Elastically active network chain EANC

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... 

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

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)... 

Weakly crosslinked epoxy-amine networks above their Tg exhibit rubbery behaviour like vulcanized rubbers and the theory of rubber elasticity can be applied to their mechanical behaviour. The equilibrium stress-strain data can be correlated with the concentration of elastically active network chains (EANC) and other statistical characteristics of the gel. This correlation is important not only for verification of the theory but also for application of crosslinked epoxies above their Tg. 

The number of elastically active network chains EANC, N, is contributed only by diepoxide units. According to the reasoning given in Section 4.2.3, the distribution of diepoxide units with respect to the number of bonds with infinite continuation is given by the pgf... 

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...

An elastically active network chain is active in the equilibrium elastic response of the network to deformation. From the topological point of view, an EANC is a chain between two active branch points. An active branch point is a imit from which at least three paths issue to infinity. In the case under consideration, only some of the chemically tetrafunctional diamine units can become active branch points. If the polyepoxide were more than bifunctional, it would also contribute to the number of EANC s. In analogy with Eq. (14), the pgf for the numbo- of bonds with infinite continuation issuing from a diamine unit T,(z) is given by... 

Fig. 17. Number of EANC elastically active network chains, N, calculated from the equilibrium modulus as a function of the gel fraction, w, in the stoichiometric mixture of azelaic acid and DGEBA. The curves are calculated theoretically for the extent of addition esterification c = % indicated...

Prediction of the elastic properties of networks using rubber elasticity theory is based upon the knowledge of concentrations of elastically active network junctions (EANJs) and chains (EANCs), respectively and [260, 261]. EANJs are the intersection of at least three chains leading to the gel, whereas EANCs are the chains linking EANJs (see Figure 3.13). 

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... 

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