Linear Relaxation Modulus

Consider a chain in a tube at time zero. Analysis of the reptation process (de Gennes 1971) shows that after a time t, only a fraction P (t) of the original tube remains unvacated, namely 

The storage and loss moduli, G and G , are obtained from the relaxation spectrum in the usual way—that is, using G = Gi[co rl/(l + co zl)] G — G,[mT /(l -P The longest relaxation mode of the relaxation modulus in Eq. (3-67) is the dominant one it accounts for 96% of the zero-shear viscosity. Thus, the reptation model predicts that for a nearly monodisperse melt, the relaxation spectrum is dominated by a single relaxation time, T = Ta. This is in reasonable accord with experimental data at low and moderate frequencies (see the dashed line in Fig. 3-29). As the frequency increases, however, there 

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Fig. 6. Reduced relaxation modulus for the Cubic array as a function of molecular weight Broken line indicates characteristic slope of the linear relaxation modulus...

No overshoot and linear limits in transient stress growth. Linear relaxation modulus in step shear strain. 

Within these three assumptions, one can identify at least two important limiting cases. In the first, the micelle breakage time tbr is long enough that the typical micelle can relax by reptation before it breaks. In this case, one obtains the linear relaxation modulus G(t)... 

Another strong indication of the entanglement effect is the observation of a clear plateau in the linear relaxation modulus amj storage modulus spectrum when the polymer has a sufficiently high molecular... 

Figure 38 Master curves of elastic storage (S, ) and viscous loss (S", o) linear viscoelastic moduli of the 12-arm 12 828 (a) and 64-am 6430 (b) star-PBd polymers in the temperature range from 150 up to -103°C, with reference temperature-83 °C. Solid arrows represent the various transitions and corresponding crossover frequencies (cos. glass to Rouse-like transition cof.. transition to rubber plateau

This expresses itself in slow power law dsmamics for both the linear relaxation modulus G(t) and the relaxation time spectrum H(X) (12-14)... 

Another symbol sometimes used in place of Pj is 6. The rubberlike liquid model (Eq. 10.5), predicts that Pj is independent of shear rate and related to the linear relaxation modulus ... 

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