Fractal space processes

The essentially different nature of transport processes with y/ t) °c r should be stressed. Processes with this type of waiting time distribution function show an absence of scale. They exhibit very sporadic behavior. Long dormancies are followed by bursts of activity. They have been desaibed as fractal time processes (Schlesinger [1984]). Fractal space processes, in which the absence of scale is present in the spatial aspects of the transport, are considered later in this section. 

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] ... 

Kozlov, G. V. Bashorov, M. T Mikitaev, A. K. Zaikov, G. E. The nanodimensional effects in curing process of epoxy polymers in the fractal space. In book Trends in Polymer Research. Ed. Zaikov, G. Jimenez, A. Monakov Yu. New York, Nova Science Publishers, Inc. 2009, 87-94. 

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] ... 

Thus, from the said above it follows, that thermooxidative degradation process of PAr and PAASO melts proceeds in the fractal space with dimension A In such space degradation process can be presented schematically as devil s staircase [33]. Its horizontal sections correspond to temporal intervals, where the reaction does not proceed. In this case the degradation process is described with fractal time t using, which belongs to Cantor s setpoints [34]. If the reaction is considered in Euclidean space, then time belongs to real numbers sets. 

The Nanodimensional Effects in the Curing Process in Fractal Space 288... 

As it has been noted above, variation D t) reason is cnring reaction proceeding in fractal space. This process by its physical significance is similar to the formation of clusters with dimension on lattice with dimension [18], In paper [44] it was supposed = D. The relation between and is given by... 

The condition in Eq. (28) allows to suppose that the largest cluster in system is the cluster, forming fractal space in the system EPS-4/DDM curing process... 

THE NANODIMENSIONAL EFFECTS IN THE CURING PROCESS IN FRACTAL SPACE... 

Hence, a solid-phase polymers deformation process is realized in fractal space with the dimension, which is equal to structure dimension d. In such space the deformation process can be presented schematically as the devil s staircase [39]. Its horizontal sections correspond to temporal intervals, where deformation is absent. In this case deformation process is described with using of fractal time t, which belongs to the points of Cantor s set [30]. If Euclidean object deformation is considered then time belongs to real numbers set. 

In some cases, when the polymerization appears, the energy distribution of micropores is negligible in comparison with the energy of polymerization. That is possible when the temperature of the treatment of the primary material (if this one can be polymerized, e.g., silica, alumina) is low (less 300-350 °C). In such cases, traditional methods of nonequilibrium thermodynamics are not effective, and the micropore formation can be considered as the result of the polymerization process which is described by methods of polymer science. However, models of macromolecular systems do not always give enough information about micropores as the empty space between polymers. For such systems, the application of fractal methods can allow us to obtain additional information, while one has to take into account the fact that they cannot be applied to very narrow pores (ultramicropores which are found, for instance, in some silica gels). 

In the real world, however, the objects we see in nature and the traditional geometric shapes do not bear much resemblance to one another. Mandelbrot [2] was the first to model this irregularity mathematically clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Mandelbrot coined the word fractal for structures in space and processes in time that cannot be characterized by a single spatial or temporal scale. In fact, the fractal objects and processes in time have multiscale properties, i.e., they continue to exhibit detailed structure over a large range of scales. Consequently, the value of a property of a fractal object or process depends on the spatial or temporal characteristic scale measurement (ruler size) used. 

Although the detailed features of the interactions involved in cortisol secretion are still unknown, some observations indicate that the irregular behavior of cortisol levels originates from the underlying dynamics of the hypothalamic-pituitary-adrenal process. Indeed, Ilias et al. [514], using time series analysis, have shown that the reconstructed phase space of cortisol concentrations of healthy individuals has an attractor of fractal dimension dj = 2.65 0.03. This value indicates that at least three state variables control cortisol secretion [515]. A nonlinear model of cortisol secretion with three state variables that takes into account the simultaneous changes of adrenocorticotropic hormone and corticotropin-releasing hormone has been proposed [516]. 

Different concentration limits of the filler arise from the CCA concept [22]. With increasing filler concentration first an aggregation limit O is reached. For >+, the distance of neighboring filler particles becomes sufficiently small for the onset of flocculation and clusters with solid fraction A are formed. Dependent on the concentration of filler particles, this flocculation process leads to spatially separated clusters or, for 0>0, a through going filler network that can be considered as a space-filling configuration of fractal CCA-clusters. The different cases for spherical filler particles are shown schematically in Fig. 1. 

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