Thermofluctuational Origin of Clusters

For a liquid in the equilibrium state, statistical mechanics gives the following expression for (Ap/p) ( V), where V is the standard volume in the limit V °o (i.e., in the thermodynamic limit) [85]  

Equation 1.37 shows that density fluctuations are due to thermal mobility of atoms with energy kT, but limited by volumetric rigidity (%.j.) h 

Based on the literary data for p and estimation of (Ap/p) (°o) at 7 temperatures ( Tj is the analogue of for semi-crystalline polymers connected with similarly to Equation 1.26) it has been shown that the mentioned value is approximately constant at these temperatures. This observation assumes that as some critical value of (Ap/p) is reached, formation of local order domains, i.e., thermofluctuational cluster networks of macromolecular entanglements, is impossible owing to high thermal mobility of macromolecules. On the contrary, below the critical (Ap/p) values local order domains (clusters) in the polymer melt are formed, which according to the present treatment are identified as nodes of the macromolecular entanglements network, i.e., a state is formed, which was defined by Boyer as a liquid with fixed structure [59]. 

Proceeding from the arguments stated above, the critical temperatures at which critical density fluctuations are reached were calculated. A (Ap/p) value for PS at 433 K was accepted. The calculation results are quoted in Table 1.3, from which it follows that T and (7 ) for a set of amorphous and semi-amorphous polymers are very close the greatest discrepancy of these temperatures does not exceed 6% [87, 88]. Let us note that in the considered case we are dealing with dynamic (shortliving) local order, which is frozen below T (T ) [9]. 

Sanditov and Bartenev [78] have shown that the (Ap/p) value is connected with 

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Figure 1.1 shows v (T) dependences for polycarbonate (PC) produced from bisphenol A and polyarylate (PAr). These dependences indicate a reduction in with an increase in T that supposes thermofluctuational origin of clusters (the local order domains). 

In Figure 1.26 the comparison of the indicated sizes is adduced, where values are accepted according to the data [94] and the calculation technique was given in paper [97]. As one can see, the temperature dependences of these structural characteristics display good correspondence. This allows the identity of such structural elements as CRR and clusters to be assumed and also confirms the thermofluctuational origin of clusters [96, 97]. 

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