Perturbation Theory Of Liquids Research Articles

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.

The evaluation of some integrals that arise in the perturbation theory of liquids

ABSTRACTThe venerable, but still very useful, perturbation theory of liquids of Barker and Henderson involves some integrals that must be obtained numerically. There is no avoiding this for potentials such as the Lennard–Jones potential. However, to obtain the pressure, Barker and Henderson also obtained the necessary density derivatives numerically. This is cumbersome and care must be used to avoid excessive error. In this paper, it is noted that the earlier and less successful perturbation theory of Frisch et al. employed an alternative method, all but forgotten, for the evaluation of similar integrals. Their method still requires numerical integration. However, it allows an investigator to obtain the integrals for the pressure by analytical differentiation. The resultant integrals must still be evaluated numerically, but the extra step and annoyance of numerical differentiation is avoided. It is noted here that this method of using density derivatives can be incorporated into the Barker–Henderson theory.

A simple derivation of random phase approximation expression in perturbation theory of fluids and its nonlinear generalization

A new simple method without reference to graph theoretical concepts is devised to derive the random phase approximation expression and its nonlinear generalization in the context of the perturbation theory of liquids.

Perturbation theory for liquids with a hard sphere plus square well potential

Numerical data have been obtained for a series of perturbation theory in the form of Barker-Henderson discrete representation. A hard sphere liquid (864 particles) was modeled by the Monte Carlo method for 106 values of occupancy from the range η = 0.005–0.530 (step 0.005); the well width was varied from 0 to 1.5 hard sphere diameters. For each occupancy, averaging was done over 60·106 ensembles. Approximating analytical expressions are given as functions of well density and width for the first three terms of the free energy decomposition. For reduced temperature T* = 1.0, the standard error in the free energy calculation (three orders of perturbation theory if the above expressions are used) is of the order of ±0.001. Analysis of four-particle correlations revealed peaks at distances of 1.35 A and 1.65 A (η = 0.53) in the high-density region of hard sphere liquids, which are absent in crystals and dense liquids. It is analyzed whether the results are useful for realization of WCA theory of simple liquids in second order perturbation theory.

An one parameter model to predict shear viscosity of dense fluids

In this work we are proposing a new density correction to correlate and to predict shear viscosity of dense fluids. The shear viscosity is calculated by means of the smooth hard sphere theory that employs a Chapman–Enskog equation corrected to high densities coupled with an effective hard sphere diameter from perturbation theory of liquids. The corrections for density effects were generated using molecular dynamics (MD) data recently published in the literature for very large systems in the whole range of density. The calculations involved simple as well as polyatomic fluids including refrigerants. According to our results there is strong evidence that the coupling parameter of the rough hard sphere theory does not play a role in correlating and predicting shear viscosity even for polyatomic fluids. Our model employed very simple algebraic expressions with one adjustable parameter and worked fairly in the sub and supercritical region. Also we studied the role of different effective hard sphere diameters in the shear viscosity. Comparisons were made with a recent model published in the literature and our results showed that at high densities the new density correction play a very important role.

Theoretical scheme for the study of thermodynamic and transport properties of simple fluids at the liquid-glass transition line.

ABSTRACTIn this work we present a theoretical scheme to study the thermodynamic and transport properties of simple fluids at the liquid-glass transition line. This scheme makes use of a recent reformulation of the classical perturbation theory of liquids [M. Robles and M. López de Haro, Phys. Chem. Chem. Phys. 3, 5528 (2001)]. Using the hard-sphere fluid as a reference system our approach requires the choice of an equation of state for the hard-sphere system and a criterion to determine an effective (density and temperature dependent) diameter. Selecting the diameter in the same way as in the Mansoori-Canfield /Rasaiah-Stell variational perturbation theory and two different equations of state for the hard-sphere system, the liquid-glass transition line in the density vs. temperature plane for a Lennard-Jones fluid derived with our approach is shown to be in very good agreement with recent numerical simulations. The transition line in the pressure vs density plane and the value of the transport coefficients in the vicinity of the liquid-glass transition for the Lennard-Jones fluid are also examined.

Adsorption isotherm of a Lennard-Jones nitrogen in a carbon slitlike pore

A density functional perturbation approximation based both on second-order perturbation theory and on the pore average density has been proposed to study the adsorption hysteresis of nitrogen in a carbon slit pore. The main advantage of the present approximation is that it is computationally much simpler than the original density functional approximation based on the second-order perturbation theory of liquids, and can be applied to several model fluids confined in a strong external field in order to study their structural and thermodynamic properties. The calculated adsorption hysteresis for the confined Lennard-Jones nitrogen is in very good agreement with computer simulation, even if its accuracy slightly deteriorates for the desorption branch. The calculated equilibrium particle density distributions also compare well with computer simulations, and are better than those of a density functional theory based on the so-called mean-field approximation.

Predicting diffusivities in dense fluid mixtures

In this work the Enskog solution of the Boltzmann equation, as corrected by Speedy, together with the Weeks-Chandler-Andersen (WCA) perturbation theory of liquids is employed in correlating and predicting self-diffusivities of dense fluids. Afterwards this theory is used to estimate mutual diffusion coefficients of solutes at infinite dilution in sub and supercritical solvents. We have also investigated the behavior of Fick diffusion coefficients in the proximity of a binary vapor-liquid critical point since this subject is of great interest for extraction purposes. The approach presented here, which makes use of a density and temperature dependent hard-sphere diameter, is shown to be excellent for predicting diffusivities in dense pure fluids and fluid mixtures. The calculations involved highly nonideal mixtures as well as systems with high molecular asymmetry. The predicted diffusivities are in good agreement with the experimental data for the pure and binary systems. The methodology proposed here makes only use of pure component information and density of mixtures. The simple algebraic relations are proposed without any binary adjustable parameters and can be readily used for estimating diffusivities in multicomponent mixtures.

Simple systems with unusual phase diagrams

Abstract A theoretical study, based on the density functional formalism, is presented to investigate different aspects related to the phase diagram of simple systems. The approximation is a perturbation theory for non-uniform systems which in the uniform limit reduces consistently to the standard liquid perturbation theory. Its application to the solid phase requires the use of recently developed approximations for the correlation function of this phase. The theory is summarized and applied to a model of a mixture of colloidal particles and non adsorbing polymer which, under certain circumstances, can be regarded as a simple system. The predictions of the theory are in good agreement with computer simulations, even quantitatively. The role of the range of the attractive interaction between particles in determining the topology of the phase diagram is discussed. Special attention is paid to two issues: The stability of the liquid phase and the recently reported isostructural solid-solid transition in simple systems.

An evaluation of density corrections for estimating diffusivities in liquids and liquid mixtures

In this work we analyse the effect of density correction on the estimation of diffusion coefficients in liquids and liquid mixtures using the Carnahan-Starling (1969, J. Chem. Phys. 51, 635–636) pair correlation function and the correlation of Speedy (1987, Molec. Phys. 62, 509–515) and Harris (1992, Molec. Phys. 77, 1153–1167) which have been proposed as models of self-diffusion coefficient of hard-sphere fluids. The hard-sphere diameters of nine liquids were estimated by fitting the experimental self-diffusion coefficients with the smooth-hard-sphere theory and the estimated diameters were used for predicting diffusivities in 13 binary and eight ternary systems. This theory with the density- and temperature-dependent hard-sphere diameter obtained from the Weeks-Chandler-Andersen (Weeks et al., 1971, J. Chem. Phys. 54, 5237–5247) perturbation theory of liquids is shown to be an excellent approach for predicting diffusivities in liquids and liquid mixtures. The calculations involved nonideal mixtures as well as systems with high molecular asymmetry. The predicted diffusivities are in good agreement with the experimental data for the binary and also for the ternary systems. The present results are much better than the estimates of Guo and Kee (1991, Chem. Engng Sci. 46, 2133–2140) for the mutual diffusion coefficient at infinite dilution for any of the density correction methods. It seems, then, that the coupling parameter of the rough hard-sphere theory has no essential role for predicting diffusion coefficients, even for large molecules, if a suitable hard-sphere diameter is used. The methodology proposed here only makes use of pure component information and density of mixtures. The simple algebraic relations proposed are without any binary adjustable parameters and can be readily used for estimating diffusivities in multicomponent liquid mixtures.

On some properties of the hard-sphere radial distribution function

We present closed-form expressions for derivatives at contact of the hard-sphere radial distribution function (RDF) in the Percus-Yevick theory. The derivatives of lower order coincide with well-known results. Corresponding expressions are also derived using the Verlet-Weis algorithm. These results are compared with values extracted from recent Monte Carlo data on the HS RDF and an accurate expression for the first-order derivative is given. Expressions for moments of the HS correlation function are also obtained in the Percus-Yevick theory. All of these quantities are useful in the perturbation theory of liquids.

Perturbation theory of liquids

Recently, Abascal and co-workers have developed an extended perturbation theory for diatomic liquids which combines the Weeks–Chandler–Andersen choice of reference potential with the Barker–Henderson choice of hard sphere diameter. Here we report results for the case of atomic liquids. This theory is shown to work well for atomic liquids at moderate densities, for ρ*<0.75. At higher densities, this theory is less accurate than the first order Barker–Henderson theory. Implications for perturbation theories involving molecular liquids are discussed.

Thermodynamics of supercooled liquid simple metals and entropic estimate of their glass transition temperatures

We calculate the thermodynamic properties of supercooled liquid Na, Rb and Mg using the earlier described pseudopotential model and the Weeks-Chandler-Andersen perturbation theory for liquids. The calculated temperature dependences of the specific heat C p ( T) and of the free energy difference Δ G( T) between liquids and crystals are similar to those observed experimentally for the polyvalent metals In, Pb, Sn, Bi which can indicate the existence of a certain “law of corresponding states” for supercooled metals. The lower limit for the glass transition temperature T g is estimated as the Kauzman point at which the excess entropy of the liquid with respect to the crystal vanishes. This gives T g ⪆ 0.4T m where T m is the melting temperature.

Shock Compression of Liquid Helium to 56 GPa (560 kbar)

The first shock-compression experiments on liquid helium are reported. With a two-stage light-gas gun, liquid He at 4.3 K and 1 atm was shocked to 16 GPa and 12 000 K and double shocked to 56 GPa and 21 000 K. Liquid perturbation theory has been used to determine an effective interatomic potential from which the equation of state of He can be obtained over a wide range of densities and temperatures.

Percus-Yevick equation for an inhomogeneous fluid: Critical region

The critical region of a locally nonuniform fluid with gaussian density inhomogeneity is investigated. Nonclassical behaviour is found; critical exponents agree very well with those obtained from the perturbation theory of liquids. The action of an external field shifts the critical region and weakens the critical behaviour. Discussed effects could be verified experimentally.

The use of the superposition approximation in perturbation theories of liquids

For hard dumbbells, the Perram-White (PW) second order perturbation term of the free energy F (2), containing a three body correlation function, is evaluated both directly by use of Monte Carlo technique and by Kirkwood's superposition approximation (SA). It is found that the PW perturbation expansion gives much better values for the free energy if F(2) is calculated exactly. At high densities a significant error is introduced by using the SA.

Iterative pertubative theory of the critical point of classical fluids

The correlations of the Lennard-Jones fluid in the critical region have been computed on the basis of a generalized “exp” approximation. The perturbation theory of liquids is generalized by considering an infinite sequence of “reference” systems and the correlation functions g( r) and S( k) are obtained from a recursion relation. On the critical isocore we find the critical exponents γ = 150 ± 0.02 for the isothermal compressibility and ν = 0.78 ± 0.04 for the correlation length. This iterative perturbative approach improves considerably the results of the usual perturbation theories.

Heats of vaporization of nonelectrolytes

The first-order perturbation theory of liquids was used for developing an approximate expression suitable for a rapid calculation of the heat of vaporization of nonpolar liquids formed by approximately spherical molecules. The calculation uses only available physico-chemical properties of substances, namely the critical data. The results are tested by a comparison with existing experimental data on a series of nonpolar substances. The agreement between calculated values and experimental data is relatively good. The maximum deviation between theoretical and experimental values does not exceed ±10%.

A note on the perturbation theory and the triangular well potential

The integrals which appear in the first order term and the macroscopic compressibility (mc) and the local compressibility (lc) approximation for the second order term in the Barker–Henderson (BH) perturbation theory of liquids are evaluated analytically for the triangular-well potential. The compressibility factors so calculated are compared with the Monte Carlo calculations.