Fractal dimensions 84, surface analysis

Besides the molecular probe method using gas adsorption,107 162 recently, the TEM image analysis method163"167 has been applied to evaluate the surface fractal dimension of porous materials. The most attractive fact in this method is that the pores in different size ranges can be extracted from the TEM images which include contributions from many different pore sizes by the inverse fast Fourier transform (FFT) operation by selecting the specific frequency range.165 167... 

Several simple relations have been proposed for the determination of the fractal dimensions from the results of such experiments as gas molecular probe method, transmission electron microscopy (TEM), small-angle X-ray scattering, neutron scattering, and laser light scattering.63,66,113 116 Among those techniques, gas molecular probe method and image analysis method have been widely used for the calculation of the surface... 

On the other hand, for the microporous carbons with pore size distribution (PSD) with pore fractality, the pore fractal dimensions56,59,62 which represent the size distribution irregularity can be theoretically calculated by non-linear fitting of experimental adsorption isotherm with Dubinin-Astakhov (D-A) equation in consideration of PSD with pore fractality.143"149 The image analysis method54,151"153 has proven to be also effective for the estimation of the surface fractal dimension of the porous materials using perimeter-area method.154"159... 

Determination of the fractal nature of a dendritic surface was first reported by Avnir and FarinJ911 Employing data obtained from solvent accessible surface area analysis (see Sect. 2.3.3.3) for the PAMAM dendrimers, a surface fractal dimension (D) was derived. Two methods were used. The first method applied the relation,... 

Various other aspects of fractal analysis have been discussed by Van Damme and Fripiat and their co-workers. For example, by extending the BET model to fractal surfaces, Fripiat et al. (1986) were able to show that the apparent fractal dimension is reduced by the progressive smoothing of a molecularly rough surface. Alternatively, the effect of a micropore filling contribution is to enhance the fractal dimension. 

In this study, the BET surface areas and pore diameters of the sludge adsorbents are 0.2 - 499 m /g and 19 - 188 A. From the point of view of the specific surface area and pore diameter, it was concluded that the optimum activation condition for the sludge adsorbent with 7MKOH and steam occurs at 7001C for Ihr.From the analysis of fractal dimensions(D) using the FHH theory, the D values could be used to describe the structure... 

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. 

Figure 8 shows that the last ten points of our experimental isotherm, corresponding to the pressure range > 0.5 yield a fairly linear plot, the tangent of which is equal to 0.323 (computer linear regression for the last ten points). This means that the fractal dimension D of our silica sample is about 2, i.e. we may safely neglect the fractal phenomena in water adsorption on that silica sample. This greatly simplifies our theoretical analysis, but it should be emphasized that adsorption on more porous surfaces 2 < D < 3, can also be analyzed in terms of a suitably modified BET model [98-100]. 

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