Fractal dimensions correlation

Our third applications example highlights the work of Nakano et al. in modeling structural correlations in porous silica. MD simulations of porous silica in the density range 2.2—0.1 g/cm were carried out on a 41,472-particle system using an iPSC/860. Internal surface area, ratio of pore surface to volume, pore size distribution, fractal dimension, correlation length, and mean particle size were determined as a function of the density, with the structural transition between a condensed amorphous phase and a low density porous phase characterized by these quantities. Various dissimilar porous structures with different fractal dimensions were obtained by controlling the preparation schedule and the temperature. 

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. 

Physically, the wetting abdity increases (the contact angle decreases) as the values of the fractal dimension of the electrode increases if the electrode material is same. However, in this study, we could not obtain a good correlation between the fractal dimensions and the wetting abilities as shown in Table 1. It means that not only the physical properties such as the surface irregularity and roughness but also the chemical interaction between electrolyte and electrode were important in wetting ability. 

The question of whether proteins originate from random sequences of amino acids was addressed in many works. It was demonstrated that protein sequences are not completely random sequences [48]. In particular, the statistical distribution of hydrophobic residues along chains of functional proteins is nonrandom [49]. Furthermore, protein sequences derived from corresponding complete genomes display a distinct multifractal behavior characterized by the so-called generalized Renyi dimensions (instead of a single fractal dimension as in the case of self-similar processes) [50]. It should be kept in mind that sequence correlations in real proteins is a delicate issue which requires a careful analysis. 

These starburst dendrimers have been subjected 47 to two different fractal analyses I48 49 (a )A c/2 D)/2, where A is the surface area accessible to probe spheres possessing a cross-sectional area, o, and the surface fractal dimension, D, which quantifies the degree of surface irregularity and (b) A = dD, where d is the object size. Both methods give similar results with D = 2.41 0.04 (correlation coefficient = 0.988) and 2.42 0.07 (0.998), respectively. Essentially, the dendrimers at the larger generations are porous structures with a rough surface. For additional information on dendritic fractality, see Section 2.3. 

Numerical simulations of the data were conducted with the algorithms discussed above, with the added twist of optimizing the model to fit the data collected in the laboratory by adjusting the collision efficiency and the fractal dimension (no independent estimate of fractal dimension was made). Thus, a numerical solution was produced, then compared with the experimental data via a least squares approach. The best fit was achieved by minimizing the least squared difference between model solution and experimental data, and estimating the collision efficiency and fractal dimension in the process. The best model fit achieved for the data in Fig. 10a is plotted in Fig. 10b, and that for Fig. 11a is shown in Fig. lib. The collision efficiencies estimated were 1 x 10-4 and 2 x 10-4, and the fractal dimensions were 1.5 and 1.4, respectively. As expected, collision efficiency and fractal dimension were inversely correlated. However, the values of the estimates are, in both cases, lower than might be expected. The lower values were attributed to the following ... 

The fractal dimension of the fat crystal network in milk fat decreased from 2.5 to 2.0 when the cooling rate was increased. Concomitantly, the particle-related constant, A, increases. These results demonstrate how a faster cooling rate leads to a less ordered spatial distribution of mass within the microstructural network, which would result in a lower value of D, and a decrease in the average particle diameter, which would result in a higher value of A, as predicted by our model. These microstructural changes were correlated with a much higher yield force value for the rapidly cooled milk fat (64.1 3.3N versus 33.0 3.9N for the samples cooled at 5.0°C/min and 0.1°C/min, respectively). 

We are drawn to the conclusion that log-log fractal plots are useful for the correlation of adsorption data - especially on well-defined porous or finely divided materials. A derived fractal dimension can also serve as a characteristic empirical parameter, provided that the system and operational conditions are clearly recorded. In some cases, the fractal self-similarity (or self-affine) interpretation appears to be straightforward, but this is not so with many adsorption systems which are probably too complex to be amenable to fractal analysis. 

Hence Fig 19 represents, in fact five correlations Other many more relations are available [64] Here, in an analogy with self-similarity the concept of fractal dimensions has been extended to graphs by... 

Figure 1. The correlation of pore diameter and fractal dimension of the sludge adsorbent from KOH activation process...

Figure 24. Linear correlations between yield force and (A) microstructural element area, (B) fractal dimension by particle-counting. (C) fractal dimension by box-counting, and (D) fractal dimension by rheology. Data shown represent all points collected at all cooling rates and storage times.

Fractal dimension was applied to characterize the internal structure of porous films made from ethylcel-lulose (EC) and diethylphthalate (DEP). Drug permeation was found to correlate with boundary fractal dimension on a semilog plot (Fig. 12). " However, Dl simply describes the ruggedness of a line and does not represent the porosity. More work is required for fractal dimensions in this case. 

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