Modulus rubbery theory

PRINT "THE OLD ELASTICALLY EFFECTIVE CROSSLINK DENSITY EXPRESSION OF BAUER" 150 PRINT "AND BUDDE. THE SECOND CALCULATES A CROSSLINK DENSITY WHICH IN" THEORY SHOULD BE PROPORTIONAL TO THE RUBBERY ELASTIC MODULUS."... 

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. 

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... 

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. 

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 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. 

Dynamic mechanical characteristics, mostly in the form of the temperature response of shear or Young s modulus and mechanical loss, have been used with considerable success for the analysis of multiphase polymer systems. In many cases, however, the results were evaluated rather qualitatively. One purpose of this report is to demonstrate that it is possible to get quantitative information on phase volumes and phase structure by using existing theories of elastic moduli of composite materials. Furthermore, some special anomalies of the dynamic mechanical behavior of two-phase systems having a rubbery phase dispersed within a rigid matrix are discussed these anomalies arise from the energy distribution and from mechanical interactions between the phases. 

The upper part of Figure 1 depicts the reduction of the modulus of the rigid phase E/Eh by incorporation of a soft rubbery phase, again with Eh/E8 as the parameter. The three theories agree well in this reverse phase position (see the curve for Eh/E8 = 100). The composite moduli can therefore be calculated easily by one of the well known equations from TakayanagFs approximative model theory (Equation 1) or from Kernels theory (Equation 2). 

Based on studies of an homologous, endlihked, epoxy/amine network series, the simple theory of rubber elasticity has proved effective for determining reasonable cross-link densities from equilibria modulus measurements in the rubbery state. 

In effect, this theory postulates a chemical reaction between promoter, polymer, and mineral substrate as in the chemical bond theory but also suggests that the presence of a region of intermediate modulus between polymer and substrate which transfers stress from the high modulus surface to the relatively low modulus polymer. Adhesive technology has long recognized this principle in specially formulated primers for use when bonding rubbery polymers to metals. 

This approach to viscoelastic theory is reasonably successful in the low modulus regions, but it requires considerable modification if the high modulus and rubbery plateau regions are to be described. 

We return now to the difference in behavior between the two types of PMMA with molecular weights of 1.5 x 10 and 3.6 x 10 . We note that the rubbery modulus of the type with the higher molecular weight reaches a plateau at /iR = 3.4 MPa. As we discuss in Chapter 6 on rubber elasticity, the entanglement molecular weight Me can be determined from this modulus through the statistical theory of rubber elasticity (Ferry, 1980) as... 

Percolation theory applied to the two-phase system It is a well known fact that the transition of modulus takes place in the two-phase system from a rubbery state to a glassy state at the critical region (not a point) as a function of the composition. Morphologically the transition is understood as the reverse of phases. Recently by introducing the concept of percolation concept, the transition composition is defined by the elastic percolation threshold. The scaling rule proposed by de Gennes [9] is applied to such a variation of modulus, using the critical composition at the elastic percolation threshold. 

Crystallites may also be considered to act as reinforcing fillers. For example, the rubbery modulus of poly(vinyl chloride) was shown by lobst and Manson (1970,1972,1974) to be increased by an increase in crystallinity calculated moduli in the rubbery state agreed well with values predicted by equation (12.9). Halpin and Kardos (1972) have recently applied Tsai-Halpin composite theory to crystalline polymers with considerable success, and Kardos et al (1972) have used in situ crystallization of an organic filler to prepare and characterize a model composite system. More recently, the concept of so-called molecular composites —based on highly crystalline polymeric fibers arranged in a matrix of the same polymer—has stimulated a high level of experimental and theoretical interest (Halpin, 1975 Linden-meyer, 1975). 

Because of the special swelling and mutual dilution effects encountered in sequential IPN s, special equations were derived for their rubbery modulus and equilibrium swelling. The new equations were used to analyze polystyrene/polystyrene homo-IPN swelling and rubbery modulus data obtained by four different laboratories. In the fully swollen state, there was no evidence for IPN related physical crosslinks, but some data supported the concept of network I domination. In the bulk state, network I clearly dominates network II because of its greater continuity in space. The analysis of the data concerning the possible presence of added physical crosslinks in the bulk state yielded inconclusive results, but this latter is of special interest for modern network theories. 

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