Dimension fractal reaction

H. Berry, Monte Carlo simulations of enzyme reactions in two dimensions Fractal kinetics and spatial segregation. Biophys. J. 83(4), 1891 1901 (2002). 

Farm and Avnir [113] were the first to use fractal geometry to determine effects of surface morphology on drug dissolution. This was accomplished by the use of the concept of fractal reaction dimension dr [114], which is basically the effective fractal dimension of the solid particle toward a reaction (dissolution in this case). Thus, (5.7) and (5.8) were modified [113] to include surface roughness effects on the dissolution rate of drugs for the entire time course of dissolution... 

Valsami, G. and Macheras, P., Determination of fractal reaction dimension in dissolution studies, European Journal of Pharmaceutical Sciences, Vol. 3, No. 3, 1995, pp. 163-169. 

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... 

Devalues, obtained for PAr and PUAr poly condensation process, showed, that the indicated processes were realized by aggre tion cluster-cluster mechanism [49], i.e., by small macromolecular coils joining in larger ones [23], Thus, polycondensation process is a fractal object with dimension D. reaction. Such reaction can be presented schematically in a form of devil s staircase [80], Its horizontal parts correspond to temporal intervals, in which reaction is not realized. In this case polycondensation process is described with irsing fractal time t, which belongs to Kantor s set points [81], If polycondensation process is considered in Euclidean space, then time belongs to a real number set. 

Let us note, that k = 0 is achieved at small, but finite quantity pNu. The calculation according to the Eqs. (76) of Chapter 1, Eqs.18 of Chapter 2 andand Eqs. (11) shows, that Dj, 2.20, i.e., fractal dimension of a branched chain in 0-conditions, conesponds to this value pN [9]. After this dimension achievement reaction rate decreases sh ly. 

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... 

One of such tendencies is polymers synthesis in the presence of all kinds of fillers, which serve simultaneously as reaction catalyst [26, 54]. The second tendency is the chemical reactions study within the framework of physical approaches [55-59], from which the fractal analysis obtained the largest application [36]. Within the framework of the last approach in synthesis process consideration such fundamental conceptions as the reaction prodrrcts stracture, characterized by their fractal (Hausdorff) dimension [60] and the reactionary medium connectivity, characterized by spectral (fracton) dimension J [61], were introduced. In its titrrt, diffusion processes for fractal reactions (strange or anomalous) differ principally from those occurring in Euclidean spaces and described by diffusion classical laws [62]. Therefore the authors [63] give transesterification model reaction kinetics description in 14 metal oxides presence within the framework of strange (anomalous) diffusion conception. 

Polycondensation reactions (eqs. 3 and 4), continue to occur within the gel network as long as neighboring silanols are close enough to react. This increases the connectivity of the network and its fractal dimension. Syneresis is the spontaneous shrinkage of the gel and resulting expulsion of Hquid from the pores. Coarsening is the irreversible decrease in surface area through dissolution and reprecipitation processes. 

According to Eq. (27), Stromme et al.125,126 developed systematically the peak-current method to determine the fractal dimension of the electrode surface by using cyclic voltammetry. It must be recalled that this method is valid when the recorded current is limited by diffusion of the electroactive species to and away from the electrode surface. Since the distribution of the reaction sites provides extensive information about the surface geometry, the fractal dimension of the reaction site distribution may agree with the fractal dimension of the electrode surface which is completely electrochemical-active. In addition, it is well known that this method is insensitive to the IR drop in the electrolyte.126... 

In practical application, it was reported that the platinum particles dispersed in highly porous carbonized polyacrylonitrile (PAN) microcellular foam used as fuel-cell electrocatalyst160 have the partially active property. The fractal dimension of the platinum particles was determined to be smaller than 2.0 by using the potentiostatic current transient technique in oxygen-saturated solutions, and it was considered to be a reaction dimension, indicating that not all of the platinum particle surface sites are accessible to the incoming oxygen molecules. 

Chapter 13 - It was shown, that limiting conversion (in the given case - imidization) degree is defined by purely structural parameter - macromolecular coil fraction, subjected evolution (transformation) in chemical reaction course. This fraction can be correctly estimated within the framework of fractal analysis. For this purpose were offered two methods of macromolecular coil fractal dimension calculation, which gave coordinated results. 

Therefore, the estimation 0 im problem brings to the question of fractal dimension Df determination. At present two methods of indicated dimension determination one exist. First method consists of using of chemical reactions fractal kinetics general relationship [9] ... 

In case of reaction course in the Euclidean spaces the value D is equal to the dimension of this space d and for fractal spaces D is accepted equal to spectral dimension ds [6], By plotting p i=( 1 -O) (where O is conversion degree) as a function of t in log-log coordinates the value D from the slope of these plots can be determined. It was found, that the mentioned plots fall apart on two linear parts at t<100 min with small slope and at PT00 min the slope essentially increases. In this case the value ds varies within the limits 0,069-3,06. Since the considered reactions are proceed in Euclidean space, that is pointed by a linearity of kinetic curves Q-t, this means, that the reesterefication reaction proceeds in specific medium with Euclidean dimension d, but with connectivity degree, characterized by spectral dimension ds, typical for fractal spaces [5],... 

At definite conditions the value H is defined by dimension Df (Euclidean or fractal) of reaction product (heptylbenzoate molecule) only [8] ... 

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 d, i.e.,... 

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... 

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