Dimension Euclidean

In case of reaction course in the Euclidean spaces the value D is equal to the dimension of this space d and for fractal spaces D is accepted equal to spectral dimension ds [6], By plotting p i=( 1 -O) (where O is conversion degree) as a function of t in log-log coordinates the value D from the slope of these plots can be determined. It was found, that the mentioned plots fall apart on two linear parts at t<100 min with small slope and at PT00 min the slope essentially increases. In this case the value ds varies within the limits 0,069-3,06. Since the considered reactions are proceed in Euclidean space, that is pointed by a linearity of kinetic curves Q-t, this means, that the reesterefication reaction proceeds in specific medium with Euclidean dimension d, but with connectivity degree, characterized by spectral dimension ds, typical for fractal spaces [5],... 

For example, the fractal dimension of the Koch curve is 1.2619 since four (m = 4) identical objects are observed (cf. levels i = 0 and i = 1 in Figure 1.1) when the length scale is reduced by a factor r = 3, i.e., dj- = In4/ln3 1.2619. What does this noninteger value mean The Koch curve is neither a line nor an area since its (fractal) dimension lies between the Euclidean dimensions, 1 for lines and 2 for areas. Due to the extremely ramified structure of the Koch curve, it covers a portion of a 2-dimensional plane and not all of it and therefore its dimension is higher than 1 but smaller than 2. 

From this relationship, we obtain A = 1/3 since the value of ds is 4/3 for A + A reactions taking place in random fractals in all embedded Euclidean dimensions [9, 19]. It is also interesting to note that A = 1/2 for an A + B reaction in a square lattice for very long times [12]. Thus, it is now clear from theory, computer simulation, and experiment that elementary chemical kinetics are quite different when reactions are diffusion limited, dimensionally restricted, or occur on fractal surfaces [9,11,20-22]. 

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... 

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... 

The values of the exponents fl, D, and a in the distribution function (71) and scaling laws (74)-(76) depend on the Euclidean dimension d of the system and satisfy hyperscaling relationships (HSR). The HSR may be different for the various models describing the various systems [221-225]. 

The average length of a nylon-6,6 polymer chain is about 50-100 pm (each polymer chain contains about 105 groups while the length of a polymer group rg is about 10 A). This length is comparable to the thickness of a sample 120-140 pm [275]. Thus, the movement of the chains is most likely occurring in the plane of the sample. This fact correlates with the values of the space fractional dimension da- For all of the samples d( C (1,2) (see Table IV). Thereby, the Euclidean dimension of the space in which chain movement occurs is dE = 2. 

FIGURE 12.4. Schematic illustration of excitation energy transfer in Euclidean dimension, (a) 1-dimension, (b) 2-dimension (c) 3-dimension. 

The fractal dimension D is used to quantify the micro structure of the fat crystal networks, where d is the Euclidean dimension, x is the backbone fractal dimension that is estimated between 1 and 1.3. The backbone fractal dimension describes the tortuosity of the effective chain of stress transduction within a cluster of particles yielding under an externally applied stress (Shih et al. 1990 Kantor and Webman 1984). 

In fact, if one measures the total number of bonds (sites) on the infinite cluster at the percolation threshold (pc) in a (large) box of linear size L, then this number or the mass of the infinite cluster will be seen to scale with L as where die (< d) is called the fractal dimension of the infinite cluster at the percolation threshold. Similar measurements for the backbone (excluding the dangling ends of the infinite cluster) give the backbone mass scaling as, de < die, where dfi is called the backbone (fractal) dimension. In fact, die can be very easily related to the embedding Euclidean dimension d of the cluster by... 

It follows from Eq. (124) that the dimension of the bond set coincides with the Euclidean dimensions only in the limit of infinitely large dimensions of the initial lattice, l0 ... 

Euclidean dimension de of the lattice spaces 3x3x3, 5x5x5, 7x ... 

For t/j = 1 (linear chains). Equation (11.9a) provides the correct value, d = 2, corresponding to a macromolecular coil at the 0-point (see Table 11.2). As noted previously, d = 4/3 for a percolation cluster, irrespective of the dimension of the Euclidean space (see Table 11.1) therefore, from Equation (11.9a), we obtain df= 4, which is consistent with the Flory-Stockmayer theory [60] for phantom chains. For three-dimensional space, d > 3 has no physical meaning because the object cannot be packed more densely than an object having a Euclidean dimension. It is evident that this discrepancy is due to the phantom nature of the polymer chains postulated by Cates [56] it is therefore, necessary to take into account self-interactions of chains due to which the dimension of a polymer fractal assumes a value that has a physical meaning. 

Polymers exhibit the fractality inherent in them at different structural levels both primary and secondary structural elements can be fractals. For example, macromolecular coils in 0 or in good solvents with df 1.66 (with the Euclidean dimension = 3) are fractals [1,29]. The fractality of a polymer at the molecular level can be demonstrated as follows. Lebedev [90] reported data on the change in the radius of gyration Rf) of the macromolecules of block polystyrene, PMMA and polyethylene with the molecular mass M. The Rg value is related to the length of a macromolecule in the following way [91] ... 

The structure of the filler particle surfaces and of the polymer surface characterised hy their fractal dimensions, affects the interfacial adhesion in composites. To explain the structural effect let us introduce the concept of the accessibility of the sites on these surfaces to form adhesion joints (physical or chemical). As a first approximation the degree of such accessibility may be defined as a difference of the fractal dimensions of two surfaces. The higher is this difference the lower is the accessibility of the surface and the less is the adhesion [21]. Suppose that the filler particle has a very rough surface with dimensions which are close to the Euclidean dimension d = 3 (for example, AI2O3 particles) [33], whereas the polymer surface is very smooth, i.e., dp = d = 2. In this case the contact between two surfaces is possible only at the apexes of the rough surface of the filler and the result could be very low adhesion. In other words, the disparity of the dimensions determines the inaccessibility of the greater part of the filler particle for the formation of adhesion bonds [21]. 

Euclidean dimension Flory formula lattice animals Monte-Carlo method phantom fractal true self-avoiding walk... 

As D approaches Euclidean dimensions, the three types of fractal become indistinguishable. 

In two Euclidean dimensions, if we require that every site and every bond of a lattice be equivalent, there are only three possible choices. The square, triangular or honeycomb lattice. On the Poincare disk, a space with negative curvature, there are infinitely many possibilities. 

The Simplest example of fractal is the contour set. We have taken the simplest example to illustrate the meaning of fractal dimension and its estimation. It should be noted that the fractal dimension is a global property of the cluster and it does not provide a deep insight into the structural details of the aggregate. Euclidean dimensions for different geometries are recorded in Table 13.1 while self-similarity dimensions of some deterministic fractals are recorded in Table 13.2. 

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