Cluster aggregate distribution

Figure 7. Silicon oxide cluster aggregate distribution below and above percolation volume fraction threshold...

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. 

For free-clusters, the cluster size distribution can be measured by the time-of-flight mass spectrometer for cluster films deposited on substrate by the cluster beam, the measurement of size distribution and observation of nanostructure are mostly done using transmission electron microscopy (TEM). In this section we will focus on the latter and pay special attention to FePt, CoPt clusters which have high anisotropy Tl0 phase after annealing [43-45]. For the TEM observations, FePt, CoPt nanoclusters, produced in a gas-aggregation chamber, in which high pressure Ar gas ( 0.5-lTorr) was applied and cooled by LN2, were directly deposited onto carbon-coated films supported by Cu grids. 

D. A. Weitz and M. Y. Lin, Dynamic scaling of cluster-mass distributions in kinetic colloid aggregation, Phys. Rev. Lett. 57 2037 (1986). 

Beside the consideration of the up-cycles in the stretching direction, the model can also describe the down-cycles in the backwards direction. This is depicted in Fig. 47a,b for the case of the S-SBR sample filled with 60 phr N 220. Figure 47a shows an adaptation of the stress-strain curves in the stretching direction with the log-normal cluster size distribution Eq. (55). The depicted down-cycles are simulations obtained by Eq. (49) with the fit parameters from the up-cycles. The difference between up- and down-cycles quantifies the dissipated energy per cycle due to the cyclic breakdown and re-aggregation of filler clusters. The obtained microscopic material parameters for the viscoelastic response of the samples in the quasi-static limit are summarized in Table 4. 

The CCA-model considers the filler network as a result of kinetically cluster-cluster-aggregation, where the size of the fractal network heterogeneity is given by a space-filling condition for the filler clusters [60,63,64,92]. We will summarize the basic assumptions of this approach and extend it by adding additional considerations as well as experimental results. Thereby, we will apply the CCA-model to rubber composites filled with carbon black as well as polymeric filler particles (microgels) of spherical shape and almost mono-disperse size distribution that allow for a better understanding of the mechanisms of rubber reinforcement. 

Although it was not shown here, a general cluster size distribution in equilibrium can be obtained using a different approach [18, 19]. It involves a stochastic description for the aggregation-fragmentation system given by the master equation of a probability balance. The equilibrium probability then follows from the detailed balance. That work is under way. 

A key feature of electrolyte systems is their tendency to associate. Figure 6 shows several clusters from a simulation of the X = 0.1 system in a box of size L = 55 SitT = 0.03. The instantaneous density is p = 0.00122. As shown in the hgure, ions form polymer-like structures whose shapes include chains, rings, and branched chains. This pronounced tendency to cluster can be rationalized by considering a simple aggregate of only four ions. Figure 7 shows the fraction of ions involved in clusters of a given size n, for X = 0.1, at r = 0.03 and (p > = 0.003 (for comparison, we also show results for the RPM model at a similar density). The cluster size distribution exhibits an... 

The cluster size distribution at an arbitrary stage of aggregation can be described by a set of number pairs (A, np) where is the number of... 

Figure 15. Evolution of the cluster-mass distribution for different time moments corresponding to the aggregation parameter = 0.2-0.8(0.2), 0.9, 0.92-0.98(0.02)...

Dj. value range for macromolecular coils in a solution [ 10] assumes that the polymerization process proceeds according to the mechanism of irreversible cluster-cluster aggregation. The function of distribution for the indicated mechanism was studied in the scientific chapters [208-210]. The authors [211-213] proposed theoretical description of MWD fimctions within the fiamework of theoretical treatment [210] and studied the factors influencing the shape of these functions on the example of polydimethyl diallyl ammonium chloride (PDMDAAC) [34]. 

In Refs. [208, 217-219] the dynamical scaling description of cluster sizes distribution in the cluster-cluster diffusion-limited aggregation (DLA) model was considered. As it was mentioned above, this model was applicable completely for PDMDAAC radical polymerization description by the following reason. The estimated according to the Eq. (4) value... 

Meakin, P Vicsek, F. Dynamic cluster-size distribution in cluster-cluster aggregation effects of cluster diffusivity. Phys. Rev. B, 1984, 31(1), 564-569. 

Type IV The physical incorporation of metal (and also semiconductor) clusters or metal complexes in macromolecules has become an important field (so-called Macromolecule Incorporated Metal Complexes and Metals , Fig. 1-7). By stabilization of metal clusters in a macromolecular environment new composite materials have been synthesized. Chapter 8 concentrates on some aspects of metal clusters in macromolecules. This chapter also describes the monomolecular or aggregated distribution of metal complexes in macromolecules. 

In another work, [177] described the particle-size distribution of sulfate lignin based on thermodynamic model. This model suggests the formation of steady-state microstructure by aggregation of primary lignin particle into metastable clusters (aggregates) closely packed. The distribution of the statistical ensembles of particles is broadened and their average size increase with temperature due to the thermal coagulation process. 

---