Infinite complex of surfaces

Today this infinite complex of surfaces would be called a fractal. It is a set of points with zero volume but infinite surface area. In fact, numerical experiments suggest that it has a dimension of about 2.05 (See Example 11.5.1.) The amazing geometric properties of fractals and strange attractors will be discussed in detail in Chapters 11 and 12. But first we want to examine chaos a bit more closely. 

Back in Chapter 9, we found that the solutions of the Lorenz equations settle down to a complicated set in phase space. This set is the strange attractor. As Lorenz (1963) realized, the geometry of this set must be very peculiar, something like an infinite complex of surfaces. In this chapter we develop the ideas needed to describe such strange sets more precisely. The tools come from fractal geometry. 

In effect, the flow is acting like the pastry transformation, and the phase space is acting like the dough Ultimately the flow generates an infinite complex of tightly packed surfaces the strange attractor. 

A new field of coordination chemistry is that of polymetallic cage and cluster complexes [Mm(p-X)xLJz with molecular (i.e. discrete) structure. They contain at least three metal atoms, frequently with bridging ligands X and terminal ligands L. These compounds link the classical complexes (m = 1) and the non-molecular (m - oo) binary and ternary compounds of the metals.1 Molecular polymetallic clusters (with finite radius) also provide a link with the surfaces (infinite radius) of metals and their binary compounds.2"5 Polymetallic complexes are known for almost all metals except the actinides. 

Another solid state reaction problem to be mentioned here is the stability of boundaries and boundary conditions. Except for the case of homogeneous reactions in infinite systems, the course of a reaction will also be determined by the state of the boundaries (surfaces, solid-solid interfaces, and other phase boundaries). In reacting systems, these boundaries are normally moving in space and their geometrical form is often morphologically unstable. This instability (which determines the boundary conditions of the kinetic differential equations) adds appreciably to the complexity of many solid state processes and will be discussed later in a chapter of its own. 

In such a representation of an infinite set of master equations for the distribution functions of the state of the surface and of pairs of surface sites (and so on) will arise. This set of equations cannot be solved analytically. To handle this problem practically, this hierarchy must be truncated at a certain level. In such an approach the numerical part needs only a small amount of computer time compared to direct computer simulations. In spite of very simple theoretical descriptions (for example, mean-field approach for certain aspects) structural aspects of the systems are explicitly taken here into account. This leads to results which are in good agreement with computer simulations. But the stochastic model successfully avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed to obtain good statistics for the results. Therefore more complex systems can be studied in detail which may eventually lead to a better understanding of such systems. 

The results obtained for the stochastic model show that surface reactions are well-suited for a description in terms of the master equations. Since this infinite set of equations cannot be solved analytically, numerical methods must be used for solving it. In previous Sections we have studied the catalytic oxidation of CO over a metal surface with the help of a similar stochastic model. The results are in good agreement with MC and CA simulations. In this Section we have introduced a much more complex system which takes into account the state of catalyst sites and the diffusion of H atoms. Due to this complicated model, MC and in some respect CA simulations cannot be used to study this system in detail because of the tremendous amount of required computer time. However, the stochastic ansatz permits to study very complex systems including the distribution of special surface sites and correlated initial conditions for the surface and the coverages of particles. This model can be easily extended to more realistic models by introducing more aspects of the reaction mechanism. Moreover, other systems can be represented by this ansatz. Therefore, this stochastic model represents an elegant alternative to the simulation of surface reaction systems via MC or CA simulations. 

For microwave radiation incident upon a slab from a direction perpendicular to its surface, a fraction of the energy will be reflected from the surface, Pr, depending upon its complex dielectric constant e. The main contribution to the magnitude of reflection however, is from the dielectric constant e. Errors due to neglecting e" are less than 5% for virtually all foods as is indicated by the 5% line in Figure 1. Neglecting the loss factor, an approximate equation for the fraction of microwave power reflected from an infinite slab food surface is given by ... 

Embedding techniques at various levels have been suggested to close the gap between the cluster and the periodic treatment. A physical approach to the electronic structure problems of solids and surfaces contrasts sharply with the intuition of chemists that local interactions dominate the properties of surfaces and adsorption complexes. Hence, it is very desirable to replace the infinite solid, which is very difficult to treat quantum-chemically, by finite models of the sites of interest. In this way it is easy to describe a local site as cluster having a relatively small number of atoms which interact with a potentially infinite number of surrounding atoms through, for example, the Madelung potential which is treated as perturbation.26,44 These lead to terms such as... 

In the framework of this description an attempt to model an effect of spatial non-uniformity of real catalytic systems was made (Bychkov et al., 1997). It was assumed that reaction proceeds in a heterogeneous system represented by two active infinite plane surfaces and in the gas gap between them. Surface chemistry was treated as for the Li/MgO catalyst (see Table III). Because of substantial complexity of the kinetic scheme consisting of several hundred elementary steps, the mass-transfer was described in this case as follows. The whole gas gap was divided into several (up to 10) layers of the same thickness, and each of them was treated as a well-stirred reactor. The rate of particle exchange between two layers was described in terms of the first-order chemical reaction with a rate constant ... 

The technique is applicable to an infinite variety of complex separations with unique attributes. The properties forming the basis of separation (surface charge, relative affinities) are unlike those of other commonly used processes and thereby... 

The Riemann surface for the cube root f(z) = z comprises three Riemann sheets, corresponding to three branches of the function. Analogously, any integer or rational power of z will have a finite number of branches. However, an irrational power such as f(z) = z = will not be periodic in any integer multiple of In and will hence require an infinite number of Riemaim sheets. The same is true of the complex logarithmic function... 

---