Isotropic fractals

The word fractal was coined by Mandelbrot in his fundamental book.1 It is from the Latin adjective fractus which means broken and it is used to describe objects that are too irregular to fit into a traditional geometrical setting. The most representative property of fractal is its invariant shape under self-similar or self-affine scaling. In other words, fractal is a shape made of parts similar to the whole in some way.61 If the objects are invariant under isotropic scale transformations, they are self-similar fractals. In contrast, the real objects in nature are generally invariant under anisotropic transformations. In this case, they are self-affine fractals. Self-affine fractals have a broader sense than self-similar fractals. The distinction between the self-similarity and the selfaffinity is important to characterize the real surface in terms of the surface fractal dimension. 

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... 

The surface fractal dimension c/slirf of the porous materials can be determined from the TEM image by using perimeter-area method54154 159. If the scaling property of the porous materials is undoubtedly isotropic, the 3-D pore surface is simply related to the projection of the 3-D pore surface onto the 2-D surface. It is well known154 155 that the area. I and the perimeter P of the self-similar lakes are related to their self-similar fractal dimension c/pss by... 

The replacement rule we have used so far to generate geometric fractals creates isotropic fractals. In other words, the property of geometric self-similarity is... 

The observed almost universal value of the surface fractal dimension ds 2.6 of furnace blacks can be traced back to the conditions of disordered surface growth during carbon black processing. It compares very well to the results evaluated within the an-isotropic KPZ-model as well as numerical simulations of surface growth found for random deposition with surface relaxation. This is demonstrated in some detail in [18]. 

The relationship between the exponent v, (v = lnp/lnfc), and the fractal dimension Dp of the excitation transfer paths may be derived from the proportionality and scaling relations by assuming that the fractal is isotropic and has spherical symmetry. The number of pores that are located along a segment of length Lj on the jth step of the self-similarity is / , — pi. The total number of pores in the cluster is S nj (pJf, where d is the Euclidean dimension... 

In order to establish the relationship between the static and dynamic fractal dimensions, the initial conditions of the classical static percolation model must be considered for the solution of differential equation (89) which can be written as 0 = Qs = 1 for D = Ds. Here the notation s corresponds to the static percolation model, and the condition s = 1 is fulfilled for an isotropic cubic hyperlattice. The solution of (89) with the above-mentioned initial conditions may be written as... 

The results of calculations of the effective Poisson s ratio vp dependence on the bulk concentration of a rigid phase p at various values of a = log i/C/Au) are shown in Fig. 53. The calculations were made for Poisson s ratios of the phases ranging from 0.1 to 0.4. It can be seen that at percolation threshold Poisson s ratio of the isotropic fractal composite is vp = 0.2, when K jK > 0 it is also independent of the Poisson s ratios of the individual components of the composite. The Poisson s ratio obtained by us near the percolation threshold is in agreement with computer simulation results and the conjecture of Arbabi and Sahimi [161]. It has been shown that an approximate theoretical treatment of percolation on a cubic lattice exactly reproduces the Poisson s ratio obtained in computer simulation at the percolation threshold. This result may encourage one to use this approximation to describe various elastic properties of composites. It is worth noting that some critical indices have been calculated recently with a high degree of accuracy in the context of the present model. 

Zeng, Y., C.J. Gantzer, R.L. Payton, and S.H. Anderson. 1996. Fractal dimension and lacunarity of bulk density determined with x-ray computed tomography. Soil Sci. Soc. Am. J. 60 1718-1724. Zhan, H., and S.W. Wheatcraft. 1996. Macrodispersivity tensor for nonreactive solute transport in isotropic and anisotropic fractal porous media Analytical solutions. Water Resour. Res. 32 3461-3474. 

Figure 2.12 Special textures arising in theory, (a] Stripes, which attain the Wiener bounds of the maximal and minimal effective slip, if oriented parallel or perpendicular to the pressure gradient, respectively (b) the HS fractal pattern of nested circles, which attains the maximal/minimal slip among all isotropic textures (patched should fill up the whole space, but their number is limited here for clarity) and (c) the Schulgasser and (d) chessboard textures, whose effective slip follows from the phase-interchange theorem (adapted from. ) Abbreviation HS, Hashin-Shtrikman.

The authors of Ref. [55] considered two main case of crack presentation by a stochastic self-affine fractal (coordinates quasihomogeneous tension occurs at scaling) and by usual isotropic fractal. For the indicated cases it can be written, respectively [54] ... 

Hence, the stated above results have shown, that the correct description of fracture process of phenylone and particulate-filled nanocomposites on its basis can be obtained within the frameworks of fiiactal model only [54], In addition the fracture crack should be simulated by an isotropic fractal [55],... 

For an isotropic ensemble of platelets, the detection of scattering takes place on a sphere of radius Q in the platelet-associated coordinate system shown in Fig. 7 [7]. Only if the Q sphere intersects the rod, will intensity contributions be expected. Thus, the detection element is diluted on the Q surface by a factor of (47rQ )" For small enough Q, this immediately yields the Q behavior of the scattering behavior from thin plates as invoked from the fractal considerations above. Such a behavior implies a divergence at Q equal zero. For finite plates this is avoided, however, by the total immersion of the very small Q sphere within the rod. There the finite inner intensity contribution of the rod is accumulated completely for all orientations. This then yields a scattering intensity proportional to the size of the aggregates. This situation is depicted in Fig. 8. 

The structure factor function is a part of the intensity function that arises from constructive scattered waves that originate from different particles. This function is a representation of the probability that a particular particle is surrounded by another particle. For isotropic systems, two types of structure factors can be distinguished (i) correlated systems that describe packing of colloidal particles, and (ii) polymeric systems that can be described by either fractalic or worm-like models. By convention, the structure function is weighted over the square of the... 

Zhan, H. and S.W. Wheatcraft. 1996. Macrodispersivity tensor for nonreactive solute transport in isotropic and anisotropic fractal porous media Analytical solutions. Water Resources Research 32(12) 3461-3474. 

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