Fractal coefficients

The determination ofsurface fractal coefficients from AFM measurements... 

The algorithms most fi equently used for calculation of fractal coefficients from the AFM results are [58] Fourier spectrum integral method, surface-perimeter method, structural function method and variable method. To determine the surface dimension by the Fourier spectrum integral method it is necessary to obtain the picture of the surface 2D FFT generating amplitude and time of the matrix (more detail s are given in paper [58]. Assuming the surface function as f(x,y), the Fourier transform in two-dimensional space can be expressed as [58] ... 

Figure 37. The dependence of specific surface areas and fractal coefficients calculated from sorptomatic (left side) and AFM methods of some nanomaterials.

The fractal dimensions of the surfaces of the materials studied have been calculated using Q-TG and independent techniques. Q-TG results are in good agreement with the results from sorptometry, porosimetry and AFM techniques and can be used for calculation of the pore-size distribution fimctions. Relationships between the specific surface areas and pore diameters and fractal coefficients calculated from sorptometry and AFM methods of selected nanoparticles have been found. 

Synthesis of a new modification of silica soluble in THF is described. At the first synthetic step, a hyperbranched polyethoxysiloxane (HBPES) is synthesized by heterofunctional condensation using triethoxysilanol previously generated in reaction mixture by neutralization of correspondent sodium salt with acetic acid. At this step, the process was monitored by IR spectroscopy, SEC, and Si NMR spectroscopy. At the second step, hydrolysis and intramolecular condensation involving silanol groups is carried out to yield silica sol macromolecules. A SAXS method was used to determine the size and fractal coefficient of trimethylsilated derivatives and silica sols obtained. An atomic-force microscopy imaging of silica sol supported on a mica substrate showed the silica sol particles to be predominantly spherical in shape. Prospects for theoretical, experimental and practical applications of silica sols are discussed. 

Johansson and coworkers [182-184] have analyzed polyacrylamide gel structure via several different approaches. They developed an analytical model of the gel structure using a single cylindrical unit cell coupled with a distribution of unit cells. They considered the distribution of unit cells to be of several types, including (1) Ogston distribution, (2) Gaussian distribution of chains, and (3) a fractal network of pores [182-184]. They [183] used the equilibrium partition coefficient... 

Obviously, the diffusion coefficient of molecules in a porous medium depends on the density of obstacles that restrict the molecular motion. For self-similar structures, the fractal dimension df is a measure for the fraction of sites that belong... 

In this section, we consider flow-induced aggregation without diffusion, i.e., when the Peclet number, Pe = VLID, where V and L are the characteristic velocity and length and D is the Brownian diffusion coefficient, is much greater than unity. For simplicity, we neglect the hydrodynamic interactions of the clusters and highlight the effects of advection on the evolution of the cluster size distribution and the formation of fractal structures. 

The foregoing treatment can be extended to cases where the electron-ion recombination is only partially diffusion-controlled and where the electron scattering mean free path is greater than the intermolecular separation. Both modifications are necessary when the electron mobility is - 100 cm2v is-1 or greater (Mozumder, 1990). It has been shown that the complicated random trajectory of a diffusing particle with a finite mean free path can have a simple representation in fractal diffusivity (Takayasu, 1982). In practice, this means the diffusion coefficient becomes distance-dependent of the form... 

Figure 5. The dependence of the coefficient p in the equation (3) on relative fractal free volume J for PAA solid state imidization. The notation is the same, that in figure 2.

Table 6. Relationship between the fractal dimension dp the exponent for the molar mass dependence of the second virial coefficient and the expected exponent m for the osmotic modulus when the scaling assumptions of Eqs. (93)-(96) are made. The experimental data were derived from the exponents for the second virial coefficient...

The electron-ion recombination in high-mobility systems has also been analyzed in terms of the fractal theory [24,25]. It was postulated that even when the fractal dimension of particle trajectories is not equal to 2, the motion of particles is still described by diffusion but with a distance-dependent effective diffusion coefficient D r) = D(H-// where the parameter / is proportional to the mean free path X [24]. However, when the fractal dimension of trajectories is not equal to 2, the motion of particles is not described by orthodox diffusion. 

This is an inner loop used in the same spirit as in traditional Julia set computations. No complex numbers are required for the computation. Hold three of the coefficients constant and examine the plane detemined by the remaining two. This code runs in a manner similar to other fractal-generating codes in which color indicates divergence rate. rmu is a quaternion constant. 

These starburst dendrimers have been subjected 47 to two different fractal analyses I48 49 (a )A c/2 D)/2, where A is the surface area accessible to probe spheres possessing a cross-sectional area, o, and the surface fractal dimension, D, which quantifies the degree of surface irregularity and (b) A = dD, where d is the object size. Both methods give similar results with D = 2.41 0.04 (correlation coefficient = 0.988) and 2.42 0.07 (0.998), respectively. Essentially, the dendrimers at the larger generations are porous structures with a rough surface. For additional information on dendritic fractality, see Section 2.3. 

If the von Smoluchowski rate law (Eq. 6.10) is to be consistent with the formation of cluster fractals, then it must in some way also exhibit scaling properties. These properties, in turn, have to be exhibited by its second-order rate coefficient kmn since this parameter represents the flocculation mechanism, aside from the binary-encounter feature implicit in the sequential reaction in Eq. 6.8. The model expression for kmn in Eq. 6.16b, for example, should have a scaling property. Indeed, if the assumption is made that DJRm (m = 1, 2,. . . ) is constant, Eq. 6.16c applies, and if cluster fractals are formed, Eq. 6.1 can be used (with R replacing L) to put Eq. 6.16c into the form... 

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]. 

In this context, Berry [277] studied the enzyme reaction using Monte Carlo simulations in 2-dimensional lattices with varying obstacle densities as models of biological membranes. That author found that the fractal characteristics of the kinetics are increasingly pronounced as obstacle density and initial concentration increase. In addition, the rate constant controlling the rate of the complex formation was found to be, in essence, a time-dependent coefficient since segregation effects arise due to the fractal structure of the reaction medium. In a similar vein, Fuite et al. [278] proposed that the fractal structure of the liver with attendant kinetic properties of drug elimination can explain the unusual... 

It has been stated that heterogeneous reactions taking place at interfaces, membrane boundaries, or within a complex medium like a fractal, when the reactants are spatially constrained on the microscopic level, culminate in deviant reaction rate coefficients that appear to have a sort of temporal memory. Fractal kinetic theory suggested the adoption of a time-dependent rate constant , with power-law form, determined by the spectral dimension. This time-dependency could also be revealed from empirical models. 

This form is very similar to the model often used when the molecules move across fractal media, e.g., the dissolution rate using a time-dependent coefficient given by (5.12) to describe phenomena that take place under dimensional constraints or understirred conditions [16]. The previous differential equation has the solution given by (9.9). 

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