Fractals infinitely ramified

This article analyzes adsorption kinetics of fractal interfaces and sorption properties of bulk fractal structures. An approximate model for transfer across fractal interfaces is developed. The model is based on a constitutive equation of Riemann-Liouville type. The sorption properties of interfaces and bulk fractals are analyzed within a general theoretical framework. New simulation results are presented on infinitely ramified structures. Some open problems in the theory of reaction kinetics on fractal structures in the presence of nonuniform rate coefficients (induced e.g. by the presence of a nonuniform distribution of reacting centres) are discussed. 

The sorption properties of interfaces and bulk fractals are discussed within the same framework, and some new simulation results on infinitely ramified fractals are also presented. Finally, we close with a brief digression on open problems in the presence of heterogeneity in kinetics. 

Figure 3 Infinitely ramified fractal structure considered in section 3.

An open problem in the theory is to ascertain whether eq. (16) also holds for infinitely ramified fractal structures. This problem can be tackled either by means of Green function renormalization or by considering lattice simulations. 

Green function renormalization can be applied to product lattices by making use of the extension theory [11]. Some preliminary calculations (performed by considering the feed from a single site, i.e. dj = 0) suggest that eq. (16) may also hold for infinitely ramified fractals as long as (d/ - dj)/da < 1, [12]. If (d/ - dj)ld > 1, then M t)/Mco t. 

It should also be observed that, in the case of an infinitely ramified fractal structure fed from all the perimeter sites, an exponent 0 fairly close to 0 = 1/2 (i.e. the regular case) is obtained, 0.40 < 0 < 0.50. The small difference between the exponent 0 obtained for infinitely ramified fractals and in the regular Euclidean case makes it difficult to discriminate between the two scaling behaviors, especially if one makes use of numerical simulations on small lattices. 

This article also discusses a general scaling theory for sorption on bulk fractals and across fractal interfaces based on the results obtained by applying Green function renormalization to finitely ramified structures. Numerical simulations of batch sorption kinetics on infinitely ramified structures confirm the validity of the scaling expression eq. (17). 

In the case of fractal substrates, one has to distinguish between two main subclasses of structures, namely deterministic and random fractals. Within the class of deterministic fractals, one additionally has a subdivision in finitely and infinitely ramified fractals. Here, (either finite or infinite) ramification refers to the number of cut operations which are required to disconnect any given subset of the structure, the upper limit of which is independent of the chosen subset [7,8]. An example of a finitely ramified structure is the Sierpinski triangular lattice, whereas the Sierpinski square lattice is an example of an infinitely ramified structure. See Figs. 2(a) and 6 in Section 4 for the respective sketches of these structures in d = 2. 

Renormalisation group (RG) techniques have been applied to several finitely ramified structures, so that results are available for some deterministic fractals including Sierpinski triangular lattices [40-47] (for a comprehensive discussion see Ref. [48]). For infinitely ramified structures, there is no RG result available and one has to rely on numerically evaluating SAWs on these fractals (note, however, the study of Taguchi [49] of SAWs on Sierpinski square lattices). Nonetheless, even in the former case when RG results are available, it is instructive to apply munerical schemes as mentioned in the Introduction. 

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