Particle-counting fractal dimension

The particle-counting fractal dimension relates the number of primary particles N in an object to the linear size of the fractal object R, and the linear size of one particle (ct) ... 

The particle-counting fractal dimension, Dj, was not sensitive to crystal shape, size, AF or the distribution orderliness. It was found that Df was affected by the radial distribution pattern of the fat crystals as shown in Figure 17.28. The simulation results were found to be consistent with experiments (Litwinenko et al. 2002 Tang and Marangoni 2006). Devalues close to 2 indicated more homogenously distributed fat crystals. It is important to note, the values of Dfm y exceed the dimensionality of the embedding space. This is not the case for the box-counting dimension or the Fourier transform fractal dimension. 

TABLE 3. Particle-Counting Fractal Dimension (Df), Box-Counting Fractal Dimension (Db), and Mean Microstructural Element Area (MEA) for Anhydrous Milkfat Crystallized at Various Rates of Cooling and Storage Times at 5°C. 

The box-counting fractal dimension is calculated from the log-log plot of the number of occupied boxes, versus side length, /,. The slope of this line is used to determine the fractal dimension. Both very small and very large box sizes should be exempt from the calculation to reduce surface artifacts (Awad and Marangoni 2005). This method is most sensitive to the degree of fill as well as particle size and it can be expected that if there are void volumes it will have a characteristic low fractal dimension. 

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.

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, D is considered as an effective number that characterises the irregular electrode surface. The term has been related to physical quantities such as mass distribution, density of vibrational stages, conductivity and elasticity. If we consider a 2-D fractal picture in its self-similar multi-steps, one can draw various spheres of known radii at various points of its structure and may thus count the number of particles, N inside the sphere by microscope, following relation will then hold good ... 

Figure 3.9 Box-counting analysis of a projection of a computer-generated diffusion-limited cluster aggregate of 10 000 particles with mass fractal dimension of 1.88. The images have box sizes L of (top left to bottom right) 1, 2,4,8,16, 32 and 64 pixels and require 205 245,59519, 17 062, 4895,1436, 462 and 135 squares respectively to cover the image.

Both porosity and fractal dimension of the generated clusters show good agreement with the predicted values resulting from (10.22) to (10.24). Slight differences arise from the accomplished random agglomerate creation process. Furthermore, the determination of the fractal dimension by the box counting method [41] has a limited accuracy for relatively low particle numbers. 

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