Self-affine fractal model

It is difficult to describe the complex forms and status of natural fracture surfaces with ordinary functions. The fractal theory can be used to describe extremely irregular geometric patterns, and many researches have indicated that it is reasonable to simulate rough fracture surfaces with fractals theory. At present, the self-affine fractal model is regarded as the best fractal model to simulate fracture surface. In the following, random Brown function method (Weierstrass- Mandelbrot function) is used to simulate a fractal fracture surface. 

Kaplan T, Gray LJ, Liu SH. Self-affine fractal model for a metal-electrol5de interface. Phys Rev B 1987 35 5379-81. 

Electric Double Layer and Fractal Structure of Surface Electrochemical impedance spectroscopy (EIS) in a sufficiently broad frequency range is a method well suited for the determination of equilibrium and kinetic parameters (faradaic or non-faradaic) at a given applied potential. The main difficulty in the analysis of impedance spectra of solid electrodes is the frequency dispersion of the impedance values, referred to the constant phase or fractal behavior and modeled in the equivalent circuit by the so-called constant phase element (CPE) [5,15,16, 22, 35, 36]. The frequency dependence is usually attributed to the geometrical nonuniformity and the roughness of PC surfaces having fractal nature with so-called selfsimilarity or self-affinity of the structure resulting in an unusual fractal dimension... 

The roughness design can be performed by virtual alteration of the surface roughness properties, approximated by analytical model PSD functions. Several PSD models exist for the description of surface roughness characteristics [15]. The fractal model, for example, is applied if roughness characteristics are assumed to be self-affine ... 

The parameters K and n are related to the vertical and fractal dimensions of the self-affine roughness components, respectively. Surfaces roughness features with characteristic mean vertical and lateral dimensions can be described using the ABC model ... 

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