Fractal dimensions characterization of textured surfaces

Rough (textured) particles do not have a unique surface. The measured surface depends upon the method of measurement and will increase as the degree of scrutiny increases. For example, corrugating a one-acre field by plowing it will increase its superficial area to V2 acres. 

Texture is difficult to define and quantify. Davies [7] for example, defined texture as the number of asperities possessed by a particle outline. 

Mandelbrot introduced a new geometry in a book, which was first published in French in 1975, with a revised English edition [64] in 1977. In 1983 he published an extended and revised edition that he considered to be the definitive text [65]. Essentially he stated that there are regions between a straight line that has a dimension of 1, a surface that has a dimension of 2 and a volume that has a dimension of 3, and these regions have fractional dimensions between these integer limits. Kaye has presented an excellent review of the importance of fractal geometry in particle characterization [66]. 

If an irregular outline is enclosed by a polygon of constant side length /I, the perimeter will increase as the side length decreases. 

Hence plots of log against log Twill have a slope of -D. The parameter D is characteristic of the texture of the particle and was called by Mandelbrot the fractal dimension. The fractal dimension for the outline of a particle lies between 1 and 2, the more irregular the outline, the higher the value. 

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