Liquid-vapor Phase Diagram Research Articles

Phase diagram of the hard-core Yukawa fluid within the integral equation method

In this study, the integral equation method proposed recently by Sarkisov [J. Chem. Phys. 114, 9496 (2001).], which has proved accurate for continuous potentials, is extended successfully to the hard sphere potential plus an attractive Yukawa tail. By comparing the results of thermodynamic properties, including the liquid-vapor phase diagram, with available simulation data, it is found that this method remains reliable for this class of models of interaction often used in colloid science.

Vapor-liquid equilibrium of hexadecapolar fluids from a perturbation-based equation of state

In this work a numerically tractable expression for the interaction potential between two point hexadecapoles with octahedral symmetry and a molecular-based equation of state derived by perturbation theory for hexadecapolar fluids are presented. The polar system is modeled by square-well particles with a point hexadecapole with octahedral symmetry at their centers. This equation of state is analytical in the state variables and in the potential parameters and allows us to study the effects of the hexadecapolar moment strength on the thermodynamic properties and liquid-vapor phase diagram. The equation presented here is applied to the thermodynamics of sulfur hexafluoride and gives very good predictions for the saturation pressures and the vapor-liquid phase diagram.

Vapor–liquid phase equilibria for mixtures containing diatomic Lennard–Jones molecules

Vapor–liquid phase diagrams are calculated for binary mixtures of diatomic Lennard–Jones molecules using Monte Carlo simulations and the Gibbs–Duhem integration method. We explore the effects of varying the molecular size ratio, intermolecular attraction parameter ratio and binary interaction parameter on the mixture’s phase behavior. We plot pressure versus composition vapor–liquid phase diagrams for the binary mixtures O 2–N 2, CO 2–C 2H 6 and N 2–C 2H 6 at different temperatures. Our results are in good agreement with experimental data, especially away from the critical point. Adding a quadrupole term to the two-center Lennard–Jones (2CLJ) potential model further improves the agreement with experimental data. We also investigate the dependence of Henry’s constant on temperature, pressure and binary interaction parameter.

Vapor-liquid equilibrium and critical behavior of the square-well fluid of variable range: A theoretical study

The vapor-liquid phase behavior and the critical behavior of the square-well (SW) fluid are investigated as a function of the interaction range, lambdain [1.25, 3], by means of the self-consistent Ornstein-Zernike approximation (SCOZA) and analytical equations of state based on a perturbation theory [A. L. Benavides and F. del Rio, Mol. Phys. 68, 983 (1989); A. Gil-Villegas, F. del Rio, and A. L. Benavides, Fluid Phase Equilib. 119, 97 (1996)]. For this purpose the SCOZA, which has been restricted up to now to a few model systems, has been generalized to hard-core systems with arbitrary interaction potentials requiring a fully numerical solution of an integro-partial differential equation. Both approaches, in general, describe well the liquid-vapor phase diagram of the square-well fluid when compared with simulation data. SCOZA yields very precise predictions for the coexistence curves in the case of long ranged SW interaction (lambda>1.5), and the perturbation theory is able to predict the binodal curves and the saturated pressures, for all interaction ranges considered if one stays away from the critical region. In all cases, the SCOZA gives very good predictions for the critical temperatures and the critical pressures, while the perturbation theory approach tends to slightly overestimate these quantities. Furthermore, we propose analytical expressions for the critical temperatures and pressures as a function of the square-well range.

Phase diagrams and capillarity condensation of methane confined in single- and multi-layer nanotubes

Using a newly developed algorithm in the Gibbs ensemble, the liquid–vapour phase diagram of methane adsorbed in single and multilayer nanotubes was considered. The method was based on improving statistical sampling by combining the cavity-bias and the parallel tempering formalisms in Gibbs ensemble Monte Carlo simulations (PTCBGMC). Two models were constructed in order to describe the liquid–vapour equilibrium of methane in nanotubes. The first model consisted of two simulated nanotube-shaped boxes, one containing the vapour and the other containing the liquid. The vapour and liquid were kept in thermodynamic equilibrium using the PTCBGMC algorithm. Cailletet–Mathias phase diagrams showed that upon adsorption of methane in the nanotubes, the critical temperature and density of methane decreased upon confinement. However, the behaviour of the liquid and vapour phases differed depending on whether the nanotube was single- or multi-layered. A second computational model was used to consider in detail the capillarity condensation of methane in the nanotubes. The results obtained were explained in terms of the difference in the strength of the intermolecular forces.

Computing Liquid–Vapor Phase Diagrams for Non-Ideal Binary Mixtures

This Mathcad worksheet is intended to help students learn about determining vapor–liquid phase diagrams for ideal and non-ideal binary mixtures.

Predicting Mixture Phase Equilibria and Critical Behavior Using the SAFT-VRX Approach

The SAFT-VRX equation of state combines the SAFT-VR equation with a crossover function that smoothly transforms the classical equation into a nonanalytical form close to the critical point. By a combinination of the accuracy of the SAFT-VR approach away from the critical region with the asymptotic scaling behavior seen at the critical point of real fluids, the SAFT-VRX equation can accurately describe the global fluid phase diagram. In previous work, we demonstrated that the SAFT-VRX equation very accurately describes the pvT and phase behavior of both nonassociating and associating pure fluids, with a minimum of fitting to experimental data. Here, we present a generalized SAFT-VRX equation of state for binary mixtures that is found to accurately predict the vapor-liquid equilibrium and pvT behavior of the systems studied. In particular, we examine binary mixtures of n-alkanes and carbon dioxide + n-alkanes. The SAFT-VRX equation accurately describes not only the gas-liquid critical locus for these systems but also the vapor-liquid equilibrium phase diagrams and thermal properties in single-phase regions.

Vapor−Liquid Equilibria and Interfacial Tensions of Associating Fluids within a Density Functional Theory

A self-consistent density functional theory is applied to investigating the phase behavior and interfacial tensions of water and low-molecular-weight normal alkanols. For the bulk phases, the density functional theory is reduced to an equation of state that provides an accurate description of saturation pressures as well as vapor-liquid phase diagrams. Near the critical region, the long-range fluctuations are taken into account using a renormalization group theory. With the same set of molecular parameters, the theory yields satisfactory results for both bulk properties and surface tensions of the associating fluids investigated at all temperatures.

A self-consistent approach for modelling the interfacial properties and phase diagrams of Yukawa, Lennard-Jones and square-well fluids

A self-consistent density-functional approach is presented for describing the phase behaviour and interfacial tensions of van der Waals fluids represented by the hard-core Yukawa (HCY), Lennard-Jones (LJ) and square-well (SW) potentials. The excess Helmholtz energy functional is formulated in terms of a modified fundamental measure theory (MFMT) for the short-ranged repulsion and a density-gradient expansion for the van der Waals attractions. Analytical expressions for the direct correlation functions of uniform fluids are utilized to take into account the effect of van der Waals’ attraction on intermolecular correlations. For bulk phases, the density functional theory is reduced to an equation of state (EOS) that provides accurate saturation pressures and vapour–liquid phase diagrams. Near the critical region, the long-range fluctuations can be corrected by using the renormalization group (RG) theory. With the same set of molecular parameters, the theory also yields satisfactory surface tensions and interfacial density profiles at all relevant temperatures.

Surface tension of quantum fluids from molecular simulations.

We present the first molecular simulations of the vapor-liquid surface tension of quantum liquids. The path integral formalism of Feynman was used to account for the quantum mechanical behavior of both the liquid and the vapor. A replica-data parallel algorithm was implemented to achieve good parallel performance of the simulation code on at least 32 processors. We have computed the surface tension and the vapor-liquid phase diagram of pure hydrogen over the temperature range 18-30 K and pure deuterium from 19 to 34 K. The simulation results for surface tension and vapor-liquid orthobaric densities are in very good agreement with experimental data. We have computed the interfacial properties of hydrogen-deuterium mixtures over the entire concentration range at 20.4 and 24 K. The calculated equilibrium compositions of the mixtures are in excellent agreement with experimental data. The computed mixture surface tension shows negative deviations from ideal solution behavior, in agreement with experimental data and predictions from Prigogine's theory. The magnitude of the deviations at 20.4 K are substantially larger from simulations and from theory than from experiments. We conclude that the experimentally measured mixture surface tension values are systematically too high. Analysis of the concentration profiles in the interfacial region shows that the nonideal behavior can be described entirely by segregation of H(2) to the interface, indicating that H(2) acts as a surfactant in H(2)-D(2) mixtures.

TOWARDS A MOLECULAR THEORY FOR THE VAN DER WAALS–MAXWELL DESCRIPTION OF FLUID PHASE TRANSITIONS

We briefly review developments of theories for phase transitions of molecular fluids and mixtures, from semi-phenomenological approaches providing equations of state with adjustable parameters to first-principles microscopic methods qualitatively correct for a variety of molecular models with realistic interaction potentials. We further present the generalization of the van der Waals–Maxwell description of fluid phase diagrams to account for chemical specificities of polar molecular fluids, such as hydrogen bonding. Our theory uses the reference interaction site model (RISM) integral equation approach complemented with the new closure we have proposed (KH approximation), successful over a wide range of density from gas to liquid. The RISM/KH theory is applied to the known three-site models of water, methanol, and hydrogen fluoride. It qualitatively reproduces their vapor-liquid phase diagrams and the structure in the gas as well as liquid phases, including hydrogen bonding. Furthermore, phase transitions of water and methanol sorbed in nanoporous carbon aerogel are described by means of the replica generalization of the RISM approach we have developed. The changes as compared to the bulk fluids are in agreement with simulations and experiment. The RISM/KH theory is promising for description of phase transitions in various associating fluids, in particular, electrolyte as well as non-electrolyte solutions and ionic liquids.

Analytical equations of state for double Yukawa and hard core double Yukawa fluids: application to simple and colloidal fluids

Analytic, explicit and non-empirical equations of state for the double Yukawa and hard core double Yukawa fluids are presented. They are based on a second-order inverse temperature expansion of the free energy within the mean spherical approximation. These equations of state yield good results for the pressure and internal energy when compared with exact mean spherical approximation calculations or Monte Carlo simulations performed on various double Yukawa fluids modelling either simple fluids or more complex systems such as sterically stabilized colloidal dispersions. Liquid–vapour phase diagrams for simple fluids and the coexistence lines between the dilute and concentrated phases of colloidal dispersions are also well predicted except around the critical points.

Monte Carlo calculations for the phase equilibria of alkanes, alcohols, water, and their mixtures

Configurational-bias Monte Carlo simulations were carried out to investigate (i) the self-aggregation of 1-hexanol in n-hexane rich binary solutions; (ii) the partitioning of normal alkane and primary alkanol solutes between water, neat or water-saturated 1-octanol, n-octane, and vapour phases, and (iii) the vapour–liquid phase diagram of the binary mixture of methanol and n-hexane at a temperature of 448 K. The optimized potentials for liquid simulations (OPLS) united-atom force field was used to describe the interactions of the alkanes and alcohols, and the transferable intermolecular potentials 4 point (TIP4P) model was used for water. Analysis of radial distribution functions and their corresponding number integrals supports a micro-heterogenous picture of the 1-hexanol/hexane solutions with the majority of the 1-hexanol molecules being part of hydrogen-bonded clusters with four to six molecules. The OPLS force field yields Gibbs free energies of transfer that are in qualitative, albeit not quantitative agreement with experimental results. Comparison of the partitioning between a helium vapour phase and dry and wet (water-saturated) 1-octanol established that water saturation affects mostly the partitioning of polar solutes, while differences for alkane partitioning were found to be negligible. The vapour pressure of OPLS n-hexane is too low and that of methanol too high, resulting in a binary vapour–liquid phase diagram that deviates significantly from experiment.

An asymptotic expression for the critical-region “bird’s beak” isotherm and adjacent isotherms on the vapor–liquid phase diagram of a simple binary mixture

For a binary mixture of a dilute nonvolatile solute in a volatile solvent, an asymptotic expression is derived for isothermal dew-bubble curves in the region just above the solvent critical point. The expression depends only on the solvent coexistence properties and the initial slopes of the continuous critical locus, with no adjustable parameters. It clarifies the mathematical behavior of these curves and shows why, for this situation, classical critical exponents can be used with relatively small error. For supercritical extraction applications, the expression does not apply to solutes with large, complex molecules, since the critical locus with carbon dioxide is usually discontinuous, but it should apply to carbon dioxide+cosolvent mixtures. The formula is in good quantitative agreement with experiment for three simple nonpolar mixtures and for carbon dioxide+acetone, but shows only qualitative agreement for carbon dioxide+ethanol.

Self-assembly of ionic surfactants and formation ofmesostructures

Self-assembly of ionic surfactants resulting in micelle formation isstudied using off-lattice Monte Carlo simulation in twodimensions (2D). We find that in addition to the Lennard-Jones (LJ)interaction, which acts among all of the particles, ascreened Coulomb interaction, acting only among the surfactant heads,makes the geometric arrangement of micelles very different comparedto that for neutral surfactants at the same density. At lowconcentrations, ionic surfactants produce spherical micelles andexhibit intermicellar ordering arising out of the longer-rangerepulsion of the surfactant head groups. For larger concentrations,they tend to form bi-layers. Motivated by recent experiments onpreparation of mesoporous sieves through surfactant-templatingroutes, we then extend these studies to micellization of ionicsurfactants in the presence of other neutral host particles. Themorphologies of the surfactant-host composites are studied as functions of host particle densities, sizes, and their interactionstrengths. Assuming the presence of micelles as quenched disorder,we obtain a qualitative understanding of the local ordering of thehost particles in terms of a dimensionless density parameter fromthe known liquid-vapour phase diagram in 2D of a LJ fluid.

Phase transitions in an associating, network-forming, Lennard-Jones fluid in slit-like pores. II. Extension of the density functional method

We study adsorption of hydrogen-bonded fluids in slit-like pores with strongly attractive walls, in the framework of the four-site associating Lennard-Jones model. The density profiles, as well as the phase behavior, are obtained by using a density functional method. We have found that, at temperatures lower than the critical temperature of the bulk fluid, the confined fluid undergoes one or more layering transitions dependent on the pore width, followed by capillary condensation. Each of the transitions is localized by analyzing the grand thermodynamic potential. The density profiles of nonbonded and differently bonded particles demonstrating changes of the structure of the fluid in the pore along the coexistence are discussed briefly. The critical temperature for capillary condensation is lower for confined fluid, compared with that for the bulk liquid–vapor transition, as expected. However, an increase of the energy of association between fluid species increases the critical temperatures for layering transitions and for capillary condensation. The envelope of the capillary condensation is narrower than the bulk liquid–vapor phase diagram. The ratio between the critical temperatures for layering transitions and capillary condensation depends on the pore width. The critical temperature for the second layering is always lower than for the first one. The triple point temperature between either the second layering transition and the capillary condensation (in wider pores) or the first layering transition and the capillary condensation (in narrower pores) increases with decreasing pore width. The triple point temperature between the layering transitions is much lower than the relevant temperature between the second layering transition and the capillary condensation. The triple point temperatures also depend on the association energy. We have shown that highly bonded fluid species prevail at triple point temperatures.

Critical properties of non-spherical molecule fluids from the virial expansion

Critical constants of pure fluids (as important reference data in constructing vapour-liquid phase diagrams and basic input of various estimation methods) were determined for systems of non-spherical Kihara molecules; values of the critical temperature, density, compression factor and pressure of fluids composed of prolate and oblate molecules were evaluated from the fourth-order virial expansion. The second and third virial coefficients of the Kihara molecules were determined by applying the recently proposed method in which the effect of molecular core geometry and functional dependence of a pair interaction on the surface-surface distance are factorized and the former contribution determined from a formula for the corresponding hard convex body virial coefficient. The virial expansion for non-spherical Kihara molecules is applied to determine the critical constants of n-alkanes (methane to octane) and cyclic hydrocarbons (cyclopentane, cyclohexane, benzene and naphthalene); a fair agreement with experimental data was found.

Structure and phase transitions in a network-forming associating Lennard-Jones fluid in a slit-like pore: a density functional approach

A study is reported of adsorption of an associating Lennard-Jones fluid with four associative sites per molecule in a slit-like pore. The density distribution of particles in the pore and thermodynamics properties are evaluated by using a density functional method. It is found that at low temperatures the fluid exhibits a set of layering transitions, followed by capillary condensation. Transitions are localized by analysing the grand canonical potential. The density profiles of particles and the distribution of unbound and differently bonded particles demonstrate changes in the structure of the fluid in the pore along the phase coexistence. The critical temperature is lower for a confined fluid, compared with the bulk counterpart. However, an increase in the energy of association increases the critical temperature. The envelope of the capillary condensation is narrower than the bulk liquid-vapour phase diagram. The dependence of the solvation force on the energy of association and on the bulk density is discussed.

Raoult's Law: Binary Liquid-Vapor Phase Diagrams: A Simple Physical Chemistry Experiment

A new physical chemistry laboratory experiment illustrating Raoult's law behavior of ideal solutions is described. The vapor composition of binary mixtures of a ketone and an aromatic solvent is measured spectrophotometrically and the results used to generate Raoult's law vapor pressure plots and binary liquid-vapor phase diagrams. The method is more straightforward and experimentally easier to carry out than the traditional distillation techniques. Student results are presented for ortho-xylene/acetone and ortho-xylene/methyl isobutyl ketone mixtures. Both of these solutions show slight positive deviations from Raoult's law. The results are discussed in terms of the intermolecular interactions occurring in the mixtures.

Liquid-vapor phase behavior of a symmetrical binary fluid mixture

Using Monte-Carlo simulation and mean field calculations, we study the liquid-vapour phase diagram of a square well binary fluid mixture as a function of a parameter $\delta$ measuring the relative strength of interactions between particles of dissimilar and similar species. The results reveal a rich variety of liquid-vapour coexistence behaviour as $\delta$ is tuned. Specifically, we uncover critical end point behaviour, a triple point involving a vapour and two liquids of different density, and tricritical behaviour. For a certain range of $\delta$, the mean field calculations also predict a `hidden' (metastable) liquid-vapour binodal.