Reactions on fractal surfaces

Formation processes often determine the dimension of the surface of the solid but the dimension of a surface also affects its chemistry. They are closely linked the growth reaction at a surface affects the surface morphology, which in turn affects the reactivity. This reciprocal influence can make the overall reaction oscillate, which was shown for two cases in Chapter 6. These are examples of positive feedback in surface reactions. 

Another chemical consequence of broken dimensions in heterogeneous catalysts is that the parameter specific surface area, expressed in square meters per gram, is not defined if the gas that is used to determine that parameter is not mentioned. One should add to the surface area that is always mentioned with catalysts the dimension that characterizes its fractal surface. The dimension does not depend on the size of the molecules that measure the surface area in the BET method but the specific 

Diffusion on fractal surfaces is less easy than in topological spaces with an integer dimension. As in the diffusion of atoms, the conductance properties also depend on the dimension. Reactions are so frequently anomalous that many reports on rate measurements, interpreted with two-dimensional surfaces, conclude with the note that the observed data invoke new questions that will have to be studied in the sequel. 

The reactions of deposition or crystal growth are surface reactions. The reactants are adsorbed, more or less mobile molecules, e.g., A and in the fictitious reaction A -h B 0. These adsorbates form the substrate surface and growth is the annihilation reaction between the adsorbed reactants. The reaction rate r is expressed as usual (Chapter 6) in the reactant concentrations as r = k[ A][ B]. This can be done if the surface (the reaction space) can be considered to be a well-stirred reactor. In other words, the mobilities of A and B are high compared to the rate of the growth reaction. If that is no longer true and there is diffusion limitation the reaction can still be fitted to the above rate equation except that the reaction rate coefficient k is replaced by kit in which h — i —jS (with S being the spectral dimension). A characteristic value for his for the case of a reaction on a percolation cluster with a spectral dimension of 

Apart from the fractal dimension D of the surface, there is a so-called reaction dimension Q (sometimes also called D ), which characterizes the way in which the rate of the reaction r on the surface scales with the size of the particles L. The power law is r oc L. The value of Q is usually between 1 and 3 but extremes such as 0.2 and 5.8 have also been observed. The distribution of chemically active sites on the surface determines the way in which the reactivity scales with the size of the crystallite. 

---

The dependence of the kinetics on dimensionality is due to the physics of diffusion. This modifies the kinetic differential equations for diffusion-limited reactions, dimensionally restricted reactions, and reactions on fractal surfaces. All these chemical kinetic patterns may be described by power-law equations with time-invariant parameters like... 

---