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3.3 Numerical Results
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The Monte Carlo techniques are used to generate the random distribution of particles of multiple sizes and to obtain the pair distribution functions gij(r) from the computer generated samples. The Monte Carlo simulation results of pair distribution functions are compared with those under PercusYevick approximation. For the computer experimental results shown below, a unit cell, g = 1, containing spheres with different sizes was chosen as our model system. The numerical results of pair distribution functions using the Metropolis technique are illustrated in Figs. 8.3.1-8.3.6, and the simulation results using the sequential addition method are shown in Fig. 8.3.7. In all the figures except Figs. 8.3.2 and 8.3.7, we have used T = 3000 configurations
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3.3 Numerical Results
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Figure 8.3.1 Pair distribution functions g(r) for media with particles of single size as a function of r normalized by 2a. The simulation parameters are (i) N = 400, f = 0.2, b. = 0.06105 and (ii) N = 600, f = 0.3, b. = 0.02757. The data points are Metropolis Monte Carlo simulation results, and the curves are obtained from Percus-Yevick calculations.
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and, assuming N c = 1, also 3000 realizations. The choice of displacement .6. is dictated by the following considerations. The choice of maximum displacement ~ in the Metropolis technique for each movement will affect the percentage of accepted displacements. If ~ is too big, almost all the displacements will be rejected, so the ensemble of system realizations will be composed of nearly the same configurations repeated many times and provides very little new information at each step. On the other hand, if ~ is too small, the displacement of a particle is always accepted, but the configuration changes very slowly and givei:i inefficient sampling in the configuration space. The size of ~ has been adjusted by trials to give about 30% to 70% acceptance of the new position. In Fig. 8.3.1 the pair distribution functions are shown for systems with particles of identical size for two cases: (i) f = 0.2, N = 400 and (ii) f = 0.3, N = 600. The pair distribution function g(r) is at its maximum at the separation equal to one diameter of particle, which shows a strong likelihood that the particles will clump together in a dense medium. The pair distribution function behaves more oscillatory and has higher peaks for the case of larger concentration of particles. The result also shows that the pair
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Figure 8.3.2 Pair distribution function g(r) for media with particles of single size as a function of r normalized by 2a. The simulation parameters are f = 0.25, N = 4000, N c = 10 configurations per realization, and T = 30 realizations. The simulated results are compared with Percus-Yevick calculations. The acceptance rate A c is 42.5%.
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distribution function asymptotically approaches unity as the interparticle distance increases. The PY results are in good agreement with Monte Carlo simulations. In Fig. 8.3.2 we consider a volume with f = 0.25 and N = 4000. We take N c = 10 configurations before we obtain a realization, but we take only T = 30 realizations. The simulated pair distribution functions, as shown in the figure, still compare very well with the Percus-Yevick approximation. This example shows that the number of required realizations can be small if the number of particles is large. In Figs. 8.3.3 and 8.3.4 we illustrate the results of pair distribution functions for a medium with particles of two different sizes. For Fig. 8.3.3 we placed N = 558 spheres inside a unit cell in which Nl = 486 of them are smaller particles and N 2 = 72 are larger in size with a2 = 1.5al. The volume fractions for each size are !I = 0.16 and 12 = 0.08, respectively. In Fig. 8.3.4 we consider a system having a higher concentration of particles with a larger size ratio, but with a smaller number of larger particles. The model parameters are N = 816, N I = 768, N2 = 48, a2 = 2al, !I = 0.2, and 12 = 0.1. In Figs. 8.3.5 and 8.3.6 the pair distribution functions are for
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