The A in SAFT: developing the contribution of association to the Helmholtz free energy within a Wertheim TPT1 treatment of generic Mie fluids

Theoretical and practical investigation of ion-ion association in electrolyte solutions

In this study, we present a new equation of state for electrolyte solutions, integrating the statistical associating fluid theory for variable range interactions utilizing the generic Mie form and binding Debye-Hückel theories. This equation of state underscores the pivotal role of ion-ion association in determining the properties of electrolyte solutions. We propose a unified framework that simultaneously examines the thermodynamic properties of electrolyte solutions and their electrical conductivity, given the profound impact of ion pairing on this transport property. Using this equation of state, we predict the liquid density, mean ionic activity coefficient, and osmotic coefficient for binary NaCl, Na2SO4, and MgSO4 aqueous solutions at 298.15 K. Additionally, we evaluate the molar conductivity of these systems by considering the fraction of free ions derived from our equation of state in conjunction with two advanced electrical conductivity models. Our results reveal that, while ion-ion association has a minimal influence on the modification of the predicted properties of sodium chloride solutions, their impact on sodium and magnesium sulfate solutions is considerably more noticeable.

Prediction of water anomalous properties by introducing the two-state theory in SAFT

Water is one of the most abundant substances on earth, but it is still not entirely understood. It shows unusual behavior, and its properties present characteristic extrema unlike any other fluid. This unusual behavior has been linked to the two-state theory of water, which proposes that water forms different clusters, one with a high density and one with a low density, which may even form two distinct phases at low temperatures. Models incorporating the two-state theory manage to capture the unusual extrema of water, unlike traditional equations of state, which fail. In this work, we have derived the framework to incorporate the two-state theory of water into the Statistical-Associating-Fluid-Theory (SAFT). More specifically, we have assumed that water is an ideal solution of high density water molecules and low density water molecules that are in chemical equilibrium. Using this assumption, we have generalized the association term SAFT to allow for the simultaneous existence of the two water types, which have the same physical parameters but different association properties. We have incorporated the newly derived association term in the context of the Perturbed Chain-SAFT (PC-SAFT). The new model is referred to as PC-SAFT-Two-State (PC-SAFT-TS). Using PC-SAFT-TS, we have succeeded in predicting the characteristic extrema of water, such as its density and speed of sound maximum, etc., without loss of accuracy compared to the original PC-SAFT. This new framework is readily extended to mixtures, and PC-SAFT-TS manages to capture the solubility minimum of hydrocarbons in water in a straightforward manner.

Modeling the Thermodynamic Properties of Saturated Lactones in Nonideal Mixtures with the SAFT-γ Mie Approach

The prediction of the thermodynamic properties of lactones is an important challenge in the flavor, fragrance, and pharmaceutical industries. Here, we develop a predictive model of the phase behavior of binary mixtures of lactones with hydrocarbons, alcohols, ketones, esters, aromatic compounds, water, and carbon dioxide. We extend the group-parameter matrix of the statistical associating fluid theory SAFT-γ Mie group-contribution method by defining a new cyclic ester group, denoted cCOO. The group is composed of two spherical Mie segments and two association electron-donating sites of type e1 that can interact with association electron-accepting sites of type H in other molecules. The model parameters of the new cCOO group interactions (1 like interaction and 17 unlike interactions) are characterized to represent target experimental data of physical properties of pure fluids (vapor pressure, single-phase density, and vaporization enthalpy) and mixtures (vapor-liquid equilibria, liquid-liquid equilibria, solid-liquid equilibria, density, and excess enthalpy). The robustness of the model is assessed by comparing theoretical predictions with experimental data, mainly for oxolan-2-one, 5-methyloxolan-2-one, and oxepan-2-one (also referred to as γ-butyrolactone, γ-valerolactone, and ε-caprolactone, respectively). The calculations are found to be in very good quantitative agreement with experiments. The proposed model allows for accurate predictions of the thermodynamic properties and highly nonideal phase behavior of the systems of interest, such as azeotrope compositions. It can be used to support the development of novel molecules and manufacturing processes.

A molecular density functional theory for associating fluids in 3D geometries

A new free-energy functional is proposed for inhomogeneous associating fluids. The general formulation of Wertheim's thermodynamic perturbation theory is considered as the starting point of the derivation. We apply the hypotheses of the statistical associating fluid theory in the classical density functional theory (DFT) framework to obtain a tractable expression of the free-energy functional for inhomogeneous associating fluids. Specific weighted functions are introduced in our framework to describe association interactions for a fluid under confinement. These weighted functions have a mathematical structure similar to the weighted densities of the fundamental-measure theory (i.e., they can be expressed as convolution products) such that they can be efficiently evaluated with Fourier transforms in a 3D space. The resulting free-energy functional can be employed to determine the microscopic structure of inhomogeneous associating fluids of arbitrary 3D geometry. The new model is first compared with Monte Carlo simulations and previous versions of DFT for a planar hard wall system in order to check its consistency in a 1D case. As an example of application in a 3D configuration, we then investigate the extreme confinement of an associating hard-sphere fluid inside an anisotropic open cavity with a shape that mimics a simplified model of zeolite. Both the density distribution and the corresponding molecular bonding profile are given, revealing complementary information to understand the structure of the associating fluid inside the cavity network. The impact of the degree of association on the preferential positions of the molecules inside the cavity is investigated as well as the competition between association and steric effect on adsorption.

Modeling Sorption of Hydrocarbons in Polyethylene with the SAFT-γ Mie Approach Combined with a Statistical-Mechanical Model to Describe Semicrystalline Polymers

A recently developed statistical-mechanical model is applied systematically to estimate the fraction of tie-molecules (polymer chains linking different crystals directly or via entanglements) in semicrystalline polyethylene (PE) samples. The amorphous domains of the polymer are divided into constrained interlamellar domains and "free" outer-lamellar domains. A set of model parameters is assigned to each sample by correlating previous experimental measurements and minimizing the difference between the predicted solubility of pure hydrocarbons in the sample and the experimental values. We show that the sorption isotherms of multiple pure fluids in each sample can be described by a single parameter set, proving that the polymer-solute interactions (described accurately by the SAFT-γ Mie EoS) are decoupled from the sample-specific properties of the polymer. We find that ∼30% of the crystalline stems in the lamellae of PE are connected to tie-molecules, within the bounds suggested by previous theoretical and computational work. The transferability of the sample-specific parameters is assessed by predicting cosolubility effects and solubility at different temperatures, leading to good agreement with experimental data.

Mixed Solvent Electrolyte Solutions: A Review and Calculations with the eSAFT-VR Mie Equation of State

In this work, mixed-solvent mean ionic activity coefficients (MIAC), vapor-liquid equilibrium (VLE), and liquid-liquid equilibrium (LLE) of electrolyte solutions have been addressed. An extended literature review of existing electrolyte activity coefficient models (eGE) and electrolyte equations of state (eEoS) for modeling mixed solvent electrolyte systems is first presented, focusing on the details of the models in terms of physical and electrolyte terms, relative static permittivity, and parameterization. The analysis of this literature reveals that the property predictions can be ranked, from the easiest to the most difficult, in the following order: VLE, MIAC, and LLE. We have then used our previously developed eSAFT-VR Mie model to predict MIAC, VLE, and LLE in mixed solvents without fitting any new adjustable parameters. The model was parameterized on MIAC of aqueous electrolyte solutions and successfully extended to nonaqueous, single solvent electrolyte solutions without any new adjustable parameters by using a salt-dependent expression for the relative static permittivity. Our approach yields excellent results for MIAC and VLE of mixed solvent electrolyte solutions, while being fully predictive. LLE is significantly more challenging, and an accurate model for the salt-free solution is crucial for accurate calculations. When the compositions of the two phases in the binary salt-free system are accurately captured, then the electrolyte extension of our model shows a lot of potential and is currently among the best eEoS for LLE prediction in the literature.

Multiple insights call for revision of modern thermodynamic models to account for structural fluctuations in water

Modern thermodynamic models incorporate the concept of association (hydrogen bonding) and they can describe very satisfactorily many properties of water containing mixtures. They have not been successful in representing water's anomalous properties and this work provides a possible explanation. We have analyzed and interpreted recent experimental data, molecular simulation results, and two-state theory approaches and compared against the predictions from thermodynamic models. We show that the dominance of the tetrahedral structure implemented in modern thermodynamic models may be the reason for their failure for describing water systems. While this study does not prove the two-state theories for water, it indicates that a high level of tetrahedral structure of water is not in agreement with water's anomalous properties when used in thermodynamic models.

Prediction of pH in multiphase multicomponent systems with ePC-SAFT advanced

Proton activity, which is usually expressed as a pH value, is among the most important properties in the design of chemical and biochemical processes as it determines the dissociation of species in aqueous mixtures. This article addresses the prediction of pH values in multiphase systems based on the IUPAC definition via proton activity. The required proton activity coefficients were predicted using the thermodynamic equation of state ePC-SAFT advanced. The developed framework considers reaction equilibria and phase equilibria (vapor-liquid and liquid-liquid) to predict pH in the equilibrated liquid phases.

Measurements and Modelling of Vapour–Liquid Equilibrium for (H2O + N2) and (CO2 + H2O + N2) Systems at Temperatures between 323 and 473 K and Pressures up to 20 MPa

Understanding the phase behaviour of (CO2 + water + permanent gas) systems is critical for implementing carbon capture and storage (CCS) processes, a key technology in reducing CO2 emissions. In this paper, phase behaviour data for (H2O + N2) and (CO2 + H2O + N2) systems are reported at temperatures from 323 to 473 K and pressures up to 20 MPa. In the ternary system, the mole ratio between CO2 and N2 was 1. Experiments were conducted in a newly designed analytical apparatus that includes two syringe pumps for fluid injection, a high-pressure equilibrium vessel, heater aluminium jacket, Rolsi sampling valves and an online gas chromatograph (GC) for composition determination. A high-sensitivity pulsed discharge detector installed in the GC was used to measure the low levels of dissolved nitrogen in the aqueous phase and low water levels in the vapour phase. The experimental data were compared with the calculation based on the γ-φ and SAFT-γ Mie approaches. In the SAFT-γ Mie model, the like parameters for N2 had to be determined. We also obtained the unlike dispersion energy for the (H2O + N2) system and the unlike repulsive exponent and dispersion energy for the (CO2 + N2) system. This was done to improve the prediction of SAFT-γ Mie model. For the (H2O + N2) binary system, the results show that the solubility of nitrogen in the aqueous phase was calculated better by the γ-φ approach rather than the SAFT-γ Mie model, whereas SAFT-γ Mie performed better for the prediction of the vapour phase. For the (CO2 + H2O + N2) ternary systems, both models predicted the experimental data for each phase with good agreement.

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.

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.

Off-Design Operation of Conventional and Phase-Change CO2 Capture Solvents and Mixtures: A Systematic Assessment Approach

Solvent-based CO2 capture technologies hold promise for future implementation but conventional solvents incur significant energy penalties and capture costs. Phase-change solvents enable a significant reduction in the regeneration energy but their performance has only been investigated under steady-state operation. In the current work, we employed a systematic approach for the evaluation of conventional solvents and mixtures, as well as phase-change solvents under the influence of disturbances. Sensitivity analysis was used to identify the impact that operating parameter variations and different solvents exert on multiple CO2 capture performance indicators within a wide operating range. The resulting capture process performance was then assessed for each solvent within a multi-criteria approach, which simultaneously accounted for off-design conditions and nominal operation. The considered performance criteria included the regeneration energy, solvent mass flow rate, cost and cyclic capacity, net energy penalty from integration with an upstream power plant, and lost revenue from parasitic losses. The 10 investigated solvents included the phase-change solvents methyl-cyclohexylamine (MCA) and 2-(diethylamino)ethanol/3-(methylamino)propylamine (DEEA/MAPA). We found that the conventional mixture diethanolamine/methyldiethanolamine (DEA/MDEA) and the phase-change solvent DEEA/MAPA exhibited both resilience to disturbances and desirable nominal operation for multiple performance indicators simultaneously.

Predictive models for the phase behaviour and solution properties of weak electrolytes: nitric, sulphuric, and carbonic acids

The distribution of ionic species in electrolyte systems is important in many fields of science and engineering, ranging from the study of degradation mechanisms to the design of systems for electrochemical energy storage. Often, other phenomena closely related to ionic speciation, such as ion pairing, clustering and hydrogen bonding, which are difficult to investigate experimentally, are also of interest. Here, we develop an accurate molecular approach, accounting for reactions as well as association and ion pairing, to deliver a predictive framework that helps validate experiment and guides future modelling of speciation phenomena of weak electrolytes. We extend the SAFT-VRE Mie equation of state [D. K. Eriksen et al., Mol. Phys., 2016, 114, 2724-2749] to study aqueous solutions of nitric, sulphuric, and carbonic acids, considering complete and partially dissociated models. In order to incorporate the dissociation equilibria, correlations to experimental data for the relevant thermodynamic equilibrium constants of the dissociation reactions are taken from the literature and are imposed as a boundary condition in the calculations. The models for water, the hydronium ion, and carbon dioxide are treated as transferable and are taken from our previous work. We present new molecular models for nitric acid, and the nitrate, bisulfate, sulfate, and bicarbonate anions. The resulting framework is used to predict a range of phase behaviour and solution properties of the aqueous acids over wide ranges of concentration and temperature, including the degree of dissociation, as well as the activity coefficients of the ionic species, and the activity of water and osmotic coefficient, density, and vapour pressure of the solutions. The SAFT-VRE Mie models obtained in this manner provide a means of elucidating the mechanisms of association and ion pairing in the systems studied, complementing the experimental observations reported in the literature.

Correction: Modelling and prediction of the thermophysical properties of aqueous mixtures of choline geranate and geranic acid (CAGE) using SAFT-γ Mie

Correction for ‘Modelling and prediction of the thermophysical properties of aqueous mixtures of choline geranate and geranic acid (CAGE) using SAFT-γ Mie’ by Silvia Di Lecce et al., RSC Adv., 2019, 9, 38017–38031. DOI: 10.1039/C9RA07057E

Modelling and prediction of the thermophysical properties of aqueous mixtures of choline geranate and geranic acid (CAGE) using SAFT-γ Mie

Deep-eutectic solvents and room temperature ionic liquids are increasingly recognised as appropriate materials for use as active pharmaceutical ingredients and formulation additives. Aqueous mixtures of choline and geranate (CAGE), in particular, have been shown to offer promising biomedical properties but understanding the thermophysical behaviour of these mixtures remains limited. Here, we develop interaction potentials for use in the SAFT-γ Mie group-contribution approach, to study the thermodynamic properties and phase behaviour of aqueous mixtures of choline geranate and geranic acid. The determination of the interaction parameters between chemical functional groups is carried out in a sequential fashion, characterising each group based on those previously developed. The parameters of the groups relevant to geranic acid are estimated using experimental fluid phase-equilibrium data such as vapour pressure and saturated-liquid density of simple pure components (n-alkenes, branched alkenes and carboxylic acids) and the phase equilibrium data of mixtures (aqueous solutions of branched alkenes and of carboxylic acids). Geranate is represented by further incorporating the anionic carboxylate group, COO⁻, which is characterised using aqueous solution data of sodium carboxylate salts, assuming full dissociation of the salt in water. Choline is described by incorporating the cationic quaternary ammonium group, N⁺, using data for choline chloride solutions. The osmotic pressure of aqueous mixtures of CAGE at several concentrations is predicted and compared to experimental data obtained as part of our work to assess the accuracy of the modelling platform. The SAFT-γ Mie approach is shown to be predictive, providing a good description of the measured data for a wide range of mixtures and properties. Furthermore, the new group-interaction parameters needed to represent CAGE extend the set of functional groups of the group-contribution approach, and can be used in a transferable way to predict the properties of systems beyond those studied in the current work.

The properties of aqueous solutions of the CAGE deep eutectic solvent are predicted with the SAFT-γ Mie approach.

A molecular equation of state for alcohols which includes steric hindrance in hydrogen bonding

In this paper, we develop the first equation of state for alcohol containing mixtures which includes the effect of steric hindrance between the two electron lone pair hydrogen bond acceptor sites on the alcohol's hydroxyl oxygen. The theory is derived for multi-component mixtures within Wertheim's multi-density statistical mechanics in a second order perturbation theory. The accuracy of the new approach is demonstrated by application to pure methanol and ethanol and binary ethanol/water mixtures. It is demonstrated that the new approach gives a substantial improvement in the prediction of the hydrogen bonding structure of both pure alcohol and alcohol/water mixtures, as compared to conventional approaches which do not include steric effects between the alcohol association sites. Finally, it is demonstrated that the inclusion of steric effects allows for more accurate binary phase equilibria and heats of mixing prediction with water.

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.