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.     

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.     

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.     

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.     

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.     

Crystal-liquid crystal binary phase diagrams

We propose a new theoretical scheme for the binary phase diagrams of crystal-liquid crystal mixtures by a combination of a phase field model of solidification, the Flory-Huggins theory for liquid-liquid mixing and Maier-Saupe-McMillan (FH-MSM) model for nematic and smectic liquid crystal orderings. The phase field theory describes the crystal phase transition of anisotropic organic crystal and/or side chain liquid crystalline polymer crystals while the FH-MSM model explains isotropic, nematic and smectic-A phase transitions. Self-consistent calculations reveal several possible phase diagram topologies of the binary crystal-liquid crystal mixtures. The calculated phase diagrams were found to accord well to the reported experimental results.     

Phase equilibria and phase separation dynamics in a polymer composite containing a main-chain liquid crystalline polymer

Various topological phase diagrams of blends of main-chain liquid crystalline polymer (MCLCP) and flexible polymer have been established theoretically in the framework of Matsuyama–Kato theory by combining Flory–Huggins (FH) free energy for isotropic mixing, Maier–Saupe (MS) free energy for nematic ordering in the constituent MCLCP, and free energy pertaining to polymer chain-rigidity. As a scouting study, various phase diagrams of binary flexible polymer blends have been solved self-consistently that reveal a combined lower critical solution temperature (LCST) and upper critical solution temperature (UCST), including an hourglass phase diagram. The calculated phase diagrams exhibit liquidus and solidus lines along with a nematic–isotropic (NI) transition of the constituent MCLCP. Depending on the strengths of the FH interaction parameters and the anisotropic (nematic–nematic) interaction parameters, the self-consistent solution reveals an hourglass type phase diagram overlapping with the NI transition of the constituent MCLCP. Subsequently, thermodynamic parameters estimated from the phase diagrams hitherto established have been employed in the numerical computation to elucidate phase separation dynamics and morphology evolution accompanying thermal-quench induced phase separation of the MCLCP/polymer mixture. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 3621-3630, 2006     

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.     

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.     

Phase and interface behaviors in type-I and type-V Lennard-Jones mixtures: theory and simulations

Density gradient theory (DGT) and molecular-dynamics (MD) simulations have been used to predict subcritical phase and interface behaviors in type-I and type-V equal-size Lennard-Jones mixtures. Type-I mixtures exhibit a continuum critical line connecting their pure critical components, which implies that their subcritical phase equilibria are gas liquid. Type-V mixtures are characterized by two critical lines and a heteroazeotropic line. One of the two critical lines begins at the more volatile pure component critical point up to an upper critical end point and the other one comes from the less volatile pure component critical point ending at a lower critical end point. The heteroazeotropic line connects both critical end points and is characterized by gas-liquid-liquid equilibria. Therefore, subcritical states of this type exhibit gas-liquid and gas-liquid-liquid equilibria. In order to obtain a correct characterization of the phase and interface behaviors of these types of mixtures and to directly compare DGT and MD results, the global phase diagram of equal-size Lennard-Jones mixtures has been used to define the molecular parameters of these mixtures. According to our results, DGT and MD are two complementary methodologies able to obtain a complete and simultaneous prediction of phase equilibria and their interfacial properties. For the type of mixtures analyzed here, both approaches have shown excellent agreement in their phase equilibrium and interface properties in the full concentration range.     

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.     

聚合物流体的范德华凝聚行为

研究了由聚合物的范德华作用导致的凝聚行为. 研究发现, 尽管聚合物同小分子的相行为的形成原因不同(聚合物体系的相行为是由动能、构象熵项和范德华作用能三项相互竞争的结果, 而小分子的相行为是由动能和范德华作用能相互竞争的结果), 但是它们表现出了极为相似的相行为.     

Mixtures of liquid-crystalline polymers

A density-functional expansion method is used to derive the free energy of a polymer mixture. The expression obtained includes the entropy of mixing, the entropy of configuration of the chains and the interactions (both isotropic and anisotropic ones). The chains are modelled as interacting elastic lines (bend curvature). The method is very general, and we only focus our attention on binary mixtures. The phase diagram and the order parameters are calculated. We show some results for two types of mixtures: a nematic polymer in a non-mesomorphic particle (polymer or solvent) and in another nematic liquid crystal (small-molecule or polymer). We discuss the influence of the molecular weights, the persistence length and the interactions on the phase separation.     

Fluid phase stability and equilibrium calculations in binary mixtures: Part II: Application to single-point calculations and the construction of phase diagrams

The methodology presented in Part I of this work is applied to a large number of pressure–temperature flash calculations, and to the automated construction of constant temperature pressure–composition phase diagrams, and constant pressure temperature–composition phase diagrams for binary mixtures modeled with an augmented van der Waals equation of state. An automated prototype implementation of the algorithm is developed for this purpose. We follow the classification of Scott and van Konynenburg [R.L. Scott, P.H. van Konynenburg, Discuss. Faraday Soc. 49 (1970) 87] and present phase diagrams corresponding to non-azeotropic mixtures of the five main types of fluid phase behavior (I–V), studying in detail representative diagrams at constant pressure and constant temperature. Special attention is given to the solution of numerically problematic equilibrium regions, such as those close to three-phase equilibria where metastable and unstable critical points can also be found. Of the order of 10⁴ flash calculations at varying temperatures and pressures, and for different intermolecular parameters of the components in the mixture, have been carried out. The algorithm provides the correct stable equilibrium state for all of the points considered. Despite the fact that our implementation is not optimised for performance, we find that the algorithm identifies the stable solution in difficult regions of the phase space without any penalty in terms of computational time, when compared to simpler regions.     

A polymer—microemulsion interaction: The coacervation model

A model which successfully predicts many of the qualitative features of the phase separation can be based upon the assumption that the microemulsion particle retains its integrity when diluted with external phase or when mixed with a polymer (which does not complex with the surfactants). In this case, the microemulsion particle behaves thermodynamically in a manner similar to a macromolecule so that one can predict that the polymer—microemulsion phase diagram will have qualitative similarities to that exhibited by the polymer—polymer₂-solvent case. Most of the experiments were conducted by following the phase boundaries by measurements of the cloud points of the polymermicroemulsion mixtures. The cloud points of the mixtures were found to be (usually) linear functions of the weight fraction of the added polymer and decreased with increasing molecular weight. Plots of the logarithm of the slopes of these curves against the logarithm of the polymer molecular weight were linear, with a slope designated as β. The values for β for twenty-five systems studied fell within the narrow range of 0.55 ± 0.15. These values were seen to be the same as the Mark—Houwink exponents in the relationship between the intrinsic viscosity and the polymer molecular weight. The reason for this correspondence appears to be that the coacervation occurs at a polymer concentration about equal to the threshold overlap concentration. Ternary phase diagrams with microemulsion particles, brine, and polymer selected as pseudocomponents also exhibit many of the features predicted by the model. The phase boundary asymmetry is in the direction expected and asymptotes with the expected independence upon microemulsion and polymer molecular weights are also demonstrated. In agreement with the model, it is shown that the controlling variable is polymer weight average, rather than z or number average molecular weight.     

The Full Pressure–Temperature Phase Envelope of a Mixture in 1000 Microfluidic Chambers

Knowing the thermodynamic state of complex mixtures—liquid, gas, supercritical or two‐phase—is essential to industrial chemical processes. Traditionally, phase diagrams are compiled piecemeal from individual measurements in a pressure–volume–temperature cell performed in series, where each point is subject to a long fluid equilibrium time. Herein, 1000 microfluidic chambers, each isolated by a liquid piston and set to a different pressure and temperature combination, provide the complete pressure–temperature phase diagram of a hydrocarbon mixture at once, including the thermodynamic phase envelope. Measurements closely match modeled values, with a standard deviation of 0.13 MPa between measurement and model for the dew and bubble point lines, and a difference of 0.04 MPa and 0.25 °C between measurement and model for the critical point.     

Multiphase coexistence in polydisperse colloidal mixtures

The authors study the phase behavior of mixtures of monodisperse colloidal spheres with a depletion agent which can have arbitrary shape and can possess a polydisperse size or shape distribution. In the low concentration limit considered here, the authors can employ the free-volume theory and take the geometry of particles of the depletion agent into account within the framework of fundamental measure theory. The authors apply their approach to study the phase diagram of a mixture of (monodisperse) colloidal spheres and two polydisperse polymer components. By fine tuning the distribution of the polymer, it is possible to construct a complex phase diagram which exhibits two stable critical points.     

Phase behavior of a DNA-based surfactant mixed with water and n-alcohols

The self-assembly behavior of a cationic surfactant (dodecyltrimethylammonium, DTA) with DNA as counterion in mixtures of water and n-alcohols (decanol, octanol, hexanol, butanol, and ethanol) was investigated. The phase diagrams were established and the different regions of the phase diagram characterized with respect to microstructure by (2)H NMR, small-angle X-ray scattering (SAXS), and other techniques. The DNA-DTA surfactant is soluble in all of the studied alcohols, showing increased solubility from decanol down to ethanol. All of the phase diagrams are analogous with respect to the occurrence of liquid crystalline (LC) regions, but the area of the LC region increases as one goes from decanol to ethanol. In all phase diagrams, hexagonal phases (of the reversed type) for the alcohol-rich side and lamellar phases for the other side were detected. For balanced proportions of the components, there is a coexistence of the lamellar and the hexagonal phase, here detected with a double quadrupole splitting in the (2)H NMR spectra. The correctness of the phase diagrams is confirmed by the fact that along the tie-lines the splitting magnitude remains nearly constant. All of the alcohols except for ethanol act as cosurfactants penetrating the DNA-DTA film. Adding salt to the ternary mixtures causes an increase in the unit cell dimension of the lamellar and the hexagonal phases. The phase diagram becomes more complicated when butanol is used for the alcohol phase. Here, there is the occurrence of a new isotropic phase with some properties analogous to those of the disordered sponge (L3) phase obtained for simple surfactant systems.