Fractals dynamic scaling models

Hence, the dynamical scaling of DMDAACh radical polymerization allows to describe quantitatively the kinetic curve Q, and can be used for its prediction. In this approach base the key physical principles and models (scaling, universahty classes, irreversible aggregation models, fractal analysis) are placed. The three key process properties, characterized reactive centers concentration (c ), diffusive characteristics of reactive medium (q) and accessibility degree of reactive centers (Df), are used for the kinetic cmves description. It is supposed, that the offered approach will be vahd for the description of radical polymerization process of any polymer. 

It was shown recently that disordered porous media can been adequately described by the fractal concept, where the self-similar fractal geometry of the porous matrix and the corresponding paths of electric excitation govern the scaling properties of the DCF P(t) (see relationship (22)) [154,209]. In this regard we will use the model of electronic energy transfer dynamics developed by Klafter, Blumen, and Shlesinger [210,211], where a transfer of the excitation... 

Strictly speaking, all naturally occurring power-laws in fractal or dynamic patterns are finite. Scale-free models nevertheless provide an efficient description of a wide variety of processes in complex systems [16,20,46,106]. This phenomenological fact is corroborated by the observation that the power-law properties of Levy processes persist strongly even in the presence of cutoffs [99]... 

Scale-invariant structures originating from growth processes have been found to be extremely widespread in nature. This observation have led to a number of careful experiments, and various growth models have been suggested to describe the fractal outcome but why did they become fractal in the first place To answer this question we must understand the spatio-temporal evolution. Dynamically, the interface is observed to be unstable, and the system eventually reaches a statistically stationary state where a rich ramified pattern is created. A major observation is that this state can be described by power laws - the pattern becomes scale invariant. 

Non-mean field corrections can be treated by renormalization group theory,which is not discussed here. In order to leave the tree approximation we turn to percolation description and model the microgels as finite clusters. The percolation theory itself is not essential for this description, but only the fractal character of the clusters on their scale of extension. Polymeric fractals have been discussed already in Section 8.2.6 and we use the properties here as well. Indeed most of the results may be applied here. In Ref. 123 the dynamics of the sol phase is discussed extensively, but we do not want to go in these details here. 

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