Primitive path fluctuations

For linear chains, only the portions near the chain ends relax by primitive path fluctuations, and we will use the symbol Tgariy( ) in our development of a model for this fast relaxation. The interior parts of the chain, however, require quite deep fluctuations to reach them, and the time required to do this is slower than the time at which these portions of the chain will have already relaxed by reptation. We will later use the symbol in our discussion of this 

The factor of two dividing Z in Eq. 6.31 is introduced because for linear molecules fluctuations occur at both ends, so only half of each molecule needs to fluctuate to relax the stress, and only Z/2 entanglements participate in the fluctuations at each end. For star polymers (discussed in Section 9.3.1), each arm has only one free end and the factor of two is not required. 

If both ends of the molecule are free to move, and so the chain can reptate, segments in the interior of the chain will relax faster by reptation than by primitive-path fluctuations, and so reptation will control the longest relaxation time of the chain. However, because primitive-path fluctuations are so much faster for the chain ends than for the chain center, the chain ends will still relax by primitive-path fluctuations. Only for very high molecular weights (MfMg 100) are the contributions of fluctuations confined to small enough portions of the chain ends that these effects can be neglected. 

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Despite these complications, there are now numerous evidences that the tube model is basically con-ect. The signatory mark that the chain is trapped in a tube is that the chain ends relax first, and the center of the chain remains unrelaxed until relaxation is almost over. Evidence that this occurs has been obtained in experiments with chains whose ends are labeled, either chemically or isotopically (Ylitalo et al. 1990 Russell et al. 1993). These studies show that the rate of relaxation of the chain ends is distinctively faster than the middle of the chain, in quantitative agreement with reptation theory. The special role of chain ends is also shown indirectly in studies of the relaxation of star polymers. Stars are polymers in which several branches radiate from a single branch point. The arms of the star cannot reptate because they are anchored at the branch point (de Gennes 1975). Relaxation must thus occur by the slower process of primitive-path fluctuations, which is found to slow down exponentially with increasing arm molecular weight, in agreement with predictions (Pearson and Helfand 1984). 

The Doi-Edwards model has been extended to allow processes of primitive-path fluctuations, constraint release, and tube stretching. These extensions of the theory allow accurate prediction of many steady-state and time-dependent phenomena, including shear thinning, stress overshoots, and so on. Predictions of strain localization and slip at walls... 

Chapter 9 presents tube models for linear viscoelasticity in systems with long-chain branching. Reptation of the molecule as a whole is suppressed by branch points, and relaxation takes place primarily by primitive path fluctuation, a relatively slow process. An alternative to the tube picture, the slip-link approach, is examined in detail. 

The exponential increase of viscosity with M is consistent with the picture in which relaxation occurs primarily by means of primitive path fluctuations (sometimes called arm retraction). In Chapter 9 we will see that this effect can be explained quantitatively by a tube model. The exp onential increase of t]q with M results from the fact that the branch point prevents reptation, so that the principal mechanism of relaxation is primitive path fluctuation, which becomes exponentially slower with increasing arm length. The energy of activation for the zero-shear viscosity is little affected by star branching, except in the case of polyethylene and its close relative, hydrogenated polyisobutylene. 

Figure 6.9 Illustration of three relaxation mechanisms in the linear viscoelastic regime, (a) Reptation of a polymer molecule out of its tube. To aid visualization, the tube of Fig. 6.4 has been straightened out. Adapted from Graessley [31].(b) Primitive path fluctuations, in which the ends of chains randomly pull away from the ends of the tube, and upon re-expansion the chain ends explore a new regions of space and creates new tube segments. Adapted from Doi and Edwards [1 ]. (c) Constraint release, in which chain "c" which presents a topological obstacle to chain A, moves, thus allowing a portion of chain Ato relax. Adapted from Doi and Edwards [1].

Figure 6.10 Linear moduli G and G" versus reduced frequency fora nearly monodisperse polybutadiene melt of molecular weight 360,000. The dashed lines are the predictions of the reptation theory.The solid lines include the effects of primitive path fluctuations. Adapted from Pearson [33].

Reptation Combined with Primitive Path Fluctuations... 

For linear polymers, primitive path fluctuations (PPF or CLF for contour length fluctuations ) occur simultaneously with reptation. At short times (or high frequencies) the ends of the chain relax rapidly by primitive path fluctuation. But primitive path fluctuations are too slow to relax portions of the chain near the center, and these portions therefore relax only by reptation. However, the relaxation of the center by reptation is speeded up by primitive path fluctuations, because the tube remaining to be vacated by reptation is shortened, since its ends have already been vacated by primitive path fluctuations. As a result, the longest reptation time Tj (i.e., the terminal relaxation time) and zero-shear viscosity, are lower than in the absence of the fluctuations and can be approximated by the following equation [ 1 ] ... 

Primitive path fluctuations (PPF or CLF) also broaden the spectrum of relaxation times. Note in Fig. 6.10 that the inclusion of PPF results in a less steep decrease in G" with frequency (at... 

Although reptation and primitive path fluctuations together provide a nearly quantitative prediction of the linear viscoelasticity of monodisperse melts of linear chains, for polydisperse melts it is clear that these are not the only important relaxation mechanisms. To develop quantitative, or even qualitative, theories for polydisperse melts, constraint release must be taken into accoimt. 

The simple replacement of P(t) by P as shown in Eq. 6.35 (which does not account for primitive path fluctuations) does not have much effect on the relaxation modulus of monodisperse linear chains, as mentioned above. This is because the relaxation spectrum for reptation is... 

As mentioned in Section 6.4.4.1 the double reptation theory contains no explicit treatment of primitive path fluctuations, and if the theoretical expression Eq. 6.28 is used for the reptation time, it will predict that a monodisperse melt will have a longest relaxation time proportional to the third power of molecular weight, in disagreement with the observed 3.4 power-law dependence. A simple way of dealing with this is to use the empirical formula, (Eq. 6.40), for the longest relaxation time. More sophisticated ways are available for dealing with this limitation [29], but here we confine ourselves to this simple fix, which is adequate for many commercial polymers with broad (but not too broad) molecular weight distributions. 

The simplest possible type of branched polymer is a monodisperse star. In some respects, monodisperse stars are actually easier to consider than linears, because for stars one can neglect reptation. This leaves only the relaxation mechanisms of primitive path fluctuations, constraint release, and high-frequency Rouse modes that need to be considered to describe the linear... 

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