Size distribution cluster

Another important characteristic of the late stages of phase separation kinetics, for asynnnetric mixtures, is the cluster size distribution fimction of the minority phase clusters n(R,z)dR is the number of clusters of minority phase per unit volume with radii between R and + cW. Its zeroth moment gives the mean number of clusters at time r and the first moment is proportional to die mean cluster size. 

For a general dimension d, the cluster size distribution fiinction n(R, x) is defined such that n(R, x)dR equals the number of clusters per unit volume with a radius between andi + dR. Assuming no nucleation of new clusters and no coalescence, n(R, x) satisfies a continuity equation... 

Figure A3.3.11 The asymptotic cluster size distribution f(x) from LS analysis for

On the other hand, whenever AV exceeds the value of AVq the formation of a dense monolayer film appears to be the continuous process. It has been demonstrated that the observed crossover between those two regimes is due to the changes in the mechanism of the adsorbate nucleation, as determined by the calculation of the nucleated cluster size distribution functions. For... 

The cluster properties of the reactants in the MM model at criticality have been studied by Ziff and Fichthorn [89]. Evidence is given that the cluster size distribution is a hyperbolic function which decays with exponent r = 2.05 0.02 and that the fractal dimension (Z)p) of the clusters is Dp = 1.90 0.03. This figure is similar to that of random percolation clusters in two dimensions [37], However, clusters of the reactants appear to be more solid and with fewer holes (at least on the small-scale length of the simulations, L = 1024 sites). 

FIGURE 22.12 Uniaxial stress-strain cycles of ethylene-propylene-diene monomer (EPDM) samples with 60 phr silica at different prestrains = 10%, 20%, 30%, 40%, and 50% (symbols) and fittings (lines) with the stress-softening model Equations 22.19-22.24. The fitting parameters are indicated. The assumed cluster-size distribution is also shown, which differs from the one in Equation 22.24. (From Kliippel, M. and Heinrich, G., Kautschuk, Gummi, Kunststojfe, 58, 217, 2005. With permission.)... 

One of the recent advances in magnetic studies is that it enables not only the estimation of the average volume v of clusters from the LF and HF approximations of the Langevln function, but also enables to compute particle size distribution based on an assumed function. By judiciously combining the parameters of the Langevln and of the "log normal function, we obtained a particle (cluster) size distribution of Y Fe203 in ZSM-5. The essential features of such computation are shown in Fig. 6. 

The cluster size distribution in the limit of small mass [x < s(f)] depends on the properties of the agglomerates undergoing fragmentation. If infinitesimal particles are formed on a single breakage, that is, b(r) r", then... 

Several works over the past decade or so (Ziff, 1991 Ziff and McGrady, 1986 McGrady and Ziff, 1987,1988 and Williams, 1990) have addressed the behavior of systems with specified breakup kernels. Certain specific forms for the breakup kernels lead to analytical solutions for the cluster size distribution. For example, Ziff (1991) obtained explicit forms of the size distribution for homogeneous breakup kernels of the form... 

A solution to this problem (Hansen and Ottino, 1996a) reveals that the cluster size distribution is bimodal, as expected, with c(x,t) for large x dependent upon the initial conditions (Fig. 35a). The distribution thus does not approach a self-similar form and the scaling results just given are not valid for this problem. This is a result of the non-homogeneous relative rate of breakup. 

In this section, we consider flow-induced aggregation without diffusion, i.e., when the Peclet number, Pe = VLID, where V and L are the characteristic velocity and length and D is the Brownian diffusion coefficient, is much greater than unity. For simplicity, we neglect the hydrodynamic interactions of the clusters and highlight the effects of advection on the evolution of the cluster size distribution and the formation of fractal structures. 

The self-similar nature of the cluster size distributions for the well-mixed flow is shown in Fig. 39c. Because of the discrete nature of cluster size in... 

Thus, we plot M(x,t)IMi vs xls(t). As noted earlier, the cluster size distribution and the first moment of the size distribution are averaged over the entire journal bearing. As indicated by the behavior of P in Fig. 39b, the cluster size distribution becomes self-similar when the average size is about 10 particles per cluster. 

All models described up to here belong to the class of equilibrium theories. They have the advantage of providing structural information on the material during the liquid-solid transition. Kinetic theories based on Smoluchowski s coagulation equation [45] have recently been applied more and more to describe the kinetics of gelation. The Smoluchowski equation describes the time evolution of the cluster size distribution N(k) ... 

Figure 11. The ion cluster size distribution obtained from computer simulations of the charged and dipolar hard sphere mixture at several states half charge 1 Molar (A) fully charged, 1 Molar (B) fully charged, 0.4 Molar (C) and half charge, 0.4 Molar (D).

In this work, the relationship between micropore filling of supercritical Xe in micropores of ACF at 300 K and cluster size distribution by cluster analysis is described. 

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