Yvon Born Green Research Articles

Radial distribution function for hard disks from the BGY2 theorya)

The Born–Green–Yvon (BGY) hierarchy of equations is truncated by using the Fisher–Kopeliovich approximation for the quadruplet distribution function. The resulting BGY2 equations are solved for hard disks at five different densities. For comparison purposes, the radial distribution functions g (r) from three first-order theories [the BGY1, the Percus–Yevick (PY1), and the convolution–hypernetted-chain (CHNC1) theories] and the Monte Carlo (MC) g (r) are also obtained. At densities less than 47% of the close-packed density, the BGY2 g (r) gives the best agreement with the MC data. At higher densities the BGY2 g (r) agrees well with the MC g (r) at short distances (r?1.7σ : σ=disk diameter), but the PY1 g (r) appears to be better at larger distances. The results at larger distances are somewhat ambiguous on account of the numerical uncertainties associated with the BGY2 solutions. The pressures predicted from the BGY2 theory are virtually identical to the MC values, demonstrating a rapid convergence of the BGY hierarchy.

On gradient theories of fluid interfacial stress and structure

The gradient approximations of the density profile equations obtained from momentum balance (the Yvon–Born–Green theory) and from minimization of the Helmholtz free energy (the van der Waals–Cahn–Hilliard theory) are compared. Tractable equations are obtained by approximating the pair correlation function as that of a homogeneous fluid at densities in the vicinity of the interacting pair. The theories are shown to be insensitive to the particular choice of such approximating pair correlation functions if the third force moments are composition and density independent. The YBG and the VDW–CH gradient approximations yield identical results for such approximating pair correlation functions if the first and third force moments are density and composition independent (in the case of a one-component fluid, only density independence of the third force moment is required). Numerical results are presented for the interfacial properties of a one-component square-well fluid.

Two-dimensional Born-Green-Yvon and other integral equations

A derivation of the Born-Green-Yvon (BGY) integral equation for two-dimensional systems is given. A form that is amenable to numerical solution for g(r) or inversion for phi (r) is obtained. To verify the result, the author shows that the integral equation for the case of rigid disks given before is derivable from the general expression. The two-dimensional forms of the Percus-Yevick (PY) and hypernetted chain (HNC) equations are also presented.

Note to Markovian approximation of Yvon–Born–Green and Kirkwood hierarchies in polymers

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The molecular structure of a liquid–vapor interface: Comments on the integral equation approach

A review of attempts to obtain interfacial density profiles by integral equation methods is presented. The first equation of the Yvon–Born–Green hierarchy is solved for ρ(1)(z) for the vapor–liquid interface, using square-well g(2)(r) data input generated by solving the second equation of the Yvon–Born–Green hierarchy. The method used in solving the first equation is discussed. Profiles are presented for three reduced temperatures spanning the vapor–liquid coexistence locus. The properties surface tension and surface excess internal energy are computed using statistical mechanical formulae and the ρ(1)(z) and g(2)(r) data generated.

Solutions of the Yvon–Born–Green and Kirkwood equations for hard spheres at very high densities

The Yvon–Born–Green and Kirkwood equations with superposition approximation for g(2)(x) for a system of hard spheres are found to have multiple solutions over broad ranges of density. These solutions are related to those of an earlier study of high density solutions to the Yvon–Born–Green equation for hard spheres. The extension of these findings to systems of square-well molecules is discussed. The results of the study are interpreted in the framework of a bifurcation theoretical analysis of the problem.

Solutions of the Yvon–Born–Green equation for the square-well fluid at very high densities

The Yvon–Born–Green equation with superposition approximation is solved for g(2)(x) for a system of square-well molecules at very high densities for the three isotherms ϑ=0, 0.2, 0.45. The solutions were obtained for a range of 30 diameters with the customary boundary condition of g(2)(x) ≡1 for x≳30 and, in one instance, for a devised periodic boundary condition. At all isotherms studied, beyond a density denoting a lower bound on the limit of stability of the pure liquid or dense fluid regime, the g(2)(x) functions demonstrated infinite periodicity. This ’’periodic’’ region extended beyond the density of close packing without any indication of termination. Throughout this region, the isothermal compressibility function is singular. Properties of these g(2)(x) functions are also discussed.

Comments on the behavior of the Yvon–Born–Green equation for the square-well fluid near the critical point

The Yvon–Born–Green equation under the superposition approximation was solved numerically for a system of molecules interacting via the square-well potential with σ2/σ1=1.85 in the vicinity of the gas–liquid critical point region. The critical point is located, and the critical exponents are determined. A reciprocal temperature perturbation treatment in the context of the Yvon–Born–Green theory is used in conjunction with previously developed stability criteria to locate the gas–liquid and solid–liquid coexistence regions. The methods employed offer encouragement for the use of perturbation approximations in delineating coexistence regions.

Properties of solutions to the Yvon–Born–Green equation for the square-well fluid

The Yvon−Born−Green equation under the superposition approximation was solved numerically for a system of molecules interacting via the square−well potential with σ2/σ1 = 1.85, for a temperature−density range bounded by 0 < ϑ < 0.55 and 0 < λ0 < 13. Solutions of the YBG equation throughout this range were found to be unique, and upon examination of the resulting thermodynamic properties, a gas−liquid coexistence region was located and evidence for a high−density, low temperature fluid transition was found. Relative to the range of ϑ − λ0 considered, the down−range properties of the pair−correlation function were used as a guide in carrying through a formal analysis of the YBG equation using basic theorems on the existence and uniqueness of solutions of nonlinear integral equations. Lower bounds on the limit of stability of the pure gas and liquid phase were identified and correlated with the thermodynamic behavior determined via numerical solution of the YBG equation.

Perturbation theory using the Yvon–Born–Green equation of state for square-well fluids

Using the Yvon−Born−Green integral equation with superposition approximation for the square−well potential with R=σ2/σ1=1.5, values of the coefficients of the reciprocal temperature expansion of the Helmholtz energy are computed. The results compare favorably up to values of n*=0.7 with the machine computations of Alder, Young, and Mark through A4/NkT despite obvious differences in the hard−sphere reference pair correlation functions between the two approaches.

Classical singlet distribution function g(1)(z) for the liquid–vapour density transition

Abstract A modified form of the singlet Yvon–Born–Green equation appropriate to the anisotropic transition zone at the surface of liquid argon is evaluated numerically. The transition profile at the triple point shows that a discrete shoulder develops in contrast to the widely reported monotonic liquid–vapour density transition.

Distribution Function of Classical Fluids of Hard Spheres. II

The pair- and triplet-correlation functions g12[≡ g(r12)] and g123[≡ g(r12, r13, r23)] are used to make a superposition approximation g123g124g134g234 / (g12g13g14g23g24g34) to the quadruplet correlation function g1234. Introduction of this approximation into the second equation of the Born–Green–Yvon (BGY) hierarchy makes it possible to truncate the hierarchy. The resulting equations are a pair of simultaneous integro-differential equations involving g12 and g123, which hereafter are referred to as the BGY2 equations. The BGY2 theory was further elucidated by the use of the hard-sphere potential to solve the corresponding BGY2 equations at two selected values (10 and 15) of λ, which is used as the independent variable and is equal to 4πρg(σ) (ρ = density, σ = sphere diameter). The densities {ρ = λ / [4πg(σ)]} corresponding to these values of λ can be established from the BGY2 solutions of g(r) at contact distance of two spheres. They are found to be, respectively, 0.298ρ0 and 0.372ρ0 (ρ0 = the close-packed density). At these densities, a separate program, which uses a Monte Carlo (MC) method, was used to evaluate the radial distribution functions for 500 hard spheres. The MC g(r) can be regarded as exact within the observed standard deviations; hence, adequacy of various approximate theories of g(r) can be tested against it. This was done here, not only for the BGY2 g(r) but also for the BGY1 g(r) obtained after truncating the BGY hierarchy by using the Kirkwood superposition approximation for g123 as well as for the Percus–Yevick (PY) g(r). At ρ = 0.298ρ0, the comparison gave values of the g(r) which are vastly improved over values calculated from either the BGY1 g(r) or the PY g(r) and which agree within each other's numerical uncertainties with the MC g(r) at most of the interparticle distances. Nearly perfect agreement (again within numerical uncertainties of the two) of the BGY2 g(r) and the MC g(r) at ρ = 0.372ρ0 was also observed for r < 1.4σ. For r > 1.4σ, the BGY2 g(r) deviates slightly (i.e., 1.6% or less) from the MC g(r) which appears to follow slightly closer to the PY g(r). The discrepancies between the BGY2 and the MC g(r) may be attributable to a large extent to the iterative scheme used in solving BGY2 equations as well as use of the inaccurate density in obtaining the MC g(r). Over-all improvement of the BGY2 g(r) over the BGY1 g(r) is striking. In particular, the pressures calculated from the BGY2 g(r) by the use of the virial theorem agree within statistical uncertainties with the MC pressures at the densities given above.

Graph-theoretic analysis of the BGY theory of classical fluids with application to the fourth-virial coefficient for the 12-6 potential

The Born-Green-Yvon (BGY) approximate theory relating the radial distribution function g(r) for a classical fluid to the pair potential of intermolecular force is analysed in terms of two-rooted graphs with Mayer f functions as links. We find that the BGY theory neglects many terms in the gradient of the sum of all simple graphs, and retains only those terms obtained from bridge graphs with an f link between root point 1 and the first nodal point by replacing the f link with its gradient. Each term in the density expansion of the gradient of g(r) may be integrated so that the BGY theory may be compared with the exact, Percus-Yevick, and hypernetted chain theories at any order of density. As an application of this analysis, the fourth-virial coefficient as a function of temperature is calculated in the BGY approximation for the Lennard-Jones 12-6 potential.

On the Consistency of the Yvon–Born–Green Hierarchy and Its Truncations

The assumption of truncation of the hierarchy of reduced distributions for fluids at thermodynamic equilibrium is discussed. We consider the question of whether a truncated hierarchy yields expressions for the mean force on a molecule, in the presence of its neighbors, which are conservative. It is concluded that in general it does not, although the Kirkwood superposition and the infinite hierarchy do. The conclusion is reached that, unless the closure satisfies a certain condition, the truncated hierarchy in general has no solutions.

Distribution Function of Classical Fluids of Hard Spheres. I

The Born–Green–Yvon (BGY) hierarchy is truncated by introducing a superposition approximation g1234≃g123g124g134g234 / [g12g13g14g23g24g34] to the quadruplet correlation function g1234. The resulting pair of two simultaneous integral equations including the triplet- and pair- correlation functions g123 and g12, which hereafter will be called the BGY2 equations, is reformulated to give mathematically simpler forms. The BGY2 theory is checked for its internal consistency by using the hard-sphere potential. The following results have been obtained: (a) The fifth virial coefficients B5 for the BGY2 theory is (0.1090 ± 0.0008) (B2)4 or (0.1112 ± 0.0005) (B2)4 depending on whether the virial theorem or the Ornstein–Zernike compressibility relation is used in the calculations of B5; the exact Monte Carlo value, B5 = 0.1103 (B2)4, lies between the two values. (b) The third coefficient g123 in the density series of g12 agrees exactly with the exact value given in the literature if r / σ ≥ 2 (σ is the sphere diameter), and it differs only slightly from the exact value if 0 ≤ r/σ ≤ 2, i.e., the deviation lies within 3%. (c) The data for the second coefficient g1232 of the density series of g123 show that the Kirkwood superposition approximation is better for a symmetric configuration than an asymmetric one when particles 1, 2, and 3 lie close to each other. (d) It is shown that the pressure calculated from the virial theorem using the BGY g(r) is exact for one-dimensional hard rods, and is independent of any approximation used to describe the g123. A proof that the BGY2 g(r) is exact for the hard-rod system is also given. These results indicate that the BGY2 theory compares favorably with any other theories for g(r) and that it satisfactorily describes low and moderately dense fluids.

Conjecture Concerning an Asymptotic Modification of the Yvon—Born—Green Equation for Fluids of Rigid Spheres and Disks

An asymptotic (long-range) modification of the Yvon—Born—Green equation for rigid spheres and disks is made which extends the validity of the equation to pressures above the Kirkwood limit. Modification of the YBG equation with both short-range and long-range terms yields equations of state in good agreement with molecular dynamics experiments up to the fluid—solid phase transitions. The application of this technique to fluids with attractive potentials and to other integral equations is briefly discussed.

Comments on the Equation of State of the Square‐Well Fluid

The simple and modified forms of the Yvon—Born—Green equation are solved for a fluid of rigid spheres with attractive square wells. The equation of state is obtained over a range of temperatures and densities. Special attention is paid to the gas—liquid and fluid—solid phase transitions predicted by the YBG equation. At the supercritical temperature kT/ε=3.33 we find nearly exact agreement with Monte Carlo pressures up to a reduced density ρσ3=0.6. At ρσ3=0.75 the disagreement with the best theoretical isotherm reaches 5%. At the critical point we find a reduced pressure (p/ρkt)c=0.50±0.02, which agrees closely with the analytic value of 0.5. In the critical region the isothermal compressibility shows strong maxima and the range of the pair correlations becomes extremely long. The YBG freezing curve shows the same qualitative features as does the Monte Carlo freezing curve.

On the Equation of State of the Rigid-Disk Fluid

The modified Yvon—Born—Green (YBG) equation of Rice and Lekner is solved for a system of rigid disks. The equation of state calculated from the solutions of the modified YBG equation is in nearly perfect agreement with the molecular-dynamics data of Alder and Wainwright up to p/ρkT=4.78, which is the Kirkwood limit for the stability of the fluid phase.

Comparison of Radial Distribution Functions from Integral Equations and Monte Carlo

The radial distribution functions g for a system of particles interacting with the Lennard-Jones 6–12 potential have been found by solving the Percus—Yevick (PY) equation, convolution approximation or hypernetted chain equation (CHNC), and the Born—Green—Yvon (BGY) equation, for four particle densities already investigated by Monte Carlo (MC) techniques by Wood and Parker. The thermodynamic quantities p/nkT, E/NkT, and nkTK have also been computed from the g's. A comparison shows that the PY quantities agree best with MC, CHNC quantities agree next best, and BGY agree most poorly, particularly at high densities. A comparison of nkTK computed by differentiating p/nkT and by direct integration involving g gives approximately consistent results for the PY g but not for the CHNC g. The asymptotic form of g for high densities for the PY and CHNC equations is found to agree with that obtained by Kirkwood.