Reactive fractal dimension

If only partial screening is present, the fractal dimension takes a value somewhere between df and df. According to this model, a crosslinker deficiency, which leads to a more open structure and therefore a lower value of du increases the value of n. Dilution of the precursor with a non-reactive species has the same effect on the relaxation exponent. 

The rate of aggregation of fully renneted micelles is very sensitive to temperature. At room temperature it is appreciably less than the diffusional collision rate, which led Payens (1977) to consider the possibility that only a fraction of the surface is reactive (so-called hot spots). The idea of hot spots is consistent with the low fractal dimension of micelle clusters formed during renneting and leads to only a proportion of all encounters between fully renneted micelles being successful. In effect, a statistical prefactor is included in the reaction kernel to reduce the diffusion rate to a level comparable with experiment. However, Payens developed the idea of hot spots only within his theory of the aggregation of fully renneted micelles. 

Relationship between reactive and surface fractal dimensions... 

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. 

In such reactive media branched clusters which do not contain fully condensed metal oxide cores are formed by kinetically limited growth processes. The structure of these clusters can be described using the fractal concept in which a mass fractal dimension D relates the cluster mass M to its radius according to... 

The chemical reaction between a solid and a reactive fluid is of interest in many areas of chemical engineering. The kinetics of the phenomenon is dependent on two factors, namely, the diffusion rate of the reactants toward the solid/fluid interface and the heterogenous reaction rate at the interface. Reactions can also take place within particles, which have accessible porosity. The behavior will depend on the relative importance of the reaction outside and inside the particle. Fractal analysis has been applied to several cases of dissolution and etching in such natural occurring caves, petroleum reservoirs, corrosion, and fractures. In these cases fractal theory has found usefulness for quantifying the shape (line or surface) with only a few parameters the fractal dimension and the cutoffs. There have been some attempts to use a fractal dimension for reactivity as a global parameter. Finally, fractal concepts have been used to aid in the interpretation of experimental results, if patterns quantitatively similar to DLA are obtained. 

The first of the conclusions that can be drawn from these calculations is that for finite planar lattices of integral or fractal dimension, in the absence of any external bias, the diffusion-reaction process is more efficient when the target molecule is localized at a site of higher valency. This conclusion is consistent with the trends noted in Section IIIA, and with the results reported in Section IIIB. Further, the conclusion is consistent with results obtained in a lattice-based study of reactivity at terraces, ledges, and kinks on a (structured) surface [28], There the reaction... 

In particular, let us consider a fractal structure Q (possessing dimension dj) and a distribution of reacting centres localized on a submanifold (the reactive manifold possessing fractal dimension dR < df). The corresponding balance equations attain the form... 

Within the frameworks of fractal analysis the notion of particles or cluster accessibility to active (reactive) sites of other particles or clusters for either reaction realization was introduced. This notion is finked with cluster fractafity notion, which is macromolecular coil in solutiom The macromolecular coil fractal dimension Df characterizes its structure opermess degree — the smaller the more intensive clusters penetration (or particles penetration) into another then cluster (it is more accessible ). This postulate is expressed analytically by the Eq. (27). Differentiating the indicated relationship by time t, let us obtain reaction realization rate (the Eq. (69)) and then it can be written [123] ... 

Let us consider the physical reasons of reactive medium ds change within the indicated above limits. Reactive medium itself presents Euclidean space with dimension d = 3, as and any solution of low-molecular substance in the same solvent [12]. The appearance of dimension d with values <3, which are specific for fractal mediums, is due to unevenness of monomer in solution distribution and c decreasing results to law-governed diffusion time increasing, which is necessary for reagent reaching one another. Let us note, that DMDAACh macromolecular coil fractal dimension in water solution is Dj. = 1.65 [13]. According to Kremer s formula the dimension dm of medium, in which such coil is formed, can be determined [14] ... 

The number of effective branching centers per one macromolecule m is controlled by four factors polymer molecular weight MM, maximum chemical density of reactive centers cch, dimension of unscreened surface du of macromolecular coil and its fractal dimension D. The Eq. (27) allows to determine the critical value... 

FIG U RE 20 The dependences of microgels fractal dimension Dj (1) and reactive medium initial viscosity q (2) on the parameter A value for the system 2DPP+HCE/DDM. 

Hence, the results quoted above demonstrated clearly, that the transition of transesterification reaction kinetic curves from an autodecelerated regime to an autoaccelerated one was defined by a structural factor, namely, by reaching of fiactal-like molecule (reaction product) fractal dimension critical value, at which the number of reactive sites in volume and on the surface of molecule became equal. Within the fiamewoik of the fractal analysis the analytic relationship is obtained, confirming this hypothesis. 

Computer simulations have shown that the value of fractal dimension largely depends on whether the aggregation process is controlled by the diffusion rate of the clusters and single particles or by their chemical reactivity at the time of collision, the latter being mainly controlled by the DLVO forces. This observation, in agreement with experimental work on aerosols and colloids, has led to a new classification of aggregation processes the reaction-limited and diffusion-limited cluster aggregation (RLCA and DLCA respectively) processes. 

Apart from the fractal dimension D of the surface, there is a so-called reaction dimension Q (sometimes also called D ), which characterizes the way in which the rate of the reaction r on the surface scales with the size of the particles L. The power law is r oc L. The value of Q is usually between 1 and 3 but extremes such as 0.2 and 5.8 have also been observed. The distribution of chemically active sites on the surface determines the way in which the reactivity scales with the size of the crystallite. 

The reaction dimension Q may be the same as the fractal dimension D of the reactive surface or it may be different. If 2 < there may screening or poisoning of reactive sites. A value of Q larger than D usually means that the reaction occurs only in the micropores. The value of Q is not a constant and may change during the course of the reaction. 

If we again adopt that / =y=0.6, then the value =1.67 accurately coincides with the dimensions of the fractal of the chain df=(d + 2)/3=5/3 and the value ) =V, coincides with the Flory index. In this case the dimensions of fractal of traps and characteristic times d according to equations (8.62) and (8.59), is equal to characteristic times set z, of macroradical relaxation is completely determined by the fractality of traps, i.e., by dimension di. In the case that the set of traps is not a fractal (for example, at the uniform distribution of traps in the reactive zone when the dimension of the f ractal of traps formally coincides with the dimension of the Euclidean space di=d), then, in accordance with equation (8.59) the set of characteristic times r, would be characterized by a fractal dimension equal to infinity and t would not depend on chain length and time of chain propagation, and would be presented by a constant value T, Tp p ,. Under this variant we have 1= 1, 1 and stretched exponential law transforms into simple exponential law. 

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