EANCs

Figure 4. See legend of Figure 2, but now the variation of the number of EANC s per monomer.

Rn, and the gel point, Figure 16 concerns the sol fraction and Figure 17 the number of EANC s per monomer. The differences are most pronounced near the gel points and they vanish with completion of the reaction. 

According to the rubber elasticity theory ( 1, 2), the equilibrium shear modulus, Ge, is proportional to the concentration of EANC s and an additional contribution due to trapped entanglements may also be considered ... 

As the reaction proceeds beyond the gel point, the molecular weight of EANCs decreases and the fraction of material in the EANCs increases. The fraction of material in dangling chains passes through a maximum but their molecular weight decreases. Figure 5.5 characterizes the behavior of simple polyurethane systems. 

Figure 5.4 Schematic representation of sol and a part of the gel DC dangling chains, EANC elastically active network chains, EAC elastically active crosslinks...

Here, [OH]0 and [NCO]0 are initial concentrations of OH and NCO groups, respectively. Figures 5.10 and 5.11 show the calculated dependences of the gel fraction, wg, and concentration of EANCs, ve on rH [30, 60-62] and their comparison with experimental results. 

However, in doing so one tests two theories the network formation theory and the rubber elasticity theory and there are at present deeper uncertainties in the latter than in the former. Many attempts to analyze the validity of the rubber elasticity theories were in the past based on the assumption of ideality of networks prepared usually by endllnklng. The ideal state can be approached but never reached experimentally and small deviations may have a considerable effect on the concentration of elastically active chains (EANC) and thus on the equilibrium modulus. The main issue of the rubber elasticity studies is to find which theory fits the experimental data best. This problem goes far beyond the network... 

Using the values of the modulus G, measured just after preparation (X = 1), one can determine the concentration of elastically active network chains (EANC), vd, related to the dry state... 

A simple mathematical description of the postgel stage will be presented for stepwise and free-radical chainwise polymerizations (in this case, the description will be limited to the range of low concentrations of the polyfunctional monomer leading to a homogeneous system). Calculations will be restricted to the evolution of sol and gel fractions, the mass fractions of pendant and elastic chains, and the concentration of crosslinks and EANC as a function of conversion. 

The mass fraction of material pertaining to elastically active network chains (EANC) is obtained from Eq. (3.34). 

This leads to the following expression for the concentration of EANC ... 

The presence of hard clusters affects mechanical properties. The major problem is the way to define elastically active network chains (EANC) and crosslinks (Chapter 3, Fig. 3.3). It has been demonstrated that hard clusters must be considered as multifunctional crosslinks (fc = 6 in Fig. 7.6a) while macrodiol chains behave as EANC. 

Fig. 13.84c, known as the Smith failure envelope, is of great importance because of its independence of the time scale. Moreover, investigations of Smith, and Landel and Fedors (1963,1967) proved that the failure envelope is independent of the path, so that the same envelope is generated in stress relaxation, creep and constant-rate experiments. As such it serves a very useful failure criterion. Landel and Fedors (1967) showed that a further generalisation is obtained if the data are reduced to ve, i.e. the number of elastically active network chains (EANCs). The latter is related to the modulus by... 

The number of elastically active chains, N, determining the equilibrium rubber elasticity, is derived from the following consideration. A chain in the gel is elastically active, if the branch points at each of its ends issue at least three paths to infinity. Such elastically active network chain (EANC) can have many long side branches but none of them may have an infinite continuation. The number of EANC s, N, is thus calculated from the number of EANC ends, i.e., branch points issuing three or more bonds with infinite continuation. The distribution of units according to the number of bonds with infinite continuation is described by a pgf T(z)... 

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