Global relaxation

In Figures 24.7 through 24.9 are shown the volume dependences of the local and global relaxation times for 1,4-polyisoprene [90], polypropylene glycol [91], and polyoxybutylene [76]. For either mode, volume does not uniquely dehne the relaxation times, as the curves for different... 

The transients shown in Figure 6 (see (9)) suggest that in the H2O/N2 phase, t O reacts with adsorbed CO to produce C0 and H2 ana that the Hj wavefront concentration is two-fold of the CO2 wavefront concentration. The overproduction of H on the wavefront seems to have been caused by the reoxidation of the catalyst by H 0. The reoxidation rate is therefore on the wavefront equal to the rate of the shift reaction, it is namely the limiting step in the global relaxation process. Furthermore, the fact that I CC is 2 1 on the wavefront suggests that presumably the shift con-... 

Where p defines the shape of the hole energy spectrum. The relaxation time x in Equation 3 is treated as a function of temperature, nonequilibrium glassy state (5), crosslink density and applied stresses instead of as an experimental constant in the Kohlrausch-Williams-Watts function. The macroscopic (global) relaxation time x is related to that of the local state (A) by x = x = i a which results in (11)... 

In the case of crosslinked polymers, the global relaxation time X has to be generalized by including a shift factor for the crosslink density (av)... 

H( P) as a function of the nondimensional relaxation time, 7 = u/x, the ratio of local to global relaxation times, and p. When Equations 3 and 5 are used simultaneously in analyzing experimental data, we have found that p= 1/2 for most amorphous polymers which will also be assumed for lightly crosslinking systems. 

The relaxation phenomenon which has been discussed so far is within the linear viscoelastic range. Under large deformation, the global relaxation time has to include the contribution from the external work Aw done on the lattice site and takes the form (20)... 

The yield occurs when the product of the applied strain rate (e) and the global relaxation time reaches the order of unity, i.e., ex 1. Thus, we obtain... 

FIGURE 18.11 Thermally stimulated depolarization currents of PVP K30 demonstrating two different global relaxation peaksPi is the (5-relaxation peak (representing molecular motion belfry, and P2 is the a-relaxation peak (representing mobility ). 

To is a terminal relaxation time describing chain motions. Other global relaxation times can be defined experimentally, for example as the inverse of the frequency of the maximum in the terminal dispersion in the loss modulus, Tmax from the time for equilibration following cessation of nonlinear shear flow, XR rj, measured by recovery of the overshoot in the transient viscosity the corresponding time for recovery of the overshoot in the second normal... 

TABLE 6.2 Comparison of Global Relaxation and Recovery Times for Entangled 1,4-Polybutadiene Solutions (Roland and Robertson, 2006)... 

Global Relaxation in Polyisoprene/Poly(vinyl ethylene)... 

For the moduli data, the time-temperature superposition fails at intermediate (0 between the segmental and global relaxation processes because these processes exhibit different Ojq at low T - (see, e.g., Adachi and Kotaka, 1993 Inoue et al., 1991, 1996 Kremer and Schonhals, 2003). (This failure is not well resolved in the compressed scale of the plots shown in Figure 3.3.) The superposition works separately at high and low ca where the viscoelastic data are dominated by one of these processes. In contrast, the dielectric data satisfy the superposition in the entire range of co because those data detect just the segmental relaxation process, although it fails in a close vicinity of... 

The above argument can be examined through direct comparison of Xa,p,Bs° and Xg,pi- Because the PtBS chains are effectively immobilized over the length scale > a in the time scale of the global relaxation of PI, Xq pi can be safely evaluated as the dielectric relaxation time of PI, Xg pi (= 10 s at 30°C cf. Figure 3.16). The time required for the cooperative Rouse equilibration process, x (= x ptBs )/ can be evaluated on the basis of a power-law-type expression of the modulus of the Rouse model (equivalent to Equation 3.35) (Osaki et al., 2001) ... 

Here, z is the local jump gate number (typically in a range of z = 2 to 4) treated as an adjustable parameter (z = 2 for the PI/PtBS blends, as shown later). Finally, the third term in Equation (3.71) corresponds to the plateau modulus sustained only by the PtBS chains before they exhibit the global relaxation. 

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