Models of phase diagrams of one-and two-component systems with heterophase equilibria in the critical and supercritical regions of the coexistence of liquid and vapor

Formal model phase diagrams of one-and two-component systems with heterophase equilibria in the region of critical vapor-liquid states are constructed. The diagrams take into account the existence of the fluid phase.     

Effect of polydispersity on the phase diagrams of linear ABC triblock copolymers in two dimensions

By using a two-dimensional (2D) real-space self-consistent field theory, we present the phase diagrams of monodisperse ABC triblock copolymers in a three-component triangle style with the interaction energies given between the distinct blocks; this system displays richer phase behavior when compared with the corresponding diblock copolymers. Polydispersity of the end or middle blocks in the ABC linear block copolymer chains results in a completely different phase diagram. The presence of a polydisperse end block may cause strong segregation to occur among the three distinct components and larger domain sizes of the dispersed phases; a polydisperse middle block may allow a connection to form between the two phases of the two end blocks.     

Phase equilibrium and insulin partitioning in aqueous two-phase systems containing block copolymers and potassium phosphate

In this work, phase diagrams of aqueous two-phase systems (ATPS) containing PEO–PPO–PEO block copolymers and potassium phosphate as well as the partitioning behavior of insulin in these systems are presented. Experiments aimed at the identification of the effects of copolymer structure (by varying the number of EO units per polymer molecule), temperature (283.15 and 298.15 K) and pH (5.0 and 7.0) on the phase behavior of these systems were carried out. The results indicated the enlargement of the two-phase region upon increasing the temperature, pH and copolymer hydrophobicity (expressed as the PO/EO ratio in the copolymer molecule). Experimental measurements of the partitioning of human insulin in these ATPS also indicated the dependency of the partition coefficients on temperature, pH, and copolymer hydrophobicity, with the latter being the most influential factor. Finally, experimental data on the phase behavior and insulin partitioning were correlated using an excess Gibbs energy virial-type model modified in order to account for coulombic interactions and ionization equilibrium between the various forms of the phosphate ion.     

Lyotropic liquid crystalline phase behaviour in amphiphile-protic ionic liquid systems

Approximate partial phase diagrams for nine amphiphile-protic ionic liquid (PIL) systems have been determined by synchrotron source small angle X-ray scattering, differential scanning calorimetry and cross polarised optical microscopy. The binary phase diagrams of some common cationic (hexadecyltrimethyl ammonium chloride, CTAC, and hexadecylpyridinium bromide, HDPB) and nonionic (polyoxyethylene (10) oleyl ether, Brij 97, and Pluronic block copolymer, P123) amphiphiles with the PILs, ethylammonium nitrate (EAN), ethanolammonium nitrate (EOAN) and diethanolammonium formate (DEOAF), have been studied. The phase diagrams were constructed for concentrations from 10 wt% to 80 wt% amphiphile, in the temperature range 25 °C to >100 °C. Lyotropic liquid crystalline phases (hexagonal, cubic and lamellar) were formed at high surfactant concentrations (typically >50 wt%), whereas at <40 wt%, only micelles or polydisperse crystals were present. With the exception of Brij 97, the thermal stability of the phases formed by these surfactants persisted to temperatures above 100 °C. The phase behaviour of amphiphile-PIL systems was interpreted by considering the PIL cohesive energy, liquid nanoscale order, polarity and ionicity. For comparison the phase behaviour of the four amphiphiles was also studied in water.     

Microphase Separation in A2B Copolymer Melts

The microphase diagrams of special A2B copolymer melts were presented by using the self-consistent field theory for star copolymer systems. Unlike the phase diagram of diblock copolymer, only three classical structures, namely spherical phase, cylindrical phase and lamellar phase were discovered in the diagram of the A2B system. The change in chain architectures allowed sufficient shifts of phase boundaries and widened the range of fB for which lamellar phase occurred, to some degree. Simply, an asymmetric architecture for copolymer allowed control of the morphology independent of the volume fraction.     

Applications of polymer rism theory to blends

Polymer RISM (reference interaction site model) theory is a theory of polymer systems in the liquid phase in which account for chemical realism can be made. Results are reported here of phase diagrams (spinodals) for blend systems calculated by means of this theory, using the mean spherical approximation as a closure. The systems investigated are an isotopic blend, a set of homopolymer/copolymer blends, and a model blend containing specific interactions.     

Theory of phase diagrams described by thermodynamic potentials with T d symmetry

Phase diagrams of crystals induced by irreducible representations with symmetry group \(L = \bar 43m\) (T _d ) are constructed within the phenomenological theory of second-order phase transitions. A model of the Landau thermodynamic potential is studied, state equations of all symmetry-conditioned phases are obtained, and general conditions for their thermodynamic stability are formulated. Equations for the boundaries of phase areas and lines of phase transitions are obtained for the fourth order of expansion of the potential via components of the order parameter. Some types of the collapse of the multicritical point of the phase diagram for the eighth order of potential expansion are studied using computer calculations. The possible existence of phase diagrams that contain one or more triple points and areas of existence of three and four phases is shown for the first time for the potentials with the above symmetry. Examples are given of crystals that undergo phase transitions in the considered symmetry of the order parameter.     

Structure formation and fractionation in systems of polydisperse attractive rods

Fractionation effects and the formation of structured domains are investigated in polydisperse systems of attractive spherocylinders with the help of Monte Carlo simulations. For sufficiently high attractive interaction strength and pressure, the large rods in the system aggregate and form a highly ordered hexatic monolayer that coexists with an isotropic fluid of smaller rods. Fractionation diminishes with decreasing interaction strength but is still observed for hard rod systems, in which the large rods form a nematic droplet rather than a monolayer. Results for polydisperse systems are accompanied by phase diagrams for monodisperse systems of attractive spherocylinders. Here, the phase behavior is shown as a function of rod length and pressure.     

Vectorizable algorithm for green function and many-body perturbation methods

The principles of an efficient, fast algorithm for the calculation of diagrams appearing in Green function and many-body perturbation methods are discussed and timing examples are given. Within the suggested algorithm, the third order-diagrams required in the Green function approach are evaluated by arranging computations in such a way that the most inner loops contain only simple scalar products and multiplication of vector by scalar operations. The molecular symmetry is taken into account for abelian groups. © 1993 John Wiley & Sons, Inc.     

Ribbon phases in surfactant systems Comparisons between experimental results and predictions of a theoretical model

Ribbon phases consist of long cylindrical aggregates that have non-circular normal sections. We have recently pointed out that scattering data for a large number of different intermediate ribbon phases of lower than hexagonal symmetry found in ionic surfactant systems indicate that these phases have a structure possessing a centred rectangular symmetry. In this communication, we have investigated the aggregate dimensions for the phases with cylindrical aggregates, i.e., the hexagonal phases and the centred rectangular ribbon phases. Previously published phase diagrams, small angle X-ray and neutron scattering data and ²HNMR data for these phases in different systems have been used for this purpose. The results are that the axial ratios of the aggregates increase when the temperature decreases, when the surfactant concentration increases, and when the average surfactant charge decreases. Models that semi-quantitatively describe the thermodynamics of the micellar, hexagonal and lamellar phases, which are based on the Poisson-Boltzmann cell model approach, have previously been presented in the literature. We have extended these models to treat also the ribbon phases. The results from the calculations show the same trends with respect to changes in the dimensions of the non-circular aggregates upon changes in temperature, surfactant concentration and average surfactant charge, as those obtained experimentally. Theoretically calculated phase diagrams with ribbon phases are also presented. Based on the predictions of the model and some previously published experimental data for hexagonal phases, it is proposed that the formation of non-circular, cylindrical aggregates is a general property of single-chain, ionic surfactant/water systems, and that these aggregates in general pack on hexagonal lattices. The normal sections of these aggregates are circular on average, on account of the fact that the degree of deformation and the orientation of deformation changes along the axis of the aggregates and with time. Only for some systems, temperatures and surfactant concentrations do the asymmetric aggregates line up and ribbon phases with centred rectangular symmetry are obtained. The driving mechanisms for the transition from the hexagonal phase with asymmetric (fluctuating) cylinders and further to the centred rectangular phase with asymmetric (stiff) cylinders is also discussed. It is argued that this phase transition is of the first order.     

Thermodynamic assessment of smectic A-nematic tricritical points through the equal gibbs energy analysis

A complete thermodynamic analysis on four two-component phase diagrams between liquid crystals belonging to the nCB and nOCB series of compounds, the so-called cyanobiphenyls, has been performed through the Oonk's equal Gibbs energy method. The binary systems dealt with in this paper show as a common feature the existence of a tricritical point at the SmA-to-N phase transition, all of which reported elsewhere. As a singular finding of the work proposed in this paper, the Oonk's method is extended in order to account for second-order phase transitions. Likewise, this extension allows performing calculations of the tricritical temperature.     

Differences of Interaction Parameter of a PS/PEO homopolymer blend and diblock copolymer in comparison to other systems

A polystyrene/polyethyleneoxide PS/PEO homopolymer blend and a diblock copolymer were investigated by small angle neutron scattering experiments. The measured susceptibilities could be described by theories which take strong fluctuations into account. The yielded Flory‐Huggins interaction parameter Γ is different between the blend and the diblock copolymer, in accordance to previous experiments [1, 2, 3]. This important difference of Γ could make the predictability of the phase boundaries worse than 50 K. When comparing the PS/PEO samples to other systems, a systematology seems to be missing.     

Statistical thermodynamics in the framework of the lattice fluid model, 1. Polydisperse polymers of special distribution

The Gibbs free energies and equations of state of polymers with special molar mass distributions, e.g., Flory distribution, uniform distribution and Schulz distribution, are derived based on a lattice fluid model. The influence of the polydispersity (or the chain length) on the close-packed mass density, the close-packed volume of a mer and the mer-mer interaction energy or the scaling temperature is discussed. The diagrams of the Gibbs free energies as a function of temperature and chain length are simulated with a computer. The results suggest that a polydisperse polymer is thermodynamically more stable than the corresponding monodisperse polymer and that the thermodynamical properties of a polydisperse polymer are identical with those of the corresponding monodisperse polymer when the average degree of polymerization is sufficiently high.     

Invited Lecture. Order-parameter theories of phase diagrams for antiferroelectric smectic-A phases

Using exact relations between Landau and molecular approaches, the symmetryinduced topologies of phase diagrams are studied for antiferroelectric smectic-A phases. In particular, the tricritical points are found for a large class of orderparameter theories of nematic-smectic A (-A₁, -A_d) and -A₂ (NA) phase transitions. These include generalizations to antiferroelectric smectic-A phases of McMillan and Meyer-Lubensky mean-field theories and the Ramakrishnan-Youssouff(RY) density-functional approach. The use of these different approaches allows study of influence of various couplings between nematic (orientational) and smectic (translational) degrees of freedom and polarization field, P₁ (cos θ), on various NA and AA phase transitions. From the results, it is of interest that the coupling between orientational degrees of freedom (P₄(cos θ) and density waves can destabilize the smectic-A phase at low temperatures-pointing to the existence of a nematic-smectic-A-reentrant-nematic phase transition. A possible relation of this result to A_d, C₂, C_d and [ctilde] phases is discussed. Some relations between Fourier components of correlation functions and order parameters at tricritical points are derived from the RY density-functional theory. Despite some limitations, the theory presented here seems to provide the simplest approach to study topologies of phase diagrams in molecular theories.     

Miscibility of poly(vinyl chloride) with poly(methyl-co-hexyl acrylate) and poly(methyl-co-2 ethyl hexyl acrylate)

The miscibility and phase behavior in blends of PVC with poly(methyl-co-hexyl acrylate)[MHA] and poly(methyl-co-2 ethyl hexyl acrylate)[MEH] were studied. It was found that PVC is miscible with MHA copolymers having a HA volume fraction from 0.30 to 0.92 and MEH copolymers having an EH volume fraction from 0.30 to 0.83 at 100°C. By applying the mean field theory to the phase diagrams of these blend systems, segmental interaction parameters which represent the binary interaction between different monomer units were estimated. The calculated values reflect the fact that the miscibility window observed for PVC/MHA and PVC/MEH blend systems was attributed to the effect of repulsion between different monomer units within the copolymer. To investigate the effect of specific interaction on the miscibility for these blend systems, an attempt was also made to describe the blend interaction parameter as a function of polar group concentration in the acrylate copolymer. The blend interaction parameter values exhibit a u-shaped curve as a function of the weight fraction of C?O group in the copolymer, and the lowest blend interaction parameter value appears at about 0.24 C?O weight fraction.     

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.     

Study of the conformational disorder in polyesters giving liquid-crystalline phases

Conformational and packing energy analyses have been performed by molecular mechanics calculations on the polyester derived from the condensation of 2-(4-hydroxyphenyl)-6-hydroxybenzoxazole with terephthalic acid, which presents a high degree of crystallinity in spite of the head-to-tail constitutional disorder. The line repetition groups in accordance with the experimental value of the c axis have been taken into account. Possible conformations of chains having constitutional and conformational disorder in the crystalline phase have been proposed. Packing energy calculations performed on pairs of chains give the close packing distances and indicate the best symmetry elements. The results are in agreement with the unit cell obtained by X-ray data, the thermodynamical properties of the polymer and the existence of liquid-crystal phases.     

Remarks on the use of the average occupation numbers in the SCF process for open shell systems

It is shown that the utilization of an average occupation number for open shell orbitals, having different occupation numbers in a degenerate wave function, tantamounts to disregard first order contributions in a CI -type expansion of the wave function. These contributions are taken into account in a SCF process that derives the differently occupied open shell orbitals as eigenfunctions of different Fock-type operators and accounts for the total symmetry.     

Fddd network phase in ABA triblock copolymer melts

Using self‐consistent field theory, we investigate the stability of the orthorhombic Fddd network phase (O⁷⁰) in ABA triblock copolymer melt systems. Consistent with previous findings, we observe that the gross topology of phase behavior is unchanged with varying chain asymmetry. However, the mean field critical point is displaced from the diblock copolymer value of f_A = 0.5 (f_A is the A segment volume fraction) to larger values as the triblock copolymer symmetry is broken with unequal A block lengths. This deviation significantly shifts the order‐order phase boundaries, resulting in an appreciable region of O⁷⁰ stability in the phase diagram of asymmetric ABA triblock copolymers. More importantly, the stability of the O⁷⁰ phase extends to the intermediate segregation regime for select chain asymmetries. Both features are desirable for achieving a synthetic realization of the phase in binary AB block copolymer systems. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016 , 54, 1112–1117