Phenomenological Models for the Generic van der Waals Equation of State and Critical Parameters

A new model approach for the near-critical point region: 1. Construction of the generalized van der Waals equation of state

To date, it has been considered that all classical equations of state (EOS) have failed to describe the properties of fluids near the critical region, where the density fluctuations have a significant influence on fluid properties. In this paper, we suggest a newly constructed equation for fluid states, the generalized van der Waals (GvdW) EOS with the highly simplified Dieterici's form P = [RT/(V - b)] - a(b/V)c by a new model potential construction describing intermolecular interactions. On the basis of the model potential construction, it is shown that the a, b, and c parameters have physical interpretations as an internal pressure, a void volume, and a dimensionless value that represents an inharmonic intermolecular cell potential, respectively. As an illustration of our model approach, we initially apply it to near the critical point (cp) region, where all classical EOS descriptions have been incorporated with experimental thermodynamic data, and we obtain a table of three parameters for 12 pure normal fluids, which precisely describes thermodynamic critical values. On the basis of the basic relations between pressure and volume at the critical point, we express the corresponding EOS in terms of the c parameter, and by this means, we also obtain a theoretical vapor-liquid equilibrium (VLE) line, which closely coincides with the experimental data for several pure normal fluids near the critical region. As a result, we show that thermodynamic properties near the critical region can be described analytically by only three parameters. In addition, to validate our EOS for the temperature-differential derivatives, we show that the calculated isochoric heat capacity (Cv) of saturated argon closely coincides with the experimental data. Moreover, the possibility of a precise description with respect to the entire fluid region is also argued, in terms of the physical cases from the triple point to the ideal gas region.

A Perturbation Method for the Ornstein−Zernike Equation and the Generic van der Waals Equation of State for a Square Well Potential Model

We calculate the generic van der Waals parameters A and B for a square well model by means of a perturbation theory. To calculate the pair distribution function or the cavity function necessary for the calculation of A and B, we have used the Percus-Yevick integral equation, which is put into an equivalent form by means of the Wiener-Hopf method. This latter method produces a pair of integral equations, which are solved by a perturbation method treating the Mayer function or the well width or the functions in the square well region exterior to the hard core as the perturbation. In the end, the Mayer function times the well width is identified as the perturbation parameter in the present method. In this sense, the present perturbation method is distinct from the existing thermodynamic perturbation theory, which expands the Helmholtz free energy in a perturbation series with the inverse temperature treated as an expansion parameter. The generic van der Waals parameters are explicitly calculated in analytic form as functions of reduced temperature and density. The van der Waals parameters are recovered from them in the limits of vanishing density and high temperature. The equation of state thus obtained is tested against Monte Carlo simulation results and found reliable, provided that the temperature is in the supercritical regime. By scaling the packing fraction with a temperature-dependent hard core, it is suggested to construct an equation of state for fluids with a temperature-dependent hard core that mimicks a soft core repulsive force on the basis of the equation of state derived for the square well model.

Excluded volume in the generic van der Waals equation of state and the self-diffusion coefficient of the Lennard-Jones fluid

In the previous papers applying the generic van der Waals equation of state the mean excluded volume was defined with the contact diameter of particles at which the potential energy is equal to zero-the size parameter in the case of the Lennard-Jones potential. This parameter appears as the upper limit of the integral for the generic van der Waals parameter B (mean excluded volume divided by the density) in the generic van der Waals equation of state. Since the choice is not unique, in this paper we reexamine the manner of defining the upper limit and propose another choice for the upper limit. We also propose an interpretation of the free volume overlap factor alpha appearing in the free volume theory of diffusion and a method of estimating it in terms of the intermolecular potential energy only. It is shown that with the so-estimated free volume overlap factor and the new choice of the upper limit of the integral for B the self-diffusion coefficient in the modified free volume theory of diffusion not only acquires a better accuracy than before, but also becomes calculable in terms of only the intermolecular interaction potential without an adjustable parameter. We also assess some of effective diameters of molecules proposed in the literature for their ability to predict the self-diffusion coefficient within the framework of the modified free volume theory of diffusion.

Molecular theory of thermal conductivity of the Lennard-Jones fluid

In this paper the thermal conductivity of the Lennard-Jones fluid is calculated by applying the combination of the density-fluctuation theory, the modified free volume theory of diffusion, and the generic van der Waals equation of state. A Monte Carlo simulation method is used to compute the equilibrium pair-correlation function necessary for computing the mean free volume and the coefficient in the potential-energy and virial contributions to the thermal conductivity. The theoretical results are compared with our own molecular dynamics simulation results and with those reported in the literature. They agree in good accuracy over wide ranges of density and temperature examined in molecular dynamics simulations. Thus the combined theory represents a molecular theory of thermal conductivity of the Lennard-Jones fluid and by extension simple fluids, which enables us to compute the nonequilibrium quantity by means of the Monte Carlo simulations for the equilibrium pair-correlation function.

Relations between Transport Coefficients and Their Density and Temperature Dependence

Nonequilibrium statistical mechanics via density fluctuation theory predicts relations between the bulk and shear viscosity, thermal conductivity, and self-diffusion coefficient of a fluid. In this Feature Article, we discuss such relations holding for fluids over wide ranges of density and temperature experimentally studied in the laboratory. It is discussed how such relations can be used to successfully compute the density and temperature dependence on the basis of intermolecular interaction potential models with the help of the modified free volume theory and the generic van der Waals equation of state once the parameters in them are determined at a low density or at a subcritical temperature. Although some approximations have been made to derive them, they represent a reliable molecular theory of transport coefficients over the entire density and temperature ranges of fluids--namely, gases and liquids--a theory hitherto unavailable in the kinetic theory of liquids and dense gases.

Pair Correlation Functions and the Self-Diffusion Coefficient of Lennard-Jones Liquid in the Modified Free Volume Theory of Diffusion

In this paper, we apply the Matteoli-Mansoori empirical formula for the pair correlation function of simple fluids obeying the Lennard-Jones potential to calculate reduced self-diffusion coefficients on the basis of the modified free volume theory. The self-diffusion coefficient thus computed as functions of temperature and density is compared with the molecular dynamics simulation data and the self-diffusion coefficient obtained by the modified free volume theory implemented with the Monte Carlo simulation method for the pair correlation function. We show that the Matteoli-Mansoori empirical formula yields sufficiently accurate self-diffusion coefficients in the supercritical regime, provided that the minimum free volume activating diffusion is estimated with the classical turning point of binary collision at the mean relative kinetic energy 3k(B)T/2, where k(B) is the Boltzmann constant and T is the temperature. In the subcritical regime, the empirical formula yields qualitatively correct, but lower values for the self-diffusion coefficients compared with computer simulation values and those from the modified free volume theory implemented with the Monte Carlo simulations for the pair correlation function. However, with a slightly modified critical free volume, the results can be made quite acceptable.

Modified Free Volume Theory of Self-Diffusion and Molecular Theory of Shear Viscosity of Liquid Carbon Dioxide

In previous work on the density fluctuation theory of transport coefficients of liquids, it was necessary to use empirical self-diffusion coefficients to calculate the transport coefficients (e.g., shear viscosity of carbon dioxide). In this work, the necessity of empirical input of the self-diffusion coefficients in the calculation of shear viscosity is removed, and the theory is thus made a self-contained molecular theory of transport coefficients of liquids, albeit it contains an empirical parameter in the subcritical regime. The required self-diffusion coefficients of liquid carbon dioxide are calculated by using the modified free volume theory for which the generic van der Waals equation of state and Monte Carlo simulations are combined to accurately compute the mean free volume by means of statistical mechanics. They have been computed as a function of density along four different isotherms and isobars. A Lennard-Jones site-site interaction potential was used to model the molecular carbon dioxide interaction. The density and temperature dependence of the theoretical self-diffusion coefficients are shown to be in excellent agreement with experimental data when the minimum critical free volume is identified with the molecular volume. The self-diffusion coefficients thus computed are then used to compute the density and temperature dependence of the shear viscosity of liquid carbon dioxide by employing the density fluctuation theory formula for shear viscosity as reported in an earlier paper (J. Chem. Phys. 2000, 112, 7118). The theoretical shear viscosity is shown to be robust and yields excellent density and temperature dependence for carbon dioxide. The pair correlation function appearing in the theory has been computed by Monte Carlo simulations.

Generic van der Waals Equation of State, Modified Free Volume Theory of Diffusion, and Viscosity of Simple Liquids

The shear viscosity formula derived by the density fluctuation theory in previous papers is computed for argon, krypton, and methane by using the self-diffusion coefficients derived in the modified free volume theory with the help of the generic van der Waals equation of state. In the temperature regime near or above the critical temperature, the density dependence of the shear viscosity can be accounted for by ab initio calculations with the self-diffusion coefficients provided by the modified free volume theory if the minimum (critical) free volume is set equal to the molecular volume and the volume overlap parameter (alpha) is taken about unity in the expression for the self-diffusion coefficient. In the subcritical temperature regime, if the density fluctuation range parameter is chosen appropriately at a temperature, then the resulting expression for the shear viscosity can well account for its density and temperature dependence over the ranges of density and temperature experimentally studied. In the sense that once the density fluctuation range is fixed at a temperature, the theory can account for the experimental data at other subcritical temperatures on the basis of the intermolecular force only; the theory is predictive even in the subcritical regime of temperature. Theory is successfully tested in comparison with experimental data for self-diffusion coefficients and shear viscosity for argon, krypton, and methane.

Theory of the viscosity of supercooled liquids and the glass transition: fragile liquids

A statistical mechanical theory is presented for viscosity of relatively low molecular weight organic liquids which are supercooled down to the glass transition temperature. In this theory a relation resembling the Stokes-Einstein relation between the viscosity and self-diffusion coefficient of supercooled liquids and an expression for the self-diffusion coefficient are augmented by a suitably constructed semiempirical generic van der Waals equation of state that makes it possible to calculate the free volume. The theory accounts in excellent accuracy for viscosities and self-diffusion coefficients of fragile liquids over the entire range of temperature experimentally investigated. According to the theory, vitrification occurs when the free volume available for translational molecular motion falls below a critical value.