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.     

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.     

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.     

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.     

Magnetic susceptibility and the spin-glass transition of Cd1−xMnxS and Zn1−xMnxS at low temperatures

Low-field magnetic susceptibility of the diluted magnetic semiconductors Cd_1?xMn_xS and Zn_1?xMn_xS was measured between 4.2 and 30 K for the Mn concentration range 0.25 < x < 0.40. When x > 0.25, both of these ternary systems show a spin-glass transition in the above temperature range, as evidenced by a somewhat rounded cusp in the susceptibility and by the presence of irreversible effects. Because these materials are insulators at low temperatures, and the interactions between the Mn ions are only antiferromagnetic, the observed spin-glass behavior is attributed to frustration inherent in the hcp lattices of these compounds. The phase diagrams for the boundary of the paramagnetic and the spin-glass phases are presented for the two alloy systems, and the difference between the two phase diagrams is discussed.     

Effect of phase behavior, density, and isothermal compressibility on the constant-volume heat capacity of ethane + n-pentane mixed fluids in different phase regions

The phase behavior, density, and constant-volume molar heat capacity (C_v,m) of ethane + n-pentane binary mixtures have been measured in the supercritical region and subcritical region at T=309.45 K. In addition, the isothermal compressibility (κ_T) has been calculated using the density data determined. For a mixed fluid with a composition close to the critical composition, C_v,m and κ_T increase sharply as the pressure approaches the critical point (CP), the dew point (DP), or the bubble point (BP). However, C_v,m is not sensitive to pressure in the entire pressure range if the composition of the mixed fluid is far from the critical composition. To tune the properties of the binary mixtures effectively by pressure, both the composition and the pressure should be close to the critical point of the mixture. The intermolecular interactions in the mixture are also discussed on the basis of the experimental results.     

Prediction of critical transitions of ternary mixtures containing ammonia and n-alkanes

The analysis of the critical transitions that occur in ternary mixtures is important to describe their physical behavior. It also enables the phase behavior of multi-component mixtures at high temperatures to be inferred. The objective of this work was to identify the critical transitions that occur in ternary mixtures containing ammonia and n  -alkane. The mixture’s critical loci were obtained and tested for stability using thermodynamic criteria expressed in terms of the Helmholtz free energy. Two equations of state were used to represent the Helmholtz free energy: the Carnahan–Starling–Redlich–Kwong (CSRK) and the Simplified Perturbed Hard Chain Theory (SPHCT). In order to identify the existing critical transitions, profiles of the critical loci were calculated along constant compositional ratios χ=x₁/x₂$\chi =x_1/x_2$. Some of the curves depict higher order critical transitions between liquid–liquid and gas–liquid critical point regions, or two different liquid–liquid critical regions. One of the critical transitions found could be considered as a new sub-class within existing classifications for ternary mixtures proposed by Sadus [R.J. Sadus, J. Phys. Chem. 96 (1992) 5197–5202].     

Critical and three phase behavior in the carbon dioxide/tridecane system

The PT behavior of the carbon dioxide/normal alkane series exhibits a distinct transition in the CO₂/nC₁₃H₂₈ system. This particular diagram is characterized by two liquid—liquid—vapor (l-l-g) loci, a lower liquid-upper liquid (l-l) critical branch extending from high pressures to the upper critical end point (UCEP) and two liquid—vapor (l-g) critical branches which cross near the lower critical end point (LCEP).An experimental PTx diagram in the vicinity of the CO₂ critical point, LCEP and K point reveals the emergene at the LCEP of a l-l region which increases in size with temperture while the upper liquid—vapor (L₂-g) region diminishes, eventually disappearing at the K point. The l-l-g surface illustrates the compositional changes of each phase with temperture.Detailed Px diagrams at three temperatures between the LCEP and K point are presented and each exhibits two critical points, a l-l-g locus and curves of constant phase volume ratio which show discrete changes in both value and slope at the l-l-g locus. Graphical and numerical methods of determining the phase densities and compositions from three phase volumetric behavior are presented.     

Phase equilibrium modeling of triglycerides in supercritical fluids

To design a reactor and separator for a supercritical biodiesel process, phase equilibria of multi-component mixtures in supercritical fluids should be determined using group contribution with association equation of state (GCA-EOS) as a thermodynamics method. The model is considered for two systems of reactants and products. System 1 is comprised of methanol and triglycerides from two sources (palm and Jatropha oils); and System 2 is unconverted methanol, FAME (product) and glycerol (by-product). Pressure and temperature diagrams were developed at different mole fraction of methanol (x_MeOH). As x_MeOH increased, the critical temperature (T_c) and pressure (p_c) increased. The increasing temperature causes the immiscibility region and the amount of methanol at the plait point to decrease. The maximum plait point pressure was observed at 19.20 MPa for palm and 19.33 MPa for Jatropha oil systems.     

Three-dimensional PTx phase diagrams through interactive computer graphics

Interactive computer graphic techniques have been developed for the display of binary mixture phase diagrams. The diagrams are defined in temperature-pressure-composition space, and are portrayed as wireframe objects with depth perception in order to provide a three-dimensional effect. The displays used were vectro refresh workstations whose transformation hardware allows real-time rotation, rescaling, and translation of the diagrams, while software allows the extraction of constant property Px, Tx, PT and x - y plots. The equilibrium surfaces and the critical lines were calculated using the Redlich-Kwong equation of state and its Soave modification.     

Phase diagrams with critical and tricritical liquid phase solution points in a ternary system

Critical phenomena of liquid phase stratification in ternary systems are considered. Models of the T-x ₁-x ₂ phase diagrams with critical and tricritical liquid solution points and isothermal diagrams over the corresponding interinvariant intervals are given.     

Extending the GC-SAFT-VR approach to associating functional groups: Alcohols, aldehydes, amines and carboxylic acids

The statistical associating fluid theory is a widely used molecular-based equation of state that has been successfully applied to study a broad range of fluid systems. It provides a framework in which the effects of molecular shape and interactions on the thermodynamics and phase behavior of fluids can be separated and quantified. In the original approach, molecules were modeled as chains composed of identical segments; the heterogeneity of molecules in terms of structure and functional groups was described implicitly through effective parameters. To overcome this limitation, in recent works [Peng et al. Fluid Phase Equilib. 227(2), 131 (2009); Ind. Eng. Chem. Res. 49(3), 1378 (2010)] the GC-SAFT-VR approach has been developed to extend the theory to model chains composed of segments of different size and/or energy of interaction and enable the development of a group-contribution approach within the SAFT-VR framework in which molecular heterogeneity and connectivity is explicitly accounted for. The parameters for several key functional groups (CH₃, CH₂, CH, CH₂CH, CO, C₆H₅, esters, ethers, cis-alkenes and trans-alkenes groups) were determined by fitting to experimental vapor pressure and saturated liquid density data for a number of small molecules containing the functional groups of interest and transferability of the parameters tested by comparing the theoretical predictions with experimental data for pure fluids not included in the fitting process and binary mixtures of both simple fluids and the VLE and LLE of small molecules in polymer systems. In this work, we further extend the applicability of the GC-SAFT-VR approach through the study of the vapor-liquid phase behavior of associating systems, such as linear and branched alcohols, primary and secondary amines, aldehydes, and carboxylic acids, and their mixtures. In the study of these new molecules several new functional groups (OH (linear and branched), HCO, NH₂, NH and COOH) are defined and their molecular parameters characterized. The transferability of the parameters is again tested by comparing the theoretical predictions with experimental data for pure fluids and binary mixtures not included in the fitting process. The GC-SAFT-VR approach is found to predict the phase behavior of the systems studied in most cases in good agreement with experimental data and accurately captures the effects of changes in structure and molecular composition on phase behavior.     

High-pressure densities and P–T–x–y diagrams for carbon dioxide+linalool and carbon dioxide+limonene

Liquid and vapor densities for carbon dioxide+linalool, and carbon dioxide+limonene were measured by using a system consisting of two vibrating tube densimeters. The P–T–x–y diagrams and saturated liquid and vapor densities for these two binary mixtures were determined at 313, 323 and 333 K, respectively, as well as at pressures up to 11 MPa. The density of the saturated CO₂ phase increased with increasing pressure. At higher pressure, the density of the liquid phase decreased with increasing pressure, corresponding to an increasing amount of carbon dioxide.     

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.     

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 behavior of dodecane–tridecane mixtures confined in SBA-15

Phase behavior of dodecane–tridecane (n-C₁₂H₂₆–C₁₃H₂₈, C₁₂–C₁₃) mixtures in bulk and confined in SBA-15 have been investigated using differential scanning calorimetry. Bulk C₁₂–C₁₃ system has a complicated behavior due to special rotator phase. It has been found that phase diagram of C₁₂–C₁₃/SBA-15 (3.8 nm) system is a straight line, C₁₂–C₁₃/SBA-15 (7.8 nm) a curve, and C₁₂–C₁₃/SBA-15 (8.9, and 17.2 nm) a loop line. The growth of the phase diagram shows size effect on phase behavior of C₁₂–C₁₃ mixtures. Moreover, in the range of 3.8–17.2 nm melting temperatures of pure C₁₂ and mixtures at mole fractions x _C13 = 0.1–0.5 have linear relation with inverse pore diameter, while C₁₃ and mixtures at x _C13 = 0.6–0.9 present curved lines. The confined mixtures show a similar melting behavior with pure C₁₂, C₁₃.     

Azeotropic line determination from the caloric data on binary stratifying systems

The experimental isochoric heat capacity values of stratifying n-hexane-water mixtures of the compositions 0.120, 0.166, 0.200, and 0.256 H₂O mole fractions were obtained using a high-temperature adiabatic calorimeter over the density ranges 244.74–498.25, 121.06–438.21, 252.02–500.00, and 208.11–398.88 kg/m³, respectively. The C _{V, x} heat capacities were tabulated for the mixture with 0.120 H₂O mole fractions. Liquid-liquid and liquid-gas phase equilibrium curves were plotted. The suggestion was made that the intersection point between these curves characterized the state of an azeotrope. The azeotropic line with the critical point at its end was constructed.     

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.