Improved models for the phase behaviour of hydrogen fluoride: chain and ring aggregates in the SAFT approach and the AEOS model

Hydrogen fluoride presents one of the strongest hydrogen bonds known. Ring aggregates exist both in the vapour and liquid phases at low temperatures resulting in an anomalously high low-temperature vapour pressure. The effect of ring-like aggregates on the vapour—liquid phase equilibria of associating fluids is studied within the framework of the statistical associating fluid theory (SAFT) and in the chemical model of Lencka and Anderko (AEOS). The SAFT approach incorporates separate contributions to describe chain formation, association (hydrogen bonding), and long range dispersion forces. The treatment of the association interactions stems from the thermodynamic perturbation theory of Wertheim. At the first level of approximation the contribution of ring-like aggregates is neglected and only chain- and treelike structures are treated. In this work an earlier extension of the approach to incorporate ring aggregates is used to model the phase behaviour of hydrogen fluoride. The chemical model of Lencka and Anderko for associating fluids is also considered together with a modification that takes into account the formation of ring aggregates. Vapour pressures and coexistence densities are examined together with heats of vapourization, and the calculations are compared with experimental data.     

Fluids with highly directional attractive forces. IV. Equilibrium polymerization

We investigate approximation methods for systems of molecules interacting by core repulsion and highly directional attraction due to several attraction sites. The force model chosen imitates a chemical bond by providing for bond saturation when binding occurs. The dense fluid is an equilibrium mixture ofs-mers with mutual repulsion. We use a previously derived reformulation of statistical thermodynamics, in which the particle species are monomeric units with a specified set of attraction sites bonded. Thermodynamic perturbation theory (TPT) and integral equations of two types are derived. The use of TPT is illustrated by explicit calculation for a molecular model with two attraction sites, capable of forming chain and ring polymers. Successes and defects of TPT are discussed. The integral equations for pair correlations between particles of specified bonding include calculation of self-consistent densities of species. Methods of calculating thermodynamic properties from the solutions of integral equations are given.Supported by the NSF under grant No. CHE-82-11236.     

A statistical associating fluid theory for electrolyte solutions (SAFT-VRE)

A general theory for electrolyte solutions is examined within the framework of the statistical associating fluid theory for potentials of variable range (SAFT-VR). A first extension of the theory (SAFT-VRE) has already been used to describe the thermodynamics and phase equilibria of aqueous solutions of alkali-halide salts [GALINDO, A., GIL-VILLEGAS, A., JACKSON, G. and BURGESS, A. N., 1999, J. phys. Chem., 103, 10272]. The approach incorporates separate contributions describing the monomer, associating and ionic interactions. In the spirit of the SAFT-VR approach the monomer contribution is written as a high-temperature perturbation expansion up to second order; the separate effects of solvent-solvent, solvent-ion and ion-ion interactions on the phase equilibria are studied. Water is taken to be the solvent throughout the study, with the same four-site model and parameters as in the previous work. The association contribution is essential to account for the hydrogen bonding interactions present in water. The effects of ion pairing and solvent-ion association are also examined. For the ionic contribution several levels of approximation are discussed. The effect of the different molecular parameters on the phase behaviour of a model aqueous solution is examined for the different choices.     

Vapour-to-Liquid Nucleation in Associating Lennard-Jones Fluids with Multiple Association Sites

The excess Helmholtz free energy functional for associating Lennard-Jones （L J） fluid is formulated in terms of a weighted density approximation for short-ranged interactions and a Weeks-Chandler-Andersen approximation for long-range attraction. Within the framework of density functional theory, phase equilibria, vapour-liquid surface tension and vapour-liquid nucleation properties including the density profile, work of formation, excess number of particles and critical supersaturation are investigated for associating LJ fluids with different numbers of association sites （M =1,2, 3, 4） per particle. The influences of association energy and association sites on phase equilibria, surface tension and vapour-liquid nucleation properties are discussed.     

Intramolecular association within the SAFT framework

A general theory for modelling intramolecular association within the SAFT framework is proposed. Sear and Jackson [Phys. Rev. E. 50 (1), 386 (1994)] and Ghonasgi and Chapman [J. Chem. Phys. 102 (6), 2585 (1995)] have previously extended SAFT to include intramolecular association for chains with two sites. We show that the resulting equations from the two approaches are equivalent, and use their work as a basis for developing a new general theory. The approach used by Ghonasgi and Chapman is based on mass balances and an infinite dilution result and provides the equations needed to determine the contribution to the Helmholtz free energy from association (inter- as well as intramolecularly) at equilibrium. Sear and Jackson rederived the contribution to the Helmholtz free energy from association from the theory by Wertheim [J. Stat. Phys. 42 (3–4), 459 (1986)] with inclusion of intramolecular association, and using this approach we obtain an expression for the Helmholtz free energy that is valid also at non-equilibrium states (with respect to hydrogen bonds), which is very useful when calculating derivatives.     

A SAFT model for associating Lennard-Jones chain mixtures

The recently proposed equation of state of statistical associated fluid theory (SAFT) is extended to associating Lennard-Jones (LJ) chain mixtures. In this extension, a new radial distribution function (RDF) for LJ mixtures is derived around the LJ potential size (σ _ij ). The RDF expression is completely analytical and real. Comparisons with computer simulation data under various conditions indicate that the RDF is very accurate up to its first peak. The new RDF, together with a previously established equation of state for LJ mixtures, is employed to study LJ chain mixtures by combining with Wertheim's first-order perturbation theory. The resulting equation of state is tested satisfactorily against computer simulation data for both non-associating and associating LJ chain mixtures, with a performance similar to its predecessors for pure LJ chains and LJ mixtures. The SAFT model is uniquely featured by being totally mixing-rule free and by being adjustable at both chain bonding and association sites. Moreover, the SAFT model is formulated very generally, so that it is applicable to both homonuclear and heteronuclear chain mixtures.     

Lennard-Jones chain mixtures: radial distribution functions from Monte Carlo simulation

Radial distribution functions are calculated for binary Lennard-Jones chain mixtures from Monte Carlo simulation. Average and end-to-end inter- and intrachain radial distribution functions are calculated, ten for a binary mixture and four for a pure component. The effects of density, concentration, temperature, chain length, Lennard-Jones size and energy parameters are investigated. It is found that intrachain radial distribution functions are largely independent of density except at very high densities, where they start to take on a structure tending towards that of a crystal lattice. In addition, the effect of using different distribution functions to calculate the associating contribution in statistical associating fluid theory (SAFT) is examined. Further, the effect of using short chain fluids rather than the monomer unit as the reference system in the calculation of the pressure and free energy of chain fluids in first-order thermodynamic perturbation theory (TPT) is examined. It is found that the choice of reference radial distribution function has a marked effect on the calculation of thermodynamic properties through the use of SAFT and TPT.     

TPT2 and SAFTD equations of state for mixtures of hard chain copolymers

We present the second-order thermodynamic perturbation theory (TPT2) and the dimer statistical associating fluid theory (SAFTD) equations of state for mixtures consisting of hetero-nuclear hard chain molecules based on extensions of Wertheim's theory for associating fluids. The second-order perturbation theory, TPT2, is based on the hard sphere mixture reference fluid. SAFTD is an extension of TPT1 (= SAFT) and is based on the non-spherical (hard disphere mixture) reference fluid. The TPT2 equation of state requires only the contact values of the hard sphere mixture site-site correlation functions, while the SAFTD equation of state requires the contact values of site-site correlation functions of both hard sphere and hard disphere mixtures. We test several approximations for site-site correlation functions of hard disphere mixtures and use these in the SAFTD equation of state to predict the compressibility factor of copolymers. Since simulation data are available only for a few pure copolymer systems, theoretical predictions are compared with molecular simulation results for the compressibility factor of pure hard chain copolymer systems. Our comparisons show a very good performance of TPT2, which is found to be more accurate than TPT1 (= SAFT). Using a modified Percus-Yevick site-site correlation function SAFTD is found to represent a significant improvement over SAFT and is slightly more accurate than TPT2. Comparison of SAFTD with generalized Flory dimer (GFD) theory shows that both are equivalent at intermediate to high densities for the compressibility factor of copolymer systems investigated here.     

Solution conformational preferences of glutaric, 3‐hydroxyglutaric, 3‐methylglutaric acid,and their mono‐ and dianions

Conformational preferences of glutaric, 3‐hydroxyglutaric and 3‐methylglutaric acid, and their mono‐ and dianions have been investigated with the aid of NMR spectroscopy. In contrast to succinic acid, glutaric acid displays essentially statistical conformational equilibria in polar and non‐polar solutions of high and low hydrogen‐bonding ability with no clear evidence for intramolecular hydrogen‐bonding interactions. The acid ionization constant ratios, K ₁/K₂, in D₂O and DMSO of glutaric, 3‐hydroxyglutaric, and 3‐methylglutaric acids also indicate that intramolecular interactions are much less important than, or indeed insignificant, for shorter‐chain acids. FTIR studies on 3‐methylglutaric acid indicate some preference for either association with solvent or dimerization, depending on the solvent, rather than intramolecular hydrogen bonding. Copyright © 2008 John Wiley & Sons, Ltd.     

Equation of state of long chain molecules and rings

An equation of state for long chain molecules has been proposed using statistical associating fluid theory (SAFT). The formalism derived here is based on the assumption that the chain is formed by the pairs of trimers. The equations of state for 48-mers and 192-mers are formulated and compared with Monte Carlo results. The theory has been developed to treat hard sphere molecules with two attraction sites to form a ring molecule. The equations of state for trimer, hexamer and 12-mer ring molecules have been formulated. There is excellent agreement with available Monte Carlo results. Second virial coefficients of tangent chain molecules and ring molecules have been determined numerically and compared with simulation results.     

Modified thermodynamic perturbation theory for fused–sphere dimer fluids

Wertheim's thermodynamic perturbation theory of first order (TPT1) is based on the approximation that the monomer–monomer distribution functions can be approximated by the reference fluid distribution functions regardless of the amount of bonding. This is remarkably accurate for chains formed by tangent spheres, but no longer valid for chains of fused spheres. This constitutes the reason for the inadequacy of TPT1 for fused sphere chains. We present a systematic modification of TPT1, the path integral perturbation method, that takes into account the variations of the distribution functions with extent of bonding. We demonstrate the accuracy of the theory for mixtures of hard spheres and diatomics over a range of extent of bonding (pure monomers to pure dimers) and degree of fusion (bond length 0–1). We found that the choice of reference fluid was decisive for the accuracy of the model's predictions. The proposed theory can accurately predict the properties of mixtures of hard spheres and diatomics, and of the pure fused diatomic fluids. The results from the path integral theory are in excellent agreement with simulation results, and compare favourably with the results from the Tildesley–Streett and the Boublík–Nezbeda equations of state.     

Structure and equation of state of interaction site models for disc-shaped lamellar colloids

We apply RISM (reference interaction site model) and PRISM (polymer-RISM) theories to calculate the site–site pair structure and the osmotic equation of state of suspensions of circular or hexagonal platelets (lamellar colloids) over a range of ratios of the particle diameter over thickness Dσ. Despite the neglect of edge effects, the simpler PRISM theory yields results in good agreement with the more elaborate RISM calculations, provided the correct form factor, characterizing the intramolecular structure of the platelets, is used. The RISM equation of state is sensitive to the number n of sites used to model the platelets, but saturates when the hard spheres, associated with the interaction sites, nearly touch; the limiting equation of state agrees reasonably well with available simulation data for all densities up to the isotropic–nematic transition. When properly scaled with the second virial coefficient, the equations of state of platelets with different aspect ratios Dσ nearly collapse on a single master curve.     

Cross-association of multi-component systems

The design and optimization of equipment in chemical industry (e.g. heat exchanger) and also process simulations require the knowledge of physical properties of mixtures, for instance the involved phase equilibria, enthalpies and the heat capacities. Most experimental data on these properties exists for pure compounds (e.g. water) and for binary mixtures (e.g. water ethanol). The database is however, very limited for mixtures of more than two species. Physically sound equations of state, like the Perturbed-Chain Statistical Associating Theory (PC-SAFT) have been used successfully to provide information about these thermodynamic properties for a wide variety of substances including systems of associating and non-associating or systems of associating and cross-associating species. One of the main challenges using this Wertheim-type equation of state is the mathematically implicit form of the underlying nonlinear system of equations, if association occurs. This article provides in depth information about our recently developed fast and stable algorithm to solve this system of equations numerically for multi-component systems, as well as a new method to find good initial values for the numerical algorithm. Furthermore, the numerical results are compared to experimental data on several properties of interest and found to be in good to excellent agreement.     

Thermodynamics of simple models of associating fluids: primitive models of ammonia,methanol, ethanol and water

Thermodynamic P-V-T properties of primitive models that descend directly from realistic Hamiltonians and reproduce the structure of real fluids have been studied both by means of theory and computer simulations. Analytic expressions for the Helmholtz free energy of four typical associating fluids, ammonia, methanol, ethanol and water, have been derived using the thermodynamic perturbation theory. Whereas for the models which allow only single bonding of each site the first-order theory is sufficient, for models in which some sites may form simultaneously up to two bonds the theory has to be extended to the second order. Comparison with simulation data shows that the theory is very accurate and has therefore also been used to determine vapour–liquid equilibria. We have found fundamental differences in the behaviour of different models; these differences are linked to the properties of the hydrogen-bond network that are discussed in detail.     

Theory and simulation of chain molecules with multiple bonding sites in an associating solvent

Although polyethylene oxide (PEO) is soluble in water, polymethylene oxide (PMO) is not, even though PMO has more association sites. Some suggest this is due to orientation effects in the water hydrogen-bond network. A simulation and theory study of the effect of bonding site density on thermodynamic properties and extent of bonding of a linear flexible chain in a hydrogen-bonding solvent is performed. Predictions from Wertheim's theory are compared against simulation results. Thermodynamic properties and extent of bonding were obtained. The solvent molecules are modeled as hard spheres with four association sites in a tetrahedral arrangement. The chains are flexible and consist of six tangent segments of hard spheres with bonding sites that interact with the solvent molecules. A solvent molecule can also form a bond with a second solvent molecule. The association interaction is modeled with an orientation-dependent square-well. The total number of bonding sites on each chain is varied and the effects studied. This is another test of the theory for the case of mixtures of associating molecules of different sizes. The Metropolis Monte Carlo technique was chosen to perform simulations in the canonical and isothermal–isobaric ensembles. Good agreement was found between theory and simulation.     

Studies of a primitive model of mixtures of nonpolar molecules with water

The paper presents calculations of the properties of binary mixtures of hard spheres and directionally associating hard spheres, a simple model for mixtures of nonpolar molecules with water that was developed by Nezbeda and his coworkers. Extensive results from Monte Carlo simulations in the isobaric, isothermal ensemble are presented for the density, configurational energy and chemical potentials in the mixtures for fluid states over a range of temperatures, pressures and compositions. A species exchange technique is used to compute the chemical potential difference between components in the mixtures. The results obtained are compared with the predictions of first-order thermodynamic perturbation theory (TPT). It is found that this theory provides an accurate picture of the system over most of the conditions considered. Calculations are also made of vapour–liquid coexistence for the model using TPT and calculations of solid–fluid coexistence for the model using TPT and existing results for the free energy of the pure component solids. It is found that the vapour–liquid coexistence for the model is pre-empted by the solid–fluid coexistence, as had previously been found for the pure component directionally associating hard sphere system.     

Continuum percolation of the four-bonding-site associating fluids

Wertheim’s integral equation theory for associating fluids is reformulated for the study of the connectedness properties of associating hard spheres with four bonding sites. The association interaction is described as a square-well saturable attraction between these sites. The connectedness version of the Ornstein-Zernike (OZ) integral equation is supplemented by the PY-like closure relation and solved analytically within an ideal network approximation in which the network is represented as resulting from the crossing of ideal polymer chains. The pair connectedness functions and the mean cluster size are calculated and discussed. The condition for the percolation transition and the analytical form of the percolation threshold are derived. The connection of the percolation with the gas-liquid phase transition is discussed.     

A new study of associating inhomogeneous fluids with classical density functional theory

ABSTRACT

A new density functional for the study of associating inhomogeneous fluids based on Wertheim's first-order thermodynamic perturbation theory is presented and compared to the most currently used associating density functionals. This functional is developed using the weighted density approximation in the range of association of hard spheres. We implement this functional within the framework of classical density functional theory together with modified fundamental measure theory to account for volume exclusion of hard spheres. This approach is tested against molecular simulations from literature of pure associating hard spheres and mixtures of non-associationg and associating hard spheres with different number of bonding sites close to a hard uniform wall. Furthermore, we compare and review our results with the performance of associating functionals from literature, one based on fundamental measure theory and the inhomogeneous version of Wertheim's perturbation theory. Results obtained with classical DFT and the three functionals show excellent agreement with molecular simulations in systems with one hard wall. For the cases of small pores where only one or two layers of fluid are allowed discrepancies between results with classical DFT and molecular simulations were found.