Entanglement Region

Therefore, for a blend solution where the viscoelastic behavior of component one is described by the Rouse theory (Eq. (7.58)) and that of 

The linear relaxation moduli and viscoelastic spectra have been studied for a series of concentrated polystyrene blend solutions consisting of two nearly monodisperse components FI = 1.03 x 10 ) as component one and X X = P7 = 6.8 x 10 FIO M = 1.00 x 10 F35 = 3.55 x 10  

the first and second terms of Eq. (11.5) are expected to have the same frictional factor, i.e. K = K. If the second term of Eq. (11.5) behaves the same in the concentrated blend solution as in the pure-melt system (i.e. when VEi = 0 in Eq. (11.5)), whose study was described in the last chapter, the same molecular-weight dependence of K jK (Fig. 10.5) should be followed with the molecular weight expressed in terms of the normalized molecular weight M /Me for the melt and Muj2/M for the blend solution. Indeed, as shown in Fig. 11.1, the molecular-weight dependences of the melt and blend-solution systems have been observed to be the same. The dependence of K /K on M/Mg (or M2/M ) can be described by the empirical equation  

Furthermore, in the line-shape analyses of these viscoelastic spectra, the molecular-weight distributions of P7, FIO, F35, and F80 included in the calculations are identical, respectively, to those extracted from the line-shape analyses of the spectra of the pure melt systems, some of which have been shown in Chapter 10. Thus, the close agreements between theory and experiment in the line-shape analyses have been achieved under the consistency of maintaining the same molecular-weight distributions of the samples between the pure-melt and blend-solution systems. Thus, in a quantitative way, the universality of viscoelastic spectrum is shown extending over the melt and blend-solution systems in accordance with Eq. (11.5) with the molecular weight normalized with respect to Mg for the melt and Mg for the blend solution. 

This universality has also been illustrated by Watanabe et al. in another way. They compared the line shapes of the viscoelastic spectra of a nearly monodisperse polystyrene melt (sample L161 with = 1.72 x 10 and Myj/Mn = 1.07) and a blend solution (consisting of two nearly monodisperse polystyrene polymers L407 with Miu = 4.27 x 10 and M /Mn = 1.05 as component two and sample LIO with = 1.05 x 10 and Mw/Mn = 1.08 as component one the weight-fraction ratio of the blend is 

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Use concentrations well below the entanglement region and as low as can be detected. 

For concentrated solutions of amorphous polymers, Bueche s mathematical model shows the ratio of zero shear viscosities of branched and linear polymer above the critical molecular weight in the entanglement region to be (28) ... 

Figure 1.4. A sketch depicting the semicrystalline nature of a polymer. The lamellae are crystalline regions while the entangled regions between them are amorphous.

For polystyrene, D = 83 SOOg.mol i, which means that beyond a mass of approximately D, the temperature T is fixed - (T ) = 49.7 0.3°C.Therefore, in the entangled region, the free volume fraction is constant at a given temperature and the iso-free volume state merges into the isothermal state. 

In entangled regions[5], polymer chains form approximately a uniform network, regardless of solvent power, so that Jo is given by Jo oc (5a)... 

Thus, JcR is expressed as a universal function of CM. Moreover, it was reported that the cross-over concentration from dilute to entangled regions for J is about 5 times higher than that from dilute to semidilute regions for 7 at the constant molecular weight[5]. 

As shown by the broken lines in Figures 5 and 6, on the other hand, the deviation from eq 8 is observed in the entangled region,... 

Figure 5. Polymer concentration dependences of Jc in the absence and presence of added-salt. Symbols are the same as in Figure 1. The closed circles denote the data from the flow birefringence measurements. The solid and broken lines are drawn to connect smoothly the data in dilute and entangled regions, respectively. (Reproduced from ref. 11).

Introducing eqs 6 and 9 into eq 10, we have the empirical formulas for Tw in the semidilute region for 7° entangled region for Jc. ... 

Since we can assume that C,(ic in the entangled region of the present samples, the polymer concentration dependence is predicted to be... 

Experimentally, is approximately 0.6. Thus, for branched polymers in the entanglement region, both rjo and may be quite large compared with the values for linear polymers of the same molecular weight. The terminal zone is inherently broader for well-entangled branched polymers than it is for linear polymers of comparable polydispersity [49]. The complex viscosities for a nearly monodisperse linear polybutadiene and three-arm polybutadiene star, shown in Fig. 3.24 for other purposes, exemplify the more gradual transition from Newtonian to power-law... 

Eq. (135) indicates that in a good solvent and entanglement region, the viscosity has a 3.5 power of cM. Similarly, below the entanglement region,... 

FIG. 9-23. Schematic delineation of dilute, semidilute, and entangled regions on a logarithmic map of M against c for polystyrenes in a good solvent. 

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