Fractal exponent

In these analyses, it has been assumed that the power law exponent (fractal dimension) is constant during the agglomeration process. This is not necessarily the case as the experimental observations di.scussed in the next section show. 

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. 

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). 

E. V. Albano. Critical exponents for the irreversible surface reaction A + B AB with B desorption on homogeneous and fractal media. Phys Rev Lett 69 656-659, 1992. 

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... 

Fractional exponent, dimensionless parameter. Fractal dimension, dimensionless parameter. Scrape method. 

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). 

Based on the fractal behavior of the critical gel, which expresses itself in the self-similar relaxation, several different relationships between the critical exponent n and the fractal dimension df have been proposed recently. The fractal dimension ds of the polymer cluster is commonly defined by [16,42]... 

Fig. 9. Relation between relaxation exponent n and fractal dimension d for a three-dimensional network. In case of complete screening of excluded volume, values of 0 < n < 1 are possible if d is chosen between 1.25 and 2.5...

If only partial screening is present, the fractal dimension takes a value somewhere between df and df. According to this model, a crosslinker deficiency, which leads to a more open structure and therefore a lower value of du increases the value of n. Dilution of the precursor with a non-reactive species has the same effect on the relaxation exponent. 

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... 

Figures 8a and 8b present the simulated current transients obtained from the self-affine fractal interfaces of r/ = 0.1 0.3 0.5 and r] = 1.0 2.0 4.0, respectively, embedded by the Euclidean two-dimensional space. It is well known that the current-time relation during the current transient experiment is expressed as the generalized Cottrell equation of Eqs. (16) and (24).154 So, the power exponent -a should have the value of - 0.75 for all the above self-affine fractal interfaces.

However, the simulated current transients in Figure 8 never exhibited the expected power exponent of -0.75 with the exception of the original self-affine fractal interface ( = 1) the... 

The above results concerning the fractal dimension as functions of the solution temperature, the CPE exponent, and the depression parameter are demonstrated again in their other investigation of the effect of the anion addition on pit morphology of Inconel alloy 600 at elevated temperatures.54... 

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