Fractal cluster size

Sorensen, C. M., Lu, N. and Cai, J. (1995). Fractal cluster size distribution measurement using static light scattering. J. Colloid Interface ScL, 174, 456-460. 

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). 

Fig. 32. Relationship between Fa, and the dimensionless cluster size (Rla) for fractal clusters (D 2.5) of polystyrene latex in a simple shear flow. Data points are experimental results and the solid line is the theoretical prediction (Sonntag and Russel, 1986,1987a).

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. 

Pig. 40. Growth of average cluster size for area conserving clusters and fractal clusters in the journal bearing flow (Hansen and Ottino, 1996b). 

Due to the characteristic self-similar structure of the CCA-clusters with fractal dimension df 1.8 [3-8, 12], the cluster growth in a space-filling configuration above the gel point O is limited by the solid fraction Oa of the clusters. The cluster size is determined by a space-filling condition, stating that, up to a geometrical factor, the local solid fraction Oa equals the overall solid concentration O ... 

The proportionality constant Nf in Eq. (21) is a generalized Flory-Number of order one (Np=l) that considers a possible interpenetrating of neighboring clusters [22]. For an estimation of cluster size in dependence of filler concentration we take into account that the solid fraction of fractal CCA-clusters fulfils a scaling law similar to Eq. (14). It follow directly from the definition of the mass fractal dimension df given by NA=( /d)df, which implies... 

For Q<0, this distribution function is peaked around a maximum cluster size (2Q/(2Q-1))< >, where < > is the mean cluster size. 2Q=a+df1 is a parameter describing details of the aggregation mechanism, where a1 is an exponent considering the dependency of the diffusion constant A of the clusters on its particle number, i.e., A NAa. This exponent is in general not very well known. In a simple approach, the particles in the cluster can assumed to diffusion independent from each other, as, e.g., in the Rouse model of linear polymer chains. Then, the diffusion constant varies inversely with the number of particles in the cluster (A Na-1), implying 2Q=-0.44 for CCA-clusters with characteristic fractal dimension d =l.8. 

The strong-, weak- and intermediate regimes are all a product of the elastic constant of the basic mechanical unit (the floe, the links between the floes, or a combination of both) and the number of these units present in the direction of the externally applied force (Shih et al. 1990). Therefore, the fractal dimension defines to the size of the clusters. A large fractal dimension represents a large cluster that translates to less cluster-cluster interactions per unit volume and a decreased elastic modulus. At high volume fractions, cluster size decreases and the number of cluster-cluster interactions increases, and thus the elastic constant also increases. 

Any discussion of fumed silica particle structure has to take into account this enormous difference. The approach of mass fractal dimension may provide a rough but helpful estimation. A real mass fractal is limited by the size of the cluster as an upper limit and the size of the particles as a lower limit. Then, the density of the cluster pduster be calculated from the true density of the particle Pparticio fhe ratio of the cluster size c/ciuster to the particle size /particle and the mass fractal dimension /) of the cluster (Eq. 2) ... 

An increase of the water content to = 12 (r v = 6.4 for the standard) did not result in significant changes. Decrease of the values shows a trend towards lower fractal dimensions and cluster sizes. At = 1 (stoichiometric water concentration is at - 2) no SAXS scattering could be observed. 

Jullien, R. and Botet, R., Aggregation and Fractal Aggregates, World Scientific, Singapore, 1987. Berry, M.V. and Percival, I.C., Optics of fractal clusters such as smoke. Opt. Acta, 33, 577, 1986. Ereltoft, T., Kjems, J.K., and Sinha, S.K., Power-law correlations and finite-size effects in silica particle aggregates studied by small-angle neutron scattering, Phys. Rev. B, 33, 269, 1986. 

Nelson, J., Test of a mean field theory for the optics of fractal clusters, J. Mod. Opt., 36, 1031, 1989. Singham, S.B. and Borhen, C.F., Scattering of nnpolarized and polarized light by particle aggregates of different size and fractal dimension, Langmuir, 9, 1431, 1993. 

There is, however, one basic principle, which derives from the fact that fractal clusters are scale invariant. When a gel is formed, clusters of size Rg make bonds with each other via strands at the periphery of the cluster, and the average number of the strands involved per cluster will not depend on cluster size. Since the apparent surface area of a cluster scales with R2 the number of junctions between clusters per unit area of cross section of the gel will scale with R 2. By using the equation for Rg we arrive at the following equation for the shear modulus of a fractal particle gel ... 

Most gels, however, form structures between these two extremes. The hydrolysis and polycondensation of metal alkoxide precursors yield an initial formation of fractal clusters, which upon reaching a critical size begin to form a continuous network or skeleton via cluster-cluster aggregation. Both the initial cluster formation and the subsequent aggregation of clusters are fractal in nature. There is a very distinct difference between these two processes, however, as is more evident in Sec. X. 

From the viscosity versus time data collected for each system, the molecular weight and cluster size can be calculated as a function of time to the gel point using the equations derived in the first few sections of this chapter. From these calculations, it is possible to model the evolution of the gel structure dynamically. In Fig. 11, the evolution of an HF-catalyzed gel as a function of time is presented, starting with the unhydrolyzed monomer molecule tetra-ethoxysilane at time zero and proceeding, in stages, to the final gel structure at 16 minutes. Also presented is the effect of normal drying, in which the fractal characteristics of the individual clusters are all but eliminated. 

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