Density fractal dimension

One characterizes a fractal cluster by its fractal dimension Df which can be determined from the integration of the radius-dependent mass density p r) from the center up to a large radius R as follows ... 

Obviously, the diffusion coefficient of molecules in a porous medium depends on the density of obstacles that restrict the molecular motion. For self-similar structures, the fractal dimension df is a measure for the fraction of sites that belong... 

In their works,51"54 the self-similar fractal dimension dF>ss of the two-dimensional distribution of the pits was determined by the analysis of the digitized SEM images using the perimeter-area method. The value of dF>ss increased with increasing solution temperature,51 and it was inversely proportional to the pit shape parameter and the pit growth rate parameter.53 Keeping in mind that dr>ss is inversely proportional to the increment of the pit area density, these results can be accounted for in terms of the fact that the increment of the pit area density is more decelerated with rising solution temperature. 

Pure nickel electrodeposits with macropores were prepared from electrolytic solutions of 0.2 mol dm NiCl2 and NH4CI with concentrations varying between 0.25 and 4 mol dm [64]. The effects of the electrodeposition current density and the NH4CI concentration on the surface morphology were determined. Surface area, faradaic efficiency, and fractal dimension... 

Hydrolysis of tetraalkoxysilanes in pure water is usually incomplete, but it is more effectively carried out in an alcohol-water mixture and can be catalyzed by H+ or weak bases such as ammonia. Polymerization occurs in the range pH 2-7 under neutral conditions, the rate of polymerization is limited by the slowness of hydrolysis, whereas in acidic media hydrolysis is complete before polymerization begins. The nanostructure of the gel (density of particles, fractal dimensions of particle clusters, and degree of cross-linking of particles to form a network) is controlled by these conditions. 

In an investigation of the spin-density (voidage) and spin-lattice relaxation time maps of many pellets, it was found that it was the heterogeneity in pore size, as characterized by the fractal dimension of the Ti map, that was consistent between images of pellets drawn from the same batch 58). The fractal dimensional of these images identifies a constant perimeter-area relationship for clusters of pixels of... 

Ferri, F., Greco, M., Arcovito, G., De Spirito, M., and Rocco, M. (2002). Structure of fibrin gels studied by elastic light scattering techniques Dependence of fractal dimension, gel crossover length, fiber diameter, and fiber density on monomer concentration. Phys. Rev. E. 66(1 Pt 1), 011913. 

Dyer, K.R., and Manning, A.J. (1999) Observation of the size, settling velocity and effective density of floes, and their fractal dimensions. Neth. J. Sea Res. 41, 87-95. 

In the following table are data on the dependence of the average floccule radius, achieved after 500 s of flocculation, on the initial primary particle number density for a transport-controlled flocculation process. Estimate the fractal dimension of the floccules formed. (Answer D 1.8, based on a log-log plot.)... 

Measurements of the average floccule radius in a suspension of colloids, using light-scattering techniques, indicated the following time dependence R(t) = Ro(l + yty, where Ro = 5 nm, y = 9.3 s 1, and j8 = 0.56. Derive this equation, and estimate the initial number density of primary particles as well as the fractal dimension of the floccules formed. (Answer p0 = 7.5 x 1017 m 3 and D = 1.78.)... 

At low relative pressures p/p0 or thin adsorbate films, adsorption is expected to be dominated by the van der Waals attraction of the adsorbed molecules by the solid that falls off with the third power of the distance to the surface (FHH-regime, Eq. 3a). At higher relative pressures p/p0 or thick adsorbate films, the adsorbed amount N is expected to be determined by the surface tension y of the adsorbate vapor interface (CC-regime, Eq. 3b), because the corresponding surface potential falls off less rapidly with the first power of the distance to the surface, only. The cross-over length zcrit. between both regimes depends on the number density np of probe molecules in the liquid, the surface tension y, the van der Waals interaction parameter a as well as on the surface fractal dimension ds [100, 101] ... 

The properties of fractal ceramic powders depend on their fractal dimensions. The density, p, of a fractal particle depends on its radius, R, and the fractal dimension ... 

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) ... 

Diffusion-limited aggregation of particles results in a fractal object. Growth processes that are apparendy disordered also form fractal objects (30). Sol—gel particle growth has also been modeled using fractal concepts (3,20). The nature of fractals requires that they be invariant with scale, ie, the fractal must look similar regardless of the level of detail chosen. The second requirement for mass fractals is that their density decreases with size. Thus, the fractal model overcomes the problem of increasing density of the classical models of gelation, yet retains many of its desirable features. The mass of a fractal, Af, is related to the fractal dimension and its size or radius, R, by equationS ... 

Our third applications example highlights the work of Nakano et al. in modeling structural correlations in porous silica. MD simulations of porous silica in the density range 2.2—0.1 g/cm were carried out on a 41,472-particle system using an iPSC/860. Internal surface area, ratio of pore surface to volume, pore size distribution, fractal dimension, correlation length, and mean particle size were determined as a function of the density, with the structural transition between a condensed amorphous phase and a low density porous phase characterized by these quantities. Various dissimilar porous structures with different fractal dimensions were obtained by controlling the preparation schedule and the temperature. 

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