Qualities of models for evaluation,representation and prediction of fluid phase equilibrium data

After defining a model for fluid phase equilibria in the framework of classical thermodynamics, the methods of building models and obtaining parameters from experimental data are briefly reviewed. A personal view is given on models and methods useful at various stages of the scientific process of understanding fluid phase equilibria, especially vapour-liquid equilibria i.e. the evaluation of experimental data and the search for physical significance. The needs for research with a view to its application are considered.     

Phasepy: A Python based framework for fluid phase equilibria and interfacial properties computation

Phasepy is a Python based package for fluid phase equilibria and interfacial properties calculation from equation of state (EoS). Phasepy uses several tools (i.e., NumPy, SciPy, Pandas, Matplotlib) allowing use Phasepy under Jupyter Notebooks. Phasepy models phase equilibria with the traditional ϕ –γ and ϕ –ϕ approaches, where ϕ (fugacity coefficient) can be modeled as a perfect gas, virial gas or EoS fluid, whereas γ (activity coefficient) can be described by conventional models (NRTL, Wilson, Redlich-Kister expansion, and the group contribution modified-UNIFAC). Interfacial properties are based on the square gradient theory couple to ϕ –ϕ approach. The available EoSs are the cubic EoS family extended to mixtures through the quadratic, modified-Huron-Vidal, and Wong-Sandler mixing rules. Phasepy allows to analyze phase stability, compute phase equilibria, interfacial properties, and optimize their parameters for vapor–liquid, liquid–liquid, and vapor–liquid–liquid equilibria for multicomponent mixtures. Phasepy implementation, and robustness are illustrated for binary and ternary mixtures.     

Pressure dependence of the vapor-liquid-liquid phase behavior in ternary mixtures consisting of n-alkanes, n-perfluoroalkanes, and carbon dioxide

Expansion of an organic solvent by an inert gas can be used to tune the solvent's liquid density, solubility strength, and transport properties. In particular, gas expansion can be used to induce miscibility at low temperatures for solvent combinations that are biphasic at standard pressure. Configurational-bias Monte Carlo simulations in the Gibbs ensemble were carried out to investigate the vapor-liquid-liquid equilibria and microscopic structures for two ternary systems: n-decane/n-perfluorohexane/CO2 and n-hexane/n-perfluorodecane/CO2. These simulations employed the united-atom version of the transferable potential for phase equilibria (TraPPE-UA) force field. Initial simulations for binary mixtures of n-alkanes and n-perfluoroalkanes showed that special mixing parameters are required for the unlike interactions of CHx and CFy pseudoatoms to yield satisfactory results. The calculated upper critical solution pressures for the ternary mixtures at a temperature of 298 K are in excellent agreement with the available experimental data and predictions using the SAFT-VR (statistical associating fluid theory of variable range) equation of state. The simulations yield asymmetric compositions for the coexisting liquid phases and different degrees of microheterogeneity as measured by local mole fraction enhancements.     

Fluid phase stability and equilibrium calculations in binary mixtures: Part I: Theoretical development for non-azeotropic mixtures

A framework is proposed for the solution of fluid phase equilibrium (P–T flash) for binary mixtures described by equations of state of general form. The framework is based on decomposing the phase equilibrium problem into sub-problems with more convenient and tractable mathematical and numerical properties. Systematic procedures are used to identify the mapping of the problem in the density and composition space, referred to as the density–composition pattern, at specified temperature and pressure. A series of stability tests is then carried out to explore the existence or non-existence of phases. Once the existence of a phase has been determined, the limits of stability and physical bounds on the problem are used to define the search area for that phase in the density–composition pattern. Finally, all available information from this detailed analysis is used for the solution of phase equilibrium between the phases identified in order to find the stable state at the specified conditions. The features of the proposed approach are exposed in detail through an algorithm for the fluid phase equilibria of the augmented van der Waals equation of state applied to non-azeotropic mixtures.     

Artificial multiple criticality and phase equilibria: an investigation of the PC-SAFT approach

The perturbed-chain statistical associating fluid theory (PC-SAFT) is studied for a wide range of temperature, T, pressure, p, and (effective) chain length, m, to establish the generic phase diagram of polymers according to this theory. In addition to the expected gas-liquid coexistence, two additional phase separations are found, termed "gas-gas" equilibrium (at very low densities) and "liquid-liquid" equilibrium (at densities where the system is expected to be solid already). These phase separations imply that in one-component polymer systems three critical points occur, as well as equilibria of three fluid phases at triple points. However, Monte Carlo simulations of the corresponding system yield no trace of the gas-gas and liquid-liquid equilibria, and we conclude that the latter are just artefacts of the PC-SAFT approach. Using PC-SAFT to correlate data for polybutadiene melts, we suggest that discrepancies in modelling the polymer density at ambient temperature and high pressure can be related to the presumably artificial liquid-liquid phase separation at lower temperatures. Thus, particular care is needed in engineering applications of the PC-SAFT theory that aims at predicting properties of macromolecular materials.     

Geometrical interpretation of crystal-vapor phase equilibria and equilibria in helium systems

A geometrical interpretation of phase equilibria was suggested and used to determine several rules governing crystal-vapor equilibria, analytically describe the transition of helium systems into the superfluid state, and reveal the general properties of the low-temperature state of matter.     

Thermodynamic calculation of supercritical-fluid equilibria: New mixing rules for equations of state

Simple cubic equations of state with conventional mixing rules have played an important role in the calculation of phase equilibria and other thermodynamic properties of non-polar fluid mixtures. In the application of supercritical fluids to separation processes, volumetric as well as phase equilibrium properties are very important for rational process design.

Heyen (1980) proposed a cubic equation of state which shows better accuracy in the calculation of volumetric properties, compared to the Peng-Robinson equation of state. In order to apply his equation to polar mixtures, Heyen recently proposed a density-independent mixing rule, but this does not obey the universally-observed quadratic mixing rule of the second virial coefficient in the low-density limit.

This paper proposes a new density-dependent mixing rule for the Heyen equation of state. The Heyen equation of state with our new mixing rule appears to calculate the phase equilibria and the volumetric properties of CO₂-containing non-polar as well as polar mixtures with good accuracy.     

Surface tension and vapor-liquid phase coexistence of confined square-well fluid

Phase equilibria of a square-well fluid in planar slit pores with varying slit width are investigated by applying the grand-canonical transition-matrix Monte Carlo (GC-TMMC) with the histogram-reweighting method. The wall-fluid interaction strength was varied from repulsive to attractive such that it is greater than the fluid-fluid interaction strength. The nature of the phase coexistence envelope is in agreement with that given in literature. The surface tension of the vapor-liquid interface is calculated via molecular dynamics simulations. GC-TMMC with finite size scaling is also used to calculate the surface tension. The results from molecular dynamics and GC-TMMC methods are in very good mutual agreement. The vapor-liquid surface tension, under confinement, was found to be lower than the bulk surface tension. However, with the increase of the slit width the surface tension increases. For the case of a square-well fluid in an attractive planar slit pore, the vapor-liquid surface tension exhibits a maximum with respect to wall-fluid interaction energy. We also report estimates of critical properties of confined fluids via the rectilinear diameter approach.     

Computational studies on thermodynamic properties, effective diameters, and free volume of argon using an ab initio potential

A quantum mechanical derived ab initio interaction potential for the argon dimer was tested in molecular simulations to reproduce the thermophysical properties of the vapor-liquid phase equilibria using the Gibbs ensemble Monte Carlo simulations as well as the liquid and supercritical equation of state using the NVT Monte Carlo simulations. The ab initio interaction potential was taken from the literature. A recently developed theory [R. Laghaei et al., J. Chem. Phys. 124, 154502 (2006)] was used to compute the effective diameters of argon in fluid phases and the results were subsequently applied in the generic van der Waals theory to compute the free volume of argon. The calculated densities of the coexisting phases, the vapor pressure, and the equation of state show excellent agreement with experimental values. The effective diameters and free volumes of argon are given over a wide range of densities and temperatures. An empirical formula was used to fit the effective diameters as a function of density and temperature. The computed free volume will be used in future investigations to calculate the transport properties of argon.     

High pressure phase equilibrium for ethylene + 1-propanol system at 283.65 K

Phase equilibria and saturated densities for ethylene+1-propanol system at high pressures were measured using a static-circulation apparatus at 283.65 K. The equilibrium composition and saturated density of each phase were determined by using gas chromatograph and vibrating tube density meters, respectively. The saturated points near the critical region are further measured by the conventional indirect method. The present experimental results include vapor–liquid equilibria (VLE), liquid–liquid equilibria (LLE), and vapor–liquid–liquid equilibria (VLLE). The experimental data were correlated with various equations of state.     

Solubilization and phase equilibria of water-in-oil microemulsions : II. Effects of alcohols, oils, and salinity on single-chain surfactant systems

The solubilization and phase equilibria of w/o microemulsions have been shown to be dependent on two phenomenological parameters, namely the spontaneous curvature and elasticity of the interfacial film, when interfacial tension is very low. The spontaneous curvature of an interface is basically determined by the geometric packing of surfactant and cosurfactant molecules at the interface, whereas the interfacial elasticity is related to the energy required to bend the interface. The droplet size and solubilization of microemulsions is mainly determined by the radius of spontaneous curvature, and is further influenced by interfacial elasticity and interdroplet interactions. A w/o microemulsion with a highly curved and relatively rigid interfacial film can exist in equilibrium with excess water at the solubilization limit due to the interfacial bending stress. Increasing the natural radius and fluidity of the interface can increase the droplet size and hence the solubilization in the microemulsion. On the other hand, a w/o microemulsion with a highly fluid interfacial film can exist in equilibrium with an excess oil phase containing a low density of microemulsion droplets due to attractive interdroplet interaction. Increasing the interfacial rigidity and decreasing the natural radius in this case can increase water solubilization in the microemulsion by retarding the phase separation process. Thus, a maximum water solubilization in a w/o microemulsion can be obtained by minimizing both the interfacial bending stress of rigid interfaces and the attractive interdroplet interaction of fluid interfaces at an optimal interfacial curvature and elasticity. The study of phase equilibria of microemulsions can serve as a simple method to evaluate the property of the interface and provide phenomenological guidance for the formulation of microemulsions with maximum solubilization capacity.     

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.     

Equations for describing liquid-vapor phase equilibria in binary mixtures

Three forms of equations for describing experimental data on liquid and vapor pressures, depending on temperature and composition at phase equilibria in binary mixtures, are proposed and evaluated. It is determined that the form of equation depends on the relationship between the temperature of a mixture and the critical temperatures of the components of the mixture. Exact data on the phase equilibria in nitrogenoxygen, nitrogen-argon, and oxygen-argon mixtures [1] are approximated to assess the effectiveness of the equations’ forms. It is found that the equations also allow us to determine the phase composition at a given temperature and pressure and temperatures of phases at a given pressure and composition.     

Molecular simulation of the phase behavior of fluids and fluid mixtures using the synthetic method

A new molecular simulation procedure is reported for determining the phase behavior of fluids and fluid mixtures, which closely follows the experimental synthetic method. The simulation procedure can be implemented using Monte Calro or molecular dynamics in either the microcanonical or canonical statistical ensembles. Microcanonical molecular dynamics simulations are reported for the phase behavior of both the pure Lennard-Jones fluid and a Lennard-Jones mixture. The vapor pressures for the pure fluid are in good agreement with Monte Carlo Gibbs ensemble and Gibbs-Duhem calculations. The Lennard-Jones mixture is composed of equal size particles, with dissimilar energy parameters (?(2)∕?(1) = 1∕2, ?(12)∕?(1) = 1∕2). The binary Lennard-Jones mixture exhibits liquid-liquid equilibria at high pressures and the simulation procedure allows us to estimate the coordinates of the high-pressure branch of the critical curve.     

Development of the PhDi software package for the calculation of phase diagrams of binary systems using parameters of state equations and thermodynamic models for solutions

The PhDi software package developed earlier for calculations of phase equilibria by the convex hull method was modified by incorporating two additional modules: “State equations” and “Local composition thermodynamic models.” The possibility of the application of the convex hull method with cubic state equations to construct phase diagrams of binary systems was shown. A new database containing parameters of the corresponding thermodynamic models has been added to the initial version of the PhDi package. The efficiency of the performance of the package in solving the problem of choosing an appropriate state equation to predict conditions of phase equilibria in binary systems was demonstrated.     

A perturbed hard-sphere-chain equation of state for nematic liquid crystals and their mixtures with polymers

A perturbed hard-sphere-chain (PHSC) equation of state is presented to compute nematicisotropic equilibria for thermotropic liquid crystals, including mixtures. The equation of state consists of an isotropic term and an anisotropic term given by the Maier-Saupe theory whose contribution disappears in the isotropic phase. The isotropic contribution is the recently presented PHSC equation of state for normal fluids and polymers which uses a reference equation of state for athermal hard-sphere chains and a perturbation theory for the squarewell fluid of variable well width. The PHSC equation of state gives excellent correlations of pure-component pressure-volume-temperature data in the isotropic region and, combined with the Maier-Saupe theory, correlates the dependence of nematic-isotropic transition temperature on the pressure. Theory also predicts a nematic-isotropic biphasic region and liquid-liquid phase separation in a temperature-composition diagram of binary mixtures containing a nematic liquid crystal and a normal fluid or polymer. Theory and experiment show good agreement for pure fluids as well as for mixtures.     

A global investigation of phase equilibria using the perturbed-chain statistical-associating-fluid-theory approach

The recently developed perturbed-chain statistical-associating-fluid theory (PC-SAFT) is investigated for a wide range of model parameters including the parameter m representing the chain length and the thermodynamic temperature T and pressure p. This approach is based upon the first-order thermodynamic perturbation theory for chain molecules developed by Wertheim [M. S. Wertheim, J. Stat. Phys. 35, 19 (1984); ibid. 42, 459 (1986)] and Chapman et al. [G. Jackson, W. G. Chapman, and K. E. Gubbins, Mol. Phys. 65, 1 (1988); W. G. Chapman, G. Jackson, and K. E. Gubbins, ibid. 65, 1057 (1988)] and includes dispersion interactions via the second-order perturbation theory of Barker and Henderson [J. A. Barker and D. Henderson, J. Chem. Phys. 47, 4714 (1967)]. We systematically study a hierarchy of models which are based on the PC-SAFT approach using analytical model calculations and Monte Carlo simulations. For one-component systems we find that the analytical model in contrast with the simulation results exhibits two phase-separation regions in addition to the common gas-liquid coexistence region: One phase separation occurs at high density and low temperature. The second demixing takes place at low density and high temperature where usually the ideal-gas phase is expected in the phase diagram. These phenomena, which are referred to as "liquid-liquid" and "gas-gas" equilibria, give rise to multiple critical points in one-component systems, as well as to critical end points and equilibria of three fluid phases, which can usually be found in multicomponent mixtures only. Furthermore, it is shown that the liquid-liquid demixing in this model is not a consequence of a "softened" repulsive interaction as assumed in the theoretical derivation of the model. Experimental data for the melt density of polybutadiene with molecular mass Mw=45,000 gmol are correlated here using the PC-SAFT equation. It is shown that the discrepancies in modeling the polymer density at ambient temperature and high pressure can be traced back to the liquid-liquid phase separation predicted by the equation of state at low temperatures. This investigation provides a basis for understanding possible inaccuracies or even unexpected phase behavior which can occur in engineering applications of the PC-SAFT model aiming at predicting properties of macromolecular substances.     

Calculation of phase equilibria in random copolymer systems

Synthesis and application of copolymers are not seldom connected with different phase equilibria. Their precise knowledge is of importance for industrial processing as well as it is a profound basis for a better understanding of the nature and thermodynamics of such systems. As a common situation today, enough experimental information is seldom available in the necessary or desired amount, and a lot of model calculation is, therefore, more or less unavoidable to cover the desired ranges of application. Different equations-of-state as well as lattice models are discussed with respect to their applicability for calculating liquid-liquid and gas-liquid phase equilibria in copolymer solutions and blends. Examples for high-pressure phase equilibria in monomer/copolymer mixtures, liquid-liquid demixing in copolymer blends and for the isotropicnematic phase equilibrium in systems with rigid rod-like copolymers characterized by distributions of rigid and flexible chain parts are given. The effects of copolymer polydispersity are included by means of continuous thermodynamics. Literature references for original sources, earlier reviews and further applications round up this paper.