Surface roughness and fractals

The one-dimensional lines, two-dimensional surfaces and three-dimensional solids of Euclidean geometry are concepts so familiar to us that we tend to regard them as common sense . Fractals involve less familiar concepts, such as curves between two points, which have infinite length, and surfaces with infinite surface area. Fractals can be characterized by a dimension, but the dimension is fractional. For example, the triadic Koch curve (see Fractals Fig. 1) has a dimension of approximately 1.26186 and the surface with an infinite succession of regular tetrahedral asperities (ibid. Figure 4), a dimension of approximately 2.58496. In this article, the meaning and calculation of fractal dimension are discussed. Texts on the mathematics of fractals introduce different kinds of fractal dimension, which are beyond the scope of this article. The dimension discussed here is strictly the self-similarity dimension. Further aspects of fractal dimensions are considered in Fractals and surface roughness. 

Fractal curves generally have a dimension, D, between 1 and 2, fractal surfaces between 2 and 3. The value of D can be worked out knowing the mathematical procedme by which the fractal is generated. It is unlikely that such a procedure will be known for a fractal observed in nature. The evaluation of fractal dimension from experimental measurements is discussed in Fractals and surface roughness. 

Fractals and surface roughness D E PACKHAM Measurement of fractal dimension, relation to roughness factor... 

Ismail, l.M.K. Pfeifer, P. (1994) Fractal analysis and surface roughness of nonporous carbon fibers and carbon blacks. Langmuir 10 1532-1538... 

Another important field of the application of fractal approach to texturology is related to surface roughness. Anvir and Pfeifer [212,213] proposed characterization of surface irregularities by adsorption and established two methods, based on Mandelbrot s fundamental equations of type 9.69. According to the first method of Dt calculation, one uses the relations that interrelate a number of molecules in a complete monolayer during physisorption, nm, or an accessible surface area, A, with a cross-sectional area, w, which correspond to one molecule in a monolayer ... 

Fractal Approach to Rough Surfaces and Interfaces in Electrochemistry... 

The structure of this review is composed of as follows in Section II, the scaling properties and the dimensions of selfsimilar and self-affine fractals are briefly summarized. The physical and electrochemical methods required for the determination of the surface fractal dimension of rough surfaces and interfaces are introduced and we discuss the kind of scaling property the resulting fractal dimension represents in Section III. 

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