Surfaces self-similar shape

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. 

In recent years much attention has been given to the application of fractal analysis to surface science. The early work of Mandelbrot (1975) explored the replication of structure on an increasingly finer scale, i.e. the quality of self-similarity. As applied to physisorption, fractal analysis appears to provide a generalized link between the monolayer capacity and the molecular area without the requirement of an absolute surface area. In principle, this approach is attractive, although in practice it is dependent on the validity of the derived value of monolayer capacity and the tacit assumption that the physisorption mechanism remains the same over the molecular range studied. In the context of physisorption, the future success of fractal analysis will depend on its application to well-defined non-porous adsorbents and to porous solids with pores of uniform size and shape. 

Note that fractals (self-similar sets with fractal dimension) were first studied and described by mathematicians long before the publications of Mandelbrot, when such fundamental definitions as function, line, surface, and shape were analyzed. 

Studies of monolayer films have included alcohols and carboxylic acids with hydrocarbon chains of varying length in the range C10-C26 [20, 21]. In the liquid condensed and solid states the 11- isotherms have similar shapes at high surface pressures. These data show that the monolayer in the self-assembled state occupies an area of 0.205 nm per monomer. This result is independent of chain length and provides evidence that the monolayer consists of a close-packed structure with all molecular units oriented with their hydrocarbon chains perpendicular to the interface. Under these circumstances, strong attractive van der Waals forces are present between the hydrocarbon tails. As a result, formation of the solid-state film can be irreversible, so that the film does not break up when the surface pressure is decreased. 

In the book [117], some data are given on the hydrodynamic characteristics of bodies of various shapes these data mainly pertain to the region of precrisis self-similarity. The influence of roughness of the cylinder surface and the turbulence level of the incoming flow on the drag coefficient is discussed in [522]. In [211], the relationship between hydrodynamic flow characteristics in turbulent boundary layers and the longitudinal pressure gradient is studied. Analysis of the transition to turbulence in the wake of circular cylinders is presented in [333]. 

We contend that the shape of large molecules in empty space is affected by the topology of the four-dimensional space-time manifold. Guided by the principle of cosmic self-similarity, it is reasonable to assume that, like many spiral galaxies, extended molecules tend to curve like the surface of a golden spiral. It lies in an... 

If the film is nonconductive, the ion must diffuse to the electrode surface before it can be oxidized or reduced, or electrons must diffuse (hop) through the film by self-exchange, as in regular ionomer-modified electrodes.9 Cyclic voltammograms have the characteristic shape for diffusion control, and peak currents are proportional to the square root of the scan speed, as seen for species in solution. This is illustrated in Fig. 21 (A) for [Fe(CN)6]3 /4 in polypyrrole with a pyridinium substituent at the 1-position.243 This N-substituted polypyrrole does not become conductive until potentials significantly above the formal potential of the [Fe(CN)6]3"/4 couple. In contrast, a similar polymer with a pyridinium substituent at the 3-position is conductive at this potential. The polymer can therefore mediate electron transport to and from the immobilized ions, and their voltammetry becomes characteristic of thin-layer electrochemistry [Fig. 21(B)], with sharp symmetrical peaks that increase linearly with increasing scan speed. 

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