Elastically active network chains, concentration

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. 

To obtain accurate values of the sol, thin specimens (1 mm) in one study (13) were kept in the solvent for six weeks in another study (14), thin specimens were extracted for more than 18 days in Soxhlet extractors. When the present experimental data were obtained (6), there was little interest in knowing the sol fraction accurately. However, as discussed subsequently, to compute the extent of the curing reactions and the concentration of elastically active network chains, the sol fraction must be known accurately. 

Note 4 Loose ends and ring structures reduce the concentration of elastically active network chains and result in the shear modulus and Young s modulus of the rubbery networks being less than the values expected for a perfect network structure. 

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 concentrations of elastically active network chains related to the dry state vd series A-F were [26] vd = 3.6, 5.7, 6.3, 7.1, 10.9 and 15.2 x 10 5 molcm 3 (structure formation at high dilution in the system. Using vd values together with other molecular parameters, the dependences of y vs ip 2 were calculated and both the extent of the collapse, A, and the critical value,... 

Assuming that no internal elastic chains are activated, the concentration of elastically active network chains per unit mass, ve, may be calculated... 

In the case of unsaturated polyesters, nondegraded samples made from a prepolymer of molar mass M and a styrene mass fraction s have a chain-ends concentration b = [2(1 — s)/M]p, where p is the density. If ve is the actual concentration of elastically active network chains, an ideal network would be obtained by welding each chain end to another one, leading to... 

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. 

Fig. 16. Time dependence of the gel fraction, w, and concentration of elastically active network chains, v, in the stoichiometric mixture of azelaic acid and DGEBA...

Along with T, further structural parameters of networks, e.g. the sol fraction, w, and the concentration of elastically active network chains in the gel, s v /(l — w, ... 

Since at long times pendant chains do not contribute to permanent elastic properties, the elastic equilibrium behavior of networks containing these chains should not differ substantially from that of regular networks. The elastic modulus from a network with pendant chains can then be obtained from the molecular theories of rubber elasticity provided that the concentration of elastically active network chains (v) can be calculated accurately. Depending on the different approaches that can be used for the rubber elasticity theory, the calculation of some other parameters, like the concentration of junctions points (p) and trapped entanglements (Te), also may be needed. 

The ideal network structure can be envisaged as a three-dimensional array of crosslink points, each crosslink point being connected to at least three other crosslink points via linear polymer segments, which are called elastically active network chains. In practice non-ideal network elements are also present, such as loops or dangling ends (Figure 16.1). Network density, or crosslink density, is expressed as the concentration of either the crosslink joints or the elastically active network chains (those chains that are part of the infinite structure and attached to crosslink junctions at both ends) per unity of volume of the unswollen material. 

To a first approximation, which neglects changes in average chain structure, the loss in elastically active junction point concentration may be translated directly into loss in concentration of elastically active chains and increase in the value of M, . For a perfect network in the dry state, the concentration of elastically active chains is given by the equations... 

An Example of Rubber Elasticity Calculations Suppose that an elastomer of 0.1 cm x 0.1 cm x 10 cm is stretched to 25 cm length at 35°C, a stress of 2 X10 dynes/cm being required. What is the concentration of active network chain segments ... 

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). 

The equilibrium shear modulus of two similar polyurethane elastomers is shown to depend on both the concentration of elastically active chains, vc, and topological interactions between such chains (trapped entanglements). The elastomers were carefully prepared in different ways from the same amounts of toluene-2,4-diisocyanate, a polypropylene oxide) (PPO) triol, a dihydroxy-terminated PPO, and a monohydroxy PPO in small amount. Provided the network junctions do not fluctuate significantly, the modulus of both elastomers can be expressed as c( 1 + ve/vc)RT, the average value of vth>c being 0.61. The quantity vc equals TeG ax/RT, where TeG ax is the contribution of the topological interactions to the modulus. Both vc and Te were calculated from the sol fraction and the initial formulation. Discussed briefly is the dependence of the ultimate tensile properties on extension rate. 

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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