Fractal transparency

A concept of mutual transparency or opacity based on the relative evolution of fractal dimension and radius of the clusters has been developed by Mandelbrot [38]. The tendency of fractal systems to interpenetrate is inversely related to the mean number of intersections 2 of two mass fractal objects of size and mass fractal dimensions Dj and D2 placed in the same region of space of dimension d ... 

The conclusion arising from these experiments was that homogeneous polymeric silica-based binary sols cam be made with the addition of a second component up to 30 mol%. Initially, the fractal dimension (-1.4) and the gyration radii (-2 run) of the polymers were foimd to be low enough to obey the concept of mutual transparency. More details are provided in Chapter 8 on the preparation of such microporous membranes for gas separation. 

Mandelbrot s relationship (eq 18) assumes that the objects are perfectly rigid and stick immediately and irreversibly at each point of intersection (chemically equivalent to an infinite condensation rate). In fact, fractal objects are more or less compliant, and the sticking probability is always 1. These factors mitigate the criterion for mutual transparency for example, if the condensation rate is reduced, screening is less effective, and objects with D > 1.5 can interpenetrate. Because the condensation rate of silica depends strongly on pH (see Figure 3), the transparency or opacity and ultimately the film porosity can be manipulated by the addition of acid or base catalyst (24). 

Three-tlimcn.sinnal representation of Ford frolh. The color of the spheres is related to their radii. Transparency is used to visualize some of the internal Chrysler spheres. See "Fractal Milkshakes and Infinite Archery" (chapter 14) for more infontiation. 

Fractals are opaque (nonleaking) one for another if increases at growth, that is, at u+ fi>d and transparent (leaking), if an intersections number decreases at enhancement [39]. In other words, for the case d=3 and two fractals with the same dimension ZT the Eq. (13) predicts transparent polymers fractal at Z) <1.5, that corresponds completely to the data of Ref [12],... 

Thirdly, e decay rate change at Z) growth at D=. 5 can be attributed to qualitative changes of the dependence with certain reservations. This bending point is not accidental and can be characterized within the framework of fractal analysis [25], where intersections number of two arbitrary fractals with dimensions and can be estimated according to the Eq. (13) of Chapter 1. The indicated equation predicts the possibility of two fractals interpenetration (their transparency and opaqueness. Fractals penetrate freely one into another (are transparent), if decreases at growth, that is, if + /2. Assuming Df=Df and =3, let us obtain the transparence (leakiness) condition for macromolecular coils D <. 5. Thus, the repulsive interactions intensification of either macromolecule elements or macromolecule and solvent results in the end in fractals (macromolecular coils) transparence one for another. 

Hence, the stated above results have shown, that the limiting conversion degree value in polycondensation process is controlled by macroniolecular coil stracture, characterized by its fractal dimension. The coiKhtionC <1.0 atDj>1.50 is defined by opacity of macromolecular coils concerning each other. For transparent macromolecular coils (D <1.50), allowing their complete interpenetration, the value Q =l 0 does not depend on macromolecular coil structure within the range of D=1.0-1.50. 

The principle of geometric transparency discussed earlier is critical to how fractal structure is encoded in images of fractal objects. For large fractal structures with a mass fractal dimension < 2, the area of the projected image will scale with exactly the same dimension as the mass scales in the real structure in three-dimensional space. When Dm > 2 the structure is geometrically opaque, which means that the projection has no holes in it and scales according to power 2 as the size of the projection increases. 

The accuracy and speed of the correlation function for measuring fractal dimension may be improved by only including pixels j which are closer to i than the aggregate edge nearest to i. This will not remove effects due to finite size and consequent increased transparency, but will eliminate gross aggregate shape effects. 

One of the easiest ways to measure fractal dimension with this technique is to capture images of slices through the structure and measure the fractal scaling of the image as discussed above. The dimension measured in this way is not the projected area dimension Dp, discussed earlier, because the image is a slice not a projection. It turns out that the dimension measured in this way is numerically equal to the mass fractal dimension minus one, by virtue of the codimension rule [58]. The measurement of fractal dimensions by this technique is not subject to the restriction of geometric transparency, as is the case with the analysis of projected images, and so fractal dimensions well over two can be measured. 

The mutual excluded volume of two objects of size R is the solid-sphere excluded volume times the average (1— (r)), as noted above. Since g r) neither approaches 0 or 1 for most of the sphere volume, its average must also be some finite, nonzero fraction. Thus these fractals have an excluded volume that is a finite, nonzero fraction of the solid-sphere excluded volume, despite their tenuous structure. We term such fractals mutually opaque. We conclude that two fractals may interact in one of two qualitatively different ways. If Di + D2 < 3 they are transparent, and have a mutual excluded volume much... 

Borderline fractals, with D D2 = 3, are neither transparent nor opaque in general and must be treated on a case-by-case basis. 

As before, it is convenient to designate a home particle on the fractal adsorber. We now consider an arbitrary point at a distance r from the home point. Despite the absorption we expect some of the walker s tracks to remain at r, as shown in Fig. 8.4. The density of such walkers relative to the initial density is the probability that the walker at r has not been removed by adsorption. This probability is the probability that the random walk representing its past has not intersected the fractal. This is just the g(r) discussed in the last section. The fractals in this case are the adsorbing fractal, with dimension D, and the random walk, with dimension 2. As we saw above, this g r) depends on the fractal dimensions. If D 4- 2 is less than 3, the two are mutually transparent g r) 1. Virtually all the walkers in the pervaded volume of the fractal never touch the fractal. Their density u is thus virtually unaffected u r) Uq. But if D-l-2 is greater than 3, the two are mutually opaque, and g r) is substantially smaller than 1 for most points r within the absorbing fractal. All connected fractals have D > 1 and thus show opaque behavior. For all such fractals, in-... 

Fracton A collective quantized vibration on a substrate with a fractal structure Free-dimensional space art The author I. Michaloudis is playing with the three-dimensional space which in the case of the indefinitive transparency of his aer( ) sculptures becomes a non-Euclidean space. Silica aerogel itself can be considered as a personification of what the French mathematician Henri Poincarre named a representative space, a space you cannot measure you just live in with all your senses for more information see loannis MICHALOU(di)S, Aer( jsculpture the enigmatic beauty of aerogel s nonentity in a pilot art and science project, Journal of Non-CrystaUine Sohds (2004) 350 61-66... 

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