Francis H. Ree

Melting Transition and Communal Entropy for Hard Spheres

In order to confirm the existence of a first‐order melting transition for a classical many‐body system of hard spheres and to discover the densities of the coexisting phases, we have made a Monte Carlo determination of the pressure and absolute entropy of the hard‐sphere solid. We use these solid‐phase thermodynamic properties, coupled with known fluid‐phase data, to show that the hard‐sphere solid, at a density of 0.74 relative to close packing, and the hard‐sphere fluid, at a density of 0.67 relative to close packing, satisfy the thermodynamic equilibrium conditions of equal pressure and chemical potential at constant temperature. To get the solid‐phase entropy, we integrated the Monte Carlo pressure–volume equation of state for a “single‐occupancy” system in which the center of each hard sphere was constrained to occupy its own private cell. Such a system is no different from the ordinary solid at high density, but at low density its entropy and pressure are both lower. The difference in entropy betwee...

Fifth and Sixth Virial Coefficients for Hard Spheres and Hard Disks

New expressions for the fourth, fifth, and sixth virial coefficients are obtained as sums of modified star integrals. The modified stars contain both Mayer f functions and f functions (f≡f+1). It is shown that the number of topologically distinguishable graphs occurring in the new expressions is about half the number required in previous expressions. This reduction in the number of integrals makes numerical calculation of virial coefficients simpler and more nearly accurate. For particles interacting with a hard‐core potential, values of the modified star integrals are shown to depend strongly on dimension; for example, several modified star integrals are identically zero for hard disks (two dimensions), but give nonzero values for hard spheres (three dimensions). Of all the modified star integrals contributing to the fourth, fifth, and sixth virial coefficients, the complete star integrals are shown to be the largest. Mayers expressions for these coefficients made the complete star integrals the small...

Seventh Virial Coefficients for Hard Spheres and Hard Disks

The seventh virial coefficient B7 is expressed as a sum of modified star integrals instead of the usual Mayer star integrals. The new graphs contain both Mayer f functions and f(≡1+f) functions. This simplifies the calculation of B7. That is, instead of the 468 star integrals that appear in Mayers formulation, only 171 modified star integrals now appear in the evaluation of B7. Furthermore, for D‐dimensional particles with a hard core, these 171 integrals are strongly dependent on the number of dimensions D. For hard rods, hard disks, and hard spheres, respectively, at most one, 78, and 164 of these integrals give a nonvanishing contribution to B7. When Monte Carlo integration is used to evaluate these integrals using hard‐sphere and hard‐disk potentials we obtain the following values of B7: hard spheres, B7/(B2)6=0.0138±0.0004; hard disks, B7/(B2)6=0.1141±0.0005. For hard spheres, the truncated seven‐term virial series for the pressure agrees within 10% with the results of the molecular dynamics data t...

Repulsive forces of simple molecules and mixtures at high density and temperature

Shock wave data for liquid Ar, Xe, N2, O2, CO2, CH4, and CO have been used to test the possibility that the repulsive pair potential of these fluids scale in accordance with the ’’Law of Corresponding States.’’ This is found to be approximately valid to a compression of about 2.5 times liquid density. The law has been applied to compressed mixtures of N2 and O2 and a theoretical prediction of the detonation velocity and pressure of liquid NO is found to be in excellent agreement with experiment.

The equation of state of molecular hydrogen at very high densitya)

In the preceding paper new shock‐wave results were reported on liquid hydrogen and deuterium up to a pressure of 76 GPa (760 kbar). In the present paper an effective pair potential is determined from these results and used in calculations of fluid and solid isotherms, melting curves, and the metallic transition pressure. The agreement with available data is good and the metallic transition is predicted to be near 300 GPa (3 Mbar).

A perturbation theory of classical equilibrium fluids

A new perturbation theory which is reliable over a wide fluid region is presented. The new theory reduces to the theory of Weeks, Chandler, and Anderson at densities near or below the triple point density of a simple fluid but it can also accurately predict thermodynamic properties at higher densities near the freezing line of the fluid. This is done by employing an optimized reference potential whose repulsive range decreases with increase in density. Thermodynamic properties for Lennard‐Jones, exponential‐6, and inverse nth‐power (n=12, 9, 6, and 4) potentials have been calculated from the new theory. Comparison of the calculated data with available Monte Carlo simulations and additional simulations carried out in this work shows that the theory gives excellent thermodynamic results for these systems. The present theory also gives a physically reasonable hard‐sphere diameter over the entire fluid range.

A statistical mechanical theory of chemically reacting multiphase mixtures: Application to the detonation properties of PETN

We present a new statistical mechanical theory of multiphase, multicomponent systems. It is based on Ross’s modification of the Mansoori–Canfield–Rasaiah–Stell hard‐sphere variational theory and the improved one‐fluid van der Waals mixture model. Next, the new theory and exponential‐6 potentials that accurately reproduce shock wave data of major detonation‐product species are used to compute the detonation properties of PETN (pentaerythritol tetranitrate). The results show satisfactory agreement with the experimental Chapman–Jouguet (CJ) data at initial densities ( ρ0) between 0.25 and 1.77 g/cm3. Small (≂9%) deviations which occur between the experimental and the theoretical CJ pressures at high ρ0 (>1.55 g/cm3) are attributed to time‐dependent processes associated with formation of solid carbon which is theoretically present in detonation products. We suggest that some CJ pressure experiments might have finished too soon to experimentally observe late‐time reactions involved in formation of solid carbon...

Radial Distribution Functions and Equation of State of the Hard‐Disk Fluid

We evaluated by a Monte Carlo method the hard‐disk radial distribution function at densities: ρ / ρ0 = 0.4, 0.5, and 0.6 (ρ0 is equivalent to the close‐packed density of hard disks). The results were used to find out the relative merits of solutions of the four integral equations (the Born–Green–Yvon, the modified Born–Green–Yvon, the Percus–Yevick, and the convolution–hypernetted‐chain equations) describing approximately the behavior of the radial distribution function. We also evaluated the first several density coefficients of the pressure and the radial distribution functions corresponding to these approximate equations. The Pade approximants formed by the calculated coefficients for the approximate equations were found to satisfactorily describe pressures for these equations over the range of low to medium density of the hard‐disk fluids. A simple analytic expression, Eq. (13), for the pressure which describes fairly well the dense hard‐disk fluid was also derived by using the modified Born–Green–Yvo...

Shock compression of liquid carbon monoxide and methane to 90 GPa (900 kbar)

Dynamic equation‐of‐state data for liquid CO and CH4 were measured in the shock pressure range 5–92 GPa (50–920 kbar) using a two‐stage light‐gas gun. The liquids were shocked from initial states near their saturation curves at 77 and 111 K for CO and CH4, respectively. The experimental technique used to double‐shock CH4 is described. The CO data were examined by using three theoretical models: (1) a chemically nonreactive model, (2) a quasi‐chemical‐equilibrium model that allows CO to dissociate into gaseous species and graphite, and (3) a chemical‐equilibrium model that also includes a dense carbon phase which exists at higher pressures and temperatures than graphite. This dense phase is assumed to be diamond. Our analysis shows that at low pressure chemical equilibrium takes much longer than a typical shock passage time. As a consequence, the experimental data initially follow the nonreactive Hugoniot to pressures well beyond the chemical dissociation limit. Both the experimental data and the Hugoniot computed with case (3) agree satisfactorily at high pressure. Further consequences of these observations to high‐explosive studies are discussed. The theoretical analysis for the CH4 data was presented in an earlier paper.

Radial Distribution Function of Hard Spheres

The third‐order density correction term g3(r) in the density series for the radical distribution function g(r) is evaluated for a system of hard spheres over all distances. Both the Monte Carlo method (r<2σ, σ=sphere diameter) and the analytic method (r≥2σ) are used to calculate the graph integrals appearing in g3(r). In carrying out the Monte Carlo integrations, calculations are simplified if gn(r) is represented by a sum of modified doubly rooted graph integrals instead of the Mayer—Montroll doubly rooted graphs. The modified doubly rooted graphs contain both Mayer f functions and f functions (f=f+1). Advantages of this representation are twofold; (a) reduction in the number of graph integrals appearing in gn(r), and (b) the strong dependence of the graph integrals both on distance and dimension for the hard‐core potentials. The analytic expression of g2(r) is obtained for r≤σ, by calculating the first‐order density correction term of the triplet distribution function.The above results are compared wi...