Harold J. Raveché

Entropy and Molecular Correlation Functions in Open Systems. I. Derivation

A method is presented for obtaining an expression for the entropy in terms of molecular correlation functions defined in the grand canonical ensemble. The procedure is for a system of a single molecular species whose dynamics are determined by classical equations of motion. The entropy is obtained as a sum of two different classes of functions each involving the correlations between n‐tuples of molecules. One class contains logarithmic terms similar to those obtained for the closed system; the other class involves isothermal activity derivatives of potentials of mean force. The latter terms, which are moments of the correlations between disjoint sets of molecules, can make appreciable contributions to the entropy. The method leads to results similar to those obtained from a different procedure by Nettleton and Green. The expression for the entropy is obtained and properties of the results are discussed for a simple fluid system.

Freezing and melting properties of the Lennard‐Jones system

Using Monte Carlo simulations, we investigate average molecular arrangements that occur with the fluid‐solid transition in a classical Lennard‐Jones system. The crystalline order of the solid phase is explicitly shown by the angularly averaged molecular correlation functions which, in the solid, exhibit a behavior not observed in the fluid. The pair and triplet correlation functions delineate the crystalline pattern of the ordered phase out to internuclear separations of many nearest‐neighbor distances. A molecular criterion for freezing is reported which claims a proportionality between the values of the pair correlation function in the fluid at the positions of the first and second nearest neighbors. The general behavior of the triplet correlation function in the fluid phase is interpreted. We also compare predictions for melting pressures and the densities of the coexisting fluid and solid phases.

Entropy and Molecular Correlation Functions in Open Systems. II Two‐ and Three‐Body Correlations

We consider the contributions from the correlation of pairs and triples of molecules to the configurational entropy per molecule of a simple fluid system. Using numerical values of the pair function and only certain moments of the triplet function, we have computed the first few terms in an expression for the entropy in terms of the correlation functions. The results, which compare favorably with experimental values over a range of densities including that of the liquid, indicate that contributions from the correlations of triples of molecules can be appreciable. The computations are done for a fluid of hard spheres and liquid neon and the results are discussed in terms of the molecular correlations in a simple fluid.

Towards a molecular theory of freezing

The subject of this article is the fluid–solid transition and, in particular, an analysis of crystallization in terms of quantities which describe the average local arrangements of molecules in a fluid. We determine whether it is possible to predict the existence of crystalline solutions for the local molecular density from a Hamiltonian which is invariant under all translations and rotations. Crystallization is studied using the singlet probability density, the pair correlation function, and the intermolecular potential energy. An integral equation is obtained for these quantities, and we pursue the existence of crystalline (i.e., periodic but nonconstant) solutions for the singlet probability which branch from the fluid (i.e., constant) solution which is the number density. The phenomenon of crystallization, that is, the existence and determination of these solutions, can then be represented as a nonlinear eigenvalue problem. The analysis is applied to hard sphere systems in one, two, and three dimensions. Crystallization to close‐packed lattices is found in two and three dimensions when the isotropic media are overcompressed by amounts which depend on the structures to which the fluids crystallize. That is, the fluid persists into a portion of the metastable region. The nature of the crystalline solutions is analyzed in the neighborhood of the branching eigenvalues, and the relation between these special eigenvalues and equilibrium freezing points is discussed. The stability of these crystalline solutions is determined by comparing the values of a free energylike functional on these solutions with its value for the fluid.The subject of this article is the fluid–solid transition and, in particular, an analysis of crystallization in terms of quantities which describe the average local arrangements of molecules in a fluid. We determine whether it is possible to predict the existence of crystalline solutions for the local molecular density from a Hamiltonian which is invariant under all translations and rotations. Crystallization is studied using the singlet probability density, the pair correlation function, and the intermolecular potential energy. An integral equation is obtained for these quantities, and we pursue the existence of crystalline (i.e., periodic but nonconstant) solutions for the singlet probability which branch from the fluid (i.e., constant) solution which is the number density. The phenomenon of crystallization, that is, the existence and determination of these solutions, can then be represented as a nonlinear eigenvalue problem. The analysis is applied to hard sphere systems in one, two, and three dimensio...

Three Atom Correlations in the Lennard‐Jones Fluid

The Monte Carlo method has been used to compute the triplet correlation function in a classical fluid with Lennard‐Jones interactions. The computations were performed for particular configurations at five thermodynamic states of high density. The structure of the triplet function is discussed in the liquid and dense gas regions. Several closure approximations, which express the triplet function in terms of the pair correlation function, are compared to the Monte Carlo results.

Three Atom Correlations in Liquid Neon

We consider the correlation of three atoms in liquid neon from the neutron diffraction measurement of the isothermal density derivative of the pair correlation function. Several closure approximations for the triplet correlation function are discussed. Three of the approximations are representations of the triplet correlation function as a functional of the pair correlation function, and the other is expressed as a simple function of the pair correlation function.

Towards a molecular theory of freezing: The equation of state and free energy from the first BBGKY equation

We determine crystalline solutions of a closed BBGKY equation for the local density in a classical hard sphere system. The inhomogeneous solutions, which bifurcate from the fluid phase, are applied to calculate the equation of state and free energy. A first order phase change is found in two and three dimensions and its existence is shown to depend on the direction of bifurcation and on global properties of the solutions. The results are discussed in terms of the closure.

Three‐Body Correlations in Simple Dense Fluids

Correlations between triples of molecules in simple fluids at thermodynamic equilibrium are studied through their contribution to the isothermal density derivative of the pair probability density. Explicit computations are performed to indicate the role of the triplet function in accounting for the structure of the density derivative of the radial distribution function. The results imply that contributions from triplet correlations are in general quite appreciable. Various consequences of the results are disscussed, and the procedure is examined in general as a method for studying correlations between triples of molecules in simple fluids.

Towards a molecular theory of freezing. II. Study of bifurcation as a function of density

We study, as a function of density, crystalline solutions for the single particle probability density which bifurcate from the fluid solution of a hard sphere system. The BBGKY equation is used to describe the fluid phase, with its closure taken from computer simulations. As the density increase from zero, crystalline solutions bifurcate from the fluid with a periodicity determined by the density. Bifurcation is found to be characteristic of metastable states, and in general it does not occur at the equilibrium coexistence of two phases. When compared to the known hard disk and hard sphere isotherms, the bifurcation points are seen to be remarkably consistent with the density at which the metastable extension of the crystalline branch meets the equilibrium fluid branch. Metastable crystalline states are also predicted in one dimension. The controversial Kirkwood instability criterion is interpreted in terms of our theory. We show that our analysis applies to a large class of potential energy functions. Th...

Computer studies of dynamics in one dimension: Hard rods

Results of molecular dynamics simulations are reported with emphasis on the relaxation of an initially ordered array of hard rods in one dimension. It is found that at high densities the pressure accurately approaches the exact value for the infinite system, which corresponds to a uniform fluid, before the singlet and pair configuration space distribution functions have completely relaxed to the equilibrium state. The velocity autocorrelation function is computed over a wide range of times, which includes the region where it is negative, and compared to the exact solution for the infinite system.