First-order Mean Spherical Approximation Research Articles

A cavity formation energy formula for hard spheres in simple electrolyte solutions.

A formula for cavity formation energy of a hard sphere in restricted primitive electrolyte solutions is derived based on the integral equation theory. Specifically, the contact values of radial distribution functions between the hard sphere and the ionic species, determined analytically from the first-order mean spherical approximation theory, are used to evaluate the cavity formation energy. In the large solute-size limit, the scaling relation of the cavity formation energy further leads to an analytical expression for the surface tension of the electrolyte solution near a curved interface. Our theory is applied to hard spheres immersed in restricted primitive electrolyte solutions, where the good agreement of the cavity formation energy with the hyper-netted chain theory demonstrates the accuracy of our theory.

Development of a fully analytical equation of state using ab initio interaction potentials. Application to pure simple fluids: Noble gases Ne, Ar, Kr, and Xe

In this work, we have extended the first-order mean spherical approximation (FMSA) equation of state (EOS) of Tang and Lu [J. Chem. Phys., 99, 9828 (1993)] in order to predict accurately fluid phase diagrams using ab initio potentials. At this level of development, we restricted to simple spherical compounds.The original equation based on hard-core two-Yukawa (HC2Y) was improved by the following:•Adding a new approximate simple correction to include higher orders in the development of Tang and Lu to get an analytical model closer to the full MSA.•Including a term to a better account for the long-range behavior of two-body potentials.•Adding a three-body interaction term (1st order perturbation) based on Axilrod-Teller potential.A general recommendation for setting the hard-sphere diameter value was also given.The predictive capability of the EOS was tested in a systematic manner. First, we checked that various simple theoretical potential (Mie n-6 and Exp n-6) phase diagrams are well reproduced. Second, the EOS was used for predicting fluid phase diagrams of the pure noble gases fluid series Ne, Ar, Kr, and Xe using realistic (i.e. derived from ab initio) two-body and three-body interaction potentials. The model is in good agreement with the experimental phase diagrams.As a conclusion, the simple statistical-mechanics-based fully analytical EOS developed in this work predicts accurately fluid phase diagrams of simple spherical compounds using ab initio potentials. No adjustable parameter was used. This EOS constitutes a strong basis for future development of a predictive model based on ab initio potentials and applicable to more complex compounds.

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.

On the phase behavior of model fluids with square-well attraction in slit-like pores. Density functional approach

The phase behavior of a model fluid with square well inter-particle attraction in slit-like pores has been studied by using a density functional theory. The mean field theory and the first-order mean spherical approximation have been applied to account for the attractive interactions. The influence of the pore width, the gas-solid interaction energy and of the square well width on the phase behavior have been explored in detail. Dependent on the values of parameters of the model, we observed capillary condensation and capillary evaporation, in some cases layering phase transition has been obtained as well. Also, a crossover between condensation and evaporation has been described. The theory and the results obtained in the present work are valuable for future theoretical exploration of water-like and other models under confinement that intrinsically involve square well as a reference ingredient.

Adsorption and phase behavior of water-like fluid models with square-well attraction and site-site association in slit-like pores: Density functional approach.

The adsorption and phase behavior of two model fluids, both with square well inter-particle attraction and site-site associative interaction, in slit-like pores have been studied in the framework of a density functional theory. The mean field approach and the first-order mean spherical approximation have been applied to account for the attractive interactions. The chemical association effects are taken into account by using the first-order thermodynamic perturbation theory of Wertheim. A set of parameters for each fluid model has been chosen according to the work of [Clark et al., Mol. Phys. 104, 3561 (2006)], to describe successfully the vapor-liquid coexistence of water in the bulk phase. The influence of the slit-like pore width and of the strength of gas-solid interaction energy on the vapor-liquid coexistence envelope under confinement has been explored in detail. The theory and the results of the present work are valuable for further exploration of a wide set of models of associating fluids and of fluids with complex molecular architecture in different adsorbents, and to deal with activated carbon surfaces.

On the theoretical description of the liquid-vapor coexistence of water-like models with square-well attraction and site-site chemical association

A liquid-vapor coexistence envelope for water-like models with square-well attraction between particles and with four attractive associating sites per particle has been investigated. The mean-field theory and the first-order mean spherical approximation for attractive interactions together with the first-order thermodynamic perturbation theory of Wertheim for association have been applied to construct the temperature - density and the temperature - chemical potential projections of the equation of state. In addition, the dependence of the fraction of non-bonded particles on the parameters of the models has been evaluated and analyzed. We explored a set of models with different square-well width and energy of associative interaction. In some cases a comparison with computer simulation data has been performed. For two specific parameter sets previously proposed to mimic water properties a comparison with experimental data for the vapor-liquid coexistence envelope and for vapor pressure has been performed. It is shown that the applied theoretical procedures are successful and can be used with confidence. On the other hand, they provide solid basis to apply in the theory of inhomogeneous associating fluids.

First-order mean spherical approximation (FMSA) for Mie(α, β) fluids

A simple mapping between the two-Yukawa (TY) and Mie(α, β) potentials parameters is obtained, which allows to extend the first-order mean spherical approximation (FMSA) to the general Mie(α, β) fluids. The FMSA first-order Helmholtz free energy f1Mie of the Mie(12, 6) and the second-order reduced pressure P⁎Mie for the Mie(32, 6), Mie(14.65, 8), Mie(9, 6) and Mie(14, 6) potentials are found in very good agreement with recent numerical simulation results. The liquid-vapour equilibrium (LVE) diagram for the Mie(32, 6) fluid is constructed and use of a third-order term in the Helmholtz free energy improves the results in the critical region.

A comprehensive comparison between thermodynamic perturbation theory and first-order mean spherical approximation: Based on discrete potentials with hard core

Using the TL (Tang and Lu, 1993) method, Ornstein-Zernike integral equation is solved perturbatively under the mean spherical approximation (MSA) for fluid with potential consisting of a hard sphere plus square-well plus square-shoulder (HS+SW+SS) to obtain first-order analytic expressions of radial distribution function (RDF), second-order direct correlation function, and semi-analytic expressions for common thermodynamic properties. A comprehensive comparison between the first-order MSA and high temperature series expansion (HTSE) to third-, fifth- and seventh-order is performed over a wide parameter range for both a HS+SW and the HS+SW+SS model fluids by using corresponding “exact” Monte Carlo results as a reference; although the HTSE is carried out up to seventh-order, and not to the first order as the first-order MSA the comparison is considered fair from a calculation complexity perspective. It is found that the performance of the first-order MSA is dramatically model-dependent: as target potentials go from the HS+SW to the HS+SW+SS, (i) there is a dramatic dropping of performance of the first-order MSA expressions in calculating the thermodynamic properties, especially both the excess internal energy and constant volume excess heat capacity of the HS+SW+SS model cannot be predicted even qualitatively correctly. (ii) One tendency is noticed that the first-order MSA gets more reliable with increasing temperatures in dealing with the pressure, excess Helmholtz free energy, excess enthalpy and excess chemical potential. (iii) Concerning the RDF, the first-order MSA is not as disappointing as it displays in the cases of thermodynamics. (iv) In the case of the HS+SW model, the first-order MSA solution is shown to be quantitatively correct in calculating the pressure and excess chemical potential even if the reduced temperatures are as low as 0.8. On the other hand, the seventh-order HTSE is less model-dependent; in most cases of the HS+SW and the HS+SW+SS models, the seventh-order HTSE improves the fifth- and third-order HTSE in both thermodynamic properties and RDF, and the improvements are very demonstrable in both the excess internal energy and constant volume excess heat capacity; for very limited cases, the seventh-order HTSE improves the fifth-order HTSE only within lower density domain and even shows a bit of inadaptation over higher density domain.

Structure factor of a hard-core fluid with short-range Yukawa attraction: analytical FMSA theory against Monte Carlo simulations

ABSTRACTAnalytical solution of the first-order Ornstein–Zernike equation known as the first-order mean spherical approximation (FMSA) theory due to Tang and Lu [J. Chem. Phys. 99, 9828 (1993)] is used to write down a closed equation for the static structure factor of the hard-sphere fluid with a short-range Yukawa attraction. Calculations are performed for a Yukawa decay exponent that corresponds to a range of attraction that does not exceed the first coordination shell of Lennard-Jones-like simple fluids. By comparison with Monte Carlo simulation data it is shown that the analytical FMSA equation for the static structure factor is of the same or even of superior accuracy as that within the seminumerical mean spherical approximation theory.

Comparison between the statistical associating fluid theory for variable range potentials (SAFT-VR) and the first-order mean spherical approximation (FMSA) equations of state for Mie (α, 6) fluids

The respective accuracies of three recent analytical equations of state (EOS) (i.e. analytical representations of the Helmholtz free energy and compressibility factor Z) for the Mie(α,6) monomer fluids are evaluated by comparing the residual Helmholtz free energy, its Barker-Henderson (BH) perturbation terms and Z with the corresponding numerical simulations for the Mie(8,6), Mie(12,6), Mie(20,6) and Mie(30,6) fluids. Two of these EOS stem from the recent enhanced statistical associating fluid theory with variable range potentials (SAFT-VR Mie 2013), with two proposed parameterizations for the free energy first-order BH perturbation term of the Sutherland potentials building up the Mie potential. The third EOS uses a new mapping between the Mie(α,6) and a two-Yukawa potential, for which the first-order mean spherical approximation (FMSA) can be used to obtain analytically the free energy and pressure of the Mie(α,6) fluids up to second order.

Theoretical approaches to the structural properties of the square-shoulder fluid

ABSTRACTA comparison of simulation results with the prediction of the structural properties of square-shoulder fluids is carried out to assess the performance of three theories: Tang–Lu's first-order mean spherical approximation, the simplified exponential approximation of the latter and the rational-function approximation. These three theoretical developments share the characteristic of being analytical in Laplace space and of reducing in the proper limit to the Percus–Yevick result for the hard-sphere fluid. Overall, the best agreement with the simulation data is obtained with the simplified exponential approximation.

Analytic perturbative FMSA equation of state and thermodynamic properties from Monte Carlo simulation of the Kihara potential with a spherical core

A new equation of state is found to reproduce analytically the thermodynamic properties of fluids interacting with a spherical Kihara potential formulated for arbitrary values of the inner hard core. The results from the expressions developed are satisfactorily compared with the Monte Carlo simulation data for a wide range of densities and temperatures. The foundation of the new equation is strictly theoretical, based on the first-order mean spherical approximation perturbation approach of Tang and Lu, and provides a good starting point for the description of polyatomic fluids within the benchmark of Wertheim's first-order perturbation theory of associating fluids.

Exponential approximation for one-component Yukawa plasma.

A theory based on the exponential approximation of the liquid-state theory is applied to study properties of several models of one-component Yukawa plasma characterized by different values of the screening parameter z. The results of the new theory are compared to the results of a conventional theory, which is based on the first-order mean spherical approximation, and to the results of a Monte Carlo simulation. The new theory shows improvements in the predictions for the thermodynamic and structural properties of Yukawa plasmas with high and intermediate values of the screening parameter, z, and coupling parameter, Γ. For low values of z and Γ, the new theory is comparable in accuracy to the conventional theory, which in turn agrees well with the results of the Monte Carlo simulation.

Communication: Fine discretization of pair interactions and an approximate analytical strategy for predicting equilibrium behavior of complex fluids

We study whether fine discretization (i.e., terracing) of continuous pair interactions, when used in combination with first-order mean-spherical approximation theory, can lead to a simple and general analytical strategy for predicting the equilibrium structure and thermodynamics of complex fluids. Specifically, we implement a version of this approach to predict how screened electrostatic repulsions, solute-mediated depletion attractions, or ramp-shaped repulsions modify the radial distribution function and the potential energy of reference hard-sphere fluids, and we compare the predictions to exact results from molecular simulations.

An improved first-order mean spherical approximation theory for the square-shoulder fluid

The theory, which utilizes an exponential enhancement of the first-order mean spherical approximation (FMSA) for the radial distribution functions of the hard-core plus square-well fluid, is adopted to study the properties of the simplest model of the core-softened fluids, i.e., the hard spheres with a square-shoulder interaction. The results for structure and thermodynamic properties are reported and compared against both the Monte Carlo simulation data as well as with those obtained within the conventional FMSA theory. We found that in the region of low densities and low temperatures, where the conventional FMSA theory fails, the exponential-based FMSA theory besides being qualitatively correct also provides with a notable quantitative improvement of the theoretical description.

Fourier space approach to the classical density functional theory for multi-Yukawa and square-well fluids

We present a Fourier space density functional approach for hard particles with attractive interactions, which is based on a previously developed two-dimensional approach [S. Hlushak, W. Rżysko, and S. Sokołowski, J. Chem. Phys. 131, 094904 (2009)] for hard-sphere chains. The interactions are incorporated by means of a three-dimensional Fourier image of the direct correlation function that is obtained from the first-order mean-spherical approximation. In order to improve the computational efficiency, we make extensive use of fast Fourier transforms for calculating density convolution integrals. A two-dimensional implementation of the new density functional approach, based on the expansion of the functional around the bulk fluid density, is used to study structure and adsorption of two model fluids in narrow cylindrical pores. We also investigate two methods that improve the accuracy of the theory as compared to the conventional DFT approach, which expands the free energy functional around the bulk fluid density: One a variant of the reference fluid density functional theory used by Gillespie et al. [Phys. Rev. E 68, 031503 (2003)], and the second a weighted density approach with energy route thermodynamics. Results from these two methods are compared to the conventional approach and also to the results of Monte Carlo simulations. We find that the method of Gillespie et al. and the weighted density approach with energy route thermodynamics yield significant improvement over the conventional approach.

Simplified exponential approximation for thermodynamics of a hard-core repulsive Yukawa fluid

Exponential approximation based on the first order mean spherical approximation (FMSA) is applied to the study of the structure and thermodynamics of hard-core repulsive Yukawa fluids. The proposed theory utilizes an exponential enhancement of the analytical solution of the FMSA due to Tang and Lu [J. Chem. Phys., 1993, 99, 9828] for the radial distribution function. From comparison with computer simulation data we have shown that at low density and low temperature conditions, where original FMSA theory fails, the FMSA-based exponential theory predicts a significant improvement.

Structure and phase behavior of two-Yukawa fluids with competing interactions in planar slit pores

A density functional perturbation theory, which is based both on the modified fundamental-measure theory and on the first-order mean-spherical approximation for long-range attractive and repulsive interactions, has been developed for studying the structure and phase behaviors of a competing system restricted to slit pores. The hysteresis loop for the adsorption and desorption curves indicates that the system exhibits vapor-cluster and cluster-liquid transitions which depend on the pair potential parameters and the slit width (H). The periodic spacing (D) of the cluster is commensurate with the periodicity of modulation in the particle density distribution and more closely related to the vapor-cluster and cluster-liquid phase transitions of the system. For the cluster phase, we find the transition from a single liquidlike slab to a multi-liquidlike slab with increasing the slit width. The multi-liquidlike slab is formed depending on the periodicity of modulation by finite-size artifacts. The cluster-related phase transitions, such as the vapor-cluster or cluster-liquid transitions occur for H>D, while for H<D the system only exhibits the vapor-liquid transition. At a low amplitude, the vapor-liquid transition disappears and the cluster-liquid transition only occurs for H<D. The coexistence curves for the confined phase diagram are contained within the corresponding bulk liquid-vapor coexistence curve. For a wide slit pore (H>D), the system exhibits two tricritical points, joined to one another by the line of second-order transition. The results support the conclusion that the confinement effect plays an important role in determining the equilibrium phase transition.

Prediction of Interfacial Structure and Tension of Binary Mixtures Containing Carbon Dioxide

In this work, a density functional approach was applied to describe the interfacial thermodynamic properties of CO2 binary mixtures including CO2 + heptane, CO2 + acetone, CO2 + ethanol, and CO2 + water. In the theoretical model, the first-order mean spherical approximation statistical associating fluid theory (FMSA-SAFT) was incorporated to describe the chemical potential of the bulk term, while, for inhomogeneous potential, the modified fundamental measure theory (MFMT) was employed to describe the hard sphere contribution, and a weighted density approximation was adopted for the attractive contribution. Without any adjustable parameter, the theoretical calculations have been well-testified by the available experimental data, showing a good predictive capability of the theory to the interfacial structure and tensions of CO2 binary mixtures with nonpolar, polar, and associating fluids.

Structure and thermodynamics of hard-core Yukawa fluids: Thermodynamic perturbation approaches

The thermodynamic perturbation theories, which are based on the power series of a coupling constant (λ-expansion), have been proposed for studying the structural and thermodynamic properties of a hard-core Yukawa (HCY) fluid: one (A1-approximation) is the perturbation theory based on the hard-sphere repulsion as a reference system. The other (A2-approximation) is the perturbation theory based on the reference system which incorporates both the repulsive and short-range attractive interactions. The first-order mean-spherical approximation (FMSA) provided by Tang and Lu [J. Chem. Phys. 99, 9828 (1993)] has been employed for investigating the thermodynamic properties of a HCY fluid using the alternative method via the direct correlation function. The calculated results show that (i) the A1 and A2 approximations are in excellent agreements with previous computer simulation results in the literature and compare with the semi-empirical works of Shukla including the higher-order free energy terms, (ii) the A1 and A2 approximations are better than the FMSA and the mean-spherical approximation, (iii) the A2-approximation compares with the A1-approximation, even though the perturbation effect of an A2-approximation is much smaller than that of an A1-approximation, and that (iv) the FMSA study is particularly of advantage in providing the structure and thermodynamics in a simple and analytic manner.