Fractal dimension linear

Fractal objects are quantified by their fractal dimension, dj. For linear-like stmctures, 1 < <2. FractaUy rough stmctures have a mass fractal... 

Experimentally accessible is D by means of scattering methods [144], The corresponding fractal analysis of scattering data is gaining special attractivity from its intriguing simplicity. In a double-logarithmic plot of I (s) v.v. s the fractal dimension is directly obtained from the slope of the linearized scattering curve. It follows from the theory of fractals that... 

It is reported116,117 that as more adsorbed layers are built up, the interface between the adsorbent and the adsorbed molecules becomes smooth, and hence the surface fractal dimension would no longer describe the interface but would describe the adsorbed molecule agglomerates. Also, Eq. (9) is only valid when the adsorbed layer exceeds monolayer coverage. Therefore, for the correct calculation, dFSP should be determined from the linear... 

Since the mean square radius of gyration requires a z-average but the molar mass a weight average the fractal dimension remains unchanged only if the ratio is independent of the molar mass or close to unity. These conditions are mostly fulfilled with polydisperse linear chains but not for the randomly branched ones. Here this ratio increases strongly with the molar mass. 

A useful parameter is the fractal dimension, D, which is the exponent in the relation between the mass, M, to a linear dimension, R ... 

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... 

On the other hand, for the microporous carbons with pore size distribution (PSD) with pore fractality, the pore fractal dimensions56,59,62 which represent the size distribution irregularity can be theoretically calculated by non-linear fitting of experimental adsorption isotherm with Dubinin-Astakhov (D-A) equation in consideration of PSD with pore fractality.143"149 The image analysis method54,151"153 has proven to be also effective for the estimation of the surface fractal dimension of the porous materials using perimeter-area method.154"159... 

Before we close this section some major, unique kinetic features and conclusions for diffusion-limited reactions that are confined to low dimensions or fractal dimensions or both can now be derived from our previous discussion. First, a reaction medium does not have to be a geometric fractal in order to exhibit fractal kinetics. Second, the fundamental linear proportionality k oc V of classical kinetics between the rate constant k and the diffusion coefficient T> does not hold in fractal kinetics simply because both parameters are time-dependent. Third, diffusion is compact in low dimensions and therefore fractal kinetics is also called compact kinetics [23,24] since the particles (species) sweep the available volume compactly. For dimensions ds > 2, the volume swept by the diffusing species is no longer compact and species are constantly exploring mostly new territory. Finally, the initial conditions have no importance in classical kinetics due to the continuous re-randomization of species but they may be very important in fractal kinetics [16]. 

Besides the compartment analysis, non-compartment models can be used. One frequently used procedure is the regression method. This method performs a linear regression fit on a voxel basis. The slope image provides information about the trapping of the tracer, while the intercept image reflects the distribution volume of the radiopharmaceutical. Another non-compartment model is based on the calculation of the fractal dimension (FD) (17). FD is a parameter for the heterogeneity and is calculated for the time-activity data of each individual VOI. The values of FD vary from 0 to 2 showing the deterministic or chaotic distribution of the tracer activity. We use a subdivision of 7 x 7 and a maximal SUV of 20 for the calculation of FD. 

For Q<0, this distribution function is peaked around a maximum cluster size (2Q/(2Q-1))< >, where < > is the mean cluster size. 2Q=a+df1 is a parameter describing details of the aggregation mechanism, where a1 is an exponent considering the dependency of the diffusion constant A of the clusters on its particle number, i.e., A NAa. This exponent is in general not very well known. In a simple approach, the particles in the cluster can assumed to diffusion independent from each other, as, e.g., in the Rouse model of linear polymer chains. Then, the diffusion constant varies inversely with the number of particles in the cluster (A Na-1), implying 2Q=-0.44 for CCA-clusters with characteristic fractal dimension d =l.8. 

Apparently the fractal dimension of the excitation paths in sample A is close to unity. Topologically, this value of Dp corresponds to the propagation of the excitation along a linear path that may correspond to the presence of second silica within the pores of the sample A. Indeed, the silica gel creates a subsidiary tiny scale matrix with an enlarged number of hydration centers within the pores. 

Disordered porous media have been adequately described by the fractal concept [154,216]. It was shown that if the pore space is determined by its fractal structure, the regular fractal model could be applied [154]. This implies that for the volume element of linear size A, the volume of the pore space is given in units of the characteristic pore size X by Vp = Gg(A/X)°r, where I), is the regular fractal dimension of the porous space, A coincides with the upper limit, and X coincides with the lower limit of the self-similarity. The constant G, is a geometric factor. Similarly, the volume of the whole sample is scaled as V Gg(A/X)d, where d is the Euclidean dimension (d = 3). Hence, the formula for the macroscopic porosity in terms of the regular fractal model can be derived from (65) and is given by... 

In the context of the SLSP model the relationship between the fractal dimension Ds of the maximal percolating cluster, the value of its size sm, and the linear lattice size L is determined by the asymptotic scaling law [152,213,220]. 

In a critical appraisal of the different methods for determining surface fractal dimensions, Neimark (1990) has stressed the importance of taking account of the different mechanisms of physisorption (e.g. at high p/p° the combination of multilayer adsorption and capillary condensation). Conner and Bennett (1993) have also warned of the risk of an oversimplistic interpretation of a linear log-log fractal plot. 

The particle-counting fractal dimension relates the number of primary particles N in an object to the linear size of the fractal object R, and the linear size of one particle (ct) ... 

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