Fractal structure, measurement

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.

The fractal dimension measures how open or packed a structure is lower fractal dimensions indicate a more open system, while higher fractal dimensions indicate a more packed system (22). Theories relating the fractal dimension to the relaxation exponent, n, have been put forward and these are based on whether the excluded volume of the polymer chains is screened or unscreened under conditions near the gd point (23). It is known that the excluded volume of a polymer chain is progressively screened as its concentration is increased, the size of the chain eventually approaching its unperturbed dimensions. Such screening is expected to occur near the... 

Note 1 w oc r in which m is the mass contained within a radius, r, measured from any site or bond within a fractal structure. 

One way of measuring the fractal dimension of aggregates is discussed in Chapter 5 (See Section 5.6a and Example 5.4). In the example below, we illustrate the relation between the fractal structure of aggregates and the surface area of the aggregates. 

Baveye, P., Boast, C. W., Gaspard, S., and Tarquis, A. M. (2008). Introduction to fractal geometry, fragmentation processes and multifractal measures Theory and operational aspects of their application to natural systems In Bio-Physical Chemistry of Fractal Structures and Process in Environmental Systems, Senesi, N., and Wilkinson, K., eds. IUPAC Series on Analytical and Physical Chemistry of Environmental Systems. Vol. 11, John Wiley Sons, Chichester, pp. 11-67. 

The third relaxation process is located in the low-frequency region and the temperature interval 50°C to 100°C. The amplitude of this process essentially decreases when the frequency increases, and the maximum of the dielectric permittivity versus temperature has almost no temperature dependence (Fig 15). Finally, the low-frequency ac-conductivity ct demonstrates an S-shape dependency with increasing temperature (Fig. 16), which is typical of percolation [2,143,154]. Note in this regard that at the lowest-frequency limit of the covered frequency band the ac-conductivity can be associated with dc-conductivity cio usually measured at a fixed frequency by traditional conductometry. The dielectric relaxation process here is due to percolation of the apparent dipole moment excitation within the developed fractal structure of the connected pores [153,154,156]. This excitation is associated with the selfdiffusion of the charge carriers in the porous net. Note that as distinct from dynamic percolation in ionic microemulsions, the percolation in porous glasses appears via the transport of the excitation through the geometrical static fractal structure of the porous medium. 

A useful (also extreme) counterpart to the also idealized linear geometry is fractal geometry which plays a key role in many non-linear processes.280 281 If one measures the length of a fractal interface with different scales, it can be seen that it increases with decreasing scale since more and more details are included. The number which counts how often the scale e is to be applied to measure the fractal object, is not inversely proportional to ebut to a power law function of e with the exponent d being characteristic for the self-similarity of the structure d is called the Hausdorff-dimension. Diffusion limited aggregation is a process that typically leads to fractal structures.283 That this is a nonlinear process follows from the complete neglect of the back-reaction. The impedance of the tree-like metal in Fig. 76 synthesized by electrolysis does not only look like a fractal, it also shows the impedance behavior expected for a fractal electrode.284... 

Flease note that the packing of fibers also has a fractal structure. Hie pecking fifectal dimension is obtained by measuring the number of holes in the fiber mat of various sizes. A log—log plot of number of hdes versus their size gives a line whose slope is the fi actal dimension of the fiber packing. See Kaye [56]. 

In Section 2.3 we studied the tent map, a schematic model for ionization that was able to produce fractal structures as a result of ionization. An important question is therefore whether the results presented in Section 2.3 are only of academic interest, or whether fractal structures can appear as a result of ionization in physical systems. In order to answer this question we return to the microwave-driven one-dimensional hydrogen atom. As we know from the previous chapter, this model is ionizing and realistic enough to qualitatively reproduce measured ionization data. Therefore this model is expected to be a fair representative for a large class of chaotic ionization processes. 

Much of the current interest in fractal geometry stems from the fact that fractal dimensions are experimentally accessible quantities. For polymers and colloids, the measurement techniques of choice are scattering experiments using X-rays, neutrons, or light. These measurements may be made on liquids or solids and can be performed readily as a function of time and temperature. Both mass and surface fractal structures yield scattering curves that are power laws, the exponents of which depend on the fractal dimension (6). For mass fractal structures the relation is... 

Essential characteristics of fine-grained solids, such as the Fe oxides, include the specific surface area, the porosity and the fractal structure. The standard procedure for measuring these properties is the Brunauer-Em-met-Teller (BET) method (Gregg Sing, 1991) This depends on the fact... 

The chaotic saddle and its manifolds are also sets of zero measure with fractal structure. The set of points, seen in Fig. 2.13 corresponding to inflow coordinates with very large, singular, escape times, typically form also a fractal set determined by the intersection of the saddle s stable manifold and the line containing the initial conditions. There is a connection between the dimension of the chaotic saddle and the dimensions of its manifolds. The trajectories on the chaotic saddle have a set of Lyapunov exponents whose number is equal to the dimension of the full space, d. The sum of the Lyapunov exponents is zero due to incompressibility and chaotic dynamics implies... 

Rice and Lin (1994) measured fractal geometry by small-angle x-ray scattering. In solid state, the fractal dimensions were between 2 and 3 which confirms the visual observations of Krasner et al (1996) (see section 2.5.4). While in the solid state the organics were surface fractals, in solution they are expected to become mass fractals. Osterberg and Mortensen (1994) measured fractal structure using small angle... 

Fig. 13.6 Log-log plot of the size D of a crumpled piece of aluminium foil versus the piece s size a. Points represent the data of our measurements. The solid line gives a best fit of the form InD -1.18+ 0.88 In a, which indicates a power law dependence D aO-88, thus manifesting the fractal structure of crumpled...

Mathematical or nonrandom fractals are scale invariant, i.e. the pattern is the same at all scales (self-similar). Natural, real or random fractals are quasi or statistically self-similar over a finite length scale that is most often determined by the characterization technique that is employed. An object or process can be classified as fractal when the length scale of the property being measured covers at least one order of magnitude. Fractal structures obey a power law, allowing the fractal dimension D to be determined from experimental data ... 

A wide range of techniques and methods have been used to measure fractal structure in environmental systems. Which approach is appropriate in given circumstances depends on whether an in-situ measurement is required or whether the samples may be taken for ex-situ analysis, on the size of the materials one is analysing and on their optical and mechanical properties. 

General Considerations in Measuring Systems with Fractal Structure... 

The measurement of fractal structure in environmental systems is accomplished by the measurement of at least two properties of the system that are related to each other through a fractal scaling law. Generally speaking, environmental systems of interest exhibit power-law scaling of mass with linear size such that... 

Even when looking at samples with rather good contrast, such as mineral oxides in water, much longer exposure times (of the order of minutes) must be used with X-ray scattering than with light (of the order of seconds) and samples must, in general, be much more concentrated. Berthon et al. [45] used USAXS to measure fractal structure in resorcinol-formaldehyde gels. 

The fractal dimensions measured for n = 1.0, 1.5 and 2.0 are respectively 2.3, 2.4 and 2.85, and do not correspond to any classical aggregation mechanism. Nonetheless, structures tend to become denser and denser as n increases. At the semi-local... 

---