Fractal concepts

Barabasi A L and Stanley H E 1995 Fractal Concepts in Surface Growth (Cambridge Cambridge University Press)... 

Similarly, the values for a =(3/d )-l follow from the Fox-Flory relationship (Eq. 26), again under the condition that (bj, does not change with the number of branching points per cluster. The assumption of constant and constant are not strictly fulfilled. Nonetheless the scafing relationship of Eq. (40), and the corresponding one for the intrinsic viscosity, lead to very reasonable results, which were indeed observed. Here the full power of the fractal concept becomes evident. 

Barabasi, A. L. Stanley H. E. Fractal Concepts in Surfaee Growth Cambridge University Press Cambridge 1995. 

See Chaps. 2 and 3 in J. Feder, op. cit.22 An introduction to fractal concepts with more mathematical detail is given by K. Falconer, Fractal Geometry, Wiley, Chichester, UK, 1990. 

Avnir, D., The Fractal Approach to Heterogeneous Chemistry, Wiley, Chichester, UK, 1989. The first three chapters of this edited treatise provide a comprehensive review of fractal concepts and experimental methodologies to measure the fractal dimension. 

Feder, J., Fractals, Plenum Press, New York, 1988. Perhaps the most accessible introduction to fractal concepts, written with the imprimatur of Benoit Mandlebrot. 

The fractal concept is based on the assumption of reproduction of the general elements of structure of porous materials at all levels—from microscopic to macroscopic ones. This assumption is valid for numerous macroporous materials, while it is too difficult to check its validity for microporous ones. However, based on general thermodynamic considerations, one may assume that fractal concepts also apply to some of microporous materials. As it is shown below, the main condition of the applicability of the fractal approach to microporous materials consists in their homogeneity. However, one has to take into account that this strict analysis does not allow the assumption of homogeneity of any microporous system, not least, because the subsystem micropore-wall of micropore is obviously heterogeneous. Therefore, the fractal concept is probably not applicable to very narrow micropores (ultramicropores, according to Dubinin s classification). 

Fractals were introduced in physical chemistry by Mandelbrot. Chemical applications of fractal concept were analyzed by some authors [1,2,6]. 

Since many of porous materials exhibit random fractal properties, one may assume that the fractal concept is sometimes applicable also to microporous systems. However, for the estimation of the validity of such an approach, we need a general explanation of the phenomenon of random fractal formation. 

Barabasi AL, Stanley HE (1996) Fractal concepts in surface growth. Cambridge University Press Barabasi AL, Araujo M, Stanley HE (1992) Phys Rev Lett 68 3729... 

A. Static Percolation in Porous Materials, Fractal Concept, and Porosity Determination... 

It was shown recently that disordered porous media can been adequately described by the fractal concept, where the self-similar fractal geometry of the porous matrix and the corresponding paths of electric excitation govern the scaling properties of the DCF P(t) (see relationship (22)) [154,209]. In this regard we will use the model of electronic energy transfer dynamics developed by Klafter, Blumen, and Shlesinger [210,211], where a transfer of the excitation... 

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... 

Diffusion-limited aggregation of particles results in a fractal object. Growth processes that are apparendy disordered also form fractal objects (30). Sol—gel particle growth has also been modeled using fractal concepts (3,20). The nature of fractals requires that they be invariant with scale, ie, the fractal must look similar regardless of the level of detail chosen. The second requirement for mass fractals is that their density decreases with size. Thus, the fractal model overcomes the problem of increasing density of the classical models of gelation, yet retains many of its desirable features. The mass of a fractal, Af, is related to the fractal dimension and its size or radius, R, by equationS ... 

Bak, R, Tang, C. and Weisenfeld, K. (1988). Physical Review A, 38, 364 Barabasi and Stanley, H. E. (1995). Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge. 

The emerging concept of fractal geometry has opened a wide area of research. The review will begin with a brief description on the methods for obtaining fractal dimension, followed by the use of fractal concept on pharmaceutical and biological applications with specific examples. 

In applying fractal concept to the protein chain, it is important to check the length scale where the... 

The problem in using fractal concept to describe the carrier-mediated drug transport is that there is no... 

The dendrite growth process may be described on the basis of cluster growth model of diffusion-limited aggregation (DLA) and fractal concepts in surface growth [83, 85],... 

The second aspect of DCM as fractal is that it allows, or encourages, multiple hypotheses. This is significant because it allows us to reframe problems and results in many ways and for many different audiences. As Nuhfer (2006) suggests, fractal concepts help deal with situations with too many variables to approach them in a strictly linear fashion or a way to track things that move through time, both apt descriptions of assessment. This is another way that DCM succeeds because it is not a self-contained metric... 

With proper manipulation of the variables in the generating algorithm, it would be possible to extend this to construct realistic porous structures that closely represent FCC catalyst particles. It would then be straight forward to model complex transport and reaction processes using partinent fractal concepts. ... 

Another approach to the problem of anomalous relaxations uses fractal concepts [187-189,200-203], Here the problem is analyzed using the mathematical language of fractional derivatives [194,200-203] based on the previously mentioned Riemann-Liouville fractional differentiation operator... 

The fractional derivative technique is used for the description of diverse physical phenomena (e.g., Refs. 208-215). Apparently, Blumen et al. [189] were the first to use fractal concepts in the analysis of anomalous relaxation. The same problem was treated in Refs. 190,194,200-203, again using the fractional derivative approach. An excellent review of the use of fractional derivative operators for the analysis of various physical phenomena can be found in Ref. 208. Yet, however, there seems to be little understanding of the relationship between the fractional derivative operator and/or differential equations derived therefrom (which are used for the description of various transport phenomena, such as transport of a quantum particle through a potential barrier in fractal structures, or transmission of electromagnetic waves through a medium with a fractal-like profile of dielectric permittivity, etc.), and the fractal dimension of a medium. 

Thus, it can be concluded that the relaxation time distribution concept applies to Debye-like relaxation (even though its frequency dependence may be smeared-out), whereas it becomes inapplicable for still slower relaxation patterns. In the latter situation, the distribution of relaxation times over a selfsimilar, fractal ensemble seems a physically more reasonable assumption. As is well known, the fractality of geometrical objects implies their non-integer dimension however, a more exact definition of the fractal concept with respect to the ensemble of relaxation times is in order. 

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