Diffusion near fractal

Laszlo7 pointed out that solids of fractal dimension D near 2.0 are usually more efficient catalysts than those of D near 3.0. An adsorbed species diffusing across the surface of a catalyst will find a target much more quickly (i.e., catalysis will be more efficient) in space of dimension near 2 than of D near 3. We find, for example, that catalytic activities of variously prepared activated charcoals increase as we go from D = 3.0 to D = 1.9. 

Giona et al. (1995) studied diffusion in the presence of a constant convective field in percolation clusters with stochastic differential equations and a coupled exit-time equation. On the basis of numerical studies on percolation clusters near the percolation threshold, they found that the volume-averaged exit time as a function ofPn did not follow the normal relationship (in which it is proportional to 1 /Pn) but instead increased monotonically with Pn. Their approach needs generalization to more realistic convective fields. They also present exit-time analyses for transport on diffusion limited aggregates and in deterministic fractals... 

Table 7.3 gives the predicted exponents n, for microbial invasion (j8 = 1) and enzyme diffusion P = V2) for two and three dimensions and fractal dimensions of percolation clusters at, above and below the percolation threshold, using equation 7.12. The fractal dimension in each case increases with p with > = at/7 = 1. The initial exponents are independent of p since the fractal pathways have not yet been developed and the starch is immediately accessible on the surface. Thus, the initial slope reflects the intrinsic behaviour A The major exponent is dominated by the fractal dimension existing near p. For the poly-disperse samples, a fractal dimension of Z) = 1.59 better describes the data for both invasion and diffusion compared to Z) 1.8 for the monodisperse samples. The difference in fractal dimension is due to the coarseness of the poly-disperse pathways (average diameter 10 im) compared to the monodisperse blend (diameter = 1 pm). In general, the theoretical exponents in Table 7.3 are in agreement with the simulation exponents. 

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