Characteristics mass fractal

Characteristic for a fractal structure is self-similarity. Similar to the mentioned pores that cover all magnitudes , the general fractal is characterized by the property that typical structuring elements are re-discovered on each scale of magnification. Thus neither the surface of a surface fractal nor volume or surface of a mass fractal can be specified absolutely. We thus leave the application-oriented fundament of materials science. A so-called fractal dimension D becomes the only absolute global parameter of the material. 

Ultra small-angle x-ray scattering (USAXS) is capable of determining all characteristics of agglomerate and primary particles from a single scattering experiment mass fractal dimension, Dj, radii of gyration (primary particles and... 

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. 

Environmental particles, microbial colonies, and even patterns of movement of organisms can be characterized in terms of several different fractal dimensions [14] (Table 1.1). The fractal dimension of the surface Ds (boundary/interface) of a solid structure is obviously an important characteristic, but many of the physical properties of solids also depend on the scaling behavior of the entire solid and/or of its pores. Systems where surface and mass scale similarly are termed mass fractal systems, those where surface and the pore volumes scale similarly are described as pore fractals and systems where only the surface is fractal are designated as surface fractals (Table 1.1 Figure 1.2). 

Baveye, P, Boast, C.W., Ogawa, S., Parlange, J.-Y. and Steenhuis, T. (1998). Influence of image resolution and thresholding on the apparent mass fractal characteristics of preferential flow patterns in fleld soils. Water Resour. Res., 34(11), 2783-2796. 

In a mass fractal, the entire bulk of mass solid is not compact but porous. For mass fractal, the mass m inside a characteristic length scale r scales according to... 

Donatti DA, VoUet DR, Ibanez Ruiz A, Mesquita A and Silva TEP (2005) Mass fractal characteristics of silica sonogels as determined by smaU-angle x-ray scattering and nitrogen adsorption. Phys Rev B 71 014203-1 014203-7... 

Another important class of morphology is fractal, which includes mass fractal and surface fractal. Although mass and surface fractals appear remarkably different, they share a common trace essential to fractal objects self-affinity. Self-affinity means that the morphology of interest looks the same under different length scales, i.e., within certain limits, the essential geometric characteristics of a fractal object are unchanged by magnification. 

Widgeon, S. J., Sen, S., Mera, G., lonescu, E., Riedel, R., Navrotsky, A. (2010). Si-29 and C-13 solid-state NMR spectroscopic study of nanometer-scale structure and mass fractal characteristics of amorphous polymer derived sibcon oxycarbide ceramics. Chemistry of Materials, 22(23), 6221-6228. doi 10.1021/cml021432. 

In the literature on particle aggregation and droplet coalescence, a mass-based NDF nmf, X, M) is often preferred because mass is a conserved quantity. For these systems, it is possible to relate the mass of a particle to its characteristic length using M = kuLPf where 1 < Df < 3 is he fractal dimension. Using the procedure outlined above, we can show that the length-based and mass-based NDF are related by l(T) =... 

A fundamental characteristic of a fractal is its fractal dimension. Suppose we draw a sphere of radius r around a point in the object. If the fractal object is a line, the mass M(r) within the sphere will be proportional to r. If it is a sheet, then M(r) oc r2. If it is a solid three-dimensional object, M r) would be proportional to r3. In a fractal, the following general relation is obeyed,... 

In Fig. II, the dependences of ratios GJG and where G and T are shear modulus and melt viscosity of nanocomposite, G and T are the same characteristics for the initial matrix polymer, on CaCOj mass contents IT for nanocomposites PP/CaCOj are adduced. Shear modulus G was calculated according to the general fractal Eq. (80) of Chapter 2. MFI reciprocal value was accepted as melt viscosity T measure. The data of Fig. 11 clearly demonstrate, that in case of the studied nanocomposites the... 

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