Pair distribution

Between the limits of small and large r, the pair distribution function g(r) of a monatomic fluid is detemrined by the direct interaction between the two particles, and by the indirect interaction between the same two particles tlirough other particles. At low densities, it is only the direct interaction that operates through the Boltzmaim distribution and... 

In fact, given that there are N(N-l) ways of choosing a pair of particles, the pair distribution fiinction,... 

The pair distribution fiinction clearly has dimensions (density), and it is nomial to introduce the pair correlation fiinction g(X, X2) defined by... 

Toby B FI and Egami T 1992 Accuracy of pair distribution function analysis applied to crystalline and noncrystalline materials Aota Crystaiiogr.k 48 336-46... 

The description of the atomic distribution in noncrystalline materials employs a distribution function, (r), which corresponds to the probability of finding another atom at a distance r from the origin atom taken as the point r = 0. In a system having an average number density p = N/V, the probability of finding another atom at a distance r from an origin atom corresponds to Pq ( ). Whereas the information given by (r), which is called the pair distribution function, is only one-dimensional, it is quantitative information on the noncrystalline systems and as such is one of the most important pieces of information in the study of noncrystalline materials. The interatomic distances cannot be smaller than the atomic core diameters, so = 0. 

R. Kjellander, S. Sarman. A study of anisotropic pair distribution theories for Lennard-Jones fluids in narrow slits. II. Pair correlations and solvation forces. Mol Phys 74 665-688, 1991. 

The pair correlation functions can be expressed directly in terms of the computed coefficients from Eq. (61) in particular, the number-number pair distribution function gN ir) and the number-number structure factor SNN k). Thus,... 

FIG. 4 Pair distribution functions (the main part of the figure) and... 

On the other hand, the connected pair distribution function for a given matrix configuration p (ri,r2 ... 

Let us proceed with the description of the results from theory and simulation. First, consider the case of a narrow barrier, w = 0.5, and discuss the pair distribution functions (pdfs) of fluid species with respect to a matrix particle, gfm r). This pdf has been a main focus of previous statistical mechanical investigations of simple fluids in contact with an individual permeable barrier via integral equations and density functional methodology [49-52]. 

FIG. 6 The fluid-fluid (a) and fluid-matrix monomer (b) pair distribution fune-tions for the matrix made of ehains with four beads (m = M = 4) at paeking fraetion r] = 0.052. The nomenelature of lines and symbols is similar to that of Fig. 5 however, the upper group of results with square symbols is for /3/xy = 2.5 and the lower group of results with eireles is for /3/xy = 0. 

We conclude, from the results given above, that both the ROZ-PY and ROZ-HNC theories are sufficiently successful for the description of the pair distribution functions of fluid particles in different disordered matrices. It seems that at a low adsorbed density the PY closure is preferable, whereas... 

Now, we would like to investigate adsorption of another fluid of species / in the pore filled by the matrix. The fluid/ outside the pore has the chemical potential at equilibrium the adsorbed fluid / reaches the density distribution pf z). The pair distribution of / particles is characterized by the inhomogeneous correlation function /z (l,2). The matrix and fluid species are denoted by 0 and 1. We assume the simplest form of the interactions between particles and between particles and pore walls, choosing both species as hard spheres of unit diameter... 

The simulations are repeated several times, starting from different matrix configurations. We have found that about 10 rephcas of the matrix usually assure good statistics for the determination of the local fluid density. However, the evaluation of the nonuniform pair distribution functions requires much longer runs at least 100 matrix replicas are needed to calculate the pair correlation functions for particles parallel to the pore walls. However, even as many as 500 replicas do not ensure the convergence of the simulation results for perpendicular configurations. 

FIG. 13 (a) A comparison of the fluid-fluid inhomogeneous pair distribution func-... 

The real space pair distributions gy(rj is the inverse Fourier transform of (Sy(Q)-l), that is ... 

In order to get the pair distribution functions gjj, which satisfy the symmetry constraints, so-called minor iterations must be converged. When we use gij(r >r j) of (fl) in (2), we obtain the minor equations regarding the rotation operator R of the angle tt/2 ... 

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