SAFT-VR Mie Equation Of State Research Articles

Classical Density Functional Theory Consistent with the SAFT-VR Mie Equation of State: Development of Functionals and Application to Confined Fluids

Classical Density Functional Theory Consistent with the SAFT-VR Mie Equation of State: Development of Functionals and Application to Confined Fluids

Molecular dynamics simulation of methane hydrates: Prediction of the phase equilibria using extracted microscopic parameters from SAFT-VR Mie EOS

This paper explores the three-phase equilibrium of structure-I methane hydrate by utilizing the extracted Mie potential parameters for non-bonded molecular interactions in the LAMMPS molecular dynamics simulator. An in-house code was developed to numerically solve the SAFT-VR Mie equation of state (EOS). To reduce numerical errors, unlike common practices, we incorporated all partial derivatives analytically in the code. The microscopic simulation parameters for methane and water molecules were selected so that saturation pressure and liquid density data from EOS were fitted on experimental data. Using this approach, the three-phase (V−Lw−H) equilibrium temperatures for pressure values of 30, 70, 100, and 300 bar were calculated. The solubility of methane in the aqueous phase at three-phase equilibrium was also determined. The results were validated against available laboratory data. To evaluate the performance of Mie potential parameters, the MD simulations were repeated using three well-known force fields for methane molecules, namely, Optimized Potentials for Liquid Simulations-United Atom (OPLS-UA), Universal force field (UFF), and General Amber force field (GAFF). The TIP4P/Ice force field was used for water molecules in all cases. The comparison with experimental data reveals that while the accuracy of the UFF force field is the highest, our simulation approach outperforms OPLS-UA and GAFF force fields in predicting methane solubility in the aqueous phase. Besides molecular dynamics simulation, the van der Waals-Platteeuw model was combined with the SAFT VR Mie EOS to account for the hydrate phase. In our case study, the SAFT-VR Mie + vdW-P model proved to be more accurate in predicting the three-phase equilibrium data compared to the MD approach.

High-pressure phase equilibrium data for the carbon dioxide + methyl esters homologous series and SAFT-VR Mie modelling

High-pressure phase transition data measured using a visual static synthetic method are presented for various CO2 + saturated fatty acid methyl ester systems in the range 308–358 K and up to 27.5 MPa. The newly measured data cover methyl esters between methyl decanoate and methyl docosanoate in the liquid-like, mixture critical and vapour-like regions, and complement previously published data. The systems studied do not suggest three-phase behaviour, liquid-liquid immiscibility or temperature inversions in the temperature range considered. An increase in temperature and molecular mass of the methyl ester leads to an increase in the phase transition pressure. The SAFT-VR Mie equation of state was able to correlate the data well with decreasing performance at higher methyl ester molecular masses and lower temperatures.

Application of machine-learning algorithms to predict the transport properties of Mie fluids.

The ability to predict transport properties of fluids, such as the self-diffusion coefficient and viscosity, has been an ongoing effort in the field of molecular modeling. While there are theoretical approaches to predict the transport properties of simple systems, they are typically applied in the dilute gas regime and are not directly applicable to more complex systems. Other attempts to predict transport properties are performed by fitting available experimental or molecular simulation data to empirical or semi-empirical correlations. Recently, there have been attempts to improve the accuracy of these fittings through the use of Machine-Learning (ML) methods. In this work, the application of ML algorithms to represent the transport properties of systems comprising spherical particles interacting via the Mie potential is investigated. To this end, the self-diffusion coefficient and shear viscosity of 54 potentials are obtained at different regions of the fluid-phase diagram. This data set is used together with three ML algorithms, namely, k-Nearest Neighbors (KNN), Artificial Neural Network (ANN), and Symbolic Regression (SR), to find correlations between the parameters of each potential and the transport properties at different densities and temperatures. It is shown that ANN and KNN perform to a similar extent, followed by SR, which exhibits larger deviations. Finally, the application of the three ML models to predict the self-diffusion coefficient of small molecular systems, such as krypton, methane, and carbon dioxide, is demonstrated using molecular parameters derived from the so-called SAFT-VR Mie equation of state [T. Lafitte et al. J. Chem. Phys. 139, 154504 (2013)] and available experimental vapor-liquid coexistence data.

Multipolar SAFT-VR Mie Equation of State: Predictions of Phase Equilibria in Refrigerant Systems with No Binary Interaction Parameter.

An extension of the SAFT-VR Mie equation of state is proposed to predict the properties of multipolar fluids. The new model, referred to as multipolar M-SAFT-VR Mie, incorporates the general multipolar term developed by Gubbins and co-workers, which accounts for dipole-dipole, quadrupole-quadrupole, and dipole-quadrupole interactions. A modification of the third order terms in the perturbation theory results in an accurate description of the simulation data for multipolar Lennard-Jones fluids. Both the M-SAFT-VR Mie and polar soft-SAFT models are extended to account for polarizability, and a good agreement is obtained with molecular simulation data. The M-SAFT-VR Mie model is applied to refrigerant systems, and it is found that including both dipole and quadrupole moments in molecular models leads to more accurate results than using only a dipole moment. The new model provides excellent predictions of the vapor-liquid equilibria of zeotropic and azeotropic refrigerant mixtures without the need for binary interaction parameters, making it a valuable tool for formulating low-GWP working fluids.

Deriving force fields with a multiscale approach: From ab initio calculations to molecular-based equations of state.

Using theoretical and computational tools for predicting thermophysical properties of fluid systems and the soft matter has always been of interest to the physical, chemical, and engineering sciences. Certainly, the ultimate goal is to be able to compute these macroscopic properties from first-principles calculations beginning with the very atomic constitution of matter. In this work, Mie potential parameters were obtained through dimer interaction energy curves derived from ab initio calculations to represent methane and substituted-methane molecules in a spherical one-site coarse-grained model. Bottom-up-based Mie potential parameters of this work were compared with top-down-based ones from the statistical associating fluid theory (SAFT) models for the calculation of thermodynamic properties and critical point by molecular dynamics simulations and SAFT-VR Mie equation of state. Results demonstrated that bottom-up-based Mie potential parameters when averaging the Mie potential parameters of a representative population of conformers provide values close to the top-down-based ones from SAFT models and predict well properties of tetrahedral molecules. This shows the level of consistency embedded in the SAFT-VR Mie family of models and confers the status of a purely predictive equation of state for SAFT-VR Mie when a reasonable model is considered to represent a molecule of interest.

The role of cross-association between carbon dioxide and hydrogen sulfide using the SAFT-VR Mie equation of state

One of the advantages of the contemporary, statistical-mechanical-based equations of state is the possibility of connecting molecular information to macroscopic behavior through an analytical mathematical model. The Statistical Associating Fluid Theory (SAFT) is the basis of a family of such equations of state, which are able, in principle, to represent the thermodynamic behavior of associating fluids such as water, alcohols, among others. In this work, we study the role of cross-association between carbon dioxide and hydrogen sulfide using the SAFT-VR Mie equation of state, which is one of the most recent and robust versions within the SAFT family. Two possibilities of cross-association are investigated: a Lewis acid-base interaction between the sulfur atom from hydrogen sulfide and the carbon atom from carbon dioxide, and a hydrogen bond between the hydrogen atoms from hydrogen sulfide and the oxygen atoms from carbon dioxide. Two cross-association parameters are fitted to vapor-liquid equilibrium data of the binary mixture for each hypothesis: the specific energy and the bonding volume. We show that taking into account cross-association leads to improved vapor-liquid equilibrium description. We also compare the different modeling strategies for the calculation of species distribution in a gravitational field for a ternary mixture of methane-carbon dioxide-hydrogen sulfide, exploring how sensitive the consideration of cross-association is in describing such a phenomenon. Both vapor-liquid equilibrium and gravitational segregation results lead us to the conclusion that the cross-interaction effects should be considered to describe the thermodynamic behavior of mixtures involving hydrogen sulfide and carbon dioxide. Performing classical equilibrium molecular dynamics simulations, we were able to show that the Lewis acid-base interaction between the sulfur atom from hydrogen sulfide and the carbon atom from carbon dioxide seems to be more predominant than the hydrogen bond between the hydrogen atoms from hydrogen sulfide and the oxygen atoms from carbon dioxide.

Ab initio development of generalized Lennard-Jones (Mie) force fields for predictions of thermodynamic properties in advanced molecular-based SAFT equations of state.

A methodology for obtaining molecular parameters of a modified statistical associating fluid theory for variable-range interactions of Mie form (SAFT-VR Mie) equation of state (EoS) from ab initio calculations is proposed for non-associative species that can be modeled as single spherical segments. The methodology provides a strategy to map interatomic or intermolecular potentials obtained from ab initio quantum-chemistry calculations to the corresponding Mie potentials that can be used within the SAFT-VR Mie EoS. The inclusion of corrections for quantum and many-body effects allows for an excellent, fully predictive description of the vapor-liquid envelope and other bulk thermodynamic properties of noble gases; this description is of similar or superior quality to that obtained using SAFT-VR Mie with parameters regressed in the traditional way using experimental thermodynamic-property data. The methodology is extended to an anisotropic species, methane, where similar levels of accuracy are obtained. The efficacy of using less-accurate quantum-chemistry methods in this methodology is explored, showing that these methods do not provide satisfactory results, although we note that the description is nevertheless substantially better than those obtained using the conductor-like screening model for describing real solvents (COSMO-RS), the only other fully predictive ab initio method currently available. Overall, the reliance on thermophysical data is completely dispensed with, providing the first extensible, wholly predictive SAFT-type EoSs.

Water–Hydrocarbon Phase Equilibria with SAFT-VR Mie Equation of State

The study of water–hydrocarbon phase equilibria is very important for both scientific and industrial purposes. This work is a thorough investigation of the predictive capability of SAFT-VR Mie equation of state (EoS) for aqueous hydrocarbon phase equilibria, including n-alkanes, alkenes, cycloalkanes, branched alkanes, and aromatic hydrocarbons. For alkenes and alkyl benzenes, induced association (solvation) with water is considered. The model is able to capture the correct phase equilibrium topology for mixtures of n-alkanes and alkyl-benzenes with water as type III according to Van Konynenburg and Scott, but it is not able to predict the barometric shift for alkanes longer than n-hexacosane due to the overshoot of the critical temperature of water. Quantitative results for the solubility of water in light hydrocarbons such as methane, ethane, and ethylene, as well as the mutual solubility of hydrocarbons and water above the solubility minimum, are possible. The latter is not predicted with SAFT-VR Mie EoS similar to other EoSs in literature. Accurate predictions are also reported for the pressure of the three-phase equilibrium and the critical lines for the binary mixtures. Henry’s law constant is often used for the prediction of low-pressure vapor–liquid equilibrium due to the low mutual solubility of hydrocarbons and water. EoS calculations are in reasonable agreement with experimental data for Henry’s constant, especially at higher temperatures. Qualitatively, SAFT-VR Mie captures the maximum of Henry’s constant with temperature, as well as the shift with the carbon number at temperatures near critical to water. SAFT-VR Mie is shown to be overall capable of straightforward prediction of phase equilibria of aqueous hydrocarbon mixtures.

SGTPy: A Python Code for Calculating the Interfacial Properties of Fluids Based on the Square Gradient Theory Using the SAFT-VR Mie Equation of State.

In this work, we showcase SGTPy, a Python open-source code developed to calculate interfacial properties (interfacial concentration profiles and interfacial or surface tension) for pure fluids and fluid mixtures. SGTPy employs the Square Gradient Theory (SGT) coupled to the Statistical Associating Fluid Theory of Variable Range employing a Mie potential (SAFT-VR-Mie). SGTPy uses standard Python numerical packages (i.e., NumPy, SciPy) and can be used under Jupyter notebooks. Its features are the calculation of phase stability, phase equilibria, interfacial properties, and the optimization of the SGT and SAFT parameters for vapor-liquid, liquid-liquid and vapor-liquid-liquid equilibria for pure fluids and multicomponent mixtures. Phase equilibrium calculations include two-phase and multiphase flash, bubble and dew points, and the tangent plane distance. For the computation of interfacial properties, SGTPy incorporates several options to solve the interfacial concentration, such as the path technique, an auxiliary time function, and orthogonal collocation. Additionally, the SGTPy code allows the inclusion of subroutines from other languages (e.g., Fortran, and C++) through Cython and f2py Python tools, which opens the possibility for future extensions or recycling tested and optimized subroutines from other codes. Supporting Information includes a review of the theoretical expressions required to couple SAFT-VR-Mie equation of state with the SGT. The use and capabilities of SGTPy are illustrated through step by step examples written on Jupyter notebooks for the cases of pure fluids and binary and ternary mixtures in bi- and three- phasic equilibria. The SGTPy code can be downloaded from https://github.com/gustavochm/SGTPy.

Modeling confined fluids with the multicomponent potential theory of adsorption and the SAFT-VR Mie equation of state

In this work, the SAFT-VR Mie equation of state is used in conjunction with the Dubinin-Radushkevich-Astakhov (DRA) and Steele potentials, according to the Multicomponent Potential Theory of Adsorption (MPTA), to model the thermodynamic behavior of fluids confined in porous media. The problem specifications are the temperature, volume, and number of moles of each component, yielding a Helmholtz energy-minimization problem. Calculation results show very good agreement with experimental adsorption data for systems of methane, nitrogen, carbon dioxide, ethane, ethylene, and isobutane adsorbed on activated carbon and molecular sieve.

Derivative Properties Data for Hydrogen–Ethylene Supercritical Mixtures Using a SAFT EoS and a SAFT Force Field

Hydrogen is a promising component for a more sustainable world, but its wide application depends upon the development of new technologies and processes. Therefore, a reliable set of thermodynamic properties is of paramount importance. Statistical mechanics provides the necessary tools to build very accurate molecular-based models for a given interaction potential. Statistical associating fluid theory (SAFT) entails a family of such models. From such molecular-based equations of state, new coarse-grained force fields can be derived for molecular simulations. In this work, we applied the SAFT-γ Mie force field to generate extensive data for derivative properties of supercritical hydrogen–ethylene mixtures at 300 K, comparing the results with the SAFT-VR Mie equation of state.

Phase behavior of CO2/toluene, CO2/n-decane and CO2/toluene/n-decane: Experimental measurements and thermodynamic modeling with SAFT-VR Mie equation of state

Experimental phase behavior data are reported for binary systems CO2/toluene and CO2/n-decane and for ternary system CO2/toluene/n-decane at (313.2 and 353.2) K with carbon dioxide composition ranging from 0.2 to 0.8 mole fraction. The equilibrium data for the binary systems were used to estimate the cross interactions parameters of the SAFT-VR Mie equation of state. The SAFT-VR Mie model correlated the temperature and composition effect on the bubble pressure with deviations close to the experimental uncertainty and lower than the ones obtained with the PC-SAFT model. The binary interaction parameters were used to calculate the bubble pressure of the ternary mixture, and the deviations are compatible to those of binary systems, confirming that SAFT-VR Mie is very useful to correlate thermodynamic properties of binary and ternary systems and promising to predict multicomponent systems properties with low deviations.

A model to predict adsorption of mixtures coupled with SAFT-VR Mie Equation of state

We extend the Statistical Associating Fluid Theory for a Mie potential with a Variable Range (SAFT-VR Mie) equation of state for the prediction of multicomponent adsorption isotherms. A generalization for adsorbed mixtures of the model proposed by Franco et al. (Langmuir, 33, 11291–11298, 2017) is obtained through the derivation of the Helmholtz free energy contribution due to the interaction between the fluid mixture and the wall, which is considered as a square-well potential. The resulting model is completely predictive for mixtures and takes into account the pore size and geometry, and the wall-fluid interaction parameters. These parameters are calculated with combining rules using the pure component wall-fluid interaction parameters. The proposed model is applied to different systems: binary mixtures of methane, ethane, ethylene, and propane adsorbed in a carbon molecular sieve, nitrogen and methane on a Metal-Organic Framework (MOF), and a ternary mixture of methane, nitrogen, and hydrogen adsorbed in activated carbon. The predicted adsorption isotherms match the experimental data reasonably well, proving that the theoretical framework on which the proposed model is based comprises a promising and adequate strategy.

Evaluation of the polar contribution in the SAFT-VR Mie equation of state for simultaneous correlation of condensed-phase density, condensed-phase speed of sound, saturated density and saturated pressure of pure polar fluids

Polar forces affect significantly the thermodynamic properties and phase equilibria of pure fluids. In this work, the original SAFT-VR Mie equation of state was modified by the introduction of an explicit term to account for multipolar effects. Two approaches were evaluated: the Polar SAFT-VR Mie (PSAFT-VR Mie) and the Truncated Polar SAFT-VR Mie (tPSAFT-VR Mie). For non-associating molecules the agreement between experimental data and model predictions are very good. The tPSAFT-VR Mie equation of state presented better results than the SAFT-VR Mie and PSAFT-VR Mie models correlating condensed-phase density, condensed-phase speed of sound, saturated density and saturated pressure with lower deviations. For associating molecules the results obtained with the tPSAFT-VR Mie model were equivalent to the ones obtained with the SAFT-VR Mie equation of state. In order to the tPSAFT-VR Mie model better predict second-derivative thermodynamic properties, such as the speed of sound and the isobaric heat capacity, some information about this type of property must be used in the pure components parameters regression. All evaluated models predicted similar values for the second virial coefficient.

A thermodynamic model for strong aqueous electrolytes based on the eSAFT-VR Mie equation of state

The latest formulation of the SAFT-VR Mie equation of state (EoS) is extended to solutions of salts which are assumed to be strongly dissociating in water – through the addition of a Born term for accurate representation of the Gibbs energy of solvation, and a Debye-Hückel term for long-range, electrostatic interactions. Two adjustable parameters are assigned to each ionic species, the ion-solvent (here water) cross dispersion energy parameter and the ion segment diameter, which are optimized against experimental data for electrolyte solution densities and mean ionic activity coefficients. Model correlations for the activity coefficients and liquid densities, as well as predictions for vapor pressure and osmotic coefficients, show that the developed model performs accurately and, in many cases, is more accurate than other recent formulations for electrolyte solutions. Finally, a methodology for equation of state parameterization is presented to capture the non-monotonic temperature dependence of activity coefficient isotherms at low temperatures.

Prediction of vapor-liquid-hydrate equilibrium conditions for single and mixed guest hydrates with the SAFT-VR Mie EOS

Abstract In this work, a thermodynamic modeling approach is utilized to predict the formation conditions of natural gas hydrates. Our approach uses van der Waals-Platteeuw (vdWP) model for the hydrate phase and the SAFT-VR Mie equation of state (EOS) for the vapor and liquid phases. We employ three different association schemes into the SAFT-VR Mie EOS to account for hydrogen bonds among water molecules using square-well (SW), Lennard-Jones (LJ) and Mie potentials. The SAFT-VR Mie EOS is used to determine the solubility of hydrate formers in liquid water. We study single and mixed guest hydrates and compare the results with experimental data. The comparisons with experimental data show that our model gives excellent to satisfactory predictions of the dissociation pressures for single and mixed guest hydrates without re-adjustment of the Langmuir constants.

SAFT-VR-Mie with an incorporated polar term for accurate holistic prediction of the thermodynamic properties of polar components

The SAFT-VR Mie equation of state is extended to polar components and their mixtures by incorporating an explicit polar term into the residual Helmholtz energy expansion. The transferability of the Jog & Chapman (JC) and Gross & Vrabec (GV) polar terms to this framework is demonstrated, yielding the SAFT-VR Mie-JC and SAFT-VR Mie-GV models. Numerical challenges common to the parameterization of both SAFT-VR Mie and polar SAFT are found to characterize the regression of parameter sets in these new models, most notably for the JC term. These challenges hampered the performance of SAFT-VR Mie-JC, which was consistently outperformed by its GV counterpart as a result. Excellent predictions of mixture behaviour for ketones, esters and ethers are in evidence if the mixture data (MD) and fixed polar parameter (FPP) regression approaches are employed to yield unique parameter sets. The performance of the latter is particularly significant, as accurate predictions for both phase equilibrium and speed of sound are in evidence from parameter sets regressed using pure component properties alone, without the need for experimental mixture data. Based on the functional groups and systems under investigation, the SAFT-VR Mie-GV model with FPP parameter sets represents a holistic, predictive approach for the thermodynamic description of polar components and their mixtures.

Statistical Mechanical Model for Adsorption Coupled with SAFT-VR Mie Equation of State.

We extend the SAFT-VR Mie equation of state to calculate adsorption isotherms by considering explicitly the residual energy due to the confinement effect. Assuming a square-well potential for the fluid-solid interactions, the structure imposed by the fluid-solid interface is calculated using two different approaches: an empirical expression proposed by Travalloni et al. ( Chem. Eng. Sci. 65 , 3088 - 3099 , 2010 ), and a new theoretical expression derived by applying the mean value theorem. Adopting the SAFT-VR Mie ( Lafitte et al. J. Chem. Phys. , 139 , 154504 , 2013 ) equation of state to describe the fluid-fluid interactions, and solving the phase equilibrium criteria, we calculate adsorption isotherms for light hydrocarbons adsorbed in a carbon molecular sieve and for carbon dioxide, nitrogen, and water adsorbed in a zeolite. Good results are obtained from the model using either approach. Nonetheless, the theoretical expression seems to correlate better the experimental data than the empirical one, possibly implying that a more reliable way to describe the structure ensures a better description of the thermodynamic behavior.

Extension of the SAFT-VR Mie EoS To Model Homonuclear Rings and Its Parametrization Based on the Principle of Corresponding States.

The statistical associating fluid theory of variable range employing a Mie potential (SAFT-VR-Mie) proposed by Lafitte et al. (J. Chem Phys. 2013, 139, 154504) is one of the latest versions of the SAFT family. This particular version has been shown to have a remarkable capability to connect experimental determinations, theoretical calculations, and molecular simulations results. However, the theoretical development restricts the model to chains of beads connected in a linear fashion. In this work, the capabilities of the SAFT-VR Mie equation of state for modeling phase equilibria are extended for the case of planar ring compounds. This modification proposed replaces the Helmholtz energy of chain formation by an empirical contribution based on a parallelism to the second-order thermodynamic perturbation theory for hard sphere trimers. The proposed expression is given in terms of an extra parameter, χ, that depends on the number of beads, ms, and the geometry of the ring. The model is used to describe the phase equilibrium for planar ring compounds formed of Mie isotropic segments for the cases of ms equals to 3, 4, 5 (two configurations), and 7 (two configurations). The resulting molecular model is further parametrized, invoking a corresponding states principle resulting in sets of parameters that can be used indistinctively in theoretical calculations or in molecular simulations without any further refinements. The extent and performance of the methodology has been exemplified by predicting the phase equilibria and vapor pressure curves for aromatic hydrocarbons (benzene, hexafluorobenzene, toluene), heterocyclic molecules (2,5-dimethylfuran, sulfolane, tetrahydro-2H-pyran, tetrahydrofuran), and polycyclic aromatic hydrocarbons (naphthalene, pyrene, anthracene, pentacene, and coronene). An important aspect of the theory is that the parameters of the model can be used directly in molecular dynamics (MD) simulations to calculate equilibrium phase properties and interfacial tensions with an accuracy that rivals other coarse grained and united atom models, for example, liquid densities, are predicted, with a maximum absolute average deviation of 3% from both the theory and the MD simulations, while the interfacial tension is predicted, with a maximum absolute average of 8%. The extension to mixtures is exemplified by considering a binary system of hexane (chain fluid) and tetrahydro-2H-pyran (ring fluid).