of the block copolymer on the domain size were interpreted as compatibilizing and incompatibilizing effects of the block copolymer on PMMA/PVAc blends based on the evaluation of changes in the stability limits of PMMA/PVAc with the addition of block copolymer using random phase approximation (RPA).     

Modeling the high-pressure phase equilibria of carbon dioxide–triglyceride systems: A parameterization strategy

The group contribution equation of state (GC-EOS) has been used in several published works to correlate or predict the high-pressure phase equilibria of a variety of systems of practical interest. Nevertheless, quantitative and even qualitative disagreement among predictions and experimental data has been detected in mixtures of CO₂ with heavy compounds, such as triglycerides, when operating at high pressure. For instance, phase split up to indefinitely high pressures has been computed, when the observed experimental behavior shows full miscibility at sufficiently high pressure. In the present work, we study the influence on calculated critical lines and solubilities (Pxy diagrams) of the group-based interaction parameters k_ij, for the interactions of CO₂ with both, the triglyceride (TG) group and the paraffinic groups. Based on such study, we propose a parameterization procedure that improves upon the conventional parameter regression practice. The distinguishing feature of such procedure is the repeated observation of the global phase equilibrium behavior, studying in particular the effect of the group–group interaction parameters on critical lines, on the composition of the phases at equilibrium along liquid–liquid–vapor lines, and on selected isothermal or isobaric phase equilibrium diagrams. For the case of the non-randomness parameter, we use a universal positive value, more consistent with its physical meaning.     

Upper critical solution temperature type phase behavior of poly(styrene‐co‐acrylonitrile) blends with poly(styrene‐co‐n‐phenyl maleimide) and their interaction energies

To enhance the heat resistance of poly(styrene‐co‐acrylonitrile‐co‐butadiene), ABS, miscibility of poly(styrene‐co‐acrylonitrile), SAN, with poly(styrene‐co‐n‐phenyl maleimide), SNPMI, having a higher glass transition temperature than SAN was explored. SAN/SNPMI blends casted from solvent were immiscible regardless of copolymer compositions. However, SNPMI copolymer forms homogeneous mixtures with SAN copolymer within specific ranges of copolymer composition upon heating caused by upper critical solution temperature, UCST, type phase behavior. Since immiscibility of solvent casting samples can be driven by solvent effects even though SAN/SNPMI blends are miscible, UCST‐type phase behavior was confirmed by exploring phase reversibility. When copolymer composition of SNPMI was fixed, the phase homogenization temperature of SAN/SNPMI blends was increased as AN content in SAN copolymer increased. To understand the observed phase behavior of SAN/SNPMI blend, interaction energies of blends were calculated from the UCST‐type phase boundaries by using the lattice‐fluid theory combined with a binary interaction model. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 1131–1139, 2008     

Investigation of the global phase behavior of polymer mixtures in the shield region

This paper is a contribution of our systematic investigation of the global phase behaviors of the chain molecules mixtures, i.e., polymer mixture solutions. The phase behavior of fluid mixtures is understood by the critical lines in fluid-gas diagrams. The critical lines of binary fluid system may, under circumstances, exhibit closed loops in the critical lines. A distinction is made between free critical loops, as described by type VI in the Scott and van Konynenburg classification, and "rooted" critical loops, as found in the shield region. We define rooted loops as closed critical lines that are attached to the critical line structure by means of unstable critical line. We obtain the rooted loops in the global phase diagrams of the polymer mixture solutions within the framework of a model that combines the lattice gas model of Schouten, ten Seldam and Trappeniers with the Flory-Huggins theory, and we present the influence of the chain length of long molecules on the rooted critical loops. We present the results in the density-density and the temperature (T)-pressure (P) planes in detail.     

Phase transition of aqueous solutions of poly(N,N-diethylacrylamide-co-acrylic acid) by differential scanning calorimetric and spectrophotometric methods

 The phase transition of aqueous solutions of poly(N,N-diethylacrylamide-co-acrylic acid) (DEAAm–AA) is studied by differential scanning calorimetry (DSC) and UV–vis spectrophotometry. The copolymer aqueous solutions are shown to have well-defined lower critical solution temperatures (LCSTs). The LCST values obtained from the maximum of the first derivatives of the DSC and optical transition curves agree well. DSC can be used to measure the phase-transition temperature of more dilute polymer solutions. On increasing the AA composition in the copolymers, the LCST values of the copolymer increase, then decrease at higher AA composition. For the aqueous solution of the copolymers, the transition curve obtained by the spectrophotometric method is highly wavelength dependent. The LCST values are found to be concentration-dependent. The changes in the heat of the phase transition of the copolymer solutions measured from DSC are lower than that of the homopolymer PDEAAm solution. This is consistent with the suggestion that the polymer chains of the copolymers collapsed only partially at temperatures above the LCST. The added salt (sodium chloride) decreases the transition temperature of the polymer solution. Received: 14 November 2000 Accepted: 15 January 2001     

Critical points of adsorbed phases using a 2D lattice gas equation of state

The types of critical phase diagrams for adsorbed binary mixtures that can be predicted by an equation of state (EOS) based on a two-dimensional lattice gas theory are investigated. The search for critical point conditions was done using the Hicks and Young algorithm, switching to the Heidemann and Khalil algorithm in the close of vicinity of a critical point. We observed that the model can predict critical points that represent the conditions in which a vapor-like and a liquid-like adsorbed phases collapse. The critical diagrams were classified using an analogy with the van Konynenburg and Scott scheme for classifying the critical behavior of binary bulk mixtures. The original classification scheme is based on the critical lines on the pressure–temperature plane; we used a straightforward extension based on the critical lines on the spreading pressure–temperature plane. Five of the six types of phase behavior classified by von Konynenburg and Scott were observed using this thermodynamic model. The transitions between the types of phase diagram were also observed in temperature–mole fraction projections. These results extend previous observations that suggested the possibility of very interesting phase behaviors for adsorbed mixtures. However, experimental data would be necessary to confirm the predicted types of critical diagrams.     

Cloud-Point Curve for the System Copoly(Ethylene-Vinyl Acetate) Plus Methyl Acetate. Measurement and Prediction by Continuous Thermodynamics

The cloud-point curve for the system copoly(ethylene-vinyl acetate) plus methyl acetate has been measured by a simple visual method. The critical point was determined by using the phase volume ratio method. The method of continuous thermodynamics was applied for thermodynamic treatment. The composition of the copolymer is described by a divariate distribution function assumed as a generalized Stockmayer distribution. The activity coefficients were obtained with the aid of the Huggins Chi -parameter concept assuming Chi to be a quadratic polynomial with respect to the weight-average chemical composition of the copolymer. The three model parameters were calculated from the critical point and the slope of the cloud-point curve at the critical point. The cloud-point curve and the shadow curve were predicted from these parameters. The cloud-point curve shows qualitative agreement with experimental data.     

Phase equilibria in ternary polymer blends

The phase states and rheological properties of blends of three polymers??polystyrene, poly(methyl methacrylate), and the styrene-acrylonitrile copolymer??in the common solvent chloroform are studied. The phase diagrams are constructed and the positions of spinodals are determined via the method of turbidity points. The effect of the third polymer on the compatibility of the binary blend obeys Prigogine??s rule; that is, it is determined by the solubility of the added polymer in the first two components. The extremum composition dependence of rheological properties of ternary polymer systems in the vicinity of the separation point (the metastable region) is found. Through the method of convex-shell construction, the phase diagrams are calculated.     

Stress–strain behavior of low‐density polyethylene/poly(methyl methacrylate) blends with modulated interfaces with a hydrogenated polybutadiene‐block‐poly(methyl methacrylate) diblock copolymer

The stress–strain diagrams and ultimate tensile properties of uncompatibilized and compatibilized hydrogenated polybutadiene‐block‐poly(methyl methacrylate) (HPB‐b‐PMMA) blends with 20 wt % poly(methyl methacrylate) (PMMA) droplets dispersed in a low‐density polyethylene (LDPE) matrix were studied. The HPB‐b‐PMMA pure diblock copolymer was prepared via controlled living anionic polymerization. Four copolymers, in terms of the molecular weights of the hydrogenated polybutadiene (HPB) and PMMA sequences (22,000–12,000, 63,300–31,700, 49,500–53,500, and 27,700–67,800), were used. We demonstrated with the stress–strain diagrams, in combination with scanning electron microscopy observations of deformed specimens, that the interfacial adhesion had a predominant role in determining the mechanism and extent of blend deformation. The debonding of PMMA particles from the LDPE matrix was clearly observed in the compatibilized blends in which the copolymer was not efficiently located at the interface. The best HPB‐b‐PMMA copolymer, resulting in the maximum improvement of the tensile properties of the compatibilized blend, had a PMMA sequence that was approximately half that of the HPB block. Because of the much higher interactions encountered in the PMMA phase in comparison with those in HPB (LDPE), a shorter sequence of PMMA (with respect to HPB but longer than the critical molecular weight for entanglement) was sufficient to favor a quantitative location of the copolymer at the LDPE/PMMA interface. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 22–34, 2005