Mass fractal dimension number

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... 

Koylu et al. used extensive TEM studies to derive the mass fractal dimension, Df, of flame-made soot particles from Equation A1 (see the appendix) by extracting the number of primary particles per agglomerate, N, the primary particle size, (ip, and the radius of gyration of the agglomerate, R. The constant fractal prefactor, A, as well as the fractal dimension, Dj, were then determined by regression. 

There are a number of different fractal dimensions commonly used to describe a specific property of a system. These fractal dimensions and methods to obtain them are explained in detail in Boundary and Surface Fractal Dimensions and Mass Fractal Dimension. Fractal dimensions used in specific applications will be shown also in the related section. 

A concept of mutual transparency or opacity based on the relative evolution of fractal dimension and radius of the clusters has been developed by Mandelbrot [38]. The tendency of fractal systems to interpenetrate is inversely related to the mean number of intersections 2 of two mass fractal objects of size and mass fractal dimensions Dj and D2 placed in the same region of space of dimension d ... 

Betti numbers can be applied to prefractal systems. For example, Fig. 3-7 shows two deterministic Sierpinski carpets with the same mass fractal dimension, dm = 1.896 and Euler-Poincare number, En = 0. The two constructions are topo-... 

We recommend that future research in this area focus on establishing theoretical and/or empirical relations between the prefactor and exponent in the power law relating K to Pn, and quantitative pore characteristics, such as the mass fractal dimension and Betti numbers. In particular, there is an urgent need for additional experimental studies, in which K is determined over a wide range of Pn on undisturbed samples of natural porous media that are well-characterized in terms of their pore space geometry. Such data are required for model testing. 

Figure 2.22 Influence of physical pixel size (image pixel size translated into actual physical length on the soil profile) on a number of mass fractal dimensions of the preferential pathway in Figure 2.21. Open symbols correspond to images thresholded with the intermeans algorithm, whereas full symbols correspond to the minimum-error algorithm. Circles, squares, and diamonds are associated with the box-counting, information and correlation dimensions respectively. (Modified from [81].)...

Figure 3.1 A log-log plot of number of particles iVo.a in an aggregate against radins of gyration of the aggregate. The data are obtained from computer simulations and show three different mass fractal dimensions, of 1.74, 2.11 and 2.41.

Figure 3.10 The number of boxes versus box size from the floe shown in Figure 3.9. The slope of the curve as log L goes to zero returns the fractal dimension. Da = 1.79 for this case, which is somewhat lower than the mass fractal dimension of 1.88 due to the finite size effects analysed by Nelson et al. [41].

Dinsmore and Weitz (2002) [60] examined a model system of polymethylmethacrylate particles dyed with rhodamine in a density- and refractive-index-matched solution of decalin and cyclohexylbromide. They used CLSM to follow the (very slow) aggregation in real time and determined the particle positions in full three-dimensional detail. From this they performed a comprehensive structural analysis of the gels, including measurement of coordination numbers and backbone fractal dimension of the structures, as well as the much more commonly measured mass fractal dimension. 

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... 

The development of new molecular closure schemes was guided by analysis of the nature of the failure of the MSA closure. In particular, the analytic predictions derived by Schweizer and Curro for the renormalized chi parameter and critical temperature of a binary symmetric blend of linear polymeric fractals of mass fractal dimension embedded in a spatial dimension D are especially revealing. The key aspect of the mass fractal model is the scaling relation or growth law between polymer size and degree of polymerization Ny cr. The non-mean-field scaling, or chi-parameter renormalization, was shown to be directly correlated with the average number of close contacts between a pair of polymer fractals in D space dimensions N /R if the polymer and/or... 

Values of d calculated according to Eq. 4 for porous aluminas equal 3.04 0.02 over the range r = 2-195 A. Pfeifer et al. interpreted this as evidence of fractally rough surfaces. Avnir [16] has compiled a table that summarizes a large number of fractal analyses of adsorption data obtained for natural and synthetic oxides. A portion of this table is listed in Table 4. It appears that many porous silicates and aluminates can be described by a surface or mass fractal dimension. 

Mass fractal dimension vs. (mean) DPA absorption number of carbon black. (Mass fractal data from C.R. Herd, G.C. McDonald, R.E. Smith, W.M. Hess., Rubb. Chem. Technol., 66,491-509,1993. DBPA data from Table 4.5.)... 

During film formation by dip-coating, primary sol species are rapidly aggregated by evaporation of solvent. The porosity of the secondary aggregate structure depends on the extent of interpenetration of the primary species. The extent of interpenetration is inversely related to the mean number of intersections Mi 2 of two polymers of radius rc and mass fractal dimension D1 and D2 confined to the same region of three-dimensional space (2) ... 

In fact, if one measures the total number of bonds (sites) on the infinite cluster at the percolation threshold (pc) in a (large) box of linear size L, then this number or the mass of the infinite cluster will be seen to scale with L as where die (< d) is called the fractal dimension of the infinite cluster at the percolation threshold. Similar measurements for the backbone (excluding the dangling ends of the infinite cluster) give the backbone mass scaling as, de < die, where dfi is called the backbone (fractal) dimension. In fact, die can be very easily related to the embedding Euclidean dimension d of the cluster by... 

The fractal dimension d of the main set of bonds (i.e., of the frame) (Fig. 12) obtained from the iteration process for po = l (all bonds colored black) can be defined from the dependence of the set mass (i.e., the number of the constituent bonds of the frame), M f at iteration stage number n on the linear dimension of the lattice ln ... 

Fractal dimension, D is considered as an effective number that characterises the irregular electrode surface. The term has been related to physical quantities such as mass distribution, density of vibrational stages, conductivity and elasticity. If we consider a 2-D fractal picture in its self-similar multi-steps, one can draw various spheres of known radii at various points of its structure and may thus count the number of particles, N inside the sphere by microscope, following relation will then hold good ... 

Problem 7-24. Sedimentation of a Colloidal Aggregate. Colloidal particles often aggregate because of London-van der Waals or other attractive interparticle forces unless measures are taken to stabilize them. The aggregation kinetics are such that the aggregate formed has a fractal dimension Df, which is often less than the spatial dimension. The fractal dimension measures the amount of mass in a sphere of radius R, i.e., mass R D<. For a fractal aggregate composed of Aprimary particles of radius Op with mass mp, estimate the sedimentation velocity of the aggregate when the Reynolds number for the motion is small. What is the appropriate Reynolds number ... 

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