Fractal scaling

Stemp, W.J., B.E. Childs, S. Vionnet, and C.A. Brown. 2008. Quantification and discrimination of lithic use-wear surface profile measurements and length-scale fractal analysis. Archaeometry XXX. 

Fractals have fascinating properties that are present in many natural objects and had not been incorporated in previous models of nature (5, 6). If any small piece of a fractal is magnified, it appears similar to a larger piece. This property is called self-similarity and is illustrated by the fractal in Figure 2. Self-similarity can occur only if structures at a small scale are related to structures at a larger scale. Fractal objects include the repeated bifurcations of the airways in the lung (7), the distribution of blood flow in the ever-smaller vessels in the heart (8), and the ever-finer infoldings of cellular membranes (9). 

Brown CA, Siegmann S (2001) Fundamental scales of adhesion and area-scale fractal analysis. Int J Mach Tools Manuf 41 1927-1933... 

A typical property of fractals is that they are locally (asymptotically) self-similar of small-length scales. Fractals are shapes that look more or less the same on all or many... 

Fractal Dimension Measure of a geometric object that can have fractional values. It refers to the measure of how fast the length, area, or volume of an object increases with a decrease in scale. Fractal dimension can be calculated by box counting or by evaluating the information dimension of an object. Generator Collection of scaled copies of an initiator. Hausdorff-Besicovitch Dimension Mathematical statement used to obtain a dimension that is not a whole number, commonly written as d = log (N)/ log (r). 

In the range of definite scales fractals have different topological structures depending on the maximum number of elements that are connected with the given system element. If each element can be connected, as a minimum, with two other ones, then the received structure has no branches. By analogy with linear polymers Family [8] calls this type of fractal linear. If branching occurs, then the resulting fractal has a network-like structure this type of fractal was called a branched fractal. 

Discuss the paradox in the wettability of a fractal surface (Eq. X-33). A true fractal surface is infinite in extent and a liquid of a finite contact angle will trap air at some length scale. How will this influence the contact angle measured for a fractal surface ... 

We have considered briefly the important macroscopic description of a solid adsorbent, namely, its speciflc surface area, its possible fractal nature, and if porous, its pore size distribution. In addition, it is important to know as much as possible about the microscopic structure of the surface, and contemporary surface spectroscopic and diffraction techniques, discussed in Chapter VIII, provide a good deal of such information (see also Refs. 55 and 56 for short general reviews, and the monograph by Somoijai [57]). Scanning tunneling microscopy (STM) and atomic force microscopy (AFT) are now widely used to obtain the structure of surfaces and of adsorbed layers on a molecular scale (see Chapter VIII, Section XVIII-2B, and Ref. 58). On a less informative and more statistical basis are site energy distributions (Section XVII-14) there is also the somewhat laige-scale type of structure due to surface imperfections and dislocations (Section VII-4D and Fig. XVIII-14). 

Wool [32] has considered the fractal nature of polymer-metal and of polymer-polymer surfaces. He argues that diffusion processes often lead to fractal interfaces. Although the concentration profile varies smoothly with the dimension of depth, the interface, considered in two or three dimensions is extremely rough [72]. Theoretical predictions, supported by practical measurements, suggest that the two-dimensional profile through such a surface is a self-similar fractal, that is one which appears similar at all scales of magnification. Interfaces of this kind can occur in polymer-polymer and in polymer-metal systems. 

Polymer-polymer fractal interfaces may result from the interdiffusion of monomers or of polymers themselves. Koizumi et al. [31] annealed the interface between polystyrene and a styrene-isoprene diblock polymer at 150 C and showed extensive roughening of the interface by mutual interdiffusion on a micron scale (Fig. 8). 

As the scale of roughness becomes finer, the effective increase in A can become enormous. Consequently Fg may be raised to very high value. Indeed, as many engineering surfaces are fractal in nature [36], we can only retain the concept of area at all, if we accept that it can be considered as indefinitely large. The practical adhesion does not become infinite, because the joint with a strong interfacial region will fail (cohesively) in some other region where Fg is smaller [89],... 

The fractal-like organization led, therefore, to conductivity measurements at three different scales (1) the macroscopic, mm-size core of nanotube containing material, (2) a large (60 nm) bundle of nanotubes and, (3) a single microbundle, 50 nm in diameter. These measurements, though they do not allow direct insights on the electronic properties of an individual tube give, nevertheless, at a different scale and within certain limits fairly useful information on these properties. 

Nelson, T. R., West, B. J., and Goldberger, A. L. (1990). The fractal lung universal and. species-related scaling patterns. Experientia 46, 251-254. 

The fractal-like organisation of CNTs produeed by elassical earbon are diseharge suggested by Ebbesen et al. [15] lead to conductivity measurements whieh were performed at various scales. 

The cluster properties of the reactants in the MM model at criticality have been studied by Ziff and Fichthorn [89]. Evidence is given that the cluster size distribution is a hyperbolic function which decays with exponent r = 2.05 0.02 and that the fractal dimension (Z)p) of the clusters is Dp = 1.90 0.03. This figure is similar to that of random percolation clusters in two dimensions [37], However, clusters of the reactants appear to be more solid and with fewer holes (at least on the small-scale length of the simulations, L = 1024 sites). 

A normal diffusion process, however, runs at a finite concentration of particles different from zero. In this situation it was found [101] that a fractal character (73) of the resulting structure is restricted to an interval a < R < if), where d is the diffusion length (67). Larger clusters have a constant density on a length scale larger than They are no longer fractal there. These observations have various consequences for crystal growth, and will be discussed in the next section. 

T. Irisawa, M. Uwaha, Y. Saito. Scaling laws in thermal relaxation of fractal aggregates. Europhys Lett 50 139, 1985. 

The fractal dimension was introduced earlier in section 2.1.1. If the minimum number of d-dimensional boxes of side e needed to eover the attractor A, N e), scales as... 

The mechanism of Self-organized criticality, a concept first introduced by Bak, Tang and Wiesenfeld [bak87a], may possibly provide a fundamental link between such temporal scale invariant phenomena and phenomena exhibiting a spatial scale invariance - familiar examples of which are given by fractal coastlines, mountain landscapes and cloud formations [mandel82],... 

Figure 21. Noise spectrum of detector amplifiers. Note that both axes have logarithmic scale. There are two main components of noise - the white noise which is present at all frequencies, and the 1// noise that is dominant at low frequencies. 1// noise has a fractal structure and is seen in many physical systems. The bandpass of a measurement decreases for slower readout, and the readout noise will correspondingly decrease. A limit to reduction in readout noise is reached at the knee of the noise spectrum (where white noise equals l/f noise) - reading slower than the frequency knee will not decrease readout noise.

Power-law scattering features will be discussed in relation to mass-fractal scaling laws. Fractal scaling concepts used to interpret the power-law decay are well published in the literature. 

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