Fractals surface profile

Non-contact laser profilometry has been used to obtain roughness parameters and surface fractal dimension (Ds) of erythromycin acistrate tablets It was expected that tablet surface fractal dimension (Ds) will depend on the compression force used for tabletting. However, tablet surface fractal dimension was found independent of the compression force ranging from 4 to 22 kN. Instead, a surface roughness parameter Rp, which is the maximum distance between the highest point and the average height of the surface profile, indicated the variation of tablet friability as a function of compression force (Fig. 7). 

Stemp, W.J., B.E. Childs, S. Vionnet, and C.A. Brown. 2008. Quantification and discrimination of lithic use-wear surface profile measurements and length-scale fractal analysis. Archaeometry XXX. 

Figure 1 Different fractals considered in environmental sciences, (a) Surface fractal colony of Bacillus subtilis on agar see Figure 8.3 for more details, (b) Mass fractal aggregate of hematite particles formed at pH 4, ionic strength 150 mM, in the presence of natural organic matter of [C] =2.8 mg dm see Chapters 4 and 5 for more details on aggregation processes, (c) Pore fractals soil profile in which a (darker) preferential pathway is visible see Figure 2.21 for more details.

The structure function method considers all points on the surface profile curve as a time sequence z(x) with fractal character. The structure function s(r) of sampling data on the profile curve can be described as s l) = [z(x + t) - z(x)] = c -2D here [z(x+z)-z(x)] expresses the arithmetic average value of difference square, and ris the random choice value of data interval. Different rand the corresponding s can be plotted versus the ron a log-log scale. Then, the fractal dimension D can be related to the slope m of a fitting line on log-log plot as D = (4 - m)/2. 

Han, J.H., Ping, S. and Shengsun, H. (2005) Fractal characterization and simulation of surface profiles of copper electrodes and aluminum sheets. Materials Science and Engineering A, 403 174-81. 

Hasegawa, M., Liu, J., Okuda, K. and Nunobiki, M. (1996) Calculation of the fractal dimensions of machined surface profiles. Wear, 192 40-5. 

Sahoo, R, Barman, T.K. and Routara, B.C. (2008) Fractal dimension modeling of surface profile and optimization in CNC end milling using response surface method. International Journal of Manufacturing Research, 3(3) 360-77. 

The question as to whether there is a general and reliable relationship between the fractal dimension of a surface and the fracture toughness of the material is considered by a few authors and it has been shown that a crack surface profile can be effectively described in terms of fractal geometry (e.g. Lange et al, 1993). 

Figure 13.2 Determination of the fractal dimension of a surface using the box-coimting method. The graph shows the number of boxes of size r necessary to approximate the measured surface profile. A planar surface would be precisely described independent of box size and the graph would be linear with slope -1. On a rough stuface, the profile would be sufficiently resolved only within certain boundaries, which are denoted I = loer. andL = logr. The fractal dimension is derived from the slope within these boundaries.

K = geometric constant G = topothesy of a surface profile, m D = fractal dimension of a surface profile K = effective electrical conductivity of two surfaces, Sm" ... 

Wool [32] has considered the fractal nature of polymer-metal and of polymer-polymer surfaces. He argues that diffusion processes often lead to fractal interfaces. Although the concentration profile varies smoothly with the dimension of depth, the interface, considered in two or three dimensions is extremely rough [72]. Theoretical predictions, supported by practical measurements, suggest that the two-dimensional profile through such a surface is a self-similar fractal, that is one which appears similar at all scales of magnification. Interfaces of this kind can occur in polymer-polymer and in polymer-metal systems. 

The main conclusion is that the fractal dimension of the distribution profile of acceptors around a donor is inversely dependent on the pore size. It is also important to notice that the same D values are obtained with all three donor/acceptor pairs. We interpret these D values as reflecting the geometry of the support as seen by an adsorbed molecule, and in particular that these D values are the surface fractal dimensions for adsorption, for the following reasons (a) The fact that the D values were found to be insensitive to the different Ro values of the three pairs and to the concentrations employed, is in keeping with the scale-invariance of the fractal model, (b) In a number of studies (21,43) it has been shown that for the same material, higher... 

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. 

---