John A. White

Renormalization group theory for fluids

Some extensions of renormalization group methods to fluids are discussed that may facilitate the development of a general renormalization group theory for real fluids that is capable of predicting their thermodynamic properties globally, including both at the critical point and away from the critical point, from a specification solely of the microscopic interactions among the constituent molecules. The extensions include application to virial series and free energies for freely moving molecules (as contrasted with Hamiltonian methods used for fixed lattices of molecules); inclusion of contributions from fluctuations of very short wavelengths, comparable to the range of the attractive forces; and evaluation of the scale factor for fluctuation amplitudes. An approximate theory incorporating these new features is formulated and illustrated in a simple application to the thermal behavior of n‐pentane in a large extended neighborhood of its critical point.

Contribution of fluctuations to thermal properties of fluids with attractive forces of limited range: theory compared with PϱT and Cv data for argon

Abstract A renormalization group theory has been developed for condensable gases whose attractive intermolecular forces have a limited range. The theory takes into account contributions to thermal properties that come from enhancements of density fluctuations at many wavelengths above expectations based on the mean field approximation. The theory begins with an expression for the Helmholtz free energy density of a gas that has only repulsive interactions plus mean field attraction. It then uses the phase-space cell method of K.G. Wilson [Phys. Rev. B 4, 3184 (1971)] to incorporate successive contributions from fluctuating attraction associated with density fluctuations of increasingly long wavelengths. Expressions are developed and evaluated numerically for the pressure, chemical potential, compressibility ratio, orthobaric densities, vapor pressure, and specific heats above and below the critical point temperature for a model gas of soft spheres. Comparison is made with specific heats, pressures near the critical point, and densities farther from the critical point measured in argon. Agreement is found over a wide range of densities, pressures, and temperatures surrounding the critical point to an accuracy ~1–3%.

Calculation of density fluctuation contributions to thermodynamic properties of simple fluids

An equation of state, expressed in the form of a recursion relation for the Helmholtz free energy density, is presented. This equation of state represents the free energy density as an initial mean field free energy density, plus a sum of density fluctuation correction terms which serve both to dress the initial mean field equation of state into a more accurate analytic equation of state away from the critical region, and to provide the characteristic asymptotic singular behavior in the critical region. Numerical solutions of the recursion relations are described and presented as pressure isotherms and the calculated density coexistence curve. A comparison is made with simple fluid data for temperatures in the range 0.7≤T/Tc≤1.5 and densities in the range 0≤ρ/ρc≤2.5. In particular, good agreement is found near the critical point. Three critical exponents, δ, γ, and β, are calculated and found to be in good agreement with simple fluid data as well as with renormalization group results for the 3D Ising model.

Renormalization theory of nonuniversal thermal properties of fluids

A recently developed renormalization group theory for fluids can be used to calculate nonuniversal as well as universal critical point properties and also to predict the changes in pressure and other thermal properties that occur out to substantial distances away from the critical point. These features of the theory are illustrated here for a model square‐well gas. The calculated pressure, temperature, and density at the critical point and transition to noncritical behavior away from the critical point for this gas are compared with measured properties of parahydrogen reported in the literature.

LENNARD-JONES AS A MODEL FOR ARGON AND TEST OF EXTENDED RENORMALIZATION GROUP CALCULATIONS

Renormalization group (RG) procedures have been extended recently in phase-space cell approximation to predict, in addition to universal thermal properties observed asymptotically close to the gas-liquid critical point of fluids, also nonuniversal and nonasymptotic properties. This “globalized” RG theory is applied here, using a Lennard-Jones potential, to calculate the temperature, density, and pressure at the critical point of argon and to calculate pressures for a wide range of densities at temperatures close to, below, and considerably above that at the argon critical point. Choices required for the Lennard-Jones parameters and the quality of fit to experimental data suggest some of the strengths and limitations of the global RG theory.

Global renormalization calculations compared with simulations for Lennard-Jones fluid

A recently developed renormalization theory for fluids, that treats nonuniversal as well as universal thermal properties both near and to far from the critical point, is applied to a Lennard-Jones potential. Predictions of volumetric properties by the theory for Lennard-Jones particles for temperatures in the range 0.53⩽T/Tc⩽4.6 and densities 0.016⩽ρ/ρc<4.0 are compared with results of molecular dynamics simulations. The comparison suggests that the theory may be capable of making volumetric predictions for the fluid accurate to ∼1%–2% (for perpendicular distance between calculated isotherms and data points obtained by simulation) for temperatures and densities throughout much of this extended neighborhood of the critical point.

Global renormalization calculations compared with simulations for square-well fluids: Widths 3.0 and 1.5

A recently developed “global” renormalization group theory for calculating thermal properties of fluids both near and to far from the critical point is applied to molecules that interact via a square-well potential. Calculations of gas-liquid coexistence curve densities and vapor pressures are compared with results of recently reported Monte Carlo simulations for hard spheres that interact with square-well potentials of range 3σ (σ=hard core diameter) and with both Monte Carlo and molecular dynamics simulations for square-well potential of range 1.5σ. Additionally, values for the effective critical point exponent βe are calculated for both square wells as a function of temperature distance from the critical point.

Equation of state for the hard‐sphere gas

Two equations are suggested to describe the hard‐sphere system interpolating between gas and close‐packed configurations. They indicate some instability and possible phase transition in the same region as that in the Adler and Wainwright molecular dynamics simulation.

Temperature variation of the correlation length of carbon dioxide at its critical density

Abstract When light scattering intensity and Rayleigh linewidth data are combined with X-ray data of Chu et al., the correlation length exponent, v , is determined to be 0.59 ± 0.02

The equation of state for hard‐sphere systems in solid phase in arbitrary dimensions

Analytic equations of state for the solid phase of hard‐sphere systems of dimensionality d=2,3,4,5 are suggested and compared with computer simulation data. Analytic expression for the heat capacity Cp is also given.