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Transfer across Random versus Deterministic Fractal Interfaces

A numerical study of the transfer across random fractal surfaces shows that their responses are very close to the response of deterministic model geometries with the same fractal dimension. The simulations of several interfaces with prefractal geometries show that, within very good approximation, the flux depends only on a few characteristic features of the interface geometry: the lower and higher cut-offs and the fractal dimension. Although the active zones are different for different geometries, the electrode reponses are very nearly the same. In that sense, the fractal dimension is the essential"universal"exponent which determines the net transfer.

A numerical study of the transfer across random fractal surfaces shows that their response are very close to the response of deterministic model geometries with the same fractal dimension. The simulations of several interfaces with prefractal geometries show that, within very good approximation, the flux depends only on a few characteristic features of the interface geometry: the lower and higher cut-offs and the fractal dimension. Although the active zones are different for different geometries, the electrode responses are very nearly the same. In that sense, the fractal dimension is the essential "universal" exponent which determines the net transfer. Many random processes such as aggregation, diffusion, fracture and percolation, build fractal objects [1,2]. Fractal geometry essentially describes hierarchical structures [3]. If properties of these random systems depend on the hierarchical character of their geometry, then the study of a deterministic structure with the same fractal dimension may provide a good approximation of the random system properties [4]. The question is significant since fractal and pre-fractal geometries are widely used in mathematical approaches or numerical simulations as a convenient model of irregularity. They are also more simply addressed by algebraic calculations and incorporated into numerical models for computer simulation. It is then an important matter to decide whether simple deterministic, artificial, fractals could help determine the properties of random, natural, fractals [5,6]. In particular, it is a question whether experiments performed on model fractal geometries [7] may help understand the behavior of real complex structures.
The property which is discussed here is the Laplacian transport to and across irregular and fractal interfaces. Such transport phenomena are often encountered in nature or in technical processes: properties of rough electrodes in electrochemistry, steady-state diffusion towards irregular membranes in physiological processes, the Eley-Rideal mechanism in heterogeneous catalysis in porous catalysts, and in NMR relaxation in porous media. In each of these examples, the interface presents a finite transfer rate, like a redox reaction, or a finite permeability, or reaction rate which is due to specific physical or chemical processes.
The mathematical formulation of the problem is simple. One considers the current flowing through an elec-trochemical cell as shown in Fig. 1. The current J is proportional to the Laplacian field ∇V , which can be viewed as an electrostatic field in electrochemistry, or a particle concentration field in diffusion problems. Then the flux and field are related by classical equations of the type J = −σ ∇V , where σ is the electrolyte conductivity (or particle diffusivity in diffusion or heterogeneous catalysis). The conservation of this current throughout the bulk yields the Laplace equation for the potential V :  The boundary presents a finite resistance to the current flow. In the simplest case, this resistance can be expressed by a linear relation linking the current density across the boundary to the potential drop across that boundary. The local flux and potential drop are then linked by transport coefficients, like the faradaic resistance in electrochemistry, the membrane permeability in physiological processes, or again the surface reactivity in catalysis. If one assumes that the outside of the irregular boundary is at zero potential, current conservation at the boundary leads to the following relation: The parameter Λ is homogeneous to a length. Given the geometry, the value of this parameter determines the behavior of the system [8,9]. The overall response of such a system is measured by one scalar quantity, its impedance Z tot , which is the ratio between the applied potential and the total flux : The contribution of the finite interface resistivity to this global impedance is given by a "spectroscopic" impedance, defined as: Z spec. = Z tot − Z 0 , Z 0 being the impedance of the cell with zero interface resistivity [9]. The main result discussed below is that the electrode impedance Z spec. , is nearly independent of the random character of the fractal interface, even though the regions where the current is concentrated are very different. This is found from a numerical comparison between impedances of deterministic and random electrodes with the same fractal dimension. Two cases are studied: (a) deterministic and random von Koch electrodes (dimension D f = ln 4/ ln 3), (b) a deterministic electrode of dimension D f = 4/3 and a self-avoiding random walk geometry with the same dimension. The deterministic von Koch curve, or classical snowflake curve, is obtained by dividing a line segment in three equal parts, removing the central segment and replacing it by two other identical segments which form an equilateral triangle. A random von Koch curve can be defined simply by choosing randomly the side of the segment where the triangle is created at each step of the building process. This is shown in Fig. 2. After three or more generations, it looks more like a realistic random boundary than a simple mathematical curve. It is then possible to automatically generate different boundaries that have the same fractal dimension and the same perimeter. By definition fractal geometries exhibit a large scale of lengths. For instance, at the sixth generation, the ratio between the smallest feature l (smaller cut-off) of the irregular boundary and the diameter L (larger cut-off) is 3 6 = 729 while the length of the perimeter is L p = 4 6 l = 4096l. Computing on a regular grid within such geometries would be very memory and time-consuming. A finite element method is then used. The standard variational formulation of the problem is discretized with a triangular mesh, obtained from a Delaunay-Voronoï tessellation and P 1 -Lagrange interpolation. The linear system obtained in such a way is solved by using the Cholesky method, from the Finite Element Library MODULEF [10]. Examples of meshes with a 6 th generation boundary are shown in Fig. 3. Computations were carried out for the two deterministic boundary geometries and the two random geometries of generation 6 shown in Fig. 4. The figure presents the isocurves of the potential for Λ = 0. Since the current density is proportional to the gradient of the potential, one can detect regions of large current density from the distance between two consecutive isocurves: the closer the equipotentials, the larger the current density. As expected, most of the current flow through the irregular interface at the tips. This gives a very different current map for each geometry. Therefore, for the different electrodes the active zones are very different. The second type of electrodes to be compared is shown in Fig. 5. The top figure shows the second generation of a deterministic fractal electrode with dimension D f = ln 16/ ln 8 = 4/3 while the bottom represents a particular self-avoiding walk with the same 4/3 fractal dimension. Both electrodes have the same perimeter and the same smaller cut-off. Here, even more than above, the active zones are totally different.
For each geometry, the impedances have been computed for an extended range of the surface resistivity r. The results are shown in Fig. 6 for two categories of geometries : 6 th generation of von Koch electrodes and the two electrodes of Fig. 5. The parameter Λ/l = σr/l ranges between 1 and 10 5 for generation 6 and between 10 −1 and 5.10 3 for the second type. The limitation of the range is due to limitations in computer time and memory.  It is striking that, despite very different current distribution in the bulk and at the interface, the impedances are very close for all values of the surface resistivity. The behavior of different interfaces are nearly indistinguishable: random and deterministic interfaces behave in the same manner. This could be considered as a partial answer to the question "Can One Hear the Shape of an Electrode?", addressed in [9,11]. In this frame, the main parameters drawn from practical impedance spectroscopy measurements would only be the size, the perimeter and the equivalent fractal dimension of the interface.
A more demanding comparison between the impedances can be made by comparing the values of r/Z as shown in Fig. 6. This quantity can be identified as an equivalent active length L eq. [12]. One finds three successives regimes, Λ <l; l< Λ < L p, and finally L p < Λ separated by smooth crossovers. These regimes can easily be compared to the so-called "land surveyor approximation" [13]. This method allows one to compute Z spec. through a finite size renormalization of the interface geometry, without solving the Laplace equation. For small r (or Λ << 1 ) there is a linear regime in which Z spec. is proportional to r, that is Z spec. = r/L eq. with L eq. ≈ L [9]. For values of Λ> l there is a fractal regime in which, in first approximation, Z spec. = (r/Λ)(l/L)(Λ/l) 1/D f and L eq. = L(Λ/l) (D f −1)/D f (for more detailed expressions of the exponents, see [14][15][16]). Finally, for values of Λ much larger than the perimeter length L p , the exact value is Z spec. = r/L p and L eq. = L p . These three asymptotic behaviors are shown in the figure and are found to match the numerical results with good accuracy.
Note that the electrodes of Fig. 5 are in some sense "poor" fractals because the range of geometrical scaling is relatively small and it has been a matter of debate recently whether the fractal concept should be of any use when the scaling range of the geometry is too small. For the phenomena considered here, one can observe that the fractal description of this limited range geometry is really useful.
In summary, one has shown on several examples that the net transfer across an irregular surface is nearly independent of the randomness of its geometry, although it depends strongly on the geometry through its fractal dimension. The fact that the overall response remains the same indicates that, buried in the fractal description, there exist the geometrical correlations that govern the overall effect of screening at different scales. In that sense the response is "universal" within a very good approximation for the category of curves considered here.