Radial Distribution Function Of Fluid Research Articles

Two-body perturbation theory versus first order perturbation theory: A comparison based on the square-well fluid.

We show that the Zwanzig first-order perturbation theory can be obtained directly from a truncated Taylor series expansion of a two-body perturbation theory and that such truncation provides a more accurate prediction of thermodynamic properties than the full two-body perturbation theory. This unexpected result is explained by the quality of the resulting approximation for the fluid radial distribution function. We prove that the first-order and the two-body perturbation theories are based on different approximations for the fluid radial distribution function. To illustrate the calculations, the square-well fluid is adopted. We develop an analytical expression for the two-body perturbed Helmholtz free energy for the square-well fluid. The equation of state obtained using such an expression is compared to the equation of state obtained from the first-order approximation. The vapor-liquid coexistence curve and the supercritical compressibility factor of a square-well fluid are calculated using both equations of state and compared to Monte Carlo simulation data. Finally, we show that the approximation for the fluid radial distribution function given by the first-order perturbation theory provides closer values to the ones calculated via Monte Carlo simulations. This explains why such theory gives a better description of the fluid thermodynamic behavior.

Mean spherical approximation for the Yukawa fluid radial distribution function

Waisman has obtained the direct correlation function (DCF) for the Yukawa fluid using the mean spherical approximation (MSA). Although his result has been used to obtain the thermodynamic functions of this fluid, the resultant radial distribution function (RDF) has not been calculated. In this study, the RDF is calculated by obtaining the Laplace transforms of the DCF and RDF of this fluid by analytic integration. These Laplace transforms are then converted to Fourier transforms that can be inverted numerically. Results are reported for a value of the decay parameter (z = 1.8) that mimics a Lennard-Jones (LJ) 12–6 fluid and for z = 5, which has been used in perturbation theory. For the cases studied, the RDF at contact is greater for the Yukawa fluid than for the hard-sphere fluid and increases monotonically with decreasing temperature. At low densities, the difference between the hard-sphere and LJ-like Yukawa RDFs is appreciable. However, at high densities this difference is much smaller. The dependence on temperature is stronger for z = 5 than for z = 1.8. A comparison of the MSA results with simulation results is made. The agreement of the MSA results with simulation is good except near contact where the MSA contact value always lies below the MC result.

Effect of cut-off distance used in molecular dynamics simulations on fluid properties

The effect of cut-off distance used in molecular dynamics (MD) simulations on fluid properties was studied systematically in both canonical (NVT) and isothermal–isobaric (NPT) ensembles. Results show that the cut-off distance in the NVT ensemble plays little role in determining the equilibrium structure of fluid if the ensemble has a high density. However, pressures calculated in the same NVT ensembles strongly depend on the cut-off distance used. In the NPT ensemble, cut-off distance plays a key role in determining fluid equilibrium structure, density and self-diffusion coefficient. The characteristic of the radial distribution function of fluid in NPT ensembles depending on the cut-off distance used in MD simulations means that the WCA theory (a perturbation theory developed by Weeks, Chandler and Andersen) is not suitable for NPT ensembles because the assumption (the effect of the attractive force in determining the liquid structure is negligible) used in the WCA theory is not valid. The dependence of fluid properties on the cut-off distance also indicates that using the WCA potential (the repulsive part of the intermolecular potential proposed in the WCA theory) to calculate fluid transport in heterogeneous systems could lead to significant errors or incorrect results.

Universality principle and the development of classical density functional theory

The universality principle of the free energy density functional and the `test particle' trick by Percus are combined to construct the approximate free energy density functional or its functional derivative. Information about the bulk fluid radial distribution function is integrated into the density functional approximation directly for the first time in the present methodology. The physical foundation of the present methodology also applies to the quantum density functional theory.

Density Functional Approach Based on Numerically Obtained Bridge Functional**The project supported by the Natural Science Foundation of Hunan Province under Grant No.01JJY3007 and Natural Science Foundation of Education Department of Hunan Province of China under Grant No.01C338

The Ornstein–Zernike equation is solved with the Rogers–Young approximation for bulk hard sphere fluid and Lennard–Jones fluid for several state points. Then the resulted bulk fluid radial distribution function combined with the test particle method is employed to determine numerically the function relationship of bridge functional as a function of indirect correlation function. It is found that all of the calculated points from different phase space state points for a same type of fluid collapse onto a same smooth curve. Then the numerically obtained curve is used to substitute the analytic expression of the bridge functional as a function of indirect correlation function required in the methodology [J. Chem. Phys. 112 (2000) 8079] to determine the density distribution of non-uniform hard sphere fluid and Lennard–Jones fluid. The good agreement of theoretical predictions with the computer simulation data is obtained. The present numerical procedure incorporates the knowledge of bulk fluid radial distribution function into the constructing of the density functional approximation and makes the original methodology more accurate and more flexible for various interaction potential fluid.

A simplified analytical expression for the first shell of the hard-sphere fluid radial distribution function

An analytical expression for the first coordination cell of the radial distribution function (RDF) of the hard-sphere fluid is derived. It is based on a series expansion of the analytical expression of the Percus–Yevick solution of the RDF of the hard-sphere fluid derived by Chang and Sandler [J. Chang, S.I. Sandler, Mol. Phys. 81 (1994) 745.]. The expansion is carried out both in terms of the packing fraction and in terms of the radial distance. Expressions are also obtained for the coordination number and its first and second derivatives as functions of radial distance and packing fraction. These expressions, which are useful in perturbation theory, are simpler to use than those obtained from the starting equation, while giving nearly the same results. They also provide close agreement with simulation data.

Density expansion (DEX) mixing rules: Thermodynamic modeling of supercritical extraction

Conformal solution theory and the density expansion expression of the radial distribution function of fluids are used to derive a set of mixing rules. The new mixing rules are composition, density, and temperature dependent. To test the new mixing rules they are used for thermodynamic modeling of supercritical extraction. Comparison of the result of calculation by the mixing rules with the van der Waals mixing rules indicates a profound improvement over the latter in prediction of properties of mixtures consisting of species with large molecular size and shape differences.

Multiparameter Conformal Solution Theory — Application to Hydrocarbon Pure Fluid and Mixture Thermodynamic Properties

AbstractThe basis for a multiparameter conformal solution theory which offers potential for applications to wide classes of fluid mixtures is discussed. The concept of conformality is invoked for functionals which are dimensionless integrals over the reference fluid radial distribution function in the Pople type of perturbation expansion rather than the usual requirement of conformality of reduced properties. For mixtures, a van der Waals one fluid type model is used to express mixture properties as hypothetical pure fluid properties with intermolecular potential parameters expressed as functions of the component potential parameters and composition. In addition to the usual separation and energy parameters, the intermolecular potential can include dipole, quadrupole, octupole and overlap potential parameters. A simplified version of the method yields accurate predictions for light hydrocarbon thermodynamic properties.

Perturbation theory and the radial distribution function of fluids with nonspherical potentials

A perturbation expansion for the radial distribution function of fluids with nonspherical pair potentials is discussed. Numerical calculations are presented for dipolar hard-sphere and quadrupolar hard-sphere fluids. Agreement with the computer simulation results is excellent, even at high dipole and quadrupole moments.

New Types of Integral Equations for Obtaining the Radial Distribution Function of Fluids

New Types of Integral Equations for Obtaining the Radial Distribution Function of Fluids

Radial Distribution Function of Fluids with the Square-Well Potential

The term χ(r), the bridge diagram in g2(r) which is proportional to the square of the density in the expansion of the radial distribution function in powers of the density, is evaluated for fluids with square-well potential. The contribution of χ(r) to g2(r) is discussed.

Radial Distribution Functions from a Linear Integral Equation

A linear integral equation derived by Mayer for the radial distribution function of fluids has been investigated numerically. To compare with other recent investigations of integral equations, a Lennard-Jones 6–12 potential was used and the 55°C isotherm of argon investigated. Results for the pressure and the radial distribution function are less satisfactory than are obtained with other integral equations. An estimate of the validity of Kirkwood's superposition approximation in the range of the calculation was also made.

Radial Distribution Function of Fluids, III

A formula for the Helmholtz free energy of a one-component fluid, classical or quantum, is obtained in terms of the radial distribution function, by attaching a coupling contant to the pair interaction potential. Compatibility of the formula is discussed in connection with the formula derived in paper II, leading to the condition to be satisfied by the radial distribution function. It is shown that the condition for compatibility is equivalent in a classical fluid to the one discussed in paper I. The present formula is applied to deal with the hyper-netted chain approximation and the condition for compatibility is shown to be satisfied in this case. The formula is generalized to a multicomponent fluid and also to the case of a grand canonical ensemble.

Radial Distribution Function of Fluids II

It is shown that the radial distribution function of fluid can be derived from the expression for the free energy in the form of a “functional derivative” with respect to the pair interaction potential. Three examples are treated on the line of the theory: (1) the simple chain approximation of classical fluid, (2) the second quantization and (3) the lowest state of the spinless bosons derived by Bogolyubov and Zubarev. A procedure, which differs from the usual one, is given in the derivation of the expressions by which the pressure and the internal energy are expressed in terms of the radial distribution function.

Radial Distribution Function of Fluids I

The consistency of several approximate radial distribution functions of fluids is examined in the sense that the pressure and the internal energy derived from them satisfy the thermodynamical relation \(\frac{\partial}{\partial T}\left(\frac{p}{T}\right){=}\frac{\partial}{\partial V}\left(\frac{E}{T^{2}}\right)\), where T , p , V and E have the usual meanings. It is found that the original form of Green's linear theory is the only one which satisfies the above relation. It is also shown that Green's theory can be improved further without breaking the above relation.