Tube Primitive path fluctuations

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].

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 ] ... 

Finally, we remark that the idea of self-consistent dynamic dilution was applied first by Marrucci [20] to the case of monodisperse linear polymers, and was then adapted by BaU and McLeish [11] to monodisperse stars. We also note that theories combining reptation, primitive path fluctuations, and constraint release by dynamic dilution have been applied successfully by Milner and McLeish and coworkers to monodisperse linear polymers [21], monodisperse stars [13], bimodal star/star blends [22], and star/linear blends [23], as well as H-branched polymers [24], and combs [25]. The approach taken for all these cases is similar at early times after a small step strain, the star arms and the tips of linear molecules relax by primitive path fluctuations and dynamic dilution. At some later time, if there are linear chains that reach their reptation time, there is a rapid relaxation of these linear chains. This produces a dilation of the effective tubes that surround any remaining unrelaxed star arms by constraint-release Rouse motion (see Section 7.3). Finally, after dilation has finished, the primitive path fluctuations of remaining portions of star arms begin again, in the dilated tube. We refer to this set of theories for stars, linears, and mixtures thereof as the Milner-McLeish theory . The details of the Milner-McLeish theory are beyond the scope of this work, but the interested reader can learn more from the original articles as well as from McLeish and Milner [26], McLeish [14], Park and Larson [27], and by Watanabe [19]. 

The case of star/linear blends is a challenging one, because the description of constraint release that works best for pure star polymers is dynamic dilution, while for pure linear polymers, double reptation , or some variant of it, seems to be the better description. However, Milner, McLeish and coworkers [23] have developed a rather successful theory for the case of star/ linear blends. In the Milner-McLeish theory, at early times after a step strain both the star branches and the ends of the linear chains relax by primitive-path fluctuations combined with dynamic dilution, the latter causing the effective tube diameter to slowly increase with time. Then, at a time corresponding to the reptation time of the linear chains, the tube surrounding the unrelaxed star arms increases rather quickly, because of the sudden reptation of the linear chains. The increase in the tube diameter would be very abrupt, if it were not slowed by inclusion of the constraint release-Rouse processes, which leads to a square-root-in-time decay in the modulus (see Section 7.3). With this formulation, the Milner-McLeish theory yields very favorable predictions of polybutadiene data for star/linear blends see Fig. 9.13, where the parameters have the same values as were used for pure linears and pure stars. 

How can a branch point move The repertoire of polymer movements that we have considered up to now reptation, primitive path fluctuations, and Rouse motion within the tube do not allow for branch-point motion, at least not directly. Yet, clearly, the branch points do move, for if they did not, branched polymers, including stars, would have zero center-of-mass diffusivity. 

The primed quantity Zb(i) is the same as Zb(i), except that the portions of the backbone that have relaxed due to primitive path fluctuations are excluded from. In Eq. 9.27, the volume fraction of unrelaxed melt T [Tjgps(gp(i)] is evaluated at the time Tr pstep(0 required for the backbone to reptate one tube diameter along the imdiluted tube ... 

The earliest tube models included only the simplest nonlinearities, that is, convective constraint release was neglected (since its importance was not clearly recognized), and the retraction was assumed to occur so fast relative to the rate of flow that the chains were assumed to remain imstretched. The linear relaxation processes of constraint release and primitive path fluctuations were also ignored, so that the model contained only one linear relaxation mechanism, namely reptation, and only the nonlinearity associated with large orientation of tube segments, but no stretch. Subsequent models added the omitted relaxation phenomena, one at a time, and in what follows we present the most important constitutive models that included these effects, starting with models for monodisperse linear polymers. 

Starting from this basic idea. Mead, Larson, and Doi [ 27 ] developed a tube model that includes chain stretch and primitive path fluctuations, as well as CCR. The MLD model was developed... 

Each monomer is constrained to stay fairly close to the primitive path, but fluctuations driven by the thermal energy kT are allowed. Strand excursions in the quadratic potential are not likely to have free energies much more than kT above the minimum. Strand excursions that have free energy kT above the minimum at the primitive path define the width of the confining tube, called the tube diameter a (Fig. 7.10). In the classical affine -and phantom network models, the amplitude of the fluctuations of a... 

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