Fractals triangular lattice

The behaviour of surface reaction is strongly influenced by structural variations of the surface on which the reaction takes place [23], Normally theoretical models and computer simulations for the study of surface reaction systems deal with perfect lattices such as the square or the triangular lattice. However, it has been shown that fractal-like structures give much better description of a real surface [24], In this Section we want to study the system (9.1.39) to (9.1.42). 

To proceed, consider two finite planar networks, a regular Euclidean triangular lattice (interior valence v — 6, but with boundary defect sites of valence v = A and v = 2) of dimension d — 2, and a fractal lattice... 

Figure 4.9. The A = 123 Sierpinski gasket, a two-dimensional uncountable set with zero measure and Hausdorff (fractal) dimension 3/ 2 = 1.584962... the companion Euclidean lattice referred to in the text is a space filling triangular lattice of (interior valency v = 6.

This result can be checked numerically for the sequence of fractal lattices defined by A = 15, 42, 123, and for the sequenee of triangular lattices defined by A = 15, 45, 153. The results of these calculations are presented... 

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. 

This subsection is devoted to the Sierpinski gasket d = 2) and its corresponding sponge d = 3), further on called Sierpinski triangular lattices. This fractal is characterized by a mass fractal dimension ds = ln(d -I- l)/ln2, which depends on the embedding spatial dimension d (see Fig. 2 for examples in d = 2 and d = 3). Note that for Sierpinski lattices in general, the Euclidean distance r between two lattice sites scales as the topological distance , i r, so that there is only one mass fractal dimension ds, M. ... 

Two features of these evolution curves are immediately noticeable. First, for both initial conditions, trapping (reaction) on the fractal lattice is distinctly slower than reaction on the triangular one. At first sight, this result would appear to be anomalous inasmuch as the space-filling triangular... 

Let us consider the case of deterministic fractals first, i.e. self-similar substrates which can be constructed according to deterministic rules. Prominent examples are Sierpinski triangular or square lattices, also called gasket or carpet (in d = 2) and sponge (in d = 3), respectively, Mandelbrot-Given fractals, which are models for the backbone of the incipient percolation cluster, and hierachical lattices (see for instance the overview in Ref. [21]). In this chapter, however, we restrict the discussion to the Sierpinski triangular and square lattice for brevity. 

This generalized des Cloizeaux relation is in very good agreement with numerically ob-tmned values [74], The second term in Eq. (43) has its origin in the (self-similar) disordered nature of the backbone of critical percolation clusters and is expected to be absent on deterministic fractals such as the Sierpinski lattice. EE results support this conclusion as shown in Section 4 for both triangular and square Sierpinski lattices. 

This is another example of a stretched exponential behavior at long times. In principle, one can apply the same calculations to any network built from domains of arbitrary internal architecture, as long as the relaxation spectrum inside the domains obeys a power-law form, see Eq. 155. For instance, the networks may belong to any type of regular lattice topology (bcc, fee, tetrahedral, triangular, hexagonal) or even be fractal structures. 

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