Percolation relationship

This is based on the following percolation relationship application ... 

The structure correct quantitative model is necessary for analytic intercommunication between polymers structure and properties obtaining. As it has been noted above, the cluster model of polymers amorphous state structure will be used with this purpose [106, 107], The notion of local (short-order) order forms the basis of this model and loeal order domains (clusters) relative fraction is connected with glass transition temperature according to the following percolation relationship [107] ... 

In its turn, value can be estimated wilh the aid of the following percolation relationship [56] ... 

For further calculations as the first approximation 6 =1 was accepted, that corresponds to perfect adhesion by Kemer [4], In Fig. 11.1, the dependence of heat release maximum rate on the sum of nanofiller (organoclay) and interfacial regions relative volume contents is adduced, calculated according to the Eqs. (3) and (4) for intercalated and exfoliated organoclay, respectively. As one can see, all data correspond to one curve, which extrapolates to d 1450 kw/m at ((p +(pp=0, that is, for unfilled polymer. This value corresponds to the indicated parameter mean value for the four matrix pol5miers, mentioned above. The sole essential deviation from the obtained curve is the data for nanocomposite on the basis of PP. This deviation can be due to the condition b= for all considered nanocomposites. The value b can be estimated more precisely with the aid of the following percolation relationship [4] ... 

The common fraction of the ordered regions (clusters and crystallites) can be determined aeeording to the percolation relationship [91] ... 

Now the value U can be calculated theoretically, proceeding fi om the known values, estimating the constant coefficient in the Eq. (10.16) fi om the best conformity of theory and experiment. The comparison of experimental U and calculated by the indicated mode LT HDPE samples with different notch length and at different testing temperatures fiiacture energy is adduced in Fig. 10.11. The data of this figure has shown the good correspondence of theory and experiment, that the percolation relationship (the Eq. (10.16)) correctness confirms. 

For semicrystalline polymers the clusters relative fraction (p j is determined according to the following percolation relationship (analog of the Eq. (4.66)) [8] ... 

Within the frameworks of the cluster model (p j estimation can be fulfilled by the percolation relationship (the Eq. (4.66)) usage. Let us note, that in the given case the temperature of polymers structure quasiequilibrium state attainment, lower of which (p j value does not change, that is, [32], is accepted as testing temperature T. The calculation (p j results according to the Eq. (4.66) for the mentioned above polymers are adduced in Table 15.2, which correspond well to other authors estimations. 

EJE for PC is adduced. As one can see, both indicated equations give a good enough correspondence with the experiment their average discrepancy makes up 5.6% in the Eq. (15.7) case and 9.6% for the Eq. (15.10). In other words, in both cases the average discrepancy does not exceed an experimental error for mechanical tests. This means, that both considered methods can be used for PC elasticity modulus prediction. Besides, it necessary to note, that the percolation relationship (the Eq. (15.7)) qualitatively describes the dependence E better, than the empirical relationship... 

Another method of the theoretical dependence EJ DJ calculation for natural nanocomposites (polymers) is given in Ref [74]. The authors of Ref [75] have shown, that the elasticity modulus E value for fractal objects, which are polymers [4], is given by the following percolation relationship ... 

The authors [8] proposed the following percolation relationship for polymer microcomposites reinforcement degree E IE deseription ... 

The authors [17] showed that the E value for fractal objects, which are HDPE/EP nanocomposites (see Figure 8.1), was given by the percolation Relationship 6.6 (for more details see Section 6.2). At the same time the cluster structure of the polymer amorphous state presents itself as a percolation system [6,18,19], for which the sum K + (pj should be accepted as p, where is the relative fraction of the local orders domains (clusters). In turn, for such a system it can be written [20] ... 

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