Phantom imperfect network

According to the theory (1,2), for phantom imperfect (cf. also (17)) network the front factor A assumes the value... 

For imperfect networks, the comparison of the ratio 2Ci/vRT with 1 — 2/q> is no longer possible because the number of effective chains v is not known, and became not all of the junctions have the same functionality. Nevertheless, 2Ci can be compared with the value of the phantom modulus, which is (v, — pj RT [see Eq. (4)], and also with (v — p) RT. The variation of the ratio 2Ci/(v, — p ) RT with M is reported in Fig. 8 and 9 for trifunctional and tetrafunctional PDMS networks, resp Aively. Here M is different from because an active chain can be formed by two or more chains bound with difunctional junctions. The constant 2Ci is always hi er than the phantom modulus and the conclusions are similar to those readied in Section 4.2. 

For imperfect epoxy-amine or polyoxypropylene-urethane networks (Mc=103-10 ), the front factor, A, in the rubber elasticity theories was always higher than the phantom value which may be due to a contribution by trapped entanglements. The crosslinking density of the networks was controlled by excess amine or hydroxyl groups, respectively, or by addition of monoepoxide. The reduced equilibrium moduli (equal to the concentration of elastically active network chains) of epoxy networks were the same in dry and swollen states and fitted equally well the theory with chemical contribution and A 1 or the phantom network value of A and a trapped entanglement contribution due to the similar shape of both contributions. For polyurethane networks from polyoxypro-pylene triol (M=2700), A 2 if only the chemical contribution was considered which could be explained by a trapped entanglement contribution. 

Equation (22) holds for phantom networks of any functionality, irrespective of their structural imperfections. In case b), fluctuations of junctions are assumed to be suppressed fully. The junctions themselves are considered to be firmly embedded in the medium and their position is transformed affinely with the macroscopic strain. This leads to the free energy expression for an f-functional network possibly containing free chain ends... 

It is worth noting that > G, which is a coni pKnce of the that the fluctuations of junctions in a phantom n wOTk are unaffected by deformation. Flory has recently pointed out that the klmtification of v, with the numb of effective chains v is valid only for perfect networks (see Sect. 6.1). For an imperfect tetra-functional network, Flory 1] shown that... 

F%. 9. Dependence of the ratio between 2C, and the phantom modulus on the number-average molecular weight of primary chains for imperfect tetrafunctional end-linked networks Mark (O) Llo-rente (x) Llorente (-I-) Meyers... 

The elastic free energy of a phantom network of Gaussian chains was obtained rigorously by Flory and is valid for networks of any functionality, irrespective of their structural imperfections. It is given in equation (101). The elastic equation of state for phantom networks may then be expressed by equation (102). Equation (84) is then recovered, as expected, because of the relationship shown in equation (103). 

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