Elastically active crosslinks

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...

Assumptions made in the derivation of these relationships are as follows, a) The fraction of the total number of junctions in the gel that contribute to the elasticity, i.e. the fraction of the number of elastically active crosslinks, is a function of concentration and not of temperature. This assumption is questionable, since the degree of intramolecular crosslinking will depend on the concentration, becoming smaller with increasing concentration. However, the variation in T with concentration is relatively small, so that the question of the strict validity of this assumption is rather irrelevant. 

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. 

Polymer networks are formed from functional precursors by covalent bond formation [1], As a result, molecular weights and polydispersity increase and the system passes through a critical point, the gel point. At this point, an infinite structure (molecule) is formed for the first time. Beyond the gel point, the fraction of the infinite structure (the gel) increases at the expense of finite (soluble) molecules (the sol). The sol molecules become gradually bound to the gel and eventually all precursor molecules can become a part of the gel - the network. This is not always the case for different reasons sometimes sol is still present after all functional groups have reacted. In passing from the gel point to the final network not only the gel fraction increases, but also the network becomes denser containing increasing amounts of crosslinks and strands between them called elastically active network chains. 

Figure 5.5 Calculated dependence of weight fractions of various substructures in the crosslinking system of H3 + I2 type on the initial molar ratio of H-groups to l-groups, ah (a polyurethane system) DC - dangling chains, BC - backbone chains, S - sol (backbone chains are elastically active network chains without dangling chains)...

The statistical theory of crosslinking used in the last section also gives the theoretical concentration of elastically-active chains, N, which in turn determines the rubbery modulus E = 3NRT (R is the gas constant and T is the absolute temperature). At 70% reaction one calculates E - 2 x 10 dyn/cm1 2 3 4 5 6 7 8 9 10, in agreement with the apparent level in Figure 1. 

Simultaneous IPN. According to the statistical theory of rubber elasticity, the elasticity modulus (Eg), a measure of the material rigidity, is proportional to the concentration of elastically active segments (Vg) in the network [3,4]. For negligible perturbation of the strand length at rest due to crosslinking (a reasonable assumption for the case of a simultaneous IPN), the modulus is given by ... 

Note 2 A model network is not necessarily a perfect network. If a non-linear polymerization is used to prepare the network, non-stoichiometric amounts of reactants or incomplete reaction can lead to network containing loose ends. If the crosslinking of existing polymer chains is used to prepare the network, then two loose ends per existing polymer chain result. In the absence of chain entanglements, loose ends can never be elastically active network chains. 

Gordon, M., Kucharuk,S., Ward,T. C. The statistics of elastically active network chains and the efficiency of crosslinking in rubbers. Collection Czech. Chem. Commun. 35,3252-3264 (1970). 

In the following, we will briefly outline the use of the link p.g.f. (l.p.g.f.) for the calculation of the gel point in /-functional polycondensation without and with cyclization including the f.s.s.e. In Chapter II, section 2.2 we will consider an application in connection with the number of elastically active network chains in random polycondensates or in a collection of randomly crosslinked chains. 

In the random crosslinking of existing chains, the primary chains are placed in the root and nodes and each repeat unit can bear part of a crosslink (Fig. 7). In this case ve is the number of elastically active chains per primary chain. 

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. 

Styrene-Divinylbenzene Networks. Using ionic polymerization methods, Rietsch et al. (1976) prepared polystyrene (PS) networks with a well-controlled length of elastically active chains and crosslinks of variable functionality. In a given series, the glass transition temperature obeys the classical free volume theory ... 

Polyurethane Networks. Andrady and Sefcik (1983) have applied the same relationship as Rietsch et al. (1976), to the glass transition temperature of networks based on poly(propylene oxide) diols with a controlled molar mass distribution, crosslinked by aromatic triisocyanates. They obtained a Kr value of 25 K kg mol-1, about twice that for PS networks. They showed that the length distribution of elastically active chain lengths, directly related to the molar mass distribution of the starting poly(propylene oxide), has practically no effect on Tg. 

How do we take into account its eventual internal rotations How do we distinguish between tetrafunctional (four elastically active chains) and trifunctional (three elastically active chains and one dangling chain) crosslinks ... 

The description of a network structure is based on such parameters as chemical crosslink density and functionality, average chain length between crosslinks and length distribution of these chains, concentration of elastically active chains and structural defects like unreacted ends and elastically inactive cycles. However, many properties of a network depend not only on the above-mentioned characteristics but also on the order of the chemical crosslink connection — the network topology. So, the complete description of a network structure should include all these parameters. It is difficult to measure many of these characteristics experimentally and we must have an appropriate theory which could describe all these structural parameters on the basis of a physical model of network formation. At present, there are only two types of theoretical approaches which can describe the growth of network structures up to late post-gel stages of cure. One is based on tree-like models as developed by Dusek7 I0-26,1 The other uses computer-simulation of network structure on a lattice this model was developed by Topolkaraev, Berlin, Oshmyan 9,3l) (a review of the theoretical models may be found in Ref.7) and in this volume by Dusek). Both approaches are statistical and correlate well with experiments 6,7 9 10 13,26,31). They differ mainly mathematically. However, each of them emphasizes some different details of a network structure. 

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 modem theory of mbber-like elasticity theory suggests that there are two types of elastically active network chains which contribute to the overall equilibrium rubbery modulus, G (1) chains attached to the network by chemical crosslinks, G and (2) chains attached by physical crosslinks or entangelements, G . That is,... 

Figure 9.2 Main elements constituting the structure of a polymer network (1) crosslink point, (2) elastically active chain, (3) dangling chain, (4) loop or cycle, (5) multiple connection between two crosslink points, and (6) permanent chain entanglements between two adjacent crosshnks.

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