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Pair Distribution Functions and Structure Factors
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Introduction Percus-Yevick Equation and Pair Distribution Function for Hard Spheres Calculation of Structure Factor and Pair Distribution Function
1.1 1.2 1.3
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Percus-Yevick Pair Distribution Functions for Multiple Sizes 411 Monte Carlo Simulations of Particle Positions
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Metropolis Monte Carlo Technique Sequential Addition Method Numerical Results
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S.ticky Particles
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Percus-Yevick Pair Distribution Function for Sticky Spheres Pair Distribution Function of Adhesive Sphere Mixture Monte Carlo Simulation of Adhesive Spheres
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References and Additional Readings
- 403-
The scattering of electromagnetic waves by random media depends on the positions of the particles. A dense medium denotes a medium where the particles are densely packed and occupy appreciable fractional volume. In the random medium, the probability density function (pdf) of particle positions p(1'i) is uniformly random. However, because of the finiteness of particle size, the joint probability density function of two particle positions cannot be independent. If p(1'i' 1'j) is the joint probability density function of two particles centered at 1'i and 1'j, then P(1'i' 1'j) = 0 for l1'i - Tj I smaller than the minimum separation which is the diameter if the two particles are spheres of the same radius. The pair distribution function is proportional to the joint probability density functions of two particles. The finiteness of the particle sizes creates nontrivial pair distribution functions. The Fourier transform of the pair distribution function is the structure factor. We have discussed pair distribution function in 4, Section 5.2 of Volume 1. In this chapter, we consider two methods of studying random particle positions. One method is based on analytic theory to derive analytic pair distribution functions. The other method is to use Monte Carlo simulations to generate particle positions. The Monte Carlo procedure of generating particle positions is also important as numerical solutions of the Maxwell equations can be computed based on the generated realizations. On the other hand, analytic pair distribution function is useful for analytic scattering theory of dense media. Analytic scattering theory of dense media will be treated in Volume III. In Section 1, we describe the pair distribution function under the PercusYevick approximation. In Section 2, we consider the case where the particles in the same medium can have different sizes. In Section 3, we describe Monte Carlo simulations of spherical particles. In Section 4, we consider the case of sticky particles. This represents the case when particles have adhesive force. The adhesive force means that the particles can form aggregates. We consider the case of collections of aggregates. In Section 5, the Monte Carlo technique is extended to particles of spheroidal shapes.
Pair Distribution Functions and Structure Factors
1.1 Introduction
Let N be the number of particles. They are centered at 1'1,1'2, ... , l'N. Let the particles be put in a volume V. Then
1.l Introduction
is the single-particle pdf, and from Volume I, Eqs. (4.5.21)-(4.5.23),
_._.)_g(ri,rj ) N p (rz,r J V2 N -1
is the joint pdf of two particles, and 9 is the pair distribution function. In the limit of large N, p(ri' rj) ~ g(ri' rj) /V 2 . Analytic theory of volume scattering is discussed in Volume III, where it is shown that in applying the quasi-crystalline approximation and the quasi-crystalline approximation with coherent potential, the pair distribution function of particle positions must be specified. In the special case of independent particle position, g(r) = 1. Another approximation to the pair distribution function is the hole-correction (He) approximation, given by g(r) = 0 for r < band g(r) = 1 for r ~ b, where b is the diameter of the circumscribing sphere of the particle. For the case of spherical particles with radius a, b is equal to 2a. The hole-correction approximation takes into account the fact that the particles cannot interpenetrate each other. Neither the independent position approximation nor the hole-correction approximation is correct when the fractional volume of scatterers, f, is appreciable. It is easier to visualize this for the case of one-dimensional scatterers. The hole-correction approximation is illustrated in Fig. 8.1.1A. Next we imagine that f is equal to unity so that the entire volume V is occupied by scatterers. In such a case, the centers of these one-dimensional particles will be separated by integral multiples of b from each other. The pair distribution function g(r) will be zero for r of. mb where m is any nonzero integer. It consists of delta functions at the position of r equal to an integral multiple of b (Fig. 8.1.1B). Thus, the hole-correction approximation is poor in such a limit. When f is not equal to 1 but appreciable, the pair distribution function will be of a form between A and B in Fig. 8.1.1. We also note that as the two-particle separation r approaches infinity, the positions of the particles should be independent of each other. Hence lim g(r) = 1 for f not equal to