Mean Spherical Approximation Research Articles

On the analogy between the restricted primitive model and capacitor circuits: Semi-empirical alternatives for over- and underscreening in the calculation of mean ionic activity coefficients

The analogy between the restricted primitive model and capacitor circuits, originally described decades ago for the Mean Spherical Approximation, is explored to demonstrate its transferability in linearized electrolyte theories. On this basis, we offer an explanation of why treating the salt diameter as a free adjustable parameter blurs differences between electrolyte theories in the calculation of mean ionic activity coefficients. Furthermore, a capacitor circuit analogy with an approximation of the Dressed Ion Theory is applied to develop a modified closest approach parameter “b” for the Pitzer-Debye-Hückel term. This modification is able to account for the qualitative effects of over- and underscreening in the calculation of mean ionic activity coefficients. This is achieved by defining “b” as a semi-empirical function that allows close resemblance with the multiple decay-length extension of the Debye-Hückel theory for high dielectric constant values and that lies close to recommended literature values of “b” for low dielectric constant values. Finally, as proof of principle, this modified semi-empirical Pitzer-Debye-Hückel term is implemented in the predictive COSMO-RS-ES model, an own reimplementation of the COSMO-RS theory developed for thermodynamic property calculations of electrolyte systems. It is shown that the modified semi-empirical Pitzer-Debye-Hückel term is an effective replacement for the recently published version of COSMO-RS-ES with explicit considerations for ion pairing. This reduces the modelling complexity by implicitly considering ion pairing and improves overall qualitative performance for the prediction of salt solubilities in mixed-solvent systems and even mean ionic activity coefficients in non-aqueous media.

Long-range interactions in model colloidal dispersions with surface charge distributions

The theoretical model, which is based on the density functional approach, has been developed for studying the electrostatic properties of colloidal dispersions (modified colloidal primitive model; MCPM) containing hard-spherical ions with a homogeneous surface charge distribution. The mean-spherical approximation for the multi-component charged ions has been used to account for the electrostatic ion correlation effects. The present study reflects the importance of the charge distribution of macroion in determining the ionic structure and mean forces acting on the macroions. Compared with the MCPM, the colloidal primitive model (CPM), on which the colloidal charge is assumed to be in the center of particle, shows very long-range electrostatic properties and mean force acting on the macroions because of the strong cross correlation between the hard-sphere contribution and the Coulomb interaction. The long-range attractions and repulsions in the charged colloidal dispersions originate from the entropic effects and are found at high packing fractions of the colloidal ions.

Self-Diffusion in Liquid Copper, Silver, and Gold

The recently developed by us semi-analytical representation of the mean spherical approximation in conjunction with the linear trajectory approximation is applied to the quantitative study of self-diffusivities in liquid Cu, Ag and Au at different temperatures. The square-well model is employed for the description of the interatomic pair interactions in metals under study. It is found that our theoretical results are in good agreement with available experimental and computer-simulation data and can be considered as a prediction when such data are absent.

Thermodynamic modeling of gas solubility in aqueous sodium chloride solution

Abstract An electrolyte Equation of State is presented by combining the Cubic Plus Association Equation of State, Mean Spherical Approximation and the Born equation. This new model uses experimental relative static permittivity, intend to predict well the activity coefficients of individual ions (ACI) and liquid densities of aqueous solutions. This new model is applied to model water + NaCl binary system and water + gas + NaCl ternary systems. The cation/anion-water interaction parameters of are obtained by fitting the experimental data of ACI, mean ionic activity coefficients (MIAC) and liquid densities of water + NaCl binary system. The cation/anion-gas interaction parameters are obtained by fitting the experimental data of gas solubilities in aqueous NaCl solutions. The modeling results show that this new model can correlate well with the phase equilibrium and volumetric properties. Without gas, predictions for ACI, MIAC, and liquid densities present relative average deviations of 1.3%, 3.6% and 1.4% compared to experimental reference values. For most gas-containing systems, predictions for gas solubilities present relative average deviations lower than 7.0%. Further, the contributions of ACI, and salting effects of NaCl on gases are analyzed and discussed.

Experimental investigation and thermodynamic modeling of equilibrium extraction of gold(III) from hydrochloric acid in 1-butyl-3-methylimidazolium hexafluorophosphate

Liquid-liquid equilibrium (LLE) characterization of ionic liquid (IL) systems is an essential method to examine ILs as potential alternative solvents for traditional extraction and separation which also affirms the significance of modeling IL containing mixtures. In the present study, extraction of gold(III) from an acidic medium was performed using 1-butyl-3-methylimidazolium hexafluorophosphate ([bmim][PF6]) at 25°C for which the reaction mechanism was investigated. In order to model the above mentioned system, the implicit version of mean spherical approximation model (MSA) was added to Statistical Associating Fluid Theory Equation of State (SAFT-HR EoS) developed by Huang and Radosz. Non-idealities of both aqueous and organic phases were taken into account. Water was assumed to have two association sites and each ion was considered as a spherical species exerting repulsive, coulombic, and dispersive interactions. Two approaches were considered to model [bmim][PF6] as an extractant; the ion-based and the salt-based. A set of EoS parameters for each approach was obtained by fitting to reliable density data of [bmim][PF6] from literature with fitting error of less than 0.31% on average. The theory is found to accurately model the experimental equilibrium concentration of gold(III) with the average error of 9.62%. The predictive capability of the model was studied, as well. The results show that the model is a powerful tool for modeling the extraction system containing water, ions and IL.

On the solution of the mean-spherical approximation (MSA) for ions in a dipolar solvent in the general case

This paper deals with the problem of a mixture of hard spherical ions and central point dipoles, described within the mean-spherical approximation. In this non-primitive model, the species have different diameters (unrestricted case). The first purpose of this work is to establish clear and valid equations for this problem and present a method to solve them. Formulas for the pressure, the internal and Helmholtz energies, and the chemical potentials of the species are given and discussed. The results are illustrated by considering a binary 1–1 electrolyte solution in a water-like dipolar solvent.

Application of ePC_SAFT-MB EoS in prediction of VLE for CO2+MDEA+H2O ternary and H2O+MDEA+CO2-CH4 quaternary mixtures

The present study aims to implement an electrolyte EoS developed in our previous work [A. Raeispour Shirazi, M. N. Lotfollahi, Fluid Phase Equilib., 502 (2019) 112289] to model the vapor-liquid equilibria (VLE) of CO2 solubility in aqueous N-methyldiethanolamine (MDEA) solution (CO2+MDEA+H2O ternary mixture) as well as H2O+MDEA+CO2-CH4 quaternary mixture. The proposed electrolyte PC-SAFT+MSA+Born equation of state ((ePC_SAFT-MB) consists of PC-SAFT equation of state, the mean spherical approximation (MSA) term and the Born term. First, an appropriate association scheme for water is specified via performing VLE calculations for H2O-CO2 binary mixture and without any adjustment in binary interaction parameter (BIP). By determining the proper association scheme for water (4C), the BIPs of H2O-CO2, MDEA-CH4 and CH4-H2O binary mixtures are adjusted with binary VLE experimental data. In what follows, the nonelectrolyte PC-SAFT EoS is applied considering the obtained BIPs to predict 162 selected data points of bubble pressure for H2O-MDEA-CO2 ternary system in a wide range of temperature (298-413 K), MDEA concentration (18.7-48.8 wt.%) and CO2 loading (0.0001-1.5). The average absolute relative deviation (AAD%) for the calculation results is obtained 25.9%. In order to consider the ions interactions in the solution and to improve the precision of calculations, Born and MSA contributions are added to PC-SAFT EoS. By adjusting two BIPs (kMDEAH+−−CO2and kMDEAH+−HCO3−) and without any readjustment of other BIPs, the AAD% is obtained 15.3%. Finally, the obtained BIPs between the molecular and ionic species for proposed ePC_SAFT-MB EoS are employed to predict the VLE of H2O-MDEA-CO2-CH4 quaternary mixture without any readjustment in BIPs. For this case, the AAD% of 28.05% indicates the reliability of proposed ePC_SAFT-MB EoS for the prediction of H2O-MDEA-CO2-CH4 quaternary VLE results.

Some remarks on the investigation of structure, thermodynamic and surface properties of liquid metals using square well potential

The shortcomings found in the paper of Mishra, Lalneihpuii and Pathak [Chem. Phys. 457 (2015) 13] are discussed. It is shown that the random phase approximation is used in the work, but not the mean spherical approximation as it is declared by the authors.

Implementing the Mean Spherical Approximation Model in the Speciation Code CHEAQS Next at High Salt Concentrations

Speciation programs all struggle with the challenge of converting equilibrium constants to non-zero ionic strength. Equations like the Davies equation are semi-empirical and do not give satisfactory results at high salt concentrations. The Mean Spherical Approximation (MSA) model is a method with a solid theoretical basis that gives good results at high salt concentrations. We implemented the MSA model in a new 64-bit version of the speciation program CHEAQS Next which is under development. It was shown that the results obtained with this research version are identical to results obtained earlier by Simonin (Ind Eng Chem Res 56:9721–9733, 2017). This is the first time the MSA model has been implemented in a production program.

Formula of compressibility and using it for air, noble gases, some hydrocarbons gases, some diatomic simple gases and some other fluids

Based on solutions of the Ornstein–Zernike equation (OZE) of Lennard–Jones potential for mean spherical approximation (MSA), we derive analytical formula for the compressibility assuming that the system is of low density, homogeneous, isotropic and composed of one component. Depending on this formula, we find the values of the bulk modulus and the compressibility of air at room temperature and the bulk modulus and the compressibility of Methane, Ethylene, Propylene and Propane at nine per ten of critical temperature of each hydrocarbon. Also, we find the speed of sound in the air at various temperatures, the speed of sound in each of Helium, Neon, Argon, Krypton, Xenon, Methane, Ethylene, Propylene, Propane, Hydrogen, Nitrogen, Fluorine, Chlorine, Oxygen, Nitrous oxide (laughing gas), Carbon dioxide, Nitric oxide, Carbon monoxide, Sulphur dioxide and dichlorodifluoromethane at room temperature. Besides, we find the speed of sound in Methane, Ethylene, Propylene and Propane at nine per ten of critical temperature of each hydrocarbons depending on the formula we find. We show that the simple formula we derive in this work is reliable and agrees with the results obtained from other studies and literatures. We believe it can be used for many systems which are in low densities and described by Lennard–Jones potential.

Perturbation approaches for describing dipolar fluids and electrolyte solutions.

This work proposes perturbation approaches for describing dipolar fluids as well as model and aqueous electrolyte solutions. The electrostatic pair potentials are split into short- and long-ranged contributions, whereas a third order perturbation expansion is applied for the short-ranged potentials. This circumvents the problem of divergent correlation integrals. The dipolar perturbation terms are represented through a [2,1]-Padé approximation to resum the poorly convergent series. For the remaining charge-charge and charge-dipole contributions, we present a new approximant, which provides a (quasi)linear dependence of the Helmholtz energy. The underlying correlation integrals are adjusted to results from molecular simulations. The long-ranged contribution to the electrostatic interactions is treated through an analytic expression developed by Rodgers and Weeks [J. Chem. Phys. 131, 244108 (2010)]. Theoretical predictions of our perturbation theory are compared to results from a widely used integral equation theory, namely, the mean spherical approximation, and we find that our perturbation theory provides much more accurate results. Furthermore, the theory shows some quantities in rather good agreement with reference data, namely, Helmholtz energies, internal energies, and densities at higher densities of solutions. Limitations of the approach, however, are observed for several other partial molar quantities, such as the mean activity coefficient.

Spherical harmonic projections of the static structure factor of the dipolar hard sphere model: Theory vs simulations.

We investigate the static correlations of a dipolar fluid in terms of the irreducible coefficients of the spherical harmonic expansion of the static structure factor. To this end, we develop a theoretical framework based on a soft-core version of Wertheim's solution of the mean spherical approximation (MSA), which renders the analytical determination of such coefficients possible. The accuracy of this approximation is tested by a comparison against the results obtained with the assistance of extensive molecular dynamics simulations at different regimes of concentration and temperature. Crucial aspects for the comparison of the results provided by the two methods are carefully discussed, concerning the different reference frames used in theory and simulations to describe rotations and orientations, and leading to important differences in the behavior of correlation functions with the same combination of spherical harmonic indices. We find a remarkable agreement between the two approaches in the fluid regime, thus providing a first stringent comparison of the irreducible coefficients of the spherical harmonic expansion of the dipolar fluid's static structure factor, provided by the MSA theory and molecular dynamics simulations.

Interfacial structure and differential capacitance of ionic liquid/graphite interface: A perturbed-chain SAFT density functional theory study

The perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) was combined with the electrostatic free energy from the mean spherical approximation (MSA) theory, and applied to represent densities of pure imidazolium ionic liquids (ILs) with anion [BF4]−. The PC-SAFT parameters of cations were linearized with their molar mass and obtained by simultaneously fitting the model predictions to experimental densities of some ILs. The PC-SAFT-MSA model provides accurate correlations and predictions of densities comparing with experimental data. Then a classical density functional theory (DFT) was developed based on PC-SAFT-MSA. The DFT model was applied to explore the structure and differential capacitance of the electrical double layer (EDL) in ILs on graphite. The model predicts similar density profiles for both cation and anion on a neutral surface, and layered structure with alternating layers of cations and anions on a charged surface. The charge inversion phenomena were also studied based on ion distributions. We further studied the effects of the alkyl chain length, temperature and non-electrostatic solid-fluid interactions on the differential capacitance of the EDL. The model provides bell-shaped differential capacitance curves. The peak positions of differential capacitance curves shift toward positive potentials as ions size asymmetry increases in agreement with previous experiments and simulations studies. The maximum capacitance decreases with increasing alkyl chain length as well as increasing temperature.

Ground-state correlation energy of counterions at a charged planar wall: Gibbs-Bogoliubov lower-bound approach

Recent simulation results imply the lowering of the ground-state correlation energy per counterion at a charged planar wall, compared with that of the 2D and 3D one-component plasma systems. Our aim is to correctly evaluate the ground-state energy of strongly-coupled counterion systems by considering a quasi-2D bound state where bound counterions are confined to a layer of molecular thickness. We use a variational approach based on the Gibbs-Bogoliubov inequality for the lower-bound free energy so that the liquid-state theory can be incorporated into the formulations. The soft mean spherical approximation demonstrates that the lowered ground-state energy can be reproduced by the obtained analytical form of a quasi-2D bound state.

Theoretical interpretation of conductivity data below and above the CMC: The case of alkaline ion decanoate solutions

In this work the conductivity data of three carboxylate surfactant systems: sodium (NaDec), potassium (KDec) and cesium (CsDec) decanoates aqueous solution in the concentration range 10 −3 − 0.25 mol dm –3 , between temperatures 278.15 and 328.15 K, are analyzed by a transport theory in the frame of the mean spherical approximation (MSA). The data in the pre-micellization region, below critical micelle concentration (CMC), are reproduced well by taking into account the formation of neutral pairs involving monomers and counterions. It was found, that the values of the thermodynamic association constants, K , for NaDec and KDec are very similar and slightly smaller than values of K for CsDec. In all cases, K decreases with increasing temperature. Above CMC, the fitting of experimental data yields the effective charge of micelles, which also turned out as counterion quite insensitive, but is bigger at higher temperatures. Thus, at higher temperatures below CMC the presence of free monomers and counterions is favorable and above CMC the binding of counterion on the micelle surface is weaker. However, at studied systems no distinctive ion specific effect could be observed, thus the dominant role of Dec- as a surfactant monomer in the micellization process could be assumed. • High precision conductivities of alkaline ion decanoate salts in water at different temperatures • Experimental data were analyzed by using a Mean Spherical Approximation transport theory. • The limiting conductance of decanoate anion was determined over temperature range. • CMC, degree of dissociation α and effective charges of micelles were estimated.

Temperature dependence of the specific volume of Lennard-Jones potential and applying in case of polymers and other materials

Based on solutions of the Ornstein–Zernike equation, we derive analytical formula for the specific volume of polymers interacting through Lennard-Jones potential. We use mean spherical approximation assuming that the system is of low density, homogeneous, isotropic and composed of one component. The specific volumes of some polymers in addition to the minimum volumes of Lennard-Jones potential and the specific volumes of proteinogenic amino acids: A, R, N, D, C, Q, E, G, H, I, L, K, M, F, P, S, T, W, Y and V, are calculated according to that formula. We show that the simple formula we derive is reliable and agrees well with results obtained from experimental and fitting reported in other studies. We believe that it can be used for many systems described by Lennard-Jones potential such as biological systems, soft matter systems, large molecules such as polymers and inert gases fluids like liquid argon.

First-order mean spherical approximation (FMSA) for the Buckingham Exp(αE, m) potential

Abstract The first-order mean spherical approximation (FMSA) is applied to the Buckingham Exp(αE,m) fluids by using a simple mapping between the Buckingham and two-Yukawa potentials. Simple analytic expressions are then obtained for the Helmholtz free energy, from which the reduced pressure, the internal energy, and the second virial coefficient can be derived. The reduced pressures for the Exp(11, 6), Exp(14, 6) and Exp(16, 6), the internal energy and compressibility factor for the Exp(14, 6) are found in good agreement with the available simulation results up to high temperatures and low to moderate densities (ρσ⁎ = ρσ3 ≤ 1). The Exp(30, 6) and Exp(14, 6) second virial coefficients are also in good agreement with the exact numerical results except at low temperatures (T⁎ ≤ 0.8). The liquid-vapour coexistence diagrams (binodals) for the Exp(12, 6), Exp(14, 6), Exp(30, 6), Exp(10, 4), Exp(14, 4) and Exp(14, 5) are also well predicted by this analytic FMSA equation of state except near the critical point.

Comments on ‘Bhatia–Thornton fluctuations, transport and ordering in partially ordered Al–Cu alloys’

A number of drawbacks found in the paper ‘Bhatia–Thornton fluctuations, transport and ordering in partially ordered Al–Cu alloys’ are considered. In particular, it is shown that this paper misleads readers confusing two different approaches—the mean spherical approximation and the random phase approximation.

The Ohm Law as an Alternative for the Entropy Origin Nonlinearities in Conductivity of Dilute Colloidal Polyelectrolytes.

We discuss the peculiarities of the Ohm law in dilute polyelectrolytes containing a relatively low concentration of multiply charged colloidal particles. It is demonstrated that in these conditions, the effective conductivity of polyelectrolyte is the linear function of . This happens due to the change of the electric field in the polyelectrolyte under the effect of colloidal particle polarization. The proposed theory explains the recent experimental findings and presents the alternative to mean spherical approximation which predicts the nonlinear I–V characteristics of dilute colloidal polyelectrolytes due to entropy changes.

Semi-clathrate hydrate phase equilibria of carbon dioxide in presence of tetra-n-butyl-ammonium chloride (TBAC): Experimental measurements and thermodynamic modeling

The promotion effect of TBAC on the dissociation condition of the studied system is determined under the present experimental data. A combination of the van der Waals–Platteeuw solid solution theory, binding mean spherical approximation electrolyte model and modified Peng–Robinson equation of state is used for calculation of the hydrate, aqueous solution and gaseous phases properties, respectively. The results are also compared with the experimental data for other salts such as tetra-n-butyl-ammonium bromide (TBAB) and tetra-hydro furan (THF). A good agreement between the model predictions and experimental data is found with an absolute average relative deviation of 13.1%.