Mass fractal dimension approach

Therefore, it appears likely that the approach considering the solid volume of primary aggregates, as evaluated from the two-dimensional cross-section area by Eq. (12), leads to an overestimation of the mass fractal dimension. A more realistic estimate is obtained with Eq. (13). By referring to Eq. (12), the data obtained by Herd et al. [108] show a successively increasing value of the mass fractal dimension from to df 2.8 with increas-... 

A very helpful approach to understand the formation of particle clusters was given by Meakin [2], who uses the concept of mass fractal dimension ) , which relates the size cf of a particle cluster to its mass M (Eq.l). 

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) ... 

Pillared clays, layered silicates whose sheets have been permanently propped open by thermally stable molecular props, were used as templates to load various organic precursors.The pore diameters of resulting carbon materials range from 8 to 22 A, and the mass fractal dimension varies from 2.5 to 2.9, which are accessible to lithium ions when the intercalation process takes place in a lithium secondary battery. An approach to pyrolyse aromatic hydrocarbons such as pyrene within a pillared clay was also reported, in which the pillared clay serves two functions.It acts as the inorganic template around which the carbon can be formed, and it also functions as an acid catalyst to promote condensation of the aromatics similar to the Scholl reaction. The resulting carbon materials have pore sizes from 15 to 50... 

Surface fractal dimensions of a number of Cambisols and Luvisols were determined using the FHH equation from data obtained from N2 and water vapor adsorption isotherms. Values were compared with those obtained from the mercury intmsion method and with mass fractal dimensions that were evaluated from particle-size distributions using a modified number-based method [108] (Figure 6.3). This method was proposed by Kozak et al. [116] in order to correct some inconsistencies of previous approaches... 

Near the origin, intensity decays first with q up to 0.4 nm . In the fractal approach, a decay of SAS intensity with a slope a and 2mass fractal behaviour characterized by the mass fractal dimension Dm=a. In the present case, the extension of the q-range where this law is observed covers only one decade so that the mass fractal behaviour may not be reasonably substantiated. We prefer to explain this power law regime with non entire exponent as the combination in the same q-range of features arising from a large polydispersity associated with a lamellar shape of scattering domains, as already observed on carbon materials. [5]. 

One of the most exciting developments of the past decade in the study of colloidal silica is the application of the fractal approach to the study of sols and gels. Fractals are disordered systems for which disorder can be described in terms of nonintegral dimension. The concept of fractal geometry, developed by Mandelbrot (II) in the early 1980s, provides a means of quantitatively describing the average structure of certain random objects. The fractal dimension of an object of mass M and radius r is defined by the relation... 

Snidaro et al. [59] used CSLM to measure the positions of bacteria in aggregates in three-dimensional space. Once the data were obtained they used a three-dimensional sandbox approach to measure the fractal dimension, with the sandbox centred on the centre of mass of the aggregate. They also used a two-dimensional sandbox method to obtain the two-dimensional fractal dimension, related to the usual through the codimension rule. Small aggregates were shown to have a compact structure using both the two- and three-dimensional approaches. 

For Ds = 2, the particles have a smooth surface and the surface is two-dimensional. When the surface roughness increases, Dg can vary between 2 and 3. The surface fractal dimension will approach three for an exceedingly rough surface that also may be described as a structure with a homogeneous distribution of mass and voids. The surface area will then be proportional to the mass R —> 3. It has become three-dimensional. 

Consider a given polymeric cluster with Gaussian size R, mass M and Gaussian fractal dimension Place this cluster in a solvent consisting of other clusters of size r and fractal dimension The screened interaction energy in the Flory approach is in this case U R. Assume now the... 

---