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.     

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.     

Excess properties of Lennard-Jones binary mixtures from computer simulation and theory

Monte Carlo simulation and theory are used to calculate the excess thermodynamic properties of binary mixtures of spherical Lennard-Jones molecules. We study the excess functions of three binary mixtures characterized by the following size and dispersive energy ratios: (1) (σ₂₂/σ₁₁)³ = 2 and ?₂₂/?₁₁ = 2; (2) (σ₂₂/σ₁₁)³ = 1 and ?₂₂/?₁₁ = 1/2 and (3) (σ₂₂/σ₁₁)³ = 1/2 and ?₂₂/?₁₁ = 2. In all cases, the unlike size parameter, σ₁₂, is kept constant and equal to the value given by the Lorentz combining rule (σ₁₂ = (σ₁₁ + σ₂₂)/2). However, different unlike dispersive energy parameter values are considered through the following combining rules: (a) ?₁₂ = (?₁₁?₂₂)^1/2 (Berthelot rule); (b) ?₁₂ = ?₁₁ (association); and (c) ?₁₂ = ?₂₂ (solvation). The pressure and temperature dependence of the excess volume and excess enthalpy is studied using the NpT Monte Carlo simulation technique for all the systems considered. Additionally, the simplest conformal solution theory is used to check the adequacy of this approach in predicting the excess properties in a wide range of thermodynamic conditions and variety of binary mixtures. In particular, we have applied the van der Waals one-fluid theory to describe Lennard-Jones binary mixtures through the use of the Johnson et al. [1993, Molec. Phys., 78, 591] Helmholtz free energy. Agreement between simulation results and theoretical predictions is excellent in all cases and thermodynamic conditions considered. This work confirms the applicability of the van der Waals one-fluid theory in predicting excess thermodynamic properties of mixtures of spherical molecules. Furthermore, since binary mixtures of spherical Lennard-Jones molecules constitute the reference fluid to be used in perturbation theories for complex fluids, such as the statistical association fluid theory (SAFT), this work shows clearly the applicability of the conformal solution theory within the framework of SAFT for predicting excess functions.     

Critical properties of homopolymer fluids studied by a Lennard-Jones statistical associating fluid theory

A Lennard-Jones statistical associating fluid theory, the soft-SAFT equation of state, is used to study the dependence of the critical properties on chain length for pure polymers. The critical constants of chains up to 10⁶ monomer units are predicted. The main advantage of these calculations, versus others found in the literature, is the fact that the equation is able to accurately predict experimental data available only for short chain lengths and, at the same time, it is able to predict the critical properties in the infinite chain length scaling regime, without any further assumptions. The equation gives quantitative agreement with experimental and Monte Carlo simulation data for normal alkanes. For very long chains, the equation predicts mean-field scaling behaviour, as expected. Moreover, taking advantage of the linearity of the molecular volumes and dispersive energies with the chain molecular weight, a value of 1/5 is found for the critical compressibility factor of an infinitely long n-alkane chain. It is demonstrated that this is not contradictory to Wertheim's theory.     

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.     

The chemical potential of Lennard-Jones associating fluids from osmotic Monte Carlo simulations

The chemical potential for a two-component Lennard-Jones fluid with associative interaction between opposite species promoting the formation of dimers is calculated using osmotic Monte Carlo (OMC) canonical ensemble simulations. Grand canonical Monte Carlo simulations also are performed to verify the accuracy of the OMC approach. The data from both methods agree very well for thermodynamic states with different degrees of dimerization. It follows that the OMC is a promising approach for the determination of the thermodynamics of and equilibria between associating and non-associating fluids and associating fluid mixtures.     

Perturbation theory for mixtures of simple liquids

The perturbation approach developed by Weeks, Chandler, and Andersen (WCA) and by Verlet and Weis (VW) for pure systems is here generalized to the case of mixtures. We study binary mixtures of molecules interacting with the 12–6 Lennard-Jones potential, for which Monte Carlo simulations are available for comparison. The work is divided into two parts: The first part presents results of Monte Carlo calculations on mixtures of hard spheres of 864 and 1000 particles. The radial distribution functions generated are used to test the VW representation for the correlation functions of hard-sphere mixtures. This representation is found to work satisfactorily within the expected error limits. The second part deals with the two-step perturbation procedure for calculating the thermodynamic quantities of the Lennard-Jones system. The Lennard-Jones potential is divided into a reference potential, which is strictly repulsive, and an attractive part. The system of the reference potential is represented by a system of hard-sphere mixture with equivalent diameters determined by the WCA rule. Analytical expressions are given for evaluating these equivalent diameters. The Lennard-Jones system is then recovered to the first order by a λ expansion over the reference system. Comparison with Monte Carlo results for a mixture of Lennard-Jones molecules, obeying the Berthelot rule, shows that the total thermodynamic properties are reproduced by the perturbation theory to 1 per cent, while the agreement in excess properties is only moderately successful, similar to some other analytical theories compared here. To reproduce these excess properties, which are extremely small, a precision of 0·1 per cent in the theory is required. The present theory is estimated to be accurate to 1 per cent in view of the successive approximations made.     

Modelling the phase equilibria of multicomponent mixtures containing CO2, alkanes,water, and/or alcohols using the quadrupolar CPA equation of state

ABSTRACT

In this work, a quadrupolar cubic plus association (qCPA) equation of state is evaluated for its ability to predict the phase equilibria of multicomponent mixtures containing CO₂ and alkanes, alcohols, and/or water. A single binary interaction parameter is employed in qCPA for all binary combinations. All parameters are based solely on pure fluid or binary mixture data and multicomponent data are used only to evaluate the predictions. The performance of qCPA is, for all mixtures, compared to CPA where CO₂ is considered to be either non-associating (inert), solvating or self-associating. In the latter two approaches, an additional adjustable parameter is employed for binary pairs of CO₂ and an associating compound. The results show that the predictions with qCPA are very similar to the best performing CPA approaches, even though the model uses fewer adjustable binary parameters. The predictions with qCPA and the best CPA approaches are typically satisfactory and predict the general behaviour of the systems. As expected, qCPA and CPA with solvation or association typically performs better than inert CPA for two- and three phase vapour–liquid and vapour–liquid–liquid equilibria. However, inert CPA yields the best results of all the models for the prediction of dew point pressures.     

NpT-ensemble Monte Carlo calculations for binary liquid mixtures

A Monte Carlo method for the calculation of thermodynamic properties in the isothermal-isobaric ensemble is described. Application is made to the calculation of excess thermodynamic properties (enthalpy, volume and Gibbs free energy) of binary mixtures of Lennard-Jones 12-6 liquids. Comparison is made with the predictions of a number of theories of liquid mixtures; the so-called van der Waals one-fluid model and the variational theory of Mansoori and Leland are both found to give excellent results. The accuracy attainable in estimates of the excess properties is discussed in terms of statistical fluctuations in various calculated quantities and the advantages and disadvantages of the method are examined in relation to calculations by the more familiar constant-volume method.     

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.     

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.     

Solvent effects on model telechelic polymers

A theory to predict the competition between intermolecular and intramolecular hydrogen bonding is extended to mixtures and applied to a model telechelic mixture. The theory is tested by comparing with simulation results for a mixture of fully flexible linear chains that can associate in a hydrogen bonding solvent. The simulation model for the telechelic is a flexible linear tetramer hard sphere chain with two hydrogen bonding sites, one on each terminal segment. The solvent is modelled as a hard sphere with four tetrahedrally arranged hydrogen bonding sites. The solvent is seen to affect the ability of the solute to bond intermolecularly and intramolecularly. The extent of hydrogen bonding and thermodynamic properties of the system were studied using Monte Carlo simulation and compared with predictions from a new statistical mechanics based theory for mixtures. Agreement of simulation and theory is good over the range of densities, temperatures and compositions studied.     

Thermodynamic calculations of linear chain molecules using a SAFT model

The recently proposed model of statistical associated-fluid theory (SAFT) by Tang, Y. and Lu, B. C.-Y. (2000, Fluid Phase Equilibria, 171, 27) is applied to phase diagram calculations of non-associating and associating linear chain molecules in which n-alkanes and n-alkenes (representing the non-associating type) and water, 1-alkanols, acids and amines (representing the associating type) are investigated. For polar molecules, the dipole-dipole interaction is taken into consideration. Overall, the proposed model yields similar accuracy to the original SAFT model, It is found that the volume and energy parameters of non-associating chain segments in the same family follow certain linear relations with the carbon number. Remarkably, these linear relations are found to hold equally well in associating chain molecules. These observations suggest that SAFT may be implemented in a more predictive manner. Furthermore, the inclusion of the contribution from dipole-dipole interaction improves the calculated values for strong polar molecules like water.     

Intramolecular bonding in a statistical associating fluid theory of ring aggregates

The first-order thermodynamic perturbation theory of Wertheim (TPT1) is extended to treat ring aggregates, formed by inter- and intramolecular association. The expression for the residual association contribution to the Helmholtz free energy for ring aggregates, incorporating the appropriate terms in Wertheim's fundamental graph sum of the TPT1 density expansion, is derived to calculate the distribution of the molecular bonding states. This requires the introduction of two new parameters to characterise each possible ring type: the ring size τ, which is equal to one in the case of intramolecular association, and a parameter W that captures the likelihood of two ring-forming sites bonding. The resulting framework can be incorporated in equations of state that account for the residual association contribution to the free energy, such as the statistical associating fluid theory (SAFT) family, or the cubic plus association (CPA) equation of state. This extends the applicability of these equations of state to mixtures with an arbitrary number of association sites capable of hydrogen bonding to form intramolecular and intermolecular rings. The formalism is implemented within SAFT-VR Mie to calculate the fluid-phase equilibria of model chain-like molecules containing two associating sites A and B, allowing for the formation of open-chain aggregates and intramolecular bonds. The effect of adding a second component that competes for the association sites that mediate intramolecular association in the chain is also examined. Accounting for intramolecular bonding is shown to have a significant impact on the phase equilibria of such systems.     

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.     

Modeling the phase equilibria of a H2O–CO2 mixture with PC-SAFT and tPC-PSAFT equations of state

Water–carbon dioxide binary mixtures are important for a number of industrial and environmental applications. Accurate modeling of the thermodynamic properties is a challenging task due to the highly non-ideal intermolecular interactions. In this work, two models based on the Statistical Associating Fluid Theory (SAFT) are used to correlate reliable experimental vapor–liquid equilibria (VLE) and liquid–liquid equilibria (LLE) data in the temperature range 298–533?K. CO₂ is modeled as a non-associating or associating component within the Perturbed Chain-SAFT (PC-SAFT) and as a quadrupolar component within the truncated PC-Polar SAFT (tPC-PSAFT). It is shown that PC-SAFT with explicit account of H₂O–CO₂ cross-association and tPC-PSAFT with explicit account of CO₂ quadrupolar interactions are the most accurate of the models examined. Saturated liquid mixture density data are accurately predicted by the two models.     

Evaluation of liquid-vapour density profiles of associating fluids: A density-functional approach

The structure of liquid-gas interface of associating Lennard-Jones particles is studied using the density-functional theory and the Monte Carlo simulation. The model with one bonding site per particle is considered. It is shown that the considered version of the density functional is quite successful in predicting the gas-liquid density profile.     

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.