Modulus rubbery

The results of the tests are plotted in Fig. 4.1 according to Fox and Flory [39] against 1> the inverse molecular mass between crosslinks, which was determined from the rubbery modulus. The two sets of results agree basically, although the DSC results are consistently 8 K lower than the temperatures TXmax, derived from... 

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. 

The glassy state modulus is of the order of 1 GPa just below Tg. One can imagine the case of a very highly crosslinked polymer in which the rubbery modulus would be equal to the glassy one (no gap at Tg). The corresponding Me value would be (at 500 K for instance) Me 16 g mol-1, which is an unrealistic value. 

The rubbery modulus may change during the measurement at high temperatures, because the polymer may participate in both postpolymerization and degradation reactions. 

Remark The model parameters are not very sensitive to variations in the rubbery modulus E0. E an be determined only by extrapolation, which is also a source of errors. 

The use of Cole-Cole plots is not very developed in practice, despite the fact that they open the way for the modeling of the viscoelastic behavior in dynamic as well as in static loading cases (through Laplace transform). By contrast, these plots could be interesting from the fundamental point of view if certain parameters would reveal a clear dependence with the crosslink density. The effects of crosslinking are difficult to detect on the usual viscoelastic properties, except for the variation of the rubbery modulus E0. 

Research on the eventual specific effects of crosslinking on characteristics other than the glass transition temperature or the rubbery modulus remains a widely open domain. 

These difficulties are again increased, sometimes considerably, by the fact that thermosets are generally used in composites or as adhesives, e.g., in applications where aging can result also from a change of the interfacial properties and in which certain key properties, e.g., the rubbery modulus, are practically inaccessible. 

Figure 14.7 Rubbery modulus against dangling chains concentration in styrene crosslinked unsaturated polyester. (After Mortaigne et a/., 1992.)...

At temperatures well below Tg, when entropic motions are frozen and only elastic bond deformations are possible, polymers exhibit a relatively high modulus, called the glassy modulus (Eg) which is on the order of 3 Gpa. As the temperature is increased through Tg the stiffness drops dramatically, by perhaps two orders of magnitude, to a value called rubbery modulus Er. In elastomers that have been permanently crosslinked by sulphur vulcanization or other means, the values of Er, is determined primarily by the crosslink density the kinetics theory of rubber elasticity gives the relation as... 

At present, there are no physical methods to measure the concentration of amines of different types in networks and thus we cannot experimentally prove the computed values. However, the computed results seem reasonable since computer simulations give many features of the real behaviour of the systems under consideration. For example, the calculations gave kinetic curves of different reacting mixtures, sol and gel fractions, and equilibrium rubbery modulus. All results showed very good correlation with experiments 6 9,13,16,ly,31). This situation allows us to correlate the structural features of networks (for example, relative amounts of defects) obtained from computer simulations with macroscopic properties of the polymers. 

The average length (or molecular weight) of network chains in a crosslinked polymer can be experimentally determined from the equilibrium rubbery modulus. This relationship is a direct result of the statistical theory of rubber-like elasticity . In the last decade or so, modem theories of rubber-like elasticity 2127) further refined this relationship but have not altered its basic foundation. In essence, it is... 

Thus, the level of sophistication which one may consider for the application of rubber-like elasticity theory to epoxy networks may depend on the application. For highly crosslinked systems (M < 1,000), a quantitative dependence of the rubbery modulus on network chain length has recently been demonstrated , but the relevance of higher order refinements in elasticity theory is questionable. Less densely crosslinked epoxies, however, are potentially suitable for testing modern elasticity theories because they form via near quantitative stepwise reactions. Detailed investigations of such networks have been reported by Dusek and coworkers in recent studies ... 

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,... 

A number of workers have treated non-Gaussian networks theoretically in terms of this finite extensibility problem. The surprising conclusion is that the effect on simple statistical theory is not as severe as might be expected. Even for chains as short as 5 statistical random links at strains of up to 0.25, the equilibrium rubbery modulus is increased by no more than 20-30 percent (typical epoxy elastomers rupture at much lower strains). Indeed, hterature reports of highly crosslinked epoxy M, calculated from equilibrium rubbery moduh are consistently reasonable, apparently confirming this mild finite extmsibiUty effect. 

The data shows the shift in soft segment T from -48 C (100 percent 1,4-BDO) to 6 C (100 percent dMPD) as expected due to phase mixing. As anticipated, hard segment crystallization occurs when the DMPD content is less than 50 percent. The shear modulus data (Figure 9) show a steady decrease in the rubbery modulus as the DMPD content of the blend increases because the degree of hard segment crystallinity is decreasing, and these results... 

If La " acts merely to increase the glass transition temperature, as a result of copolymerization then the "rubbery modulus of the material should not necessarily change if, on the other hand, it does crosslink the material, then the rubbery modidus should change in a manner predicted by the kinetic theory. 

Unfortunately the materials do not have a sufficiently well-developed rubbery modulus for use in calculations. One therefore resorts to the equivalent ultimate Maxwell element from which the maximiun relaxation time was computed, and utilizes the modulus corresponding to that ultimate element for subsequent computations. Now if La" " " ions act as crosslinks, then the values should be directly proportional to their concentration, c, since both and c are inversely proportional to the molecular weight between crosslinks. Mg. The former relationship is due to the kinetic theory of rubber elasticity (E = 03qRTIMc where 0 is the front factor, q is the density, and R the gas constant), and the latter to simple stoichiometry (c = g/2Mj) for tetrafunctional crosslinks. A plot of vs. c was shown in Fig. 9, both for La" " " " and for Ca++ indicating that both ions act as crosslinks, at least at low concentrations and only for the ultimate Maxwell element. 

The lower temperature crystalline transition has a greater heat of fusion in every case than the higher transition. If the molar heats of fusion for the two crystalline endotherms were equal, the low melting crystal would be 86% and 96% respectively of the total crystallinity of the 1200 and 1000 EW SSC polymer. Evidently, since the rubbery modulus in the dynamic mechanical data is controlled by crystallinity, it is the low temperature crystalline component which has the greatest influence. 

The elastic modulus of the polymer can be calculated from Ea with knowledge of a from thermal mechanical analysis (TMA), when stress relaxation effects are negligible. It should also be possible to calculate crosslink density from the rubbery modulus (above Tg) by using rubber elasticity theory. 

While ionomers of many types have been made and characterized [1,2,3], there is little work on the overall relaxation mechanisms. For polymers with low ionic concentrations, there is general agreement on the fundamental relaxation step. The stress relaxes by detachment of an ion pair from one cluster and reattachment to another. For the styrene/methacrylic acid Na salt (ST/-MAA-Na) system, there is a secondary plateau in the relaxation modulus which depends on the ionic content and can be described as a rubbery modulus [4], While a rubbery modulus with stress relaxation due to ionic interchange has been invoked earlier, it does not adequately describe the relaxation curves. A different approach is taken here. 

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