Long-range oscillations of a single-particle distribution function for a molecular system of hard spheres near a solid surface in the Percus–Yevick approximation

The properties of molecular systems are defined by asymptotic behavior of distribution functions that have descending character with long-range oscillating order. We consider the molecular system of hard spheres near hard wall. The decrement and period of oscillation are calculated for the distribution function which defines the local density profile.


Introduction
In present work we consider the long-range oscillating correlations of the liquid's local density near the hard surface. In our previous study we used the singlet approximation of statistical physics of liquids to consider the linear Fredholm equation of the second kind for the one-particle distribution function, which depends on the distance from the surface [1]. The boundary condition is defined by transition to macroscopic liquid far from the surface. The structure characteristics of the macroscopic liquid are determined by pair correlation function which depends on the distance between the particles [2]. The core and the right part of the Fredholm equation are also evaluated with this function.
The pair correlation function is related to the direct correlation function by the Percus-Yevick approximation [3]. The relationship (closure) between these two functions produces approximate nonlinear equation integral equation, which should be solved numerically. The Percus-Yevick approximation is the only exception, which has the analytical solution for the hard spheres system. The direct correlation function depends on the distance between particles as the third-degree polynomial function [4].
The Percus-Yevick closure allows to analytically evaluate the core and right part of the Fredholm equation. With this approach we get the analytical solution for the one-particle distribution function for the liquid near the hard surface. This solution can be generalized with statistical physics methods using Random First Order Transition Theory (RFOT) to describe the metastable conditions such as surface amorphization of supercooled liquids. For the last twenty years RFOT approach become popular to describe amorphization phenomena [5].

Common equations
The Born-Green-Yvon (BGY) equations system may be transformed to equations for one-and twoparticle distribution functions, which may be written [3] as Ornstein-Zernike equations: Here we integrating on coordinates of i-particle: ( ) one-particle distribution function, which describes particle position; i potential energy in external field; i one-particle thermal potential; aactivity coefficient, which is defined by condition of passing to isotropic system. The pair correlation function − is connected with two particle distribution function by the expression: G12(r1, r2) are critical ones since they describe internal structure. Let us obtain thermodynamic parameters of the system. Equations (1) and (2)  should approximate these series by simple expressions (closures). In such a manner we can obtain approximated equations for high density systems.

Molecular system of liquid near hard surface
Space-heterogeneous systems (liquid near a hard surface) are described by one-and two-particle distribution functions: G1(r1) and G12(r1, r2). Boundary condition for these equations is a transition from a hard surface to a liquid.
Let's form the equations for molecular system near hard surface. When solving first equation of (1) origin is located in the center of the particle, which contacts with hard surface. The Z axis is perpendicular to the surface; thus, the whole liquid is placed in upper half-space ( 0 z  ). Bottom half-space ( 0 z  ) is unavailable for the particles. Such a system has axial symmetry. We assume that all irreducible diagrams in (1) are calculated in such way to compensate the nonlinearities. With this approach we get first equation of (1) as: where r12 is measured in particle diameter units, 0 i z particle distance from the surface. This is the second kind of Fredholm equation.

Percus-Yevick approximation
The internal integrals of equation (3)      = + − (5) In our previous work [1] we have proposed that function (1,0) 12 12 () Cr may be substituted with its mean value in the range 12 01 r  . In this case () z  will be the polynomial function of degree 2 and equation (3) has an analytical solution. In the present work we won't use this approximation, and taking the direct definition of () z  given in [4] write equation (3) in the form: