Surface fractal dimension interfaces

Diffusion-limited electrochemical techniques as well as physical techniques have been effectively used to determine the surface fractal dimensions of the rough surfaces and interfaces made by electrodeposition, " fracture, " vapor deposition, ... 

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. 

In Section IV, from the studies on diffusion towards self-affine fractal interface, the surface fractal dimension as determined by the electrochemical method is characterized as being self-similar, even though the rough surfaces and interfaces show the self-affine scaling property. Finally, in Section V, we exemplified the application of fractal geometry in electrochemical systems in view of the characterization of rough surfaces and interfaces by the surface fractal dimension. 

It is reported116,117 that as more adsorbed layers are built up, the interface between the adsorbent and the adsorbed molecules becomes smooth, and hence the surface fractal dimension would no longer describe the interface but would describe the adsorbed molecule agglomerates. Also, Eq. (9) is only valid when the adsorbed layer exceeds monolayer coverage. Therefore, for the correct calculation, dFSP should be determined from the linear... 

The present article summarized the fractal characterization of the rough surfaces and interfaces by using the physical and the electrochemical methods in electrochemistry. In much research, both the physical and the electrochemical methods were used to evaluate the fractal dimension and they are complementary to each other. It should be stressed that the surface fractal dimension must be determined by using the adequate method, according to the inherent scaling properties of the rough surfaces and interfaces. 

III. CHARACTERIZATION OF ROUGH SURFACES AND INTERFACES BASED UPON FRACTAL GEOMETRY METHODS NEEDED FOR THE DETERMINATION OF THE SURFACE FRACTAL DIMENSION... 

At low relative pressures p/p0 or thin adsorbate films, adsorption is expected to be dominated by the van der Waals attraction of the adsorbed molecules by the solid that falls off with the third power of the distance to the surface (FHH-regime, Eq. 3a). At higher relative pressures p/p0 or thick adsorbate films, the adsorbed amount N is expected to be determined by the surface tension y of the adsorbate vapor interface (CC-regime, Eq. 3b), because the corresponding surface potential falls off less rapidly with the first power of the distance to the surface, only. The cross-over length zcrit. between both regimes depends on the number density np of probe molecules in the liquid, the surface tension y, the van der Waals interaction parameter a as well as on the surface fractal dimension ds [100, 101] ... 

For smooth surface, the parameter S is assumed to be equal to -1/3, while for a fractal surface, S is a function of the surface fractal dimension, D. If the van der waals attractive forces are dominant between adsorbent and adsorbate, then S is equal to (Ds-3)/3 [9]. For higher surface coverage where the adsorbent-adsorbate interface is... 

The interface between the crystalline fibrils and the interstitial amorphous regions has a sharp boundary, giving rise to the power law with the exponent Odi = 4.0 as predicted by the Porod law. The exponent suggests the corresponding surface fractal dimension of Dj = 2 from eqn [5], indicating a smooth and flat interface at the relevant length scale. The upper and... 

The chemical reaction between a solid and a reactive fluid is of interest in many areas of chemical engineering. The kinetics of the phenomenon is dependent on two factors, namely, the diffusion rate of the reactants toward the solid/fluid interface and the heterogenous reaction rate at the interface. Reactions can also take place within particles, which have accessible porosity. The behavior will depend on the relative importance of the reaction outside and inside the particle. Fractal analysis has been applied to several cases of dissolution and etching in such natural occurring caves, petroleum reservoirs, corrosion, and fractures. In these cases fractal theory has found usefulness for quantifying the shape (line or surface) with only a few parameters the fractal dimension and the cutoffs. There have been some attempts to use a fractal dimension for reactivity as a global parameter. Finally, fractal concepts have been used to aid in the interpretation of experimental results, if patterns quantitatively similar to DLA are obtained. 

It is worth noting that at the minimum value of d = 2, value A becomes negative. That means, in accordance with Equation (12.1), that tan < tan 8m- In other words, at small d (smooth surfaces of the filler particles) the packing of the polymer molecules at the interface may be more dense compared with the bulk. This fact leads to the diminishing molecular mobility in the interfacial layer [1, 37, 38] Extrapolation of the dependence of A on d to maximum value of d = 3 gives the limiting value of A 4.5. This value meets the value A = 4.2, that was derived from the extrapolation of the volume of the interfacial layer on d to d = 3 [39]. Thus, this analysis allows an estimation of the structural factors influencing formation of the adhesion joints. The main factors are fractal dimensions of the particle surface, d and of the polymer df, which determine adhesion at the polymer-filler particle interface. 

Environmental particles, microbial colonies, and even patterns of movement of organisms can be characterized in terms of several different fractal dimensions [14] (Table 1.1). The fractal dimension of the surface Ds (boundary/interface) of a solid structure is obviously an important characteristic, but many of the physical properties of solids also depend on the scaling behavior of the entire solid and/or of its pores. Systems where surface and mass scale similarly are termed mass fractal systems, those where surface and the pore volumes scale similarly are described as pore fractals and systems where only the surface is fractal are designated as surface fractals (Table 1.1 Figure 1.2). 

Most surfaces are fractal and their chemistry differs from that of flat crystal faces. The chapters comprising this volume are concerned with the chemistry of molecules, solids, surfaces, and fractal matter. Not only the morphology of materials controls many of their properties but the fractal dimension of the matter distribution in materials also strongly affects the result of chemical processes in or on those distributions of atoms. Dimensions of interfaces pervade materials properties and it is difficult to ignore them when optimizing polycrystalline composites. 

There are two different types of fractals in solid state chemistry (a) mass fractals, sets of solid particles that form aggregates and have as measure their mass that scales as I with 0 < D 3 and (b) surface fractals that consist of interfaces between solids and the vacuum and that have as measure the surface, which also scales as IP with 0 < D < 3. The fractal dimension of an object, composite, or aggregate affects the values of the heat capacity, heat conductivity, electric conductivity, mechanical resistance against deformation, specific mass, and light scattering. 

In all the cases examined so far, it is the matter distribution of the object that has exhibited the property of self-similarity. These objects are called mass fractals. Other situations are encountered, where it is not the matter distribution which has self-similarity, but rather the pore distribution in these cases, we speak of pore volume fractals. Some structures are found in which only the contour or surface manifests scale invariance these are called boundary or surface fractals, and the exponent we need to know is the boundary fractal dimension. To obtain the corresponding exponents, we calculate the autocorrelation function, the mass distribution or the number of boxes, restricting ourselves to the relevant subsets (the points occupied by matter, the points in the pore volume, or the points lying in the interface). 

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