Surface fractal nature

Although the rate of dissolving measurements do thus give a quantity identified as the total surface area, this area must include that of a film whose thickness is on the order of a few micrometers but basically is rather indeterminate. Areas determined by this procedure thus will not include microscopic roughness (or fractal nature). 

We have considered briefly the important macroscopic description of a solid adsorbent, namely, its speciflc surface area, its possible fractal nature, and if porous, its pore size distribution. In addition, it is important to know as much as possible about the microscopic structure of the surface, and contemporary surface spectroscopic and diffraction techniques, discussed in Chapter VIII, provide a good deal of such information (see also Refs. 55 and 56 for short general reviews, and the monograph by Somoijai [57]). Scanning tunneling microscopy (STM) and atomic force microscopy (AFT) are now widely used to obtain the structure of surfaces and of adsorbed layers on a molecular scale (see Chapter VIII, Section XVIII-2B, and Ref. 58). On a less informative and more statistical basis are site energy distributions (Section XVII-14) there is also the somewhat laige-scale type of structure due to surface imperfections and dislocations (Section VII-4D and Fig. XVIII-14). 

Wool [32] has considered the fractal nature of polymer-metal and of polymer-polymer surfaces. He argues that diffusion processes often lead to fractal interfaces. Although the concentration profile varies smoothly with the dimension of depth, the interface, considered in two or three dimensions is extremely rough [72]. Theoretical predictions, supported by practical measurements, suggest that the two-dimensional profile through such a surface is a self-similar fractal, that is one which appears similar at all scales of magnification. Interfaces of this kind can occur in polymer-polymer and in polymer-metal systems. 

The word fractal was coined by Mandelbrot in his fundamental book.1 It is from the Latin adjective fractus which means broken and it is used to describe objects that are too irregular to fit into a traditional geometrical setting. The most representative property of fractal is its invariant shape under self-similar or self-affine scaling. In other words, fractal is a shape made of parts similar to the whole in some way.61 If the objects are invariant under isotropic scale transformations, they are self-similar fractals. In contrast, the real objects in nature are generally invariant under anisotropic transformations. In this case, they are self-affine fractals. Self-affine fractals have a broader sense than self-similar fractals. The distinction between the self-similarity and the selfaffinity is important to characterize the real surface in terms of the surface fractal dimension. 

There are some very special characteristics that must be considered as regards colloidal particle behavior size and shape, surface area, and surface charge density. The Brownian motion of particles is a much-studied field. The fractal nature of surface roughness has recently been shown to be of importance (Birdi, 1993). Recent applications have been reported where nanocolloids have been employed. Therefore, some terms are needed to be defined at this stage. The definitions generally employed are as follows. Surface is a term used when one considers the dividing phase between... 

Farin, D. Avnir, D. (1989) The fractal nature of molecule-surface interactions and reactions. In Avnir, D. (ed.) The fractal approach to heterogenous chemistry. Wiley, New York, 271-294... 

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,... 

Earin, D. Avnir, D. The fractal nature of molecule-surface chemical activities and physical interactions in porous materials. In Characterisation of Porous Solids Unger, K.K., Rouquerol, J., Sing, K.S.W., Krai, H., Eds. Elsevier Science Amsterdam, 1988 421 32. 

The surface property of a solid is characterized by the nature of the surface boundary. The surface boimdary is expected to be related to the underlying geometric nature of the surface, hence its fractal dimension. Many properties of the solid depend on the scaling behavior of the entire solid and of the pore space. The distribution of mass in the porous solid and the distribution of pore space may also reflect the fractal nature of the surface. If the mass and the surface scale are alike, that is, have the same power-law relationship between the radius of a particle and its mass, then the system is referred to as a mass fractal. In a similar manner if the pore volume of porous material has the same power-law relationship between the pore volume and radius as that of the surface, then it is described as a pore fractal. 

In modern language assumption (L3) states that the surface reconstructs in a self-similar way. Though the exact mathematical specification of this self-similarity is an open problem, the appearance of the Elovich equation seems essentially related to a fractal nature of the reconstructed surface this property is immediately understood by observing that the number of molecules accomodated on a given finite geometric surface area diverges as t —> -poo. 

The fractal nature of the porous silica surface is clearely demonstrated by the fact that neither a two-dimensional nor a three-dimensional model could be applied to simulate the time-correlated fluorescence decay curves. The observed fractal dimensions are in very good agreement with other published results. 

Effects of the fractal nature of soil on water and solute transport can be further complicated by the differences among fractal dimensions in soils at different scales. Avnir et al. (1986) found at least two ranges of radii with different surface fractal dimensions in studied soils. Dependence of fractal dimension on pore radii was demonstrated (Wu et al., 1993 Perfect et al., 1993). Pachepsky et al.( 1995a) found three or four distinct scaling intervals with different fractal dimensions in the range of pore radii from 4 nm to 5 pm. Fractal scaling of soil water retention is usually well pronounced at capillary potentials lower than -30 kPa. 

Physisorption or chemisorption of molecules and atoms has been employed to determine the fractal nature of surfaces. Two approaches can be adopted. One uses a probe molecule or atom to measure the surface area as a function of particle size. The other probes the surface area at constant particle size but with molecules or atoms of different sizes. Figure (5.20) illustrates the first approach for the chemisorption of... 

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