Perturbed-chain Polar Statistical Associating Fluid Theory Research Articles

ML-SAFT: A machine learning framework for PCP-SAFT parameter prediction

The Perturbed Chain Polar Statistical Associating Fluid Theory (PCP-SAFT) equation of state (EoS) is widely used to predict fluid-phase thermodynamics, but parameterization of PCP-SAFT for individual molecules is often challenging. We propose a machine learning framework called ML-SAFT that can turn experimental data in predictive models of PCP-SAFT parameters. We demonstrate methods for automated large scale regression of PCP-SAFT parameters and thus create a large PCP-SAFT parameter dataset in the literature. We then evaluate several machine learning architectures for predicting PCP-SAFT parameters. We find that our best model provides accurate predictions for a wider range of molecules than existing predictive methods with 40 % average absolute deviation (% AAD) in vapor pressure predictions and 8 % AAD in density predictions.

Predicting the density and viscosity of deep eutectic solvents at atmospheric and elevated pressures

Deep eutectic solvents (DESs) are promising and sustainable substitutes for organic solvents in various applications. The knowledge of their density and viscosity at different temperatures and pressures is crucial for the corresponding process design. However, most literature primarily focuses on predicting these properties at atmospheric pressure. In this study, we employed the perturbed chain polar statistical associating fluid theory (PCP-SAFT) model and entropy scaling method to predict DESs’ density and viscosity at atmospheric and elevated pressures. 2831 density data points of 38 DESs and 1018 viscosity data points of 31 DESs at different temperatures and pressures were used to evaluate the accuracy of the applied methods. The pure component PCP-SAFT parameters of choline chloride (ChCl) were regressed using density data of ChCl-based DES at various temperatures and pressures. It was found that the PCP-SAFT model can provide satisfactory density predictions for DESs at atmospheric pressure, even without using the system-specific binary interaction parameter (kij=0). However, accurate viscosity prediction at atmospheric pressure for DESs necessitated the inclusion of the binary interaction parameter (kij=aijT+bij) in the PCP-SAFT model. The system-specific coefficients aij andbij, derived from experimental data at atmospheric pressure, were used to predict the density or viscosity of DES at elevated pressures. Accurate predictions for density were observed across the entire pressure range, whereas effective viscosity predictions were confined to a specific pressure range.

High-Pressure-Mediated Thiourea-Organocatalyzed Asymmetric Michael Addition to (Hetero)aromatic Nitroolefins: Prediction of Reaction Parameters by PCP-SAFT Modelling.

Thiourea-organocatalyzed Michael additions of diethyl malonate to various heteroaromatic nitroolefins (13 examples) have been studied under high-pressure (up to 800 MPa) and ambient pressure conditions. High pressure was conducive to enhanced product yields by a factor of 2-12 at a given reaction time, high reaction rates (reaction times were decreased from 72-24 h down to 4-24 h) and high enantioselectivity. Elucidating the effects of solvents for maximizing reaction rates and yields has been carried out using the Perturbed-Chain Polar Statistical Associating Fluid Theory (PCP-SAFT), allowing for the first time a prediction of the kinetic profiles under high-hydrostatic-pressure conditions.

Tolman lengths and rigidity constants from free-energy functionals-General expressions and comparison of theories.

The leading order terms in a curvature expansion of surface tension, the Tolman length (first order), and rigidities (second order) have been shown to play an important role in the description of nucleation processes. This work presents general and rigorous expressions to compute these quantities for any nonlocal density functional theory (DFT). The expressions hold for pure fluids and mixtures and reduce to the known expressions from density gradient theory (DGT). The framework is applied to a Helmholtz energy functional based on the perturbed chain polar statistical associating fluid theory (PCP-SAFT) and is used in an extensive investigation of curvature corrections for pure fluids and mixtures. Predictions from the full DFT are compared to two simpler theories: predictive DGT, which has a density and temperature dependent influence matrix derived from DFT, and DGT, where the influence parameter reproduces the surface tension predicted from DFT. All models are based on the same equation of state and predict similar Tolman lengths and spherical rigidities for small molecules, but the deviations between DFT and DGT increase with chain length for alkanes. For all components except water, we find that DGT underpredicts the value of the Tolman length but overpredicts the value of the spherical rigidity. An important basis for the calculation is an accurate prediction of the planar surface tension. Therefore, further work is required to accurately extract Tolman lengths and rigidities of alkanols because DFT with PCP-SAFT does not accurately predict surface tensions of these fluids.

Experimental Data of Fluid Phase Equilibria- Correlation and Prediction Models: A Review

The examples of phase equilibria in binary systems, solid/liquid (SLE), liquid/liquid (LLE), vapor/liquid (VLE), as well as liquid/liquid equilibria in ternary systems mainly containing ionic liquids (ILs), or the infragrance materials, or pharmaceuticals with molecular organic solvents, such as an alcohol, or water, or hydrocarbons, are presented. The most popular correlation methods of the experimental phase equilibrium data are presented, related to the excess Gibbs free energy models such as Wilson, universal-quasichemical, UNIQUAC and non-random two-liquid model, NRTL as well as several popular theories for the modeling of the phase equilibria and excess molar enthalpy, HE in binary or ternary mixtures are presented: the group contribution method (Mod. UNIFAC) and modified UNIFAC model for pharmaceuticals and lattice theory based on non-random hydrogen bonding (NRHB). The SLE, LLE, or VLE and HE of these systems may be described by the Perturbed-Chain Polar Statistical Associating Fluid Theory (PC-SAFT), or a Conductor-like Screening Model for Real Solvents (COSMO-RS). The examples of the application of ILs as extractants for the separation of aromatic hydrocarbons from alkanes, sulfur compounds from alkanes, alkenes from alkanes, ethylbenzene from styrene, butan-1-ol from water phase, or 2-phenylethanol (PEA) from water are discussed on the basis of previously published data. The first information about the selectivity of extrahent for separation can be obtained from the measurements of the limiting activity coefficient measurements by the gas–liquid chromatography technique. This review outlines the main research work carried out over the last few years on direct measurements of phase equilibria, or HE and limiting activity coefficients, the possibility of thermodynamic modeling with emphasis on recent research achievements and potential for future research.

Experimental measurements and thermodynamic modeling of Coumarin-7 solid solubility in supercritical carbon dioxide: Production of nanoparticles via RESS method

For the first time, solubility of coumarin-7 dye (C7) in supercritical carbon dioxide (SC-CO2) was measured in temperature and pressure ranges of (308–338) K and (90–330) bar, respectively, for binary (C7-CO2) and ternary systems (C7-CO2-methanol). Mole fractions were determined to range from 0.415 × 10−5 to 1.009 × 10−5 and 0.1089 × 10−4 to 1.2007 × 10−4 for binary and ternary systems, respectively. The solubility data was correlated using two thermodynamic models, namely quadrupolar cubic plus association theory (qCPA) and perturbed-chain polar statistical associating fluid theory (PCP-SAFT). Continuing with the research, the rapid expansion of supercritical solution (RESS) process was used to produce coumarine-7 nanoparticles. Effects of temperature, pressure, nozzle diameter, and spraying distance on the particle size were investigated. For this purpose, unprocessed and processed particles were characterized by FTIR, XRD, DSC, SEM, and DLS. A significant decline (21.37 ≤ x50 ≤ 224.13 nm) was observed in the final particle size, as compared to the original one (40.35 μm).

Self-Diffusion Coefficients from Entropy Scaling Using the PCP-SAFT Equation of State

This study proposes a model for self-diffusion coefficients of pure substances from entropy scaling using the perturbed-chain polar statistical associating fluid theory (PCP-SAFT) equation of state. In accordance with the entropy scaling approach proposed by Y. Rosenfeld [Rosenfeld, Y. Phys. Rev. A 1977, 15, 2545–2549], we observe that the self-diffusion coefficient of real substances, once made dimensionless with an appropriate expression, only depends on residual entropy. The proposed model requires 3 parameters for each pure substance. For substances with scarce experimental data, however, a scheme is proposed to estimate one or two of these parameters. We study 133 substances from more than 14 different chemical families and find the average absolute deviation of 8.2% between the proposed model and experimental data (9992 data points). The model shows satisfying robustness for extrapolating self-diffusion coefficients to conditions rather distant from the state points where experimental data are avail...

Identifying Pure-Component Parameters of an Analytic Equation of State Using Experimental Surface Tension or Molecular Simulations with a Transferable Force Field

A meaningful parametrization of an analytic equation of state can be obtained by adjusting the pure-component parameters to the vapor-pressure data and liquid-density data of the considered pure substance. For substances with no vapor-pressure data available, the parametrization of an equation of state is more subtle. In this study, we investigate two different routes to retrieve pure-component parameters for the perturbed-chain polar statistical associating fluid theory (PCP-SAFT) for such cases. In the first route, classical density-functional theory is applied, and PCP-SAFT parameters are adjusted to experimental liquid density as well as to data of surface tensions. The second approach uses results for liquid density as well as enthalpies obtained from molecular-dynamics simulations using transferable force fields. It is shown that through this route, pure-component parameters for polymeric compounds with ‘vanishing’ vapor pressure can be obtained without experimental data as input. We assess the approaches for several n-alkanes, 1-alkenes, and ethers as training substances. For both routes, rather satisfactory results are obtained for the correlated thermodynamic properties as well as for the predicted ones.

Pure Substance and Mixture Viscosities Based on Entropy Scaling and an Analytic Equation of State

This study proposes a simple model for viscosities, based on entropy scaling, for real substances and mixtures. The residual entropy is calculated with the perturbed chain polar statistical associating fluid theory (PCP-SAFT). The model requires two or three pure component parameters, noting, however, that an entirely predictive group contribution approach as proposed in our previous work [Loetgering-Lin O.; Gross J. Ind. Eng. Chem. Res. 2015, 54, 7942–7952] gives also very good results. Overall, 140 real substances are considered with relative mean deviations from experimental data of about 5% (without excluding “outliers”). We performed molecular simulations for mixtures of simple model fluid in order to determine a suitable mixture model. A completely predictive approach for viscosities of real mixtures is thereby obtained. The model is evaluated for 566 mixtures with about 34,500 experimental data points of various complexity (i.e., nearly ideal systems as well as highly asymmetric mixtures). Mixtures...

Modeling properties of the one-dimensional vapor-liquid interface: Application of classical density functional and density gradient theory

This study compares surface tensions as calculated from the classical density functional theory (DFT) and from density gradient theory (DGT) to experimental data. This comparison is inevitably not on equal ground, because the DFT is purely predictive for interfacial properties, both for pure substances and for mixtures, whereas DGT requires an adjustable influence parameter for each pure component (adjusted to surface tension data) and possibly a further adjustable parameter for each binary pair. In that sense, our comparison takes the perspective of a user who, because experimental data is available, finds it acceptable to correlate interfacial properties with adjustable parameters but who can alternatively decide for applying the DFT method. The perturbed-chain polar statistical associating fluid theory (PCP-SAFT) is used to determine phase equilibrium properties as well as the local Helmholtz energy density for DGT. For DFT, a Helmholtz energy functional consistent with PCP-SAFT is applied. DGT correlations and DFT predictions of surface tension for pure components as well as results for mixtures agree very well and exhibit excellent agreement to reference data for non-associating non-polar and polar molecules. Only for pure associating compounds, the adjustable parameter of DGT leads to significant improvements over DFT results. For mixtures, depending on the system, results can be better for either method. In the case of alkane-alcohol mixtures, DGT with the geometric combining rule for the cross-wise influence parameter leads to non-physically steep gradients in the density profiles. Adjusting a binary correction parameter for the influence parameter to experimental mixture surface tension data helps to alleviate this problem. However, the practical utility of this binary correction parameter to improve mixture surface tension results is very limited. We conclude that DFT is for most applications a preferable approach.

Transferability of cross-interaction pair potentials: Vapor-liquid phase equilibria of n-alkane/nitrogen mixtures using the TAMie force field

The van der Waals contribution of force fields designed for calculating phase equilibria and thermodynamic properties are usually adjusted to experimental data of pure components. Using these force fields for mixtures by applying combining rules for the cross-wise interaction potentials does not always lead to satisfactory results. This study considers the cross-wise van der Waals energy parameters εαβ among individual pairs of (united-)atom groups as adjustable (but subsequently transferable) parameters. The cross-energy parameters are simultaneously adjusted to an experimental training set of several mixtures. An analytic equation of state, the Perturbed-Chain Polar Statistical Associating Fluid Theory (PCP-SAFT), is used for ensuring very swift convergence of the optimization procedure. As an example, we consider the phase equilibria of n-alkane/nitrogen mixtures. The cross-energy parameters (-CH3 to N and -CH2- to N) are adjusted to three mixtures as the training set. We explore the transferability of the so-obtained force field for varying temperatures and for mixtures that were not considered in the training set. Phase equilibria of nine n-alkane/nitrogen mixtures (each at various temperatures) are determined using Monte Carlo simulations in the grand canonical ensemble. Results for vapor-liquid equilibrium calculations show very good agreement with experimental data.

Modeling of interfacial properties of multicomponent systems using density gradient theory and PCP-SAFT

The perturbed-chain polar statistical associating fluid theory (PCP-SAFT) and density gradient theory are used to calculate interfacial properties of multicomponent systems. The constant influence parameters are adjusted to experimental surface tension data of pure fluids and the cross-influence parameters are determined from a geometric combining rule. The influence parameters of simple branched alkanes, in the absence of experimental surface tension data, can be estimated from a correlation for n-alkanes. Results for surface tension of hydrocarbon mixtures of up to twenty components as well as systems including polar and supercritical species are in good agreement with experiments as long as bulk densities are reproduced accurately and molecular interactions between the components are symmetric or only weakly asymmetric.

Experimental measurements and thermodynamic modeling of the cloud point pressure for solubility of copolymers of vinyl acetate and dibutyl maleate in supercritical CO2

In this study the cloud point pressure for the solubility of copolymers of vinyl acetate and dibutyl maleate in supercritical carbon dioxide were measured and modeled. The copolymers investigated have different molecular weights, with varying structures. The cloud point pressure data were modeled using the Perturbed Chain Polar Statistical Associating Fluid Theory (PCP-SAFT) equation of state (EoS) that was developed for the prediction of the phase behavior of polar polymers which are soluble in supercritical carbon dioxide (J. of Supercritical Fluids 97 (2015) 45–50). Using this model, the cloud point pressure of the polymer-supercritical CO2 systems were correlated by taking into account quadrupole-quadrupole interactions between carbon dioxide and the polymer segments through the PCP-SAFT EoS. Thus, the new parameters were determined for the copolymers studied in this work. Moreover, we applied PC-SAFT EoS for the same systems. It is shown that the results are obtained using the PCP-SAFT EoS is superior respect to PC-SAFT so that the present EoS can be applied to calculate the cloud point pressure of the copolymer-CO2 systems and the deviation of the calculated values from experiment values are low.

Group Contribution Method for Viscosities Based on Entropy Scaling Using the Perturbed-Chain Polar Statistical Associating Fluid Theory

In this work, we propose a new predictive entropy-scaling approach for Newtonian shear viscosities based on group contributions. The approach is based on Rosenfeld’s original work [Rosenfeld, Y. Phys. Rev. A 1977, 15, 2545−2549]. The entropy scaling is formulated as third order polynomial in terms of the residual entropy as calculated from a group-contribution perturbed-chain polar statistical associating fluid theory (PCP-SAFT) equation of state. In this study, we analyze the course of entropy scaling parameters within homologous series and suggest suitable mixing rules for the parameters of functional groups. The viscosity of nonpolar, of polar, and of self-associating (hydrogen bonding) components are considered. In total, 22 functional groups are parametrized to viscosity data of 110 pure substances, from 12 different chemical families. The mean absolute relative deviations (MADs) to experimental viscosity data are typically around 5%. For three chemical families, namely, branched alkanes, 1-alcohols,...

Effect of the quadrupolar contribution of CO2 on the vapor–liquid equilibria and surface tensions of CO2–hydrocarbon binary mixtures

An equation of state (EOS) was established by combining the perturbed-chain polar statistical associating fluid theory (PCP-SAFT) with density-gradient theory (DGT). The bulk and interfacial properties of carbon dioxide (CO2)–butane, CO2–decane, CO2–benzene, CO2–cyclohexane and CO2–tetradecane binary mixtures were investigated. To illustrate the effect of the quadrupolar contribution of CO2 on the bulk and interfacial properties, the calculated vapor–liquid equilibria (VLE) and surface tensions were compared with those calculated from the perturbed-chain statistical associating fluid theory (PC-SAFT) and DGT, in which CO2 was treated as non-polar molecule. Results showed that by taking into account the quadrupolar contribution of CO2, both the VLE and surface tensions of CO2–hydrocarbon binary mixtures can be significantly improved compared with the experimental data.

Comparison between a Homo- and a Heterosegmented Group Contribution Approach Based on the Perturbed-Chain Polar Statistical Associating Fluid Theory Equation of State

Depending on the molecular model, group contribution (GC) approaches based on the statistical associating fluid theory (SAFT) can be classified in homosegmented and heterosegmented GC approaches. In homosegmented GC approaches, molecules are modeled as chains composed of identical segments. Heterosegmented GC approaches, on the other hand, consider molecular chains composed of different segment types and thus maintain a more detailed picture of real molecules. Therefore, heterosegmented GC approaches are arguably more physically realistic and ought to give more accurate descriptions of thermodynamic properties. In this work, we evaluate the performance of a homosegmented and a heterosegmented GC approach based on the perturbed-chain polar SAFT (PCP-SAFT) equation of state (EoS). To ensure a meaningful comparison between both GC approaches, a dipole term for the heterosegmented GC approach is formulated. Group parameters of 22 functional groups were adjusted to pure component property data. The comparison ...

Modeling of Liquid–Liquid Interfacial Properties of Binary and Ternary Mixtures

The Perturbed Chain Polar-Statistical Associating Fluid Theory (PCP-SAFT) equation of state was for the first time combined with the Density Gradient Theory (DGT) to calculate liquid–liquid (L–L) interfacial properties of ternary mixtures containing compounds of different polarity. The components involved were decane, dodecene, linear aldehydes (butanal, octanal, dodecanal), N,N-dimethylformamide and propylene carbonate. Besides ternary mixtures, also interfacial properties of binary systems consisting of the aforementioned compounds were studied. As a supplement, liquid densities of the equilibrium bulk phases of the binary and ternary mixtures were investigated. In addition to the modeling of data, L–L interfacial tensions were measured using the drop volume method. Experimental density data were generated using a vibration tube densitometer. Investigating the applicability of the model to describe the L–L interfacial tensions of binary and ternary mixtures revealed very good agreement between calculate...

A Density Functional Theory for Vapor–Liquid Interfaces of Mixtures Using the Perturbed-Chain Polar Statistical Associating Fluid Theory Equation of State

A density functional theory based on the perturbed-chain polar statistical associating fluid theory (PCP-SAFT) is extended to mixtures. The Helmholtz energy functional is suitable for inhomogeneous fluid phases and is here applied to vapor–liquid systems. The attractive branch of the van der Waals attractions are treated with a perturbation theory of first order in a nonmean field approach. The radial distribution function is considered in the Percus–Yevick closure for chain fluids with a three-fluid theory approach. This perturbation term is brought to consistency with the PCP-SAFT equation of state using the approach of Gloor et al. [Gloor, G. J.; Jackson, G.; Blas, F. J.; del Rio, E. M.; de Miguel, E. J. Chem. Phys. 2004, 121, 12740]. The approach is applied to vapor–liquid interfaces of mixtures and the resulting surface tension is compared to experimental data. Binary mixtures with two subcritical components as well as mixtures with one species above its critical point, for the considered temperature...

Interfacial tension of binary mixtures exhibiting azeotropic behavior: Measurement and modeling with PCP-SAFT combined with Density Gradient Theory

This work focuses on modeling and experimental investigation of temperature dependent interfacial properties of binary DMF/n-alkane (C7, C10, C12) mixtures. The systems consisting of solvents with very different polarity show azeotropic behavior. New experimental vapor–liquid and liquid–liquid interfacial tension data are provided between 298.15 and 328.15K using the drop volume method. The Perturbed Chain Polar Statistical Associating Fluid Theory (PCP-SAFT) equation of state was combined with the Density Gradient Theory (DGT) to calculate phase equilibria and interfacial properties. Modeling results are in good agreement with the corresponding experimental data. Thereby, the binary parameter βij within the DGT framework does not equal one. Investigating density and concentration profiles in the interface revealed characteristic trends which are related to the azeotropic behavior of the mixtures.

Liquid–Liquid Equilibria of Systems with Linear Aldehydes. Experimental Data and Modeling with PCP-SAFT

Liquid–liquid equilibrium (LLE) data were measured for the binary system propylene carbonate/decane (288.15 to 403.15 K) as well as for the ternary systems propylene carbonate/decane + linear aldehyde (C4, C8, C10, C12, C13) and DMF/decane + linear aldehyde (C4, C8, C10, C13) at 298.15 K using the analytic method. The reliability of the ternary LLE data is ascertained by employing Othmer–Tobias plots. Concerning the aldehyde distribution coefficients in the ternary systems, a distinctive dependency on the chain length of the aldehyde was observed. The LLE data were modeled with the Perturbed Chain Polar Statistical Associating Fluid Theory (PCP-SAFT) equation of state using a heterosegmented approach for describing the aldehyde molecules.