Reactions on fractals

Another important parameter which appears in connection with dynamical properties of fractals (such as diffusion) is the spectral (fracton) dimension d. Thus, in the diffusion-limited reactions, one has to replace d in (2.1.78) by J, i.e., 

In this Section following [9], we analyse the A + B 0 reaction with immobile reactants on the so-called Sierpinski gasket described below. We will proceed to show that in this case equation (6.1.1) with a = d/2 transforms into 

In simulations [9] Sierpinski gaskets on the 12th stage, containing 177147 or 265722 sites, were used respectively. The number Nq of randomly distributed A or B particles was 10 percent of the total number of sites. The random mutual annihilation of dissimilar particles was simulated through a minimal process method [10] from all AB pairs at each reaction step one pair was selected randomly, according to its reaction rate (3.1.2) the time 

In order to extend the analytical equations to a fractal lattice, we will need the radial distribution function rdf(r) of the Sierpinski gasket, rdf(r) dr being the average number of sites with distance between r and r + dr from a given site. For fractal lattices one has 

In order to determine the constant 7, we computed the radial distribution functions for the two types of the Sierpinski gaskets under consideration. In Fig. 6.6 these functions are plotted, as averaged over all sites of the finite gaskets at the 11th stage. Due to the finite size of the structures, deviations 

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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). 

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... 

Ardyralds [12] showed that at the study of ohemieal reactions on fractal objects the cor-rections on small clusters availability in the system were necessary. Just such corrections require the usage in theoretical estimations not generally accepted spectral (fracton) dimension ds [13], but its effective value. For percolation system two cases are possible [12] ... 

E. V. Albano. Finite-size effects in kinetic phase transitions of a model reaction on a fractal surface Scahng approach and Monte Carlo investigation. Phys Rev B 42 R10818-R10821, 1990. 

E. V. Albano. Irreversible phase transitions in the dimer-monomer surface reaction process on fractal media. Phys Lett A 765 55-58, 1992. 

In summary, we have shown that the kinetics of the bimolecular reaction A + B —> 0 with immobile reactants follows equation (6.1.1), even on a fractal lattice, if d is replaced by d, equation (6.1.29). Moreover, the analytical approach based on Kirkwood s superposition approximation [11, 12] may also be applied to fractal lattices and provides the correct asymptotic behaviour of the reactant concentration. Furthermore, an approximative method has been proposed, how to evaluate integrals on fractal lattices, using the polar coordinates of the embedding Euclidean space. 

Lastly, Argyrakis and Kopelman [33] have simulated A + B -4 0 and A + A —> 0 reactions on two- and three-dimensional critical percolation clusters which serve as representative random fractal lattices. (The critical thresholds are known to be pc = 0.5931 and 0.3117 for two and three dimensions respectively.). The expected important feature of these reactions is superuniversality of the kinetics independent on the spatial dimension and... 

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]. 

However, there is another typ>e of confinement that can be imposed on a reactive system, namely, by a reduction in the effective dimensionality. The simplest examples are those in which the motions of the reactive species are confined to a flat surface or a one-dimensional chain. However, in many systems the connectivity of the configuration space is such that it has effectively a fractal dimension d. The Hausdorf dimension is defined from the behavior of the pair distribution function at sufficiently large R, which varies as that is, the probability of finding the pair with a separation between R and R + dR is proportional to dR. The reduction of the encounter problem from d dimensions to the one dimension R is studied in Section VII A. The important case of reactions on surfaces is considered separately in Section VIIB. 

Examples of the complex plane plots obtained for fractal electrodes are presented in Fig. 33. With a decrease in parameter ([), the semicircles become deformed (skewed). The complex plane impedance plots obtained from Eq. (183) are formally similar to those found by Davidson and Cole " in their dielectric studies. Kinetic analysis of the hydrogen evolution reaction on surfaces displaying fractal ac impedance behavior was... 

Two features of these evolution curves are immediately noticeable. First, for both initial conditions, trapping (reaction) on the fractal lattice is distinctly slower than reaction on the triangular one. At first sight, this result would appear to be anomalous inasmuch as the space-filling triangular... 

This article analyzes adsorption kinetics of fractal interfaces and sorption properties of bulk fractal structures. An approximate model for transfer across fractal interfaces is developed. The model is based on a constitutive equation of Riemann-Liouville type. The sorption properties of interfaces and bulk fractals are analyzed within a general theoretical framework. New simulation results are presented on infinitely ramified structures. Some open problems in the theory of reaction kinetics on fractal structures in the presence of nonuniform rate coefficients (induced e.g. by the presence of a nonuniform distribution of reacting centres) are discussed. 

Hence, the fractal reactions of polymerisation can be divided, as a minimum, into two classes reactions of fractal objects (homogeneous) whose kinetics are described similarly to the curves shown in Figure 10.5, and reactions in a fractal space (nonhomogeneous) whose kinetics are described similarly to the curves shown in Figure 10.6. The reactions of the second class correspond to the formation of structures on fractal lattices [34]. The basic distinction of the pointed classes of reactions is the... 

As has been mentioned before [1], the reason for a variation of D(t) is of course the curing reaction in fractal space. This process in the physical sense is similar to the formation of a cluster with dimension D on fractal lattices with dimension Dj, [7]. In paper [1] it was supposed that D[3t=D. The relationship between D and Dj is given by the following equation [7] ... 

It is accepted to call fractal reactions either fractal objects reactions or reactions in fractal spaces [135], The characteristic sign of such reactions is autodeceleration, that is, reaction rate reduction with its proceeding duration f [136]. Let us note, that for Euclidean reactions the linear kinetics and respectively the condition =const are typical [137], The fiactal reactions in wide sense of this term are very often found in practice (synthesis reactions, sorption processes, stress-strain curves and so on) [74]. The following relationship is the simplest and clearest for the indicated effect description [136] ... 

As it was noted above, at present it becomes clear, that polymers in all their states and on different structural levels are fractals [16, 17]. This fundamental notion in principle changed the views on kinetics of processes, proceeding in polymers. In case of fractal reactions, that is, fractal objects reactions or reactions in fractal spaces, their rate fr with time t reduction is observed, that is expressed analytically by the Eq. (106) of Chapter 2. In its turn, the heterogeneity exponent h in the Eq. (106) of Chapter 2 is linked to the effective spectral dimension d according to the following simple equation [18] ... 

In Fig. 21 the kinetic curves conversion degree—reaction duration Q-t for two polyols on the basis of ethyleneglycole (PO-1) and propylene-glycole (PO-2) are adduced. As it was to be expected, these curves had autodecelerated character, that is, reaction rate was decreased with time. Such type of kinetic curves is typical for fractal reactions, to which either fractal objects reactions or reactions in fractal spaces are attributed [85], In case of Euclidean reactions the linear kinetics (i> =const) is observed. The general Eq. (2.107) was used for the description of fractal reactions kinetics. From this relationship it follows, that the plot Q t) construction in double logarithmic coordinates allows to determine the exponent value in this relationship and, hence, the fractal dimension value. In Fig. 3.22 such dependence for PO-1 is adduced, from which it follows, that it consists of two linear sections, allowing to perform the indicated above estimation. For small t t 50 min) the linear section slope is higher and A =2.648 and for i>50 min A =2.693. Such A increase or macromolecular coil density enhancement in reaction course is predicted by the irreversible... 

Either fractal objects reactions or reactions in fractal spaces are accepted to call the fractal reactions [174]. The autodeceleration, i.e., reaction rate reduction 9 with it duration t, is a characteristic sign of such reactions. Let us note, that for reactions in Euclidean space the linear kinetics and, accordingly, the condition 9 = const are typical [176]. The fractal reactions in broad sense of this term veiy often occur in practice (a synthesis reactions, sorption curves, curves stress-strain and so on) [177, 178], The simplest and clearest relationship for this effect description is the following one [175] ... 

As it is well-known [12], fractal objects are characterized by strong screening of internal regions by fractal surface. Therefore accessible for reaction (in our case - for branching formation) sites are either on fractal (macromolecular coil) surface, or near it. Such sites ntrmber is scaled with coil gyration radius R as follows [12] ... 

Hence, the fractal reactions of polymerization can be divided, as a minimum, into two classes fractal objects reaction whose kinetics is described similarly to the curve 1 in Fig. 4 and reactions in fractal space whose kinetics is described by the curve 2 in Fig. 4. The second class reactions correspond to stractures formation on fractal lattices and the first class - on Euclidean ones [18, 19]. 

Hence, the stated above results have shown that fractal reactions at cross-linked polymers cnring can be of two classes fractal objects reactions and reactions in fractal space. The main distinction of the two indicated reaction classes is the dependence of their rates on fractal dimension D. of reaction products. Such... 

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