Systematic investigation of global phase behavior of polymer mixtures in the pressure-temperature plane

We investigate the critical lines of polymer mixtures in the presence of their vapor phase at the mathematical double point, where two critical lines meet and exchange branches, and its environment. The model used combines the lattice gas model of Schouten, ten Seldam and Trappeniers with the Flory-Huggins theory. The critical line structure is displayed for various combinations of the chain length and system parameters in the pressure (P)-temperature (T) plane, as is usually done with experimental results. This type of work sheds light on the essential transition mechanism involved in the phase diagram's change of character, such as multi-critical points and mathematical double points, which are of great practical importance in supercritical fluid extraction processes. The P, T diagrams are discussed in accordance with the Scott and van Konynenburg binary phase diagram classification. We found that our P, T plots were in agreement with type II, type III, or type IV phase diagram behaviors. We also found that some of our phase diagrams represent the liquid-liquid equilibria in polymer solutions and mixtures.     

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.     

Molecular theory of phase equilibria in model and real associated mixtures III. Binary solutions of inert gases and n-alkanes in ammonia and methanol

Using new molecular models of ammonia and methanol and thermodynamic perturbation theory, the global phase diagrams of model mixtures of these compounds with a van der Waals fluid, representing a simple nonpolar fluid, have been calculated. The global phase diagram of these mixtures is much richer than that of corresponding aqueous mixtures. More types of critical line behavior are found, including the presence of van Laar points and a small region where the mixtures exhibit a closed liquid-liquid immiscibility loop (Type VI phase behavior). The individual mixture components are characterized by two molecular parameters, which can be adjusted to their critical temperature and critical volume; the mixture model itself contains no adjustable parameters. It is shown that the theory gives qualitatively correct predietions of mixtures with n-alkanes. This includes the prediction of Type III critical line behavior for small and large values of the ratio of the critical temperatures of the components, and Type II over a large range of conditions, including the presence or absence of absolute or limited azeotropy, and temperature and pressure extrema of critical lines and their dependence on the number of carbon atoms.     

Liquid-liquid equilibria of polymer solutions: Applicability of extended Redlich-Kister expansion

We developed a simple and improved expression for the Helmholtz energy of mixing which uses a Taylor series of an exponential function based on extending the Redlich-Kister expansion. This model incorporates the chain-length dependence of polymers and specific interactions such as hydrogen bonds. The proposed model can accurately predict most phase diagrams of various binary polymer solutions including upper critical solution temperature (UCST), lower critical solution temperature (LSCT), both UCST and LCST, and closed miscibility loops. Our model fits experimental data of the complex phase behavior of polymer solutions well.     

Critical lines for an unequal size of molecules in a binary gas-liquid mixture around the van Laar point using the combination of the Tompa model and the van der Waals equation

We combine the modified Tompa model with the van der Waals equation to study critical lines for an unequal size of molecules in a binary gas-liquid mixture around the van Laar point. The van Laar point is coined by Meijer and it is the only point at which the mathematical double point curve is stable. It is the intersection of the tricritical point and the double critical end point. We calculate the critical lines as a function of χ(1) and χ(2), the density of type I molecules and the density of type II molecules for various values of the system parameters; hence the global phase diagrams are presented and discussed in the density-density plane. We also investigate the connectivity of critical lines at the van Laar point and its vicinity and discuss these connections according to the Scott and van Konynenburg classifications. It is also found that the critical lines and phase behavior are extremely sensitive to small modifications in the system parameters.     

Equation of state for square-well chain molecules with variable range II. Extension to mixtures

An equation of state (EOS) developed in our previous work for square-well chain molecules with variable range is further extended to the mixtures of non-associating fluids. The volumetric properties of binary mixtures for small molecules as well as polymer blends can well be predicted without using adjustable parameter. With one temperature-independent binary interaction parameter, satisfactory correlations for experimental vapor–liquid equilibria (VLE) data of binary normal fluid mixtures at low and elevated pressures are obtained. In addition, VLE of n-alkane mixtures and nitrogen + n-alkane mixtures at high pressures are well predicted using this EOS. The phase behavior calculations on polymer mixture solutions are also investigated using one-fluid mixing rule. The equilibrium pressure and solubility of gas in polymer are evaluated with a single adjustable parameter and good results are obtained. The calculated results for gas + polymer systems are compared with those from other equations of state.     

Phase behavior of polymer mixtures with nonadditive hard-sphere potential

The phase behavior of a binary mixture of homopolymers in which macromolecules are composed of tangent hard spheres was studied. The interaction of unlike units is characterized by the contact distance (1/2)(σ_A + σ_B)(1 + Δ), where σ _i is the diameter of the ith sphere (unit) and Δ is the nonadditivity parameter. The effect of nonadditivity was taken into account by means of the perturbation theory relative to the additive system (Δ = 0) considered earlier (Polymer Science, 47, 2146 (2005)) in terms of the Percus-Yevick approximation. The theoretical consideration presented is completely analytical. It was found that a polymer mixture experiences phase separation with an increase in pressure; the two-phase region extends with an increase in both the size ratio between the units α = σ_A/σ_B and the length of the chain per se. Closed phase diagrams were first predicted for athermal mixtures; such diagrams appear at Δ < 0 and certain values of α. It was shown that the thermodynamics of an incompressible mixture of hard-chain molecules at α = 1 follows the Flory-Huggins theory with the temperature-independent interaction parameter. Phase separation in polymer solutions with the nonadditive hard-sphere potential was also analyzed.     

Phase diagrams of binary crystalline-crystalline polymer blends

A thermodynamically self-consistent theory has been developed to establish binary phase diagrams for two-crystalline polymer blends by taking into consideration all interactions including amorphous-amorphous, crystal-amorphous, amorphous-crystal, and crystal-crystal interactions. The present theory basically involves combination of the Flory-Huggins free energy for amorphous-amorphous isotropic mixing and the Landau free energy of polymer solidification (e.g., crystallization) of the crystalline constituents. The self-consistent solution via minimization of the free energy of the mixture affords determination of eutectic, peritectic, and azeotrope phase diagrams involving various coexistence regions such as liquid-liquid, liquid-solid, and solid-solid coexistence regions bound by liquidus and solidus lines. To validate the present theory, the predicted eutectic phase diagrams have been compared with the reported experimental binary phase diagrams of blends such as polyethylene fractions as well as polycaprolactone/trioxane mixtures.     

Phase behavior of n-alkanes in supercritical solution: a Monte Carlo study

We present a coarse-grained model for n-alkanes in a supercritical solution, which is exemplified by a mixture of hexadecane and CO2. For pure hexadecane, the Monte Carlo simulations of the coarse-grained model reproduce the experimental phase diagram and the interfacial tension with good accuracy. For the mixture, the phase behavior sensitively depends on the compatibility of the polymer with the solvent. We present a global phase diagram with critical lines, which is in semiquantitative agreement with experiments. In this context we developed two computational schemes: The first adopts Wang-Landau sampling to the off-lattice grand canonical ensemble, the second combines umbrella sampling with an extrapolation scheme to determine the weight function. Additionally, we use Wertheim's theory (TPT1) to obtain the equation of state for our coarse-grained model of supercritical mixtures and discuss the behavior for longer alkanes.     

Phase equilibria of poly(ethylene-co-vinyl acetate) copolymers in subcritical and supercritical ethylene and ethylene---vinyl acetate mixtures

Experimental high-pressure phase equilibrium data (cloud point and coexistence data) are reported for solutions of commercial poly(ethylene---co-vinyl acetate) samples in supercritical ethylene and ethylene---vinyl acetate (VA) mixtures. These data are correlated with an equation of state rooted in statistical associating fluid theory (SAFT). SAFT captures the effects of polymer MW, incorporated VA%, and free VA on the cloud point pressure, and on the size of the fluid-liquid miscibility gap, over a broad range of temperatures (50–250°C) and polymer concentrations. Free VA is found to behave as a cosolvent (lowering the cloud point pressure), except at low temperatures (< 100°C) and with low VA-containing copolymers, in which case free VA, if present in large excess (> 70 wt.%), behaves as a polar antisolvent due to favorable self-interactions among the free VA molecules. SAFT predicts a shift in the phase transition type, from upper-critical-solution-temperature (UCST), to upper-lower-critical-solution-temperature (U-LCST) with increasing free VA in the monomer mixture.     

The structure and phase transitions in polymer blends,diblock copolymers and liquid crystalline polymers: The Landau-Ginzburg approach

The polymer systems are discussed in the framework of the Landau-Ginzburg model. The model is derived from the mesoscopic Edwards Hamiltonian via the conditional partition function. We discuss flexible, semiflexible and rigid polymers. The following systems are studied: polymer blends, flexible diblock and multi-block copolymer melts, random copolymer melts, ring polymers, rigid-flexible diblock copolymer melts, mixtures of copolymers and homopolymers and mixtures of liquid crystalline polymers. Three methods are used to study the systems: mean-field model, self consistent one-loop approximation and self consistent field theory. The following problems are studied and discussed: the phase diagrams, scattering intensities and correlation functions, single chain statistics and behavior of single chains close to critical points, fluctuations induced shift of phase boundaries. In particular we shall discuss shrinking of the polymer chains close to the critical point in polymer blends, size of the Ginzburg region in polymer blends and shift of the critical temperature. In the rigid-flexible diblock copolymers we shall discuss the density nematic order parameter correlation function. The correlation functions in this system are found to oscillate with the characteristic period equal to the length of the rigid part of the diblock copolymer. The density and nematic order parameter measured along the given direction are anticorrelated. In the flexible diblock copolymer system we shall discuss various phases including the double diamond and gyroid structures. The single chain statistics in the disordered phase of a flexible diblock copolymer system is shown to deviate from the Gaussian statistics due to fluctuations. In the one loop approximation one shows that the diblock copolymer chain is stretched in the point where two incompatible blocks meet but also that each block shrinks close to the microphase separation transition. The stretching outweights shrinking and the net result is the increase of the radius of gyration above the Gaussian value. Certain properties of homopolymer/copolymer systems are discussed. Diblock copolymers solubilize two incompatible homopolymers by forming a monolayer interface between them. The interface has a positive saddle splay modulus which means that the interfaces in the disordered phase should be characterized by a negative Gaussian curvature. We also show that in such a mixture the Lifshitz tricritical point is encountered. The properties of this unusual point are presented. The Lifshitz, equimaxima and disorder lines are shown to provide a useful tool for studying local ordering in polymer mixtures. In the liquid crystalline mixtures the isotropic nematic phase transition is discussed. We concentrate on static, equilibrium properties of the polymer systems.     

Quantitative effect of nonionic surfactant partitioning on the hydrophile-lipophile balance temperature

Phase behaviors of water/nonionic surfactants/isooctane systems are determined experimentally in temperature-global surfactant concentration diagrams. The surfactants are monodistributed polyoxyethylene glycol n-dodecyl ether. They are used as model mixtures of two, three, or five compounds or as constituents of a commercial surfactant. It is found that the phase diagrams of these systems are bent gradually toward the highest temperatures as the global surfactant concentration decreases. Each phase diagram is well-characterized by the curve of the HLB (hydrophile-lipophile balance) temperature versus the global surfactant concentration. For any fixed global surfactant concentration, this temperature is the middle temperature of the three-phase region; it can be calculated from an additive rule of the HLB temperatures of the surfactants weighted by their mole fractions at the water/oil interface. These mole fractions are determined through the pseudophase model using surfactant partitioning. Calculations require the knowledge of the critical micelle concentration, the partition coefficient between water and oil, and the HLB temperature of each surfactant of the mixture. This treatment can be used to correctly predict the variation of the HLB temperatures of the surfactant mixtures studied versus the global surfactant concentration. Furthermore, these calculations show that the observed curvature of the phase diagrams at the lowest global concentrations is due to the most favorable partitioning toward the oil of the lowest ethoxylated surfactant molecules.     

Spinodal decomposition in asymmetric polymer mixtures

We present a preliminary numerical study of spinodal decomposition in an asymmetric polymer mixture, i.e., of polymer with different chain lengths, in three dimensions with full Flory-Huggins-de Gennes free energy, numerically integrating the time evolution equations for the conserved order parameter. For the sake of comparison, we also present a numerical study of the symmetric polymer mixture. The results indicate that the scaled structure factor for the asymmetric polymer mixture is much broader than that of a symmetric polymer mixture. It is interesting that the growth exponents are not symmetric around the critical quench, i.e., growth exponents on the two sides of the critical composition are different. In addition to that, the magnitudes of the pair correlation functions of asymmetric mixtures are very small for x larger than the characteristic domain size r_g and the oscillations seen in the symmetric mixture are almost absent. We have attributed this finding to the rough interfaces and broader domain size distribution in the phase separated asymmetric polymer mixtures. Therefore, the simulation reveals that the asymmetry plays an important role for the spinodal decomposition dynamics of polymer mixtures.     

Automated calculation of complete Pxy and Txy diagrams for binary systems

An algorithm for the calculation of global phase equilibrium diagrams has been recently developed [M. Cismondi, M.L. Michelsen, Global phase equilibrium calculations: critical lines, critical end points and liquid–liquid–vapour equilibrium in binary mixtures, J. Supercrit. Fluids 39 (2007) 287–295]. It integrates the calculation of critical lines, liquid–liquid–vapour (LLV) lines and critical end points, and was implemented in the software program GPEC: global phase equilibrium calculations [M. Cismondi, D.N. Nuñez, M.S. Zabaloy, E.A. Brignole, M.L. Michelsen, J.M. Mollerup, GPEC: a program for global phase equilibrium calculations in binary systems, in: Proceedings of the CD-ROM EQUIFASE 2006, Morelia, Michoacán, Mexico, October 21–25, 2006; www.gpec.plapiqui.edu.ar]. In this work we present the methods and computational strategy for the automated calculation of complete Pxy and Txy diagrams in binary systems. Being constructed from the points given by the global phase diagram at a specified temperature or pressure, their calculation does not require the implementation of stability analysis.     

Liquid-liquid equilibria of polymer solutions: a closed miscibility loop phase behavior

In the phase behavior of binary polymer/solvent mixtures, a lower critical solution temperature (LCST) and hour-glass shaped and closed miscibility loop phase behavior are encountered. The closed miscibility loop phase behavior may be mainly due to highly oriented interactions such as hydrogen bonding. The purpose of this study is to describe closed miscibility loop phase behavior in the liquid-liquid equilibria of polymer solutions. To consider highly oriented interactions (or specific interactions), we employed a secondary lattice concept as a perturbation term.     

A simple statistical mechanical theory for the mutual solubility of fluid mixtures of water and organic solvents under high pressure

A simple statistical mechanical theory is presented to explain phase diagrams of fluid mixtures with both a lower critical solution temperature and an upper critical solution temperature under pressure. By postulating a temperature dependence for the interaction free energy parameter of the constituent molecules and a pressure dependence for the excess volume, phase diagrams with both lower critical solution temperature, and upper critical solution temperature and their pressure dependence can be reproduced by quadratic surfaces in temperature-concentration-pressure space. The topological aspects of the observed phase diagrams in this space have been related to our theoretical model, and the thermodynamical meaning of the topologies has been interpreted based on our model. Experimental data for the mutual solubility of water and 2-butanol under pressure and that of water and 3-methylpyridine with added salts have been analyzed quantitatively and theoretical parameters are determined.     

Estimation of the Thermodynamic Quality of Alkylbenzenes with Respect to Low Density Polyethylene

The original optical method was used for the first time to obtain complete diagrams of the state of mixtures of low-density polyethylene with o-xylene and ethylbenzene. The diagrams are of the type describing the phase state of systems partially crystalline polymer–good solvent. In contrast to the diagrams described in the literature, the diagrams described here contain an additional boundary curve for the solubility of the liquid in the polymer. Using the coordinates of the intersection of this line with the curve of complete amorphization of the polymer, five alkyl benzenes were ranked according to their thermodynamic affinity to low-density polyethylene.     

水溶性高分子分子量对其与电荷稳定的胶体分散体系的混合体系相行为的影响

利用同步辐射小角X射线散射技术,对不同相对分子质量的水溶性高分子聚氧化乙烯(PEO)与电荷稳定的聚甲基丙烯酸甲酯(PMMA)乳胶的混合体系的相行为进行了研究。 PEO与PMMA乳胶混合体系的相行为与体系中乳胶粒子的体积分数和PEO的浓度相关。 在一定乳胶粒子体积分数下,在较低PEO浓度下,混合体系保持均匀分散性。 而当PEO浓度高于某一临界浓度时,混合体系将发生相分离,生成集团相或者形成面心立方(FCC)晶体结构。 PEO相对分子质量的大小也是影响混合体系相行为的重要因素。 当PEO的相对分子质量较高时,混合体系发生相分离所对应的临界PEO浓度较低。 除此,PEO相对分子质量对混合体系的结晶行为也有影响。 在低乳胶粒子体积分数下,较高相对分子质量的PEO容易使乳胶粒子结晶。 相反的,在较高乳胶粒子体积分数下,较低相对分子质量的PEO容易使乳胶粒子堆积形成结晶结构。     

Thermoreversible gelation and phase separation in aqueous methyl cellulose solutions

We derived typical phase diagrams for aqueous solutions of methyl cellulose (MC) of different molecular weights via micro‐differential scanning calorimetry, small‐angle X‐ray scattering, and visual inspection. The phase diagrams showed the cooccurrence of gelation and phase separation and qualitatively agreed with the theoretically calculated diagrams. The sol–gel transition line and phase separation line of a lower critical solution point type shifted toward lower temperatures and lower concentrations with an increase in the MC molecular weight. The sol–gel transition line intersected at a temperature higher than the critical point of the phase separation; therefore, both sol–gel phase separation and gel–gel phase separation were possible, depending on the temperature. Specifically, through visual inspection of a high molecular weight MC sample in the critical temperature region, we observed phase separation into two coexisting gels with different polymer concentrations. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 91–100, 2001