Fractal interfaces

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. 

Polymer-polymer fractal interfaces may result from the interdiffusion of monomers or of polymers themselves. Koizumi et al. [31] annealed the interface between polystyrene and a styrene-isoprene diblock polymer at 150 C and showed extensive roughening of the interface by mutual interdiffusion on a micron scale (Fig. 8). 

Polymer-metal fractal interfaces may result from processes such as vacuum deposition and chemical vapour deposition where metal atoms can diffuse con-... 

A high modulus gradient at the interface is also be avoided in materials Joined as a result of the interdiffusion of materials to form a fractal surface [32]. The effect is to produce an interfacial composite region. This strengthens the interface and leads to a more gradual change in modulus and avoids the sharp concentrations of stress which would occur at a smooth interface. 

Wool, R.P, Dynamics and fractal structure of polymer interfaces. In Lee, L.-H. (Ed.), New Trends in Physics and Physical Chemistry of Polymers. Plenum Press, New York, 1989, p. 129. 

FHH (Frenkel-Halsey-Hill) theory is valid for multi molecules adsorption model of the flat surfrtce material. When this model is applied for the surface fractal in the range of capillary condensation, in other words, in the state of interface which was controlled by the surface tension between liquid and gas, the modified FHH equation can be expressed as Eq. (3). 

Johans et al. derived a model for diffusion-controlled electrodeposition at liquid-liquid interface taking into account the development of diffusion fields in both phases [91]. The current transients exhibited rising portions followed by planar diffusion-controlled decay. These features are very similar to those commonly observed in three-dimensional nucleation of metals onto solid electrodes [173-175]. The authors reduced aqueous ammonium tetrachloropalladate by butylferrocene in DCE. The experimental transients were in good agreement with the theoretical ones. The nucleation rate was considered to depend exponentially on the applied potential and a one-electron step was found to be rate determining. The results were taken to confirm the absence of preferential nucleation sites at the liquid-liquid interface. Other nucleation work at the liquid-liquid interface has described the formation of two-dimensional metallic films with rather interesting fractal shapes [176]. 

In situ SAXS investigations of a variety of sol-gel-derived silicates are consistent with the above predictions. For example, silicate species formed by hydrolysis of TEOS at pH 11.5 and H20/Si = 12, conditions in which we expect monomers to be continually produced by dissolution, are dense, uniform particles with well defined interfaces as determined in SAXS experiments by the Porod slope of -4 (non-fractal) (Brinker, C. J., Hurd, A. J. and Ward, K. D., in press). By comparison, silicate polymers formed by hydrolysis at pH 2 and H20/Si = 5, conditions in which we expect reaction-limited cluster-cluster aggregation with an absence of monomer due to the hydrolytic stability of siloxane bonds, are fractal structures characterized by D - 1.9 (Porod slope — -1.9) (29-30). 

The present volume gives a general and at the same time rather detailed review on main research developments in the field of dendrimers (oligomer and polymer) during the past several years, but also offers views and visions of the future - of what could soon be achieved in this area at the interface between small organic molecules and macromolecules (polymers). We are sure that the rapid development of fractal-shaped molecules will continue in academic institutes as well as in industry - there is still more to come. 

Using refinement, you can model objects and interactions at all levels of granularity. This arrangement provides a fractal view of architecture, from the business roles and processes to large-grained interacting architectural components including a system to individual interfaces and classes. Any refinement has associated architectural decisions. 

Electrochemical processes usually take place on rough surfaces and interfaces and the use of fractal theory to describe and characterize the geometric characteristics of surfaces and interfaces can be of significant importance in electrochemical process description and optimization. Drs. Joo-Young Go and... 

Fractal Approach to Rough Surfaces and Interfaces in Electrochemistry... 

Diffusion-limited electrochemical techniques as well as physical techniques have been effectively used to determine the surface fractal dimensions of the rough surfaces and interfaces made by electrodeposition, " fracture, " vapor deposition, ... 

The structure of this review is composed of as follows in Section II, the scaling properties and the dimensions of selfsimilar and self-affine fractals are briefly summarized. The physical and electrochemical methods required for the determination of the surface fractal dimension of rough surfaces and interfaces are introduced and we discuss the kind of scaling property the resulting fractal dimension represents in Section III. 

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. 

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