Optimized Random Phase Approximation Research Articles

Theoretical investigation of collective motion in some liquid metals using pseudopotential approach

The collective motions are studied using a recently created pseudopotential and some elastic constants. This paper involves the calculations of the collective motions like phonon dispersion curves (PDCs) for longitudinal and transverse frequencies ([Formula: see text] and [Formula: see text]), longitudinal sound velocity ([Formula: see text]), transverse sound velocity ([Formula: see text]), isothermal bulk modulus (B), modulus of rigidity (G), Young’s modulus (Y), Poisson’s ratio [Formula: see text], Debye temperature ([Formula: see text]) of some monovalent (Na), divalent (Mg) and polyvalent liquid metals (Al, Pb, Bi). This investigation is based upon the theory of pseudopotential of second-order approach via the equations of Hubbard and Beeby (HB). [Formula: see text] The pair correlation function is calculated from the different mean spherical approximation (MSA), one-component plasma (OCP) and optimized random phase approximation (ORPA) structure factors employed in this work. Three different types of local field correction (screening) functions proposed by Hartree (H), Taylor (T) and Nagy (N) are utilized in this paper. The current findings are found to be in qualitative accord with experimental and theoretical data.

Comparative studies of role of different structure factors in various physical properties of some simple liquid metals

In the current work, the comparison of the structure factors and pair correlation functions produced by using eight different theoretical models based on the Perckus-Yevick Hard Sphere (PYHS), Hard Sphere Yukawa (HSY), Mean Spherical Approximation (MSA), Generalized Mean Spherical Approximation (GMSA), Soft Sphere (SS), One-Component Plasma (OCP), Optimized Random Phase Approximation (ORPA) and Charged Hard Sphere (CHS) models for liquid metals viz. Li, Na, K, Rb, Cs, Mg, Zn, Ca, Al, Ga, In, Pb, Sn, Bi and Sb are carried out. Our own model potential is used with the Taylor (TY) screening function in the present computation. With this, certain physical properties such as electrical transport (electrical resistivity), vibrational property (phonon dispersion), dynamical property (velocity autocorrelation function (VACF)) and static (long wavelength of structure factor) properties has also been calculated. When the several theoretical models of the structure factors of the researched simple liquid metals are compared, it is discovered that the experimental data is consistent and in good agreement with the theoretical models.

Theoretical investigation of atomic dynamics of molten metals at melting points

In molten metals, the atomic dynamics have been studied in the present study with the help of equations of motion in terms of the velocity autocorrelation function (VACF), power spectrum G ( ω ) , and mean square displacement r 2 ( t ) . The pair correlation function g ( r ) is used to produce VACF for molten metals in the current study with the help of three different types of structure factor methods i.e. hard sphere Yukawa (HSY) method, optimised random phase approximation (ORPA) method and charged hard sphere (CHS) method. The presently obtained results for molten metals of different groups like Na (Z = 1), Mg (Z = 2), Al (Z = 3), Sn (Z = 4), and Sb (Z = 5) are compared with the molecular dynamic (MD) study and also with the other theoretical data available in the literature. For the present study, our own developed pseudopotential has been used for the calculations of the above-mentioned properties. The screening effect is also been observed over and beyond the bare ion potential with Hartree (H) and the effects of exchanges and correlation were computed using procedures given by Taylor (T) and Nagy (N).

Augmentation of the Optimised random phase approximation by use of mayer function as a new perturbation theory of simple fluids

ABSTRACT The Optimised Random Phase Approximation in the thermodynamic perturbation theory of liquids is augmented by use of Mayer function. The direct correlation function of a hardcore potential is assumed to be represented by A Mayer function instead of THE one used in the Random Phase Approximation and Mean Spherical Approximation. Correcting the unphysical behaviour of the corresponding pair correlation function in the hardcore region will give rise to a new perturbation theory as a natural generalisation of the Optimised Random Phase Approximation. To assess its accuracy and reliability, the theory is used to find the structure and thermodynamics of hard-core Yukawa fluids. The new-resulted theory is shown to be more accurate than the Optimised Random Phase Approximation both in predicting the thermodynamic and structure functions of hard-core Yukawa potentials.

Variants of the optimised random phase approximation: from one direct correlation function to many different radial distribution functions

It is shown that there are an infinite number of radial distribution functions (RDFs), corresponding to only one direct correlation function (DCF) of the optimised random phase approximation (ORPA). This observation in the thermodynamic perturbation theory is in sharp contrast to that of integral equation theories in which they uniquely correspond. By devising a new method we will be able to introduce various perturbation theories of simple liquids all coming from one DCF. Among all, we will only present analytically seven variants of the ORPA in the thermodynamic perturbation theory of liquids. The DCF of hard-core potential for all variants is assumed to be the same as the ORPA. However, interestingly enough the resulted expressions for the Helmholtz free energies and the RDF are obtained very differently. Furthermore, the resulted thermodynamic properties come out somehow the same, whereas the structural functions of some variants are found to behave much better than the standard ORPA.

Phase behaviour in ionic solutions: Restricted primitive model of ionic liquid in explicit neutral solvent

We study fluid-fluid equilibrium in the simplest model of ionic solutions where the solvent is explicitly included, i.e., a binary mixture consisting of a restricted primitive model (RPM) and neutral hard-spheres (RPM-HS mixture). First, using the collective variable method we find free energy, pressure and partial chemical potentials in the random phase approximation (RPA) for a rather general model that takes into consideration solvent-solvent and solvent-ion interactions beyond the hard core. In the special case of a RPM-HS mixture, we consider two regularizations of the Coulomb potential inside the hard core, i.e., the Weeks-Chandler-Andersen (WCA) regularization leading to the WCA approximation and the optimized regularization giving the optimized RPA (ORPA) or the mean spherical approximation (MSA). Furthermore, we calculate the phase coexistence using the associative mean spherical approximation (AMSA). In general, the three approximations produce qualitatively similar phase diagrams of the RPM-HS mixture, i.e., a fluid-fluid coexistence envelope with an upper critical solution point, a shift of the coexistence region towards higher total number densities and higher solvent concentrations with increasing pressure, and a small increase of the critical temperature with an increase of pressure. As for a pure RPM, the AMSA leads to the best agreement with the available simulation data when the association constant proposed by Olaussen and Stell is used. We also discuss the peculiarities of the phase diagrams in the WCA approximation.

Structural and thermodynamic properties of argon liquid as a square-well fluid based on a modification of the ORPA theory

The structural and thermodynamic properties of the argon fluid were investigated by means of a modification of the optimised random phase approximation (ORPA) theory and the square-well (SW) potential. The results were obtained at the SW potential values 1.25 < λ < 2, and along different isotherms at a wide range of densities. The radial distribution function, g(r), was calculated, and the results obtained were extrapolated to give the appropriate contact values g(σ), g(λσ −) and g(λσ +). The results were compared with the simulation and experimental data available in the literature. The results calculated for the Helmholtz free energy (A) and internal energy (E) are in good agreement with the experimental data at the sub- and super-critical temperatures. The ORPA version considered here provided good results for the compressibility factor Z from the Virial route with the SW attractive range λ = 1.7 at high temperatures and low densities.

Determination of the structure factor for the rare gas fluids based on the ORPA theory and LIR equation of state

Linear isotherm regularity is applied to generate an expression for the direct correlation function (DCF), based on the optimized random-phase approximation theory. We use the Lennard-Jones potential in the modelling of real fluids, in which its molecular parameters are state-dependent. Based on the perturbation theory, we assume that the core contribution of the DCF is related to the geometric effect via a linear form, which arises from the excluded volume, and the tail contribution is related to the long-range intermolecular interactions of the system via a non-linear form. We use this model to predict the behaviour of the structure factor, S(k), in a wide range of k values for the xenon and krypton liquids. Finally, we compare our results with the experimental and theoretical data available in literature. The model is also successful in the presentation of the Ornstein–Zernike behaviour of S(k) at the low-k region.

Optmizied random phase approximation for the phase diagram of C&lt;sub&gt;60&lt;/sub&gt; material

This paper determines the phase diagram of C60 fluid by an efficient and robust optimized random phase approximation (ORPA) method of Pastore et. al (1995), imposes physical requirements as in the original ORPA scheme with a view to achieving consistency within the liquid structure factor. Our perturbation/variational approach for the Helmholtz free energy of the C60 molecules is based on the Lennard-Jones intermolecular interaction. We observe that higher accuracy is obtainable by treating all the grid points within the exclusion hole of the pair distribution function as independent variables. Our numerical results show appreciable improvement in both the thermodynamic functions and the structure factor. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 21-26

MODELING OF ADSORPTION IN PORES BY MEANS OF THIRD ORDER + SECOND ORDER PERTURBATION DENSITY FUNCTIONAL THEORY AND MONTE CARLO SIMULATION

Results for the inhomogeneous structure of the hard-core repulsive Yukawa (HCRY) fluid and of the hard-core attractive Yukawa (HCAY) fluid in planar and in spherical micropores are presented. The density profiles are obtained by the recently proposed third order + second order perturbation density functional approximation (DFA) and compared with the results of open ensemble Monte Carlo simulation. As known from other recent studies, the reliability of the DFA theory considered in this work is very sensitive to the accuracy of the required bulk direct correlation function (DCF) obtained by the solution of the Ornstein–Zernike (OZ) equation combined by a suitable closure relation. Comparison between the DFA results and simulation data shows that for the HCRY fluid, the DFA theory utilizing the DCF obtained by the OZ equation supplemented by Malijevsky–Labik approximation satisfies the required accuracy for a broad range of conditions. The results for the nonuniform HCAY fluid obtained in our previous work via the mean spherical approximation (MSA)/OZ DCF showed larger disagreement in some cases. In addition, the MSA/OZ equation for the HCAY model failed to have a physical solution at subcritical temperatures when approaching the vapor–liquid coexistence curve. For this reason, an improved version of the theory incorporating nonlinear optimized random phase approximation (ORPA) in the OZ equation for the RDF calculation of the HCAY fluid is applied. This leads to much better agreement of the DFA predictions with the simulation data. In addition, the calculations can also be performed at the conditions at which the MSA/OZ equation has no physical solution.

Gas–liquid critical point in ionic fluids

Based on the method of collective variables, we develop thestatistical field theory for the study of a simple charge-asymmetric1:z primitive model (SPM). It is shown that the well-known approximations for the free energy,in particular the DH limiting law (DHLL) and the optimized random phase approximation(ORPA), can be obtained within the framework of this theory. In order to study the gas–liquidcritical point of the SPM we propose a method for the calculation of chemical potentialconjugate to the total number density which allows us to take into account the higher-orderfluctuation effects. As a result, the gas–liquid phase diagrams are calculated forz = 2–4. The results demonstrate a qualitative agreement with Monte Carlo(MC) simulation data: the critical temperature decreases whenz increases and the critical density increases rapidly withz.

A simple approximation for fluids with narrow attractive potentials

A simple modification of the optimized random phase approximation (ORPA) has been made, aimed at improving the performance of the theory for interactions with a narrow attractive well by taking into account contributions to the direct correlation function that are nonlinear with respect to the interaction. The theory is applied to a hard core Yukawa and a square-well potential. Results for the equation of state, the correlations, and the critical point have been obtained for attractions within several ranges, and compared with Monte Carlo simulations. When the attractive interaction is narrow, the modified ORPA is a significant improvement over the plain one, especially with regard to the consistency between different routes to the thermodynamics, the two-body correlation function, and the critical temperature. However, although the spinodal curve of the modified theory is accessible, the liquid-vapour coexistence curve is not. A possible strategy to overcome this drawback is suggested.

Thermodynamics and Structure of Liquid Metals from a New Consistent Optimized Random Phase Approximation

Abstract We study thermodynamics and structural properties of several liquid metals to assess the validity of the generalized non-local model potential (GNMP) of Li et al. [J. Phys. F16, 309 (1986)]. By using a new thermodynamically consistent version of the optimized random phase approx­ imation (ORPA), especially adapted to continuous reference potentials, we improve our previous results obtained within the variational approach based on the Gibbs -Bogoliubov inequality. Hinging on the unified and very accurate evaluation of structure factors and thermodynamic quantities provided by the ORPA, we find that the GNMP yields satisfactory results for the alkali metals. Those for the polyvalent metals, however, point to a substantial inadequacy of the GNMP for high valence systems.

Structure and thermodynamic properties of a binary liquid in a porous matrix: the formalism

Using the replica trick we derive a formalism to describe the structure and the thermodynamic properties of a binary liquid in equilibrium with a porous medium. We present the replica Ornstein-Zernike equations for the general case of a k-component liquid inside a porous matrix; besides the usual liquid-state closure relations, we consider in particular the optimized random phase approximation (ORPA) restricting ourselves at present to hard-core potentials exclusively. We present furthermore several thermodynamic relations: the Gibbs-Duhem equation, the compressibility, and the viral equation. Within the framework of the ORPA (mean spherical approximation), closed expressions for the perturbation contribution to the free energy and the chemical potentials can be presented. Finally, we offer suggestions for numerical implementations.

Structure and thermodynamics of square-well and square-shoulder fluids

We have reinvestigated the structural and thermodynamic properties of the square-well and the square-shoulder system using two different theoretical frameworks, i.e., the optimized random-phase approximation and the Rogers-Young integral equation. We discuss the limits of applicability of the respective concepts and compare the results with `exact' Monte Carlo simulation results and with data obtained from a semi-analytic method proposed by Nezbeda for narrow wells. Using these correlation functions, we study in the framework of the modified-weighted-density approximation the isostructural solid-solid transition predicted for narrow wells and narrow shoulders.

Thermodynamic perturbation theory for polydisperse colloidal suspensions using orthogonal polynomial expansions.

We present a method for calculating the thermodynamic and structural properties of a polydisperse liquid by means of a thermodynamic perturbation theory: the optimized random phase approximation (ORPA). The approach is an extension of a method proposed recently by one of us for an integral equation application [Phys. Rev. E 54, 4411 (1996)]. The method is based on expansions of all sigma-dependent functions in the orthogonal polynomials p(i)(sigma) associated with the weight function f(Sigma)(sigma), where sigma is a random variable (in our case the size of the particles) with distribution f(Sigma)(sigma). As in the one-component or general N-component case, one can show that the solution of the ORPA is equivalent to the minimization of a suitably chosen functional with respect to variations of the direct correlation functions. To illustrate the method, we study a polydisperse system of square-well particles; extension to other hard-core or soft-core systems is straightforward.

Optimized random-phase approximations for arbitrary reference systems: Extremum conditions and thermodynamic consistence

The optimized random-phase approximation (ORPA) for classical liquids is reexamined in the framework of the generating functional approach to the integral equations. We show that the two main variants of the approximation correspond to the addition of the same correction to two different first order approximations of the homogeneous liquid free energy. Furthermore, we show that it is possible to consistently use the ORPA with arbitrary reference systems described by continuous potentials and that the same approximation is equivalent to a particular extremum condition for the corresponding generating functional. Finally, it is possible to enforce the thermodynamic consistence between the thermal and the virial route to the equation of state by requiring the global extremum condition on the generating functional.

Phase diagrams of single-component fluids in disordered porous materials: Predictions from integral-equation theory

We present the calculation of phase diagrams for fluids in disordered porous materials using theories based on the replica symmetric Ornstein–Zernike equations. We consider molecular models in which the porous medium is described by quenched disordered configurations of spheres and the fluid-fluid and matrix intermolecular potentials are the sum of a hard-sphere core and an attractive tail. Such models account for the combined effect of confinement, wetting, and disorder that are expected to be important to describe recent experimental observations. We use the replica method to derive the expressions relating the thermodynamic properties of the fluid inside the porous material to the pair distribution functions within the mean-spherical approximation and the optimized random-phase approximation (ORPA). We also consider higher-order corrections within the optimized cluster theory developed by Andersen and Chandler for bulk fluids. In most cases a vapor–liquid coexistence curve, similar to that observed for the bulk fluid, although displaced and somewhat narrowed, is obtained. The improved ORPA+B2/EXP approximation also predicts the appearance of a second fluid–fluid phase transition at a lower temperature.

Numerical solution of the optimized random phase approximation

An accurate, efficient and robust numerical method for the solution of the optimized random phase approximation (ORPA) of classical liquids is presented. The uniqueness of the solution of the ORPA is proved rigorously. The method, hinging on the characterization of the generating functional, improves significantly on previous algorithms. Higher accuracy is obtained by using the values of the unknown functions on the grid points as independent variables instead of the usual coefficients of an expansion in orthogonal polynomials. It is shown that minimizing a suitably modified functional with a conjugate-gradient algorithm results in a very efficient and robust algorithm.

Variational random phase approximation using the one-component-plasma reference

Abstract We propose a simplified variational random phase approximation (VRPA) for the structure factor of liquid alkali metals using the one-component-plasma (OCP) reference. Our formulation combines the standard random phase approximation (RPA) for the structure factor with the variational form of the mean spherical approximation (MSA) for the reference system. Our formula is similar to the optimized random phase approximation (ORPA), but can avoid the introduction of the fictitious hard core to the reference system. Application of the method to liquid Na and K indicates that the accuracy of this method is comparable with the ORPA using the hard sphere reference.