Mean Spherical Approximation Model Research Articles

Response to "Comment on 'Binding Debye-Hückel theory for associative electrolyte solutions'" [J. Chem. Phys. 159, 154503 (2023)

This Response addresses critiques raised about the Binding Debye-Hückel (BiDH) theory [Naseri Boroujeni et al., J. Chem. Phys. 159, 154503 (2023)] by Simonin and Bernard [J.-P. Simonin and O. Bernard, J. Chem. Phys. (2024)]. The critiques questioned the foundational framework of the Debye-Hückel (DH) theory, the relevance of ion pairing in primitive model fluids, and the accuracy of the BiDH model compared to mean spherical approximation model. Through a systematic rebuttal, supported by extensive literature review and comparison with Monte Carlo simulation data, this Response addresses these concerns. It demonstrates the efficacy of DH theory in describing real electrolyte solutions, validates the relevance of ion pairing in primitive model fluids, and establishes the BiDH model's accuracy in describing electrolyte properties.

Measurement and modeling of the electrical conductivity of organic electrolyte solutions and their mixtures with compressed CO2

The electrochemical carbon dioxide reduction reaction (CO2RR) is of increasing importance for the development of a closed carbon cycle. Here, electrolytes are often used to increase the electrical conductivity and thus the current density. In addition, the use of compressed CO2 and electrolyte-containing organic solvents suppresses hydrogen evolution. However, the electrical conductivity of CO2-saturated electrolyte solutions is mostly unknown. Therefore, their electrical conductivity was investigated in this work at different electrolyte concentrations, pressures (up to 150 bar) and temperatures (298.15 K and 343.15 K). In addition, the simplified Mean Spherical Approximation (MSA-simple) model, supported by ePC-SAFT for density modeling, was used to successfully model the conductivity. The results show that CO2-saturated electrolyte solutions generally exhibit lower conductivities than CO2-free electrolyte solutions. However, at very low electrolyte and CO2 concentrations, the conductivities were higher in CO2-saturated electrolyte solutions than in CO2-free electrolyte solutions, which is a new finding in the literature.

A Theoretical and Computational Study of Structure‐Property Relationship in Five Binary Liquid Alloys

AbstractA theoretical model is proposed in this work for the evaluation of composition‐dependent thermophysical properties such as the isothermal compressibility, velocity of sound and Debye temperature of liquid binary alloys. Using a square‐well (SW) model potential function under mean spherical model approximation, the composition‐dependent atomic‐level structural functions of Al1‐xCux, Ag1‐xCux, Au1‐xCux, Ag1‐xAlx, and Au1‐xAgx alloys have been determined and compared with available experimental data. An extensive computation was done to calculate the composition‐dependent isothermal compressibility through the structure factor in the long wavelength limit (k→0), and then we employed this data to investigate the velocity of sound in the considered melts. This yields the fundamental maximum frequency with which the constituent particles are vibrating in the melts and is used to determine the composition‐dependent Debye temperature, ΘD. The computed Debye temperature of considered melts using the Square‐well potential function is found to be in satisfactory agreement with the available DFT and MD simulation results.

Modeling of dilution enthalpies within implicit-solvent models for electrolytes

This work deals with the calculation of dilution enthalpies for strong aqueous electrolyte solutions when their thermodynamic properties are described within implicit-solvent models. The solution is modeled as a collection of charged hard spheres in a continuum solvent. A new general expression is obtained that accounts for the variation of the solution permittivity, of the size of the ions, and of the specific volume of the solution, with temperature and concentration. This expression is used to compute the enthalpies of LiCl aqueous solutions at temperatures in the range of 25∘C to 100∘C for concentrations up to 6 mol kg−1. The mean spherical approximation (MSA) model with implicit continuum solvent is employed to describe the thermodynamic properties of these solutions.

Calculation of specific loss power of magnetic fluids by mean spherical approximation (MSA) model

According to various theoretical models the heating performance, i.e. the volumetric power dissipation of magnetic fluids under alternating magnetic field is proportional to the imaginary part of the complex susceptibility. However, the magnetic field strength dependence of the susceptibility is usually taken into account only by the Langevin theory. The main goal of our work was to propose a theoretical model based on the series expansion of the mean spherical approximation (MSA) theory to calculate the magnetic properties, and derive the power dissipation by including the dipole-dipole interactions. The predictions of the theory for the field strength dependence of the power dissipation was verified by comparison with experimental data. It was shown that compared to the MSA theory the generally used Langevin limiting case underestimates the power dissipation. The contribution of the higher-order terms in the approximated dynamic susceptibility at different field strengths was also investigated.

Correlation between structures and atomic transport properties of compound forming liquid Cu-In alloys

The Lebowitz solution of the Percus-Yevick hard-sphere mixture is invoked with the square-well potential function to determine Ashcroft-Langreth type three partial correlation functions in binary Cu-In liquid alloys. These computed partial correlation functions have been employed to generate Bhatia-Thornton fluctuations in the considered melts. Using these structural parameters we investigate the temperature and composition-dependent diffusion coefficients of considered melts. The computed values of concentration-concentration fluctuations, in the long-wavelength limit, and the Warren-Cowley short-range order parameter, show the formation of chemical compounds between unlike atoms in the In-rich region of Cu-In melts. We find a very good agreement between theoretically formulated and computed data of with the corresponding experimental values. Computed results of and suggest that hetero coordination is favorable over homo coordination in the investigated melts. The validity of the Stokes-Einstein relation was observed over a wide range of temperatures and compositions in the investigated melts. Further, a new correlation between two body pair excess entropy and the Stoke-Einstein relation has been formulated for square-well binary liquid alloys. Surface tension and compressibility as a function of In composition have been computed through microscopic structural functions of the alloys. Computed surface tension and ratio of mutual to intrinsic diffusion (D m /D id ) are found comparable to available simulated data. The computed values of shear viscosity are in good agreement with the experimental data. Calculated results suggest that the combination of hard sphere potential with square-well tail under mean spherical model approximation is one of the good method for determining the structures and transport coefficients of compound forming binary liquid alloys.

A Simplified Mean Spherical Approximation Model for the Description of Activity Coefficients in Electrolyte Mixtures

A Simplified Mean Spherical Approximation Model for the Description of Activity Coefficients in Electrolyte Mixtures

Nonlinear contributions to the dynamic magnetic susceptibility of magnetic fluids

• Higher order harmonic susceptibilities of magnetic fluids are obtained theoretically. • Mean spherical approximation is used with field strength expansion of magnetization. • Theoretical prediction is compared to the Langevin and Debye-Weiss limiting cases. • Field amplitude and frequency range of validity of the expanded MSA model is mapped. Based on our earlier analytical results for the magnetization of magnetic fluids with respect to the magnetic field strength, we propose an expansion method within the framework of mean spherical approximation (MSA) to obtain the coefficients of different nonlinear terms. Through a Fourier expansion of the frequency-dependent magnetic susceptibility the harmonic coefficients corresponding to the linear and nonlinear dynamic susceptibilities are calculated from the field expansion of magnetization. The frequency dependence of the higher order susceptibilities is determined on the basis of the Debye relaxation of magnetic dipoles. Our MSA based results are in line with the corresponding limiting case of the Debye-Weiss theory. We mapped the range of applicability of the expansion method concerning the field strength and frequencies. Our results show that under weak fields a 7th order expansion is sufficient to predict the magnitudes of the susceptibility components up to the 4th harmonic relevant for magnetic fluids.

Non-linear boundary condition for non-ideal electrokinetic equations in porous media

ABSTRACT This paper studies the partial differential equation describing the charge distribution of an electrolyte in a porous medium. Realistic non-ideal effects are incorporated through the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. The main novelty is the consideration of a non-constant surface charge density on the pore walls. Indeed, a chemical equilibrium reaction is considered on the boundary to represent the dissociation of ionizable sites on the solid walls. The surface charge density is thus given as a non-linear function of the electrostatic potential. Even in the ideal case, the resulting system is a new variant of the famous Poisson–Boltzmann equation, which still has a monotone structure under quantitative assumptions on the physical parameters. In the non-ideal case, the MSA model brings in additional non-linearities which break down the monotone structure of the system. We prove existence, and sometimes uniqueness, of the solution. Some numerical experiments are performed in 2d to compare this model with that for a constant surface charge.

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.

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.

On the "Born" term used in thermodynamic models for electrolytes.

In the literature, many expressions for the Helmholtz or Gibbs energy of electrolyte solutions have included a term that takes into account the variation of the solution permittivity with the composition of solution (e.g., within the statistical-associated fluid theory formalism). This contribution is often called the "Born" term because it was inspired by the classic expression established by Born to describe the solvation energy of an ion. The present work is an attempt to get more physical insight into this semiempirical "Born" term. The way in which it has been used in the literature is briefly examined, and its typical magnitude is evaluated. Next, it is proposed to use the nonprimitive mean spherical approximation model to calculate the chemical potential of an ion in a solution composed of charged hard spheres (HSs) (the ions) and dipolar HSs (the solvent). The cation and the anion are monovalent monoatomic ions of equal diameter. The dipoles have a different size and mimic water molecules. The theoretical expressions for this model were found to fulfill the Gibbs-Duhem relation, which suggests that they are correct. A rescaled ion-dipole contribution is introduced, in a form that is suitable for inclusion in electrolyte models. It is compared with a "Born" term expressed in the same framework. It is found that the former is in general not well estimated by the latter. The two might even be of opposite signs in the case of ions of sufficiently small size.

Bhatia–Thornton fluctuations, transport and ordering in partially ordered Al–Cu alloys

The Bhatia–Thornton (BT) correlation functions namely the number–number, concentration–concentration and number-concentration correlation functions in liquid binary melts are important parameters for understanding the complexities in binary liquids. In this paper, the microscopic BT correlation functions of liquid Al–Cu alloys are investigated using square well (SW) potential under mean spherical model approximation. These correlation functions are linearly related to the Ashcroft–Langreth type partial structure factors. Thermodynamically important microscopic function i.e. concentration–concentration correlation function in the long-wavelength limit, SCC(0) is analytically calculated, which is well-known in BT formalism and is often used to characterized the mixing behavior in binary melts. The Warren–Cowley chemical short range order parameters, as function of Cu concentration is also computed in the liquid Al–Cu alloys through structural studies in the long wavelength limit, which gives valuable information regarding ordering in melts. The compound forming behavior in Al–Cu alloys is evidenced by the results obtained for the value of SCC(0) and . Further, transport properties in binary mixture have been determined by using BT-correlation functions and its long wavelength limit values. Kinetic as well as thermodynamic factors were taken care in the computation of dynamic properties of alloys.

Determination of Thermodynamic Complexity Constants and Speciation for Multicomplexing Electrolytes within the Mean Spherical Approximation Model

It has been shown in previous work that the determination of speciation in aqueous solutions containing self-complexing electrolytes requires caution as regards the thermodynamic model used to esti...

Non-additivity of ionic radii in electrolyte solutions: Hofmeister effect on mixtures modeled by an Associated MSA model

By considering the solution of the Mean Spherical Approximation (MSA) given by Lesser Blum, we addressed the problem of specificity in aqueous electrolyte solutions. The reference diameter of the cations is defined as the biggest possible diameter of the cation (obtained with the less associating anion). Then the specificity is taken into account by an association term similar to Bjerrum theory. The resulting activity coefficients can be used to describe pure electrolyte and electrolyte mixtures up to molar concentrations. Hydration appear to be relatively stable with small cations but it strongly depends on the anion for big cations. The validity of Zdanovskii-Stokes-Robinson mixing rule has also been tested and it appears to be valid with typically a 1% accuracy for molar concentrations.

Concentration dependent structural parameters of liquid Al-Fe alloys

Square well potential is perturbed over Lebowtiz solution of hard sphere mixtures to determine direct correlation function,C(0)ij(r) in repulsive and attractive regions under Mean Spherical Model Approximation [1]. Obtained direct correlation functions were employed to derive partial structure factors and then total structure factor, S(k) in liquid Al-Fe alloy at different atomic percent of Al. Fourier transform of partial and total structure factors gives partial and total radial distribution functions, g(r) from which partial and total coordination numbers and the partial nearest-neighbor distances were computed.

Modeling of activity coefficients of amino acid and electrolyte in aqueous solutions

The amino acid exists in the zwitterion form with positive (amino group, −NH3+) and negative (carboxyl group, COO−) charge in water, and it is able to form the ion-pair complexes physically bonded with ions. The ion-pair complex between amino acid and ions has a strong effect on the variation of activity coefficient of amino acid in aqueous electrolyte solution. In this work, the PC-SAFT EOS combined with primitive mean spherical approximation (PM-MSA) model is applied to the 32 amino acid-electrolyte aqueous systems (a total of 1305 data points) for modeling the activity coefficients of amino acids and electrolyte in aqueous solutions. The ion-pair complex is considered as a cross-association between amino acid and ions, and this method accurately represents the activity coefficient of an amino acid and electrolyte over a wide range of concentrations. Based on the calculation of activity coefficient ratio of amino acid, the solubility of an amino acid in electrolyte-water system and density of solution are predicted, and the predicted results show qualitative agreement with experimental results. Therefore, the model proposed in this work is expected to be useful for describing activity coefficients of an electrolyte and amino acid in amino acid-electrolyte-water system.

Evaluation of compressibility factor and mean ionic activity coefficient for aqueous electrolyte solutions with hard sphere equations of state in the MSA model and artificial neural network method

In this study, twenty-five hard sphere equations of state (EOSs) were applied to evaluate the compressibility factor of fifty different single electrolyte solutions (Z1). Then by using two well-known mixing rules (Barrio Solana (BS) and Santos et al. (S)) the compressibility factor (Zm) was calculated for 1086 binary mixtures of electrolytes. The obtained results were compared with the simulation data and it was concluded that the Dehghani–Modarress (DM) EOS when applied along with the Santos et al. mixing rule has the highest accuracy (ARD%Z1DM=0.124 and ARD%ZmDM−S=0.466). Also, the mean ionic activity coefficients of a number of 1:1 electrolytes were calculated by applying the hard sphere EOSs with the mixing rules based on the mean spherical approximation (MSA) model where the concentration of cation was considered a function of electrolyte concentration. By comparing these results with reported experimental data it was found that the Ree–Hoover (RH) EOS with Santos et al. mixing rule predicts the mean ionic activity coefficients with the highest accuracy (ARD%γ(RH−S)=0.206). In continuation of this study the compressibility factor and mean ionic activity coefficient were recalculated by means of the artificial neural network (ANN) method. The optimum architectures of ANN models for predicting the compressibility factor of single and binary mixtures of electrolytes as well as the mean ionic activity coefficient of 1:1 electrolytes were respectively obtained as 2:4:1, 3:3:1 and 3:5:1. The calculated error in the results showed that the mathematical models of the ANN method predicted the abovementioned parameters with better accuracy in comparison with the EOSs and MSA models (ARD%Z1ANN=0.047, ARD%ZmANN=0.022 and ARD%γ(ANN)=0.014).

Role of non-ideality for the ion transport in porous media: Derivation of the macroscopic equations using upscaling

This paper is devoted to the homogenization (or upscaling) of a system of partial differential equations describing the non-ideal transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. Realistic non-ideal effects are taken into account by an approach based on the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. We first consider equilibrium solutions in the absence of external forces. In such a case, the velocity and diffusive fluxes vanish and the equilibrium electrostatic potential is the solution of a variant of the Poisson–Boltzmann equation coupled with algebraic equations. Contrary to the ideal case, this nonlinear equation has no monotone structure. However, based on invariant region estimates for the Poisson–Boltzmann equation and for small characteristic value of the solute packing fraction, we prove existence of at least one solution. To our knowledge this existence result is new at this level of generality. When the motion is governed by a small static electric field and a small hydrodynamic force, we generalize O’Brien’s argument to deduce a linearized model. Our second main result is the rigorous homogenization of these linearized equations and the proof that the effective tensor satisfies Onsager properties, namely is symmetric positive definite. We eventually make numerical comparisons with the ideal case. Our numerical results show that the MSA model confirms qualitatively the conclusions obtained using the ideal model but there are quantitative differences arising that can be important at high charge or high concentrations.

Temperature Effect on the Nanostructure of SDS Micelles in Water.

Sodium dodecyl sulfate (SDS) surfactants form micelles when dissolved in water. These are formed of a hydrocarbon core and hydrophilic ionic surface. The small-angle neutron scattering (SANS) technique was used with deuterated water (D2O) in order to characterize the micelle structure. Micelles were found to be slightly compressed (oblate ellipsoids) and their sizes shrink with increasing temperature. Fits of SANS data to the Mean Spherical Approximation (MSA) model yielded a calculated micelle volume fraction which was lower than the SDS surfactant (sample mixing) volume fraction; this suggests that part of the SDS molecules do not participate in micelle formation and remain homogeneously mixed in the solvent. A set of material balance equations allowed the estimation of the SDS fraction in the micelles. This fraction was found to be high (close to one) except for samples around 1 % SDS fraction. The micelle aggregation number was found to decrease with increasing temperature and/or decreasing SDS fraction.