Effective spectral dimension

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] ... 

Then by the dependences (t.) in double logarithmic coordinates construction the value h can be determined from their slope and the experimental values of effective spectral dimension d], can be calculated according to the Eq. (6). In Fig. 4 the comparison of theoretical values d calculated according to the Eq. (11), and evaluated according to the described method magnitudes is adduced. As one can see, a theory and e5q)eriment good correspondence is obtained (Ihe mean discrepancy of (dly and d] values makes up 4.5% and maximiun one 9.3%). 

Thus, theoretical technique of effective spectral dimension d estimation was elaborated for pol mier melts. The experimental and theoretical values of this parameter comparison has shown their good correspondence. 

Such reaction rate is described by the Eq. (106) of Chapter 2 and the parameters h and d], intercommunication gives the Eq. (6). The effective spectral dimension d is linked to spectral (fiacton) dimension d according to the subordination theorem as follows [22] ... 

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. 

FIGURE 24 The depen dence of reactionary medium effective spectral dimension on region size r(t), visited by reagents, for the system 2DPP+HCE/DDM. 

Therefore, the results quoted above have shown that for the transesterification reaction catalyzed by nanoparticles of metal inorganic compound autoaccelerated regime of kinetics is defined by the time-dependent transition from the fractal behavior to Euclidean one. This transition is due to the increase of effective spectral dimension of reactionary medium owing to time interval raising between collisions of reacting molecules. 

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 ... 

Let us consider physical principles of reactionary system connectivity degree change, characterized by effective spectral dimension, at imidization temperature... 

FIGURE 4 The dependence of reactionary medium effective spectral dimension on macromolecular coil fractal dimension in imidization process. The notation is the same, as in Figure. 3 [8],... 

FIGURE 5 The comparison of the calculated according to the relationship (2) d], and the equations (7)-(9) effective spectral dimension values of reactionary medium in... 

Therefore, the data considered above demonstrated that the main parameter, controlling solid-state imidization rate, is the reactionary system connectivity degree characterized by its effective spectral dimension. In its turn, this dimension is a function of macromolecular coil structure that is polymeric reaction specific feature. Imidization temperature raising defines reactionary medium heterogeneity reduction and corresponding increase of its cormectivity degree [8,9]. 

Using Relationship 3.1, the D value can be determined, replotting the indicated relationship in double logarithmic coordinates, after which the D value is calculated by the obtained linear plot slope [51]. D values calculated by the indicated method for the system 2DPP+HCE/DDM are adduced in Table 3.4 together with the effective spectral dimension magnitudes, estimated according to the linear plots slope in... 

---