Polymers effective spectral dimensions

Table 11.5 Effective spectral dimensions d, for some polymers [10] ...

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] ... 

Dolbin, 1. V. Kozlov, G. V. Zaikov, G. E. The theoretical estimation of effective spectral dimension for polymer melts. Proceedings of International Interdisciplinary seminar Fractals and Applied Synergetics, FaAS-01. Moscow, Pubhshers MSOU, 2001,41 2. 

It is obvious, that molecular weight distribution availabihty in real polymers requires the usage of just effective spectral dimension value. The authors [14] offered the following formula for the value determination in case of polymer melts ... 

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. 

Hence, the results stated above have shown that the macromolecular coil fractal dimension D can serve as volume effects measure. The value D for coil in 0-solvent characterizes such effects absence (D=2.0, e=0). Dj.<2.0 means repulsive interactions availability, >2.0—attractive interactions between randomly drawing closer to one another chain links and also between chain links and solvent molecules. The repulsive interactions weakening and, respectively, attractive interactions intensification means macromolecular coil coimectivity degree enhancement, characterized by the spectral dimension d. Thus, the dimensions D, and d variation (at fixed d) characterizes completely enough biopol5miers (and polymers at all) macromolecular coil behavior in diluted solutions [39]. 

The fractal dimension Df of macromolecular coil in solution with excluded volitme effects appreciation was determined according to the Eq. (39). Since all the considered polyarylates are linear polymers, then for them the spectral dimension is ds = 1.0 [72]. The gelation transition or transition from solution up to the condensed state is characterized by macromolecitlar coil environment change and now instead of solvent molecules it is in similar coils environment. This results to fractal dimension change and now its value df for the condensed state is determined according to the eqiration [17] ... 

Fig. 2. Traces along the proton dimension of 2D-WISE experiments performed on 10% hydrated (left) and 35% hydrated (right) onion cell wall material. The corresponding carbon resonances are given. Proton spectral widths of 150 and 70 kHz were used for the 10 and 35% hydrated samples respectively. Reprinted from Carbohydr. Res., Vol. 322(1-2), S. Hediger, L. Emsley and M. Ficher, Solid-state NMR characterization of hydration effects on polymer mobility in onion cell wall material , pp. 102-112, Copyright 1999, with permission from Elsevier Science.

Arg5riakis [24] has shown that at the investigation of chemical reactions on fiactal objects the corrections by small clusters in system availability are necessary. Just such corrections require the usage in theoretical calculations of not generally accepted spectral (fiacton) dimension r/ [23], but its effective value d. It is obvious, that the availability in real polymers molecular weight distribution by virtue ofdie indicated above causes requires application of just dimension. The authors [21] offered the Eq. 

Within the framewoiks of fractal analysis fractal (macromolecular coil) branching degree is characterized by spectral (fraction) dimension J, which is object connectivity degree characteristic [24]. For linear polymer J = 1.0, for statistically branched one = 1.33 [24]. For macromolecular coil with arbitrary branching degree the value varies within the limits of 1.0-1.33. Between dimensions D and the following relationship exists, that takes into consideration the excluded volume effects [17] ... 

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