Fractal space

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],... 

L. Nottale, Fractal Space-Time and Microphysics, Chap. 3. World Scientific, Singapore, 1993. 

Evesque on energy migration in fractal spaces such as those found in polymers and similarly organized systems. 

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 to reactions in fractal spaces, here the situation is quite opposite. As is known [38], if we consider a trajectory of diffusive movement of oligomer and curing agent molecules... 

Change of Microgel Structure on Curing Epoxy Polymers in Fractal Space... 

This chapter considers the reasons for a variation of microgel structure characterised by its fractal dimension, D, formed in the cure of epoxy resin systems. Quantitatively, change of D during the increase of reaction time is well described within the framework of mechanism of aggregation cluster - cluster. The fractal space, in which the reaction curing proceeds, is formed by a structure of the greatest cluster in system. 

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] ... 

DDM which also has appeared to be linear. Together with the previously stated reasons, the data of Figure 13.2 assume that the curing reaction of system EPS-l/DDM proceeds in fractal space with dimension Dj. We should note that att = 3xl0 sa deviation is observed (given in Figure 13.2) of dependences on linearity. From the comparison with the data of Figure 13.1 it is seen that this deviation corresponds to Dj = 3, i.e., transition to nonfractal behaviour at D[ (=d, as should have been expected. 

Up to now we considered pol5meric fiiactals behavior in Euclidean spaces only (for the most often realized in practice case fractals structure formation can occur in fractal spaces as well (fractal lattices in case of computer simulation), that influences essentially on polymeric fractals dimension value. This problem represents not only purely theoretical interest, but gives important practical applications. So, in case of polymer composites it has been shown [45] that particles (aggregates of particles) of filler form bulk network, having fractal dimension, changing within the wide enough limits. In its turn, this network defines composite polymer matrix structure, characterized by its fractal dimension polymer material properties. And on the contrary, the absence in particulate-filled polymer nanocomposites of such network results in polymer matrix structure invariability at nanofiller contents variation and its fractal dimension remains constant and equal to this parameter for matrix polymer [46]. 

The Eq. (22) gives unsatisfactory correspondence to numerical calculations for precise fiactals and the authors [51] expressed doubt about the general Flory approximation derivation possibility for fractal spaces (lattices), which would be both simple and exact at the same time. 

The authors [52] obtained the following Flory approximant for fractal spaces ... 

The Eq. (27) application gave an exact enough (within the limits of 6%) description of composites polymier matrix fractal dimension change in fractal space, created by filler particles (aggregates of particles) network [45]. 

Rammal, R. Toulouse, G. Varmimenus, J. Self-avoiding walks on fractal spaces exact results and Floiy approximation. J. Phys. France, 1984, 45(3), 389-394. 

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] ... 

The condition of Eq. (117) is confirmed experimentally (k value does not change at increasing) [156], For the curing reaction proceeding in fractal space the situation differs completely from the described above. This aspect attains special meaning within the framework of nanochemistry [161], therefore deserves consideration in more detail. 

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