Peak fractal dimension

According to Eq. (27), Stromme et al.125,126 developed systematically the peak-current method to determine the fractal dimension of the electrode surface by using cyclic voltammetry. It must be recalled that this method is valid when the recorded current is limited by diffusion of the electroactive species to and away from the electrode surface. Since the distribution of the reaction sites provides extensive information about the surface geometry, the fractal dimension of the reaction site distribution may agree with the fractal dimension of the electrode surface which is completely electrochemical-active. In addition, it is well known that this method is insensitive to the IR drop in the electrolyte.126... 

Keeping in mind that the dc sputter-deposited Pt films have the completely electrochemical-active surface, the fractal dimensions of the rough film surfaces were calculated from Eq. (27) according to the peak-current method by taking the slopes of the (log 7peak - log v) plots within the scan rate range of v0 to... 

Fractal Dimensions of Self-Affine Fractal Electrodes Determined by the Perimeter-Area Method (2nd Column), the Peak-Current Method (3rd Column), and the Triangulation... 

Therefore, we can define a length scale range where the rough surface shows the fractal behavior by determining the inner and outer cutoffs using Eq. (28).121 In other words the fractal dimensions determined from the power relation between /peak and... 

For Q<0, this distribution function is peaked around a maximum cluster size (2Q/(2Q-1))< >, where < > is the mean cluster size. 2Q=a+df1 is a parameter describing details of the aggregation mechanism, where a1 is an exponent considering the dependency of the diffusion constant A of the clusters on its particle number, i.e., A NAa. This exponent is in general not very well known. In a simple approach, the particles in the cluster can assumed to diffusion independent from each other, as, e.g., in the Rouse model of linear polymer chains. Then, the diffusion constant varies inversely with the number of particles in the cluster (A Na-1), implying 2Q=-0.44 for CCA-clusters with characteristic fractal dimension d =l.8. 

A second method for determining the singularity spectrum, the one we use here, is to numerically determine both the mass exponent and its derivative. In this way we calculate the multifractal spectrum directly from the data using Eq. (86). It is clear from Fig. 9b that we obtain the canonical form of the spectrum that is, f(h) is a convex function of the scaling parameter h. The peak of the spectrum is determined to be the fractal dimension, as it should. Here again we have an indication that the interstride interval time series describes a multifractal process. We stress that we are only using the qualitative properties of the spectrum for q < 0, due to the sensitivity of the numerical method to weak singularities. 

For the sample PPX -f Cu the calculated fractal dimension Df is equal to 2.609 [70]. It should be noted that the above-mentioned size distribution of metal nanoparticles leads to the mutual charging of such particles in the percolation cluster. This effect is discussed in the following section in coimection with catalysis by nanoparticles. As stated in reference 70, the specific low-temperature peak of dielectric losses in the synthesized composite samples PPX -t-Cu is probably due to the interaction of electromagnetic field with mutually charged Cu nanoparticles immobilized in the PPX matrix. The minor appearance of this peak in PPX -i- Zn can be explained by oxidation of Zn nanoparticles. 

The surface profilometer software calculates numerous parameters from the surface roughness profile. As the Advanced Processing Program continues, the research effort will be focused on detennining which of these roughness parameters or additional spectral analysis (max peak/unit distance, fractal dimension, etc.) are predictive of mechanical behavior, and relating them to fabrication variables. 

Fractal dimensions can be estimated from peak height and spacing such as provided by atomic force microscopy micrographs, yielding much information... 

Low-frequency dielectric measurements (0.1 Hz-1 MHz) of hydrated lysozyme, ovalbumin, and pepsin were used to estimate the fractal dimension for the random walk of protons through the hydrogen-bonded network of water molecules by fitting the shape of the dielectric loss peak [600, 601]. In all three systems, a crossover from 2D to 3D water network occurs within the interval of hydration from 0.05 to 0.10 g/g. For lysozyme, these values are noticeably below the value of about 0.17 g/g, reported in [592]. This difference may be attributed to the presence of about 0.07 g/g of strongly bound water, which presumably was not taken into account in [601]. When hydration exceeds the threshold value, dielectric losses increase almost Unearly with temperature. With approaching T 310 to 330 K, this increase slows down and dielectric losses turn to decrease with temperature. This behavior may reflect thermal break of the spanning hydrogen-bonded water network, which wdl be considered in Section 8.1. 

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