Rouse-Mooney theory

The compliances are probably more instructive than the moduli in describing the viscoelastic behavior of networks, because of the maximum in the loss compliance, J", which is characteristic of networks as pointed out in Chapter 2. This is shown for the Rouse-Mooney theory, equations 35 to 37, in Fig. 10-7 together with some modifications to be described in the following section. Here J" is calculated simply as G" equation 28 of Chapter 1, arid then normalized... 

If the cross-links are considered as fixed, the Rouse-Mooney theory can be readily modified by exactly the same procedures as for molecular weight distribution in uncross-linked polymers (Section A3 above) or dilute solutions (Section B7, Chapter 9). The result is a somewhat more gradual frequency or time dependence... 

FIG. 12-9. Logarithmic plots of G and G" against frequency near the onset of the transition zone, enlarged from curves IV of Figs. 2-3 and 2-4, for poly(n-octyl methacrylate) at 100 C. Dashed curves drawn from Rouse-Mooney theory (Section B-l of Chapter 10). Reproduced, by permission, from the... 

The onset of the transition zone on the frequency scale can be defined as in Fig. 12-9 except that for a cross-linked polymer the left side of the curve for G goes into the equilibrium modulus Ge instead of the plateau G% (except for the very lightly cross-linked systems discussed in Section B4 below). The boundary frequency (j>tr is given by equation 9 of Chapter 12 according to the Rouse-Mooney theory. Various cross-linked rubbery polymers show similar frequency dependences of G as the transition zone is entered, as shown in Fig. 14-1 if the logarithmic scales are arbitrarily shifted to make Gg and o),r coincide for all four polymers shown, the curves nearly coincide. ... 

It has already been pointed out in Chapter 13, Section Al, that maxima in the loss compliance and the retardation spectrum are characteristic of network structures as predicted from the Rouse theory, suitably modified, in Fig. 10-7. Such maxima appear in moderately cross-linked polymers as well as in uncross-linked polymers of high molecular weight, and their shapes are remarkably similar. In Fig. 14-3, J" is compared for styrene-butadiene copolymer without cross-links and cross-linked to a value of Gg = 7.3 X 10 dynes/cm, characteristic of a well-vulcanized soft rubber e.g., curve VI of Figs. 2-1 to 2-8). The curves are both close in shape to that predicted by the Rouse-Mooney theory for a most probable distribution of network strands, i.e., curve D of Fig. 10-7, as already evident from Fig. 13-1. 

Rouse-Mooney Theory for Networks with Uniform Strand Length... 

FIG. 10-5. Storage and loss shear moduli, normalized by the equilibrium modulus, plotted logarithmically against frequency for the Mooney modification of the Rouse theory. 

FIG. 10-7. Storage and loss shear compliances, normalized by the equilibrium compliance, plotted logarithmically against frequency for various network theories. (A) Mooney-Rouse theory, corresponding to Fig. 10-5 (B) Blizard model with trifurcate branching corresponding to tetrafunctional connectivity (C) tetrafunctional model of Chompff and Duiser ( >) Mooney-Rouse theory with most probable distribution of strand lengths. 

FIG. 13-1. Plots of J" ft against co/(Om fof four polymers black circles, 1,2-polybutadiene open circles, 1,4-polybutadiene, cis trans - 43 SO half black, styrene-butadiene copolymer crossed circles, polyisobutylene. 7 from equation 4, except for 1,4-polybutadiene (chosen to match the other curves) ti) from position of maximum on frequency scale. Solid curve is curve D of Fig. 10-7, with linear ordinate (Mooney modification of Rouse theory with most probable distribution of strand lengths). (Sanders, Valentine, and Ferry. )... 

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