Experimental Determination of Fractal Dimensions

The results of investigation of the transfer of energy between molecules acting as electron density donors and acceptors have been reported [83, 84]. The Raman scattering method gives actually the fractal dimension of the surface on which the energy transfer takes place [84]. The fractal dimensions found for silicates are in the range of 2.23-2.82 [84] that for a polystyrene film is 2.2 [83]. 

Kopelman and co-workers [70] have studied the annihilation of the triplet excitons in naphthalene aggregates located in the pores of various materials including polymers. A common feature of these systems is non-classical annihilation kinetics, which resemble the annihilation kinetics of fractals. The coefficient of the annihilation rate k, varies with time t  

Kopelman and co-workers [10] also measured as a function of temperature. At low temperatures, all the polymers studied exhibit properties similar to the properties of fractals. As the temperature increases, h decreases, i.e., the d value increases. In some specimens, h 0 as the temperature is raised. This implies that all the effects described by fractional dimensions are associated with disorder [85]. A number of specimens also behave as fractals at room temperature. It is noteworthy that the d values for the polymers studied vary over wide limits, from 0.8 to 1.8. In the case of PMMA, d exactly corresponds to the spectral dimension determined by Raman scattering measurements [22, 35] it is 1.8 in both cases. 

The theoretical concepts outlined previously are confirmed rather accurately by experimental data. Nevertheless, it is worth noting that the concept of polymeric (macromolecular) fractal [56] has been developed for polymer solutions, while the Vilgis theory [61, 62] is applicable only to rubbers and polymer melts, although it has been developed for condensed media. Therefore, it can be expected that extension of these concepts to the vitreous state of polymers would require some corrections taking into account the specific features of this state. 

Some possible approximations have been considered by Cates [56], who concentrated attention on macromolecular entanglements, which play an important role in the description of the behaviour of block polymers [86-89]. Cates believes that the fact that the concept of polymer fractal neglects the effects of macromolecular entanglements is the main drawback of this theory. Nevertheless, Cates [56] introduced several simplifications that make it possible to ignore these effects for dilute solutions and relatively low molecular masses. However, in the opinion of Cates, even in the case of predominant influence of entanglements, theoretical interpretation of this phenomenon is impossible without preliminary investigation of the properties of the system in terms of Rouse-Zimm dynamics, which can serve as the basis for a more complex theory. It was assumed [56] that the effects of entanglement can be due to the substantially enhanced local friction of macromolecules. 

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