Fractal lattice

J. Mai, A. Cashes, W. von Niessen. A model for the catalytic oxidation of CO on fractal lattices. Chem Phys Lett 796 358-362, 1992. 

I. Jensen. Non-equilibrium critical behavior on fractal lattices. J Phys A (Math Gen) 24 L1111-L1117, 1991. 

In order to extend the analytical equations to a fractal lattice, we will need the radial distribution function rdf(r) of the Sierpinski gasket, rdf(r) dr being the average number of sites with distance between r and r + dr from a given site. For fractal lattices one has... 

Now we want to apply the above described formalism, Section 5.1, to fractal lattices. Here one has to be careful when performing ensemble averages. Due to the lack of translational invariance, the ensemble averaged concentration (n(f, t)) on a fractal is no longer independent of the position vector r, as it is in normal dimensions. In order to avoid a further complication of the formalism, however, we neglect in the following analytical approach this... 

In summary, we have shown that the kinetics of the bimolecular reaction A + B —> 0 with immobile reactants follows equation (6.1.1), even on a fractal lattice, if d is replaced by d, equation (6.1.29). Moreover, the analytical approach based on Kirkwood s superposition approximation [11, 12] may also be applied to fractal lattices and provides the correct asymptotic behaviour of the reactant concentration. Furthermore, an approximative method has been proposed, how to evaluate integrals on fractal lattices, using the polar coordinates of the embedding Euclidean space. 

Lastly, Argyrakis and Kopelman [33] have simulated A + B -4 0 and A + A —> 0 reactions on two- and three-dimensional critical percolation clusters which serve as representative random fractal lattices. (The critical thresholds are known to be pc = 0.5931 and 0.3117 for two and three dimensions respectively.). The expected important feature of these reactions is superuniversality of the kinetics independent on the spatial dimension and... 

On the fractal lattice, Dv is a constant, and holes exhibit the Gaussian characteristics, The self-similarity of the fractal has dilation symmetry shown in Eq. (8). Using the Fourier-Laplace transformation in time,... 

In accordance with the anomalous diffusion on fractal lattice, we expect [12,15]... 

By looking at Eqs. (27) and (28), Eq. (32) confirms the customary way of relating P of the stretched exponential function, Eq. (19), to the relaxation time spectrum. The glassy state relaxation is dominated by the part of the spectrum having longer relaxation times. The fractal dynamics of holes are diffusive, and the diffusivity depends strongly on the tenuous structure in fractal lattices, v is the exponent in the power-law relationship between local diffusivity and diffusion length ... 

The fractal dynamics of holes are diffusive, and the diffusivity depends strongly on the tenuous structure in fractal lattices. The fractal dimension defines the self-similar connectivity of hole motions, the relaxation spectrum, and stretched exponential... 

P2VN(70,000)/ PS(2200), [17,18] has a higher value of E. Resolution of this problem will require development of a quasi one-dimensional model, perhaps bai. d on a random walk on a fractal lattice. Research along these lines is being pursued by Webber. [26]... 

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... 

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... 

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... 

Hence, the fractal reactions of polymerisation can be divided, as a minimum, into two classes reactions of fractal objects (homogeneous) whose kinetics are described similarly to the curves shown in Figure 10.5, and reactions in a fractal space (nonhomogeneous) whose kinetics are described similarly to the curves shown in Figure 10.6. The reactions of the second class correspond to the formation of structures on fractal lattices [34]. The basic distinction of the pointed classes of reactions is the... 

As has been mentioned before [1], the reason for a variation of D(t) is of course the curing reaction in fractal space. This process in the physical sense is similar to the formation of a cluster with dimension D on fractal lattices with dimension Dj, [7]. In paper [1] it was supposed that D[3t=D. The relationship between D and Dj is given by the following equation [7] ... 

Up to now we considered pol5meric fiiactals behavior in Euclidean spaces only (for the most often realized in practice case fractals structure formation can occur in fractal spaces as well (fractal lattices in case of computer simulation), that influences essentially on polymeric fractals dimension value. This problem represents not only purely theoretical interest, but gives important practical applications. So, in case of polymer composites it has been shown [45] that particles (aggregates of particles) of filler form bulk network, having fractal dimension, changing within the wide enough limits. In its turn, this network defines composite polymer matrix structure, characterized by its fractal dimension polymer material properties. And on the contrary, the absence in particulate-filled polymer nanocomposites of such network results in polymer matrix structure invariability at nanofiller contents variation and its fractal dimension remains constant and equal to this parameter for matrix polymer [46]. 

The authors [51] obtained the following relationship for Floiy exponent determination in case of SAW on fractal lattices ... 

Vannimenus, J. Phase transitions for polymer on fractal lattices. Physica D, 1989, 38(2), 351-355. 

Hence, the fractal reactions of polymerization can be divided, as a minimum, into two classes fractal objects reaction whose kinetics is described similarly to the curve 1 in Fig. 4 and reactions in fractal space whose kinetics is described by the curve 2 in Fig. 4. The second class reactions correspond to stractures formation on fractal lattices and the first class - on Euclidean ones [18, 19]. 

Gefen, Y, Meir, Y, Mandelbrot, B.B. and Aharony, A. (1983). Geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices. Phys. Rev. Lett., 50(3), 145-148. 

Finally, one can treat the fractal lattice as a simple model for a disordered substrate on which the polymer is adsorbed, and use the exact results found for polymers on fractals to develop some understanding about real experimental systems. But for this, this article... 

We note that the collapse transition on fractal lattices corresponds to a new fixed point, intermediate between swollen (SAW) phase and the collapsed phase and cannot be viewed as a perturbation of the Gaussian fixed point describing random walks. 

Knezevic and Vannimenus realized that real-space renormahzation method for studying linear polymers on fractals can be extended directly to the case of branched polymers [42,44], They considered asymptotic properties of branched polymers with attractive selfinteraction on fractal lattices, restricting the attractive interactions to bonds within first order units of the fractal lattices. We summarize their findings here. 

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