Interfaces self-similarity

Wool [32] has considered the fractal nature of polymer-metal and of polymer-polymer surfaces. He argues that diffusion processes often lead to fractal interfaces. Although the concentration profile varies smoothly with the dimension of depth, the interface, considered in two or three dimensions is extremely rough [72]. Theoretical predictions, supported by practical measurements, suggest that the two-dimensional profile through such a surface is a self-similar fractal, that is one which appears similar at all scales of magnification. Interfaces of this kind can occur in polymer-polymer and in polymer-metal systems. 

In Section IV, from the studies on diffusion towards self-affine fractal interface, the surface fractal dimension as determined by the electrochemical method is characterized as being self-similar, even though the rough surfaces and interfaces show the self-affine scaling property. Finally, in Section V, we exemplified the application of fractal geometry in electrochemical systems in view of the characterization of rough surfaces and interfaces by the surface fractal dimension. 

It should be stressed here that the specific power dependences from the above self-affine fractal interfaces are maintained even during the relatively long time (or number of random jumps) interval. This implies that the morphology of the self-affine fractal interfaces tested is possibly characterized by the self-similar fractal dimension within a relatively wide spatial cutoff range. 

Under the assumption that the morphology of the self-affine interface has the self-similar scaling property, the apparent selfsimilar fractal dimension d ss of the electrode was calculated... 

From the above results, it is noted that the self-similar scaling property investigated by the triangulation method can be effectively utilized to analyze the diffusion towards the self-affine fractal interface. This is the first attempt to relate the power dependence of the current transient obtained from the self-affine fractal curve to the self-similar scaling properties of the curve. 

An interesting class of exact self-similar solutions (H2) can be deduced for the case where the newly formed phase density is a function of temperature only. The method involves a transformation to Lagrangian coordinates, based upon the principle of conservation of mass within the new phase. A similarity variable akin to that employed by Zener (Z2) is then introduced which immobilizes the moving boundary in the transformed space. A particular case which has been studied in detail is that of a column of liquid, initially at the saturation temperature T , in contact with a flat, horizontal plate whose temperature is suddenly increased to a large value, Tw T . Suppose that the density of nucleation sites is so great that individual bubbles coalesce immediately upon formation into a continuous vapor film of uniform thickness, which increases with time. Eventually the liquid-vapor interface becomes severely distorted, in part due to Taylor instability but the vapor film growth, before such effects become important, can be treated as a one-dimensional problem. This problem is closely related to reactor safety problems associated with fast power transients. The assumptions made are ... 

A useful (also extreme) counterpart to the also idealized linear geometry is fractal geometry which plays a key role in many non-linear processes.280 281 If one measures the length of a fractal interface with different scales, it can be seen that it increases with decreasing scale since more and more details are included. The number which counts how often the scale e is to be applied to measure the fractal object, is not inversely proportional to ebut to a power law function of e with the exponent d being characteristic for the self-similarity of the structure d is called the Hausdorff-dimension. Diffusion limited aggregation is a process that typically leads to fractal structures.283 That this is a nonlinear process follows from the complete neglect of the back-reaction. The impedance of the tree-like metal in Fig. 76 synthesized by electrolysis does not only look like a fractal, it also shows the impedance behavior expected for a fractal electrode.284... 

Earlier speculations about the effect of the curvature of space on elemental synthesis and the stability of nuclides (2.4.1) are consistent with the interface model. The absolute curvature of the closed double cover of projective space, and the Hubble radius of the universe, together define the golden mean as a universal shape factor [233], characteristic of intergalactic space. This factor regulates the proton neutron ratio of stable nuclides and the detail of elemental periodicity. The self-similarity between material structures at different levels of size, such as elementary particles, atomic nuclei, chemical... 

Consider dilute chains with N monomers of size b restricted to the air water interface in an athermal good solvent (two-dimensional polymers). These chains are a dsorbed to a contact line with monomer-surface interaction of (5kr=0.1 kT. Calculate the density profile of the de Gennes self-similar carpet. Calculate the thickness of the adsorbed layer and the linear coverage P (the total number of monomers in the adsorbed layer per unit length of the contact line). 

However, there is no obvious length scale in the parameters of the problem. The radius of the tube is irrelevant because the temperature fields are independent of r, and the distance from the bottom of the container to the interface,, varies with time. In fact, this is another example in which the solution of the problem exhibits a self-similar form. In view of this, we... 

The prediction of Riemann-Liouville approximate models may be compared with Monte Carlo simulation data, obtained on a self-similar fractal interfaces (Koch-like curve) possessing a fractal dimension = log 8/log 4 = 3/2. Figure 1 shows the comparison of... 

Rogak, S.N. and Flagan, R.C. (1990). Stokes drag on self-similar clusters of spheres. J. Colloid Interface Sci., 134, 206-218. 

---