Pair Distribution Functions of Adhesive Sphere Mixture in .NET

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4.2 Pair Distribution Functions of Adhesive Sphere Mixture
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Figure 8.4.4 Percus-Yevick pair distribution function for a binary mixture of sticky spheres with a2 = 0.6a1, h = 0.1, h = 0.15, and Tl1 = T12 = T22 = 0.2.
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Figure 8.4.5 Percus-Yevick pair distribution function for a binary mixture of sticky spheres with a2 = OAa1, h = 0.2,12 = 0.04, and Tl1 = 1, T12 = 0.2, and T22 = 00.
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8 PARTICLE POSITIONS FOR DENSE MEDIA CHARACTERIZATIONS
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4.3 Monte Carlo Simulation of Adhesive Spheres
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The Metropolis shuffling technique as described in Section 3.1 is usually employed for particles without surface adhesion. This conventional method generates new configurations by giving random displacements to particles and then accepting or rejecting the configuration based on particle overlap. However, this method cannot be directly applied to particles interacting via the adhesive potential as given by (8.4.1). By taking the limit, S - t d in the exponential of (8.4.1), we have
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d) for r ~ d (8.4.37) 1 for r > d here d is the particle diameter and T is the stickiness parameter related to the adhesion strength. Thus, the adhesive potential (8.4.1) implies that there exists a finite probability of contact, proportional to d/(12T), for adhesive particle pairs. As T decreases, the adhesion strength increases, and favors more the formation of clusters. This feature cannot be captured by using the shuffling technique since it is very unlikely to sample any surfaces of other particles while each particle is displaced in a three-dimensional space. In order to explore the available contact states of adhesive particles, Seaton and Glandt [1986, 1987] and Kranendonk and Frenkel [1988] have developed alternative sampling procedures which allow a particle to break or form bonds with other particles during its Monte Carlo displacement. They have considered only transitions between four binding states a = 0,1,2,3, corresponding to the unbounded, single-bond, double-bond, and triple-bond configurations, respectively. The a-bond state indicates that a particle is in contact with 0: number of other particles. The maximum number of bonds is 12. The binding states a = 0,1,2,3 of a particle, represented by a filled circle, are illustrated in Fig. 8.4.6. We will describe the Kranendonk-Frenkel algorithm in this section for the three-dimensional simulations of adhesive particles. The description of Seaton-Glandt algorithm will be given in 9 for the two-dimensional simulations. In the Kranendonk-Frenkel algorithm, a total effective volume Ve~) is assigned to the a-bond state of the test particle k as given by
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= { l~T<5(r -
(a) 0 d v:,ff = '" ( 12T )aJ
(8.4.38)
where dqi represents all degrees of freedom of the test particle k, remaining after making the a bonds. The summation in (8.4.38) is over all possible combinations of the test particle with all the other particles that can realize the type of a-bond. Possible excluded volume effects are ignored in com-
4.3 Monte Carlo Simulation of Adhesive Spheres
Figure 8.4.6 The particle binding states: (a) unbounded, (b) single-bond, (c) double-bond, and (d) triple-bond.
puting the integral in (8.4.38). The effective volume ~~) represents a direct measure of the probability of the a-bond configuration, with a contributing factor C~T)a. The total effective volume associated with the unbounded state of a test particle k is
v:,~) = V
(8.4.39)
which is simply the total volume available to the test particle, including overlapping regions. Next we examine the single bond (Fig. 8.4.7a). Let r be the distance between particle i and test particle k. Then
127rl~2 sin ()d()d
is the surface area that the center Tk of particle k can be located such that particle k is attached to particle i. The total effective volume of the singlebond state of a test particle k is
~~) = i# (~) Jo Jo rr 12T
r 2 sin () d()d
L v~~i
(N - 1)
(l~T)
(47rd
(8.4.40)
As shown in Fig. 8.4.7a, the test particle k can be placed anywhere on the surface of another particle i. Thus, as indicated in (8.4.40), V~~i is the effective volume associated with the single bond between particles k and i.