Resistivity, fractal conductivity

The fractal concept has proved to the helpful in describing such systems. In this connection our attention has been focused on using the fractal concept to make predictions concerning the physical properties of inhomogeneous media with a random structure. We remark that numerous other examples of fractal behavior than those treated here appear in the literature. We should mention resistance capacitance transmission lines [240] and fractal models for the alternating current response at a rough interface between materials of very dissimilar conductivities [241,242] and how a resistance capacitance line may be used as a semiintegrator [243]. 

Equation (6.12) faces one main difficulty, namely the determination of the explicit functional form of D(s). It is important to stress that D s) does not have exactly the same properties as the classical diffusion coefficient, and we refer to it here as the conductivity. Likewise, we define the resistivity of the medium as R = l/D. We expect the resistivity to be proportional to the number of steps of the particle, and arguments from random walks on fractals should be useful to determine R. Walks on fractals are characterized by the existence of two scales. Divide the medium into small blocks of size, such that the diffusion is normal within the small blocks, f. At scales larger than f, the effect of the heterogeneities becomes important, and motion depends on the fractal parameters. The self-invariance properties of the fractal are not valid at short distances. Similarly, the idealized concept of self-similarity at all scales does not hold for fractals in practice. 

There are two different types of fractals in solid state chemistry (a) mass fractals, sets of solid particles that form aggregates and have as measure their mass that scales as I with 0 < D 3 and (b) surface fractals that consist of interfaces between solids and the vacuum and that have as measure the surface, which also scales as IP with 0 < D < 3. The fractal dimension of an object, composite, or aggregate affects the values of the heat capacity, heat conductivity, electric conductivity, mechanical resistance against deformation, specific mass, and light scattering. 

Interfacial contact resistance depends not only on contact (clamping) pressure, but also on the surface characteristics and the effective conductivities of the two surfaces in contact. Using the fractal geometric descrip-... 

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