Diffusion on Fractals

In accordance with the anomalous diffusion on fractal lattice, we expect [12,15]... 

As it is known [11], for strange (anomalous) diffusion on fractal objects its two main types can be select slow and rapid diffusion. At the basis of such division the dependence of mobile reagent displacement s on time t [11] was appointed ... 

As far as transport properties of a fractal structure are concerned, the mean square displacement (MSD) of a particle follows a power law, (r ) where r is the distance from the origin of the random walk and is known as the random walk dimension. In other words, diffusion on fractals is anomalous, see Sect. 2.3. Recall that for normal diffusion in three-dimensional space the MSD is given by (r ) = 6Dt. For fractals, dy, > 2, and the exponent of t in the MSD is smaller than 1. We introduce a dimensionless distance by dividing r by the typical diffusive... 

In the previous section we presented some of the equations proposed in the literature for describing diffusion on fractal structures. These equations must meet three requirements to be considered valid. First, the MSD must display subdiffusion,... 

One of the important issues is the possibility to reveal the specific mechanisms of subdiflFusion. The nonlinear time dependence of mean square displacements appears in different mathematical models, for example, in continuous-time random walk models, fractional Brownian motion, and diffusion on fractals. Sometimes, subdiffusion is a combination of different mechanisms. The more thorough investigation of subdiffusion mechanisms, subdiffusion-diffiision crossover times, diffusion coefficients, and activation energies is the subject of future works. 

Diffusion on fractal surfaces is less easy than in topological spaces with an integer dimension. As in the diffusion of atoms, the conductance properties also depend on the dimension. Reactions are so frequently anomalous that many reports on rate measurements, interpreted with two-dimensional surfaces, conclude with the note that the observed data invoke new questions that will have to be studied in the sequel. 

De Gennes [121] considered the problem of anomalous diffusion on fractal networks in an attempt to understand the conductivity threshold of a percolation cluster. In normal diffusion the mean-square displacement, d, is related to the diffusion coefficient, D, according to ... 

In our opinion, this book demonstrates clearly that the formalism of many-point particle densities based on the Kirkwood superposition approximation for decoupling the three-particle correlation functions is able to treat adequately all possible cases and reaction regimes studied in the book (including immobile/mobile reactants, correlated/random initial particle distributions, concentration decay/accumulation under permanent source, etc.). Results of most of analytical theories are checked by extensive computer simulations. (It should be reminded that many-particle effects under study were observed for the first time namely in computer simulations [22, 23].) Only few experimental evidences exist now for many-particle effects in bimolecular reactions, the two reliable examples are accumulation kinetics of immobile radiation defects at low temperatures in ionic solids (see [24] for experiments and [25] for their theoretical interpretation) and pseudo-first order reversible diffusion-controlled recombination of protons with excited dye molecules [26]. This is one of main reasons why we did not consider in detail some of very refined theories for the kinetics asymptotics as well as peculiarities of reactions on fractal structures ([27-29] and references therein). 

From this relationship, we obtain A = 1/3 since the value of ds is 4/3 for A + A reactions taking place in random fractals in all embedded Euclidean dimensions [9, 19]. It is also interesting to note that A = 1/2 for an A + B reaction in a square lattice for very long times [12]. Thus, it is now clear from theory, computer simulation, and experiment that elementary chemical kinetics are quite different when reactions are diffusion limited, dimensionally restricted, or occur on fractal surfaces [9,11,20-22]. 

The dependence of the kinetics on dimensionality is due to the physics of diffusion. This modifies the kinetic differential equations for diffusion-limited reactions, dimensionally restricted reactions, and reactions on fractal surfaces. All these chemical kinetic patterns may be described by power-law equations with time-invariant parameters like... 

The description of these phenomena in complex media can be performed by means of fractal geometry, using the spectral dimension ds. To express the kinetic behavior in a fractal object, the diffusion on a microscopic scale of an exploration volume is analyzed [278]. A random walker (drug molecule), migrating within the fractal, will visit n (t) distinct sites in time t proportional to the number of random walk steps. According to the relation (2.9), n(t) is proportional to tdA2j so that diffusion is related to the spectral dimension. 

The reduced value of the scaling exponent, observed in Fig. 29 and Fig. 30a for filler concentrations above the percolation threshold, can be related to anomalous diffusion of charge carriers on fractal carbon black clusters. It appears above a characteristic frequency (O when the charge carriers move on parts of the fractal clusters during one period of time. Accordingly, the characteristic frequency for the cross-over of the conductivity from the plateau to the power law regime scales with the correlation length E, of the filler network. 

An explanation of the observed relaxation transition of the permittivity in carbon black filled composites above the percolation threshold is again provided by percolation theory. Two different polarization mechanisms can be considered (i) polarization of the filler clusters that are assumed to be located in a non polar medium, and (ii) polarization of the polymer matrix between conducting filler clusters. Both concepts predict a critical behavior of the characteristic frequency R similar to Eq. (18). In case (i) it holds that R= , since both transitions are related to the diffusion behavior of the charge carriers on fractal clusters and are controlled by the correlation length of the clusters. Hence, R corresponds to the anomalous diffusion transition, i.e., the cross-over frequency of the conductivity as observed in Fig. 30a. In case (ii), also referred to as random resistor-capacitor model, the polarization transition is affected by the polarization behavior of the polymer matrix and it holds that [128, 136,137]... 

Aerosol particles used for inhalation deposit within the lower respiratory tract mainly by inertial impaction, sedimentation, and diffusion. Loose fractal aerosols were found to settle slower and therefore had more time to increase gravitational coagulation with other floes leading to much more rapid particle growth. This will increase the chance of the aerosol floes settling on the airway walls before reaching the end of the airways. 

Anderson et al. (1996) used Eq. [14], in conjunction with digitized images of thin sections, to investigate the influence of pore space geometry on diffusion in soil systems. Giona et al. (1996) applied renormalization analysis to study diffusion and convection on fractal media. Coppens (1997), Santra et al. (1997), and Levitz (1998) have studied the effects of geometrical confinement on diffusion in the Knudsen regime, in which particle collisions with a fractal internal surface dominate over particle-particle collisions. 

Mazo (1998) studied Taylor dispersion in fractal media and found that the proportionality constant between the spatial spreading of a solute pulse and the time depended on both the fractal dimension of the medium and the dimension of the random walk through it. In normal diffusion the average particle position is directly proportional to the time. Diffusion in fractal media is anomalous with proportional to f2/dt, where dt is the random walk dimension. 

Cushman, J.H. 1991. On diffusion in fractal porous media. Water Resour. Res. 27 643-644. 

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... 

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. 

---