Fractal dimension determination

Table 1 Surface fractal dimension determined by nitrogen adsorption...

Here, the surface fractal dimension determined by using the MP method is denoted by dF MP. The probe molecules need not be... 

Therefore, we can define a length scale range where the rough surface shows the fractal behavior by determining the inner and outer cutoffs using Eq. (28).121 In other words the fractal dimensions determined from the power relation between 7peak and... 

The self-similar fractal dimension determined from Figure 9 by Eq. (35) is listed in Table 1 (3rd column). The values of d7 ss... 

Regarding the electrochemical method, the generalized forms of the Cottrell relation and the Randles-Sevcik relation were theoretically derived from the analytical solutions to the generalized diffusion equation involving a fractional derivative operator under diffusion-controlled constraints and these are useful in to determining the surface fractal dimension. It is noted that ionic diffusion towards self-affine fractal electrode should be described in terms of the apparent self-similar fractal dimension rather than the self-affine fractal dimension. This means the fractal dimension determined by using the diffusion-limited electrochemical method is the self-similar fractal dimension irrespective of the surface scaling property. 

Here, Nmono is the number of adsorbed molecules to form a monolayer for each probe molecule and surface fractal dimension determined by using the MP method. The probe molecules need not to be spherical, provided they belong to a homologous series for which the ratio [linear extent rm]2 to molecular cross-sectional area Ac is the same for all members, i.e., an isotropic series. In this case, Eq. (13) turns into... 

Apore.sp Pore fractal dimension determined by using the... 

It is important to note that in some cases, the fractal dimensions estimated by different methods are not the same. It is therefore necessary that one be aware of the particular method used and its interpretation this is mandatory for comparisons of dimensions of different data sets (65). One of the future goals within our laboratory is to continue investigating and quantifying the fink between the fractal dimensions determined by rheology and microscopy. 

The nonlinear form of the mass exponent in Fig. 9a, the convex form of the singularity spectmm/(/i) in Fig. 9b, and the fit to f(q) in Fig. 13, are all evidence that the interstride interval time series are multifractal. This analysis is further supported by the fact that the maxima of the singularity spectra coincide with the fractal dimensions determined using the scaling properties of the time series using the allometric aggregation technique. 

For imprinted mesoporous carbons, the overall fractal dimension, determined from gas adsorption data, indicate that these materials are composed of two groups of pores. The surface fractal dimension of the carbonization-induced pores surface and that of the silica-imprinted pores surface has been obtained from TEM image analysis [38]. 

Strucmre and fractal dimension determine packing and water permeability of a deposit. 

Surface fractal dimension determined by using the multiprobe method using adsorption Surface fractal dimension determined by using the single-probe method using gas adsorption... 

Shustov, G. B. Kozlov, G. V. The Techniques of Fractal Dimensions Determination for Polymeric Materials. (Preprint). Nal chik, Publishers KBSU, 2006, 53 p. 

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