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

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. 

With the help of fractional calculus, Dassas and Duby123 have worked on the problem of diffusion towards the fractal interfaces. They have proposed the following generalized diffusion equation involving a fractional derivative operator ... 

IV. INVESTIGATION OF DIFFUSION TOWARDS SELF-AFFINE FRACTAL INTERFACE... 

Recently, Pyun et al.43,45 gave a clear interpretation of diffusion towards self-affine fractal interface. They investigated theoretically how the diffusing ions sense the self-affine fractal interface during the diffusion-limited process43 and then provided successful experimental evidence of the theoretical investigation.45 Here, let us explore their works in detail. 

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. 

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... 

Dassas, Y, and Duby, P. 1995. Diffusion toward fractal interfaces. Potentiostatic. galvanos-tatic, and linear sweep voltammetric techniques. Journal of the Electrochemical Society 142,4175-4180. 

Go, J.-Y, and Pyun, S.-I. 2007. A review of anomalous diffusion phenomena at fractal interface for diffusion-controlled and non-diffusion-controlled transfer processes. Journal of Solid State Electrochemistry 11, 323-334. 

Sapoval, B., Ross, M., and Gouyet, J.F. Fractal interfaces in diffusion, invasion a corrosion. The Fractal Approach to Heterogeneous Chemistry. Surfaces, Colloids, Polymers, D. Avnir, ed., Wiley, London, pp. 227-245,1989. 

In this section we discuss in some detail a macroscopic approximate model for adsorption kinetics on fractal interfaces. Further details can be found in [8. In diffusion-limited conditions, the balance equations for adsorption on flat surfaces take the form... 

In the case of fractal interfaces, the anomalies in diffusive motion are localized within a thin section located close to the interface and referred to as the fractal layer, since diffusion is perfectly regular (Fickian) in the bulk fluid phase. 

In the second model, referred to as the fractal layer model, the presence of the fractal layer is explicitly considered. The model is therefore characterized by the presence of three phases 1) a bulk region, far from the fractal interface, in which the diffusional phenomena are regular, i.e. not influenced by the fractality of the solid surface, and Fick s law applies 2) a fractal region (the fractal layer), in which anomalous diffusion occurs, described by means of model eqs. (2)-(3) 3) an adsorbed surface phase. In this way, a finite thickness of the fractal interface Lp is introduced, delimiting the region in which anomalous diffusion o.ccurs. The resulting balance equations should therefore read... 

Figure 2 shows the comparison of the fractal-layer (solid line a) and two-timescale (solid line b) models with the simulations in terms of effective diffusivity, eq. (13). Both the models furnish a satisfactory level of agreement with simulation data. We may therefore conclude that approximate models based on a Riemann-Liouville constitutive equation are able to furnish an accurate description of adsorption kinetics on fractal interfaces. These models can also be extended to nonlinear problems (e.g. in the presence of nonlinear isotherms, such as Langmuir, Freundlich, etc.). In order to extend the analysis to nonlinear cases, efficient numerical sJgorithms should be developed to solve partied differential schemes in the presence of Riemann-Liouville convolutional terms. 

Equation (16) also holds for transfer across fractal interfaces. In the latter case, the fractal dimension dj refers to the (Euclidean) bulk in which particles diffuse, and is given hy d] = d = dx + I, d = 2, while dj equals the fractal dimension d of the interface itself. 

Metal-polymer fractal interfaces may result from processes such as vacuum deposition and chemical vapor deposition where metal atoms can diffuse considerable distances into the polymer. Mazur et al. [76,77] electrodeposited silver within a polyimide film. The Silver [I] solution was able to diffuse into the polymer film where it... 

The structure of the diffuse weld interface resembles a box of width X, with fractal edges containing a gradient of interdiffused chains as shown by Wool and Long. When the local stress at a crack tip exceeds the yield stress, the deformation zone forms and the oriented craze fibrils consist of mixtures of fully entangled matrix chains and partially interpenetrated minor chains. 

It is worth mentioning that although Eqs (20)-(27) may be interpreted as involving nonuniform diffusion (Schrama [1957]) either in bulk or at an interface, another allied but somewhat different approach which also leads to Eq. (22) has been proposed (Le Mehaute and Crepy [1983]) without reference to its earlier history and use. This theory involves mass transfer at a fractal interface, one with apparent fractal dimensionality d, with d= y/ = (1 - a) A more solidly based treatment of a fractal interface has been published by Liu [1985]. Unfortunately, neither of these approaches provides a quantitative interpretation in terms of microscopic parameters of why y/, determined from data fitting on solids or liquids, often depends appreciably on temperature. 

A general transport phenomenon in the intercalation electrode with a fractal surface under the constraint of diffusion mixed with interfadal charge transfer has been modelled by using the kinetic Monte Carlo method based upon random walk approach (Lee Pyim, 2005). Go and Pyun (Go Pyun, 2007) reviewed anomalous diffusion towards and from fractal interface. They have explained both the diffusion-controlled and non-diffusion-controlled transfer processes. For the diffusion coupled with facile charge-transfer reaction the... 

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]. 

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, ... 

---