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maximum concentration. The study of pair distribution functions is a subject of interest in statistical mechanics [McQuarrie, 1976]. Based on the form of the pair distribution function, substances can be classified into three different types: (1) gas; (2) liquid and amorphous solid; and (3) crystalline solid. The three forms of the pair distribution function are illustrated in Fig. 8.1.2. The case of gas with particles sparsely distributed is considered to be a system of extreme disorder, so that the hole-correction approximation or the independent position approximation is a good description of the pair distribution function. In the opposite extreme, the case of crystalline solid with relative positions
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b (I) Particles g(1') 9(1')
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Figure 8.1.1 Pair distribution function for one-dimensional particles: (A) Small f; particles and pair function. (B) f = 1; particles and pair function.
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of the particles fixed is a case of extreme order. The pair distribution function exhibits sharp peaks (Fig. 8.1.2c). The case of a liquid and amorphous solid is a system of partial order and is an interpolation between the two extreme cases of gas and crystalline solid (Fig. 8.1.2d). Extensive experimental and theoretical investigations have been carried out for the pair distribution function of liquid and amorphous solid [Waseda, 1980]. To the first-order approximation, the bistatic scattering intensity is proportional to the structure factor which is related to the Fourier transform of the pair distribution functions. The study of pair distribution functions is an important subject in molecular theory of fluids as well as in random media [Wertheim, 1963, 1964; McQuarrie, 1976; Ziman, 1979; Perram et al. 1984; Perla et al. 1986; Penders and Vrij, 1990; Shi et al. 1993; Zurk et al. 1997].
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1.2 Percus-Yevick Equation and Pair Distribution Function for Hard Spheres
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A total influence h of a particle 1 on another particle 2 can be defined as
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(8.1.3)
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The total influence is decomposed into a sum of direct and indirect correlation functions. The direct correlation function or influence is denoted by
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(8.1.4)
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The direct correlation function is defined such that it satisfies the following
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1.2 Percus-Yevick Equation and Pair Distribution Function
g(l~==,
a) Independent
b) Gas (Hole correction)
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c) Crystalline solid
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d) Liquid or amorphous solid
Figure 8.1.2 Pair functions for (a) independent particle position, (b) gas, (c) crystalline solid, and (d) liquid and amorphous solid.
integral equation:
(8.1.5a)
From (8.1.3),
h(r12) = c(r12)
+ no
dr3c(r13) [g(r32) - 1]
(8.1.5b)
which is known as the Ornstein-Zernike equation. The physical interpretation of the second term of (8.1.5a), which is the indirect correlation function, is that the indirect influence of particle 1 on particle 2 is a result of particle 1 acting directly on a particle at r3, which in turn exerts total influence on particle 2. The indirect influence is averaged over particle positions r3 and weighted by the number of particles per unit volume no as indicated in (8.1.5a) and (8.1.5b). The Ornstein-Zernike equation consists of two unknowns c(r) and h(r) in one equation. An approximation is to be made on the relation between c(r) and h(r), reducing (8.1.5) to one equation and one unknown. The PercusYevick approximation [Percus and Yevick, 1958] can be introduced in the following heuristic manner. The potential energy between two particles is governed by u(r) where r is their separation. For the case of hard sphere potential, we have
u(r) =
{oo o
for r < bb or r ~
(8.1.6)
Equation (8.1.6) says that in the absence of other particles, the potential energy between the two particles is infinite when they overlap each other
8 PARTICLE POSITIONS FOR DENSE MEDIA CHARACTERIZATIONS
(thus disallowing interpenetration) and is zero otherwise. For this case, we define y(r) so that
g(r) = { y(r)
for r < b for r 2: b
(8.1.7)
The function y(r) is defined for both r < band r > b. Equation (8.1.7) defines it for r 2: b. Later we will define it for r < b. In (8.1.7), we let g(r) = y(r) for r 2: b. Then, for hard-sphere potential we have -1 for r < b h(r) = g(r) - 1 = { y(r) _ 1 for r 2: b (8.1.8) When y = 1, there is no indirect influence. Thus y - 1 is a measure of indirect influence. Also h - c is equal to to indirect influence. The PercusYevick approximation consists of equating h - c to Y - 1 for all r.
h(r) - c(r) = y(r) - 1
This equation then extends the definition of y(r) to r < b. From (8.1.8) to (8.1.9) and (8.1.3) we obtain
(8.1.9)
c(r) = h(r) _ y(r)
+1 -
y(r) = g(r) - y(r)