Fractals and fractal dimension

An alternative log/log plot of total measured area versus tile size will have a gradient of 1 - DU (Fig. 2b). This can easily be seen for the tetrahedral fractal (total area A and tile size a, see Table 3 in Fractals and fractal dimension) ... 

Fractals and fractal dimension D E PACKHAM Concept of fractal dimension... 

The surface of a crack depends on the properties of the material and on loading characteristics. The surface may be more or less rough and developed. It appears that the surface of the crack in cement-based materials has fractal nature, which indicates that the effective determination of its area is related to the scale of magnification. General remarks about fractals and fractal dimension may be found in a book by Mandelbrot (1983). The methods on how to characterize the cracks and the fracture surfaces using the notion of fractal dimension is briefly described in Section 10.5. 

B. B. Mandelbrot, Self-affine fractals and fractal dimension, Phys. Ser. 32, 257-260 (1985). 

The currently useful model for dealing with rough surfaces is that of the selfsimilar or fractal surface (see Sections VII-4C and XVI-2B). This approach has been very useful in dealing with the variation of apparent surface area with the size of adsorbate molecules used and with adsorbent particle size. All adsorbate molecules have access to a plane surface, that is, one of fractal dimension 2. For surfaces of Z> > 2, however, there will be regions accessible to small molecules... 

The monolayer amount adsorbed on an aluminum oxide sample was determined using a small molecule adsorbate and then molecular-weight polystyrenes (much as shown in Ref. 169). The results are shown in the table. Calculate the fractal dimension of the oxide. 

Chaotic attractors are complicated objects with intrinsically unpredictable dynamics. It is therefore useful to have some dynamical measure of the strength of the chaos associated with motion on the attractor and some geometrical measure of the stmctural complexity of the attractor. These two measures, the Lyapunov exponent or number [1] for the dynamics, and the fractal dimension [10] for the geometry, are related. To simplify the discussion we consider tliree-dimensional flows in phase space, but the ideas can be generalized to higher dimension. 

A primary sol particle in an acid-cataly2ed sol has radius between 1 and 2 nm (3). The secondary fractal particle has a radius, R, of 5 to 20 nm as seen from saxs (3). For the TMOS-based sols investigated by saxs, ( increases with time, as does the Guinier radius, R. The stmcture reaches a fractal dimension around 2.3 at the gelation point. 

Polycondensation reactions (eqs. 3 and 4), continue to occur within the gel network as long as neighboring silanols are close enough to react. This increases the connectivity of the network and its fractal dimension. Syneresis is the spontaneous shrinkage of the gel and resulting expulsion of Hquid from the pores. Coarsening is the irreversible decrease in surface area through dissolution and reprecipitation processes. 

The first detailed book to describe the practice and theory of stereology was assembled by two Americans, DeHoff and Rhines (1968) both these men were famous practitioners in their day. There has been a steady stream of books since then a fine, concise and very clear overview is that by Exner (1996). In the last few years, a specialised form of microstructural analysis, entirely dependent on computerised image analysis, has emerged - fractal analysis, a form of measurement of roughness in two or three dimensions. Most of the voluminous literature of fractals, initiated by a mathematician, Benoit Mandelbrot at IBM, is irrelevant to materials science, but there is a sub-parepisteme of fractal analysis which relates the fractal dimension to fracture toughness one example of this has been analysed, together with an explanation of the meaning of fractal dimension , by Cahn (1989). 

For a fractal surface D > 2, and usually D < 3. In simple terms the larger D, the rougher the surface. The intuitive concept of surface area has no meaning when applied to a fractal surface. An area can be computed, but its value depends on both the fractal dimension and the size of the probe used to measure it. The area of such a surface tends to infinity, as the probe size tends to zero. 

Obviously the roughness factor is similarly arbitrary, but it is of interest to use Eq. 25 to compute its value for some trial values of D and a. This is done in Table 2. In order to map the surface features even crudely, the probe needs to be small. It can be seen that high apparent roughness factors are readily obtained once the fractal dimension exceeds 2, its value for an ideal plane. 

Roughness factor calculated for a fractal surface, according to the fractal dimension D and probe area a... 

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). 

That Dfractai givcs the expected result for simple sets in Euclidean space is easy to see. If A consists of a single point, for example, we have N A, e) = 1, Ve, and thus that / fractal = 0. Similarly, if A is a line segment of length L, then N A,e) L/e so that Dfraciai = 1. In fact, for the usual n dinien.sional Euclidean sets, the fractal dimension equals the topological dimension. There are nrore complicated sets, however, for wliic h the two measures differ. 

Although it is not the only such measure, the fractal dimension docs quantify the intuitive belief that the Cantor set is somewhere in-between a point and a line. We will consider generalizations of fractal needed in later chapters,... 

Pig. 2.2 First three steps in the construction of the THadic Koch Curve. The fractal curve is obtained in the limit N 00 and has a fractal dimension Dfractal = In 4/ln 3 1.26. 

Fig. 3.3 First, three st( ps in the recursive geometric construction of the large-time pattern induced by R90 when starting from a simple nonzero initial state. The actual final pattern would be given as the infinite time limit of the sequence shown hero, and is characterized by a fractal dimension Df,actai = In 3/In 2.

Since the phase space of a dissipative dynamical system contracts with time, we know that, in the long time limit, t oo, the motion will be confined to some fixed attractor, A. Moreover, becaust of the contraction, the dimension, D, of A, must be lower than that of the actual phase space. While D adds little information in the case of a noiichaotic attractor (we know immediately, and trivially, for example, that all fixed-points have D = 0, limit cycles have D = 1, 2-tori have D = 2, etc.), it is of significant interest for strange attractors, whose dimension is typically non-integer valued. Three of the most common measures of D are the fractal dimension, information dimension and correlation dimension. 

The three dimensions introduced above, Dp, Di and Dq, are actually three members of an (uncountably infinite) hierarchy of generalized fractal dimensions introduced by Heiitschel and Procaccia [hent83]. The hierarchy is defined by generalizing the information function /(e) (equation 4.88) used in defining Dj to the i/ -order Renyi information function, Io e) -... 

Lyapunov Dimension An interesting attempt to link a purely static property of an attractor, - as embodied by its fractal dimension, Dy - to a dynamic property, as expressed by its set of Lyapunov characteristic exponents, Xi, was, first made by Kaplan and Yorke in 1979 [kaplan79]. Defining the Lyapunov dimension, Dp, to be... 

Generalized Renyi Entropies and Dimensions A hierarchy of generalized entropies and dimensions, SQ B,t), s[ B,t), S B,t),. .. - analogous to the hierarchy of fractal dimensions, Dq, Di,. .., introduced earlier in equation 4.94 for continuous systems may also be defined ... 

The value a = 1 corresponds to ideal capacitive behavior. The fractal dimension D introduced by Mandelbrot275 is a formal quantity that attains a value between 2 and 3 for a fractal structure and reduces to 2 when the surface is flat. D is related to a by... 

The power, [P], in the fractal power-law regime gives as the fractal dimension, d(. P = —df for each level of the fit, the parameters obtained using the unified model are G, Rg, B, and P. P is the exponent of the power-law decay. When more than one level is fitted, numbered subscripts are used to indicate the level—i.e., G —level 1 Guinier pre-factor. The scattering analysis in the studies summarized here uses two-level fits, as they apply to scattering from the primary particles (level 1) and the aggregates (level 2). 

FIGURE 17.4 USAXS and SALS results for samples Al, A2, and A3 described in the text. Each sample shows four structural levels with the i g for some of the levels indicated in the graph. The power-law value for the second level, corresponding to the mass-fractal dimension, d(, is also indicated. 

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