Fractal dimension increment

In their works,51"54 the self-similar fractal dimension dF>ss of the two-dimensional distribution of the pits was determined by the analysis of the digitized SEM images using the perimeter-area method. The value of dF>ss increased with increasing solution temperature,51 and it was inversely proportional to the pit shape parameter and the pit growth rate parameter.53 Keeping in mind that dr>ss is inversely proportional to the increment of the pit area density, these results can be accounted for in terms of the fact that the increment of the pit area density is more decelerated with rising solution temperature. 

Figure 11. Schematic diagram shows the incremental decreases in box size used for the particle counting method for the determination of the microscopic fractal dimension Df.

Successive increments of mathematical fractal random processes are independent of the time step. Here D = 1.5 corresponds to a completely uncorrelated random process r = 0, such as Brownian motion, and D = 1.0 corresponds to a completely correlated process r= 1, such as a regular curve. Studies of various physiologic time series have shown the existence of strong long-time correlations in healthy subjects and demonstrated the breakdown of these correlations in disease see, for example, the review by West [56]. Complexity decreases with convergence of the Hurst exponent H to the value 0.5 or equivalently of the fractal dimension to the value 1.5. Conversely, system complexity increases as a single fractal dimension expands into a spectrum of dimensions. 

Aoki also considers the stochastic aspects of phase propagation mechanism and relates his analysis to the theory of percolation and the fractal dimension of the system. In this approach the Nemst equation for charge transfer at the substrate/film interface is used to compute the probability of the presence of a conductive seed or nucleus. When the potential is incremented, this seed can then grow in a one-dimensional manner governed by the propagation rate constant kp or the kinetic parameter to form a conductive pillar of a definite length. New nuclei can also form at the support electrode/film interface during the potential... 

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