Modification of the singlet equation for a molecular system of solid spheres near a solid surface in the Percus-Yevick approximation

An analysis of theoretical methods for studying the molecular structure of liquids bordering a solid surface was carried out. It was established that the currently existing nonlinear integral equations for partial distribution functions do not have an analytical solution for spatially inhomogeneous liquids. A linear integral equation having an analytical solution for a system of solid spheres near a solid surface was proposed.


Introduction
The microscopic description of close order in liquids, as a rule, is based on integral equations for oneand two-particle distribution functions. The core of such equations contains the infinite functional series of irreducible diagrams. Cut off of these infinite series, usually performed, produces various nonlinear integral equations. One of them is the Percus-Yevick (PY) equation for uniform spatial liquid [1] allows one to get an analytical solution for hard sphere system. Other similar equations don't have analytical solution.
For space-heterogeneous systems, such as a liquid near a hard surface, two-particle distribution function is replaced by its value taken far from surface. We call such substitution as singlet approximation. In this approximation we obtain the equation for one-particle distribution function, which depends on single variabledistance from surface. This equation can be solved only numerically [2] and even for modern computers such calculations take much time.
We suggest to determine the contribution of PY approximation into irreducible diagrams so that other series terms will compensate all non-linear behavior. As a result, we obtain the linear Fredholm integral equation of the second kind, having an analytical solution for hard sphere system near hard surface. This equation can be generalized for molecular systems with realistic potentials, which is important for melting amorphization study. The BGY-equations system may be transformed to equations for one-and two-particle distribution functions, which may be written as Ornstein-Zernike [3]:

Common equations
Here we integrating over coordinates of i-particle: is one-particle distribution function, which describes particle position; i  is potential energy in external field; i  is one-particle thermal potential; a is activity coefficient, which is defined by condition of passing to isotropic system. Pair correlation function connected with two particle distribution function by expression: ij  is two-particle thermal potential, which takes into account indirect interaction of two particles;   k ij C are direct correlation functions: Functions G1(r1) and G12(r1, r2) describing internal structure are critical ones. Let's obtain thermodynamic parameters of the system. Equations (1) and (2)  M contains infinite series of distribution functions. To use the following equations in practice one should approximate these series by simple expressions (closures). In such a manner we can obtain approximated equations for high density systems.
Space-homogeneous systems (isotropic liquids in absence of external fields and surfaces) are of special interest. In these systems the first equation in (1) As a result, an exponential non-linearity lowers to polynomial one for hard sphere system, i.e. the non-linearity becomes weaker. That provides possibility to get analytical solution.
For space-heterogeneous systems, such as a liquid near a hard surface, functions , CC  rr describe microstructure of the substance and allow to obtain all thermodynamic parameters.

Singlet approximation
Direct solution of many variables' equations (1-2) is possible only numerically [2]. It requires a lot of computational time. One can simplify this task using boundary value (1)  nG n  rr near hard surface. All of them contain non-linear terms and have to be solved numerically. Review of solutions is presented in [2]. It should be stressed that analytical solution for one-particle distribution function is possible for special cases, for example, one-and two-dimension problem [4].

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 conditions for these equations take into account 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 halfspace ( 0 z  ) is unavailable for the particles. Such a system has axial symmetry, Here r12 is measured in particle diameter units, 0 i z particle distance from the surface. Boundary conditions for G1(r1) and G12 (r1, r2) are defined as follows

Modification of the singlet equation
We suggest modification of singlet approximation, which allows analytical solution for hard sphere system near hard surface. In the first equation (1) for a single-particle distribution function, we combine the starting point with the center of the particle 1. After integration performed in a cylindrical coordinate system,  (1) (1,0) (1,0) 1 1 2 12 12 1 2 12 12 1 12 12 12 12 12 12 12 12 0 | | ( ) 2 ( ) ( , , ) ( )) 2 ( ) z z z z n dz r dr G z C r z z C r n dz r dr C r We modify equation (6). The unknown function in the first integral can be written as (1) (1,0) (1) 1 2 12 12 1 12 12 12 12 12 1 12 where (  z power and assuming (17), we obtain the algebraic equations system (1) (2) Numerical solution of this system allows us to fully define the function on the interval 1 1 z  . Solving equations (13), (14) with closure (12) gives qualitatively correct results on the interval 1 0 z    . To close Percus-Yevick (10) it is necessary to decompose the function of the desired function (16) up to the sixth-order summands. The result gives a slightly more complex analytical expression.

Conclusions
We have combined the Percus-Yevick closure relation (12) with OZ relations (1-2) to determine the one-particle function for a molecular system of liquid near hard surface. We have proposed a new approximation for irreducible diagrams of the direct correlation function in a single particle equation. This approximation leads to a linear Fredholm integral equation of the second kind. The right part and core of the integral equation are expressed in terms of a direct correlation function in the Percus-Yevick approximation. An analytical solution of the equation for a molecular system of solid spheres has been obtained. For realistic intermolecular interaction potentials, the numerical solution of a linear equation requires much less computational resources compared to nonlinear integral equations.