Constraint-Release Rouse Relaxation

In this chapter and Chapter 9, we wish to introduce more advanced constraint-release concepts, which can be applied to cases for which the double reptation model works poorly, including monodisperse and bidisperse, linear polymers. We will show that when the advanced concepts of constraint release Rouse relaxation and dynamic dilution are introduced into the tube model, then successful predictions of the linear rheology of bidisperse melts can be achieved. While bidisperse melts are not of great commercial interest, the concepts we will introduce in this chapter are also important for polymer with long side branches, which are of great commercial interest, and are discussed in Chapter 9. The reader not interested in the details of advanced tube theories may want to focus on the comparisons of predictions of these models with experimental data in Figures 7.9 through 7.13. However, where needed, results from this chapter will be used in Chapter 9, which covers branched polymers. 

Very long P-mers have the constraint release time [Eq. (9.85)] shorter than their reptation time. Such very long P-mers relax and diffuse by constraint release (Rouse motion of their tubes) before they get a chance to reptate out of their confining tubes. For shorter P-mers, the reptation time Tep( ) is shorter than the constraint release time Ttube and reptation dominates... 

Each of these entanglements has a mean lifetime corresponding to the reptation time of the LP. In modeling the constraint release (CR) relaxation of the CP, we visualize the entanglements as Rouse beads with a frictional drag proportional to T[,. 

Figure 7.6 Relaxation moduli of entangled binary blends of long and short chains according to the double reptation theory (bold dashed line) and the constraint release Rouse picture (bold solid line) when the reptation time of the long chain exceeds its constraint-release Rouse time /.e.,TjL > cr,l both (a) and (b), double reptation predicts two "step" decreases...

Figure 7.8 Relaxation modulusof entangled polydisperse meltsaccording to thedouble reptation theory (bold dashed line) and the constraint release Rouse theory (bold solid line).

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. 

As discussed in Section 9.3.2, when matrix chains relax, they cease to act as constraints on test chains. However, the wider tube that is created by rapid relaxation of matrix chains can only gradually be explored via the constraint release Rouse process. Thus, the contribution of constraint release to stress relaxation is controlled not by P t) directly, but through a constraint release volume fraction Pc it), which is eq ual to P t) whenever P t) is relaxing no faster than as but if P(f) is relaxing faster than t (for example, exponentially fast), than cr( ) i given by (see Section 7.3.3)... 

The advanced molecular models described in this chapter, namely the Milner-McLeish model and the hierarchical model, involve combinations of multiple relaxation mechanisms reptation, primitive path fluctuations, and constraint release described by both constraint release Rouse motion and dynamic dilution. However, all these mechanisms can be captured in algorithms in which entanglements are viewed as slip links between two chains see for example Fig. 9.22. 

If constraint release were the only process for conformational rearrangement, the initial path motions would be the same as the chain motions of the N-element Rouse model. Equation 4 relates the longest relaxation time Ti to the diffiision coefficient for Rouse chains. The diffusion cxieffident from constraint release is gjven by Eq. 90 with Ns = N. With Eqs. 1, 2, 4 and 9,... 

The constraint release process for the P-mer can be modelled by Rouse motion of its tube, consisting of P/A e segments, where is the average number of monomers in an entanglement strand. The average lifetime of a topological constraint imposed on a probe P-mer by surrounding A -mers is the reptation time of the A -mers Trep(A ). The relaxation time of the tube... 

The reptation time of the P-mer is Tep(P) and the constraint release time Tube given in Eq. (9.85). The faster of the two types of motion controls the diffusion of the P-mer. For constraint release to significantly affect terminal dynamics, the Rouse relaxation time of the confining tube Ttube must be shorter than the reptation time of the P-mer Tep( ) ... 

Constraint release has a limited effect on the diffusion coefficient it is important only for the diffusion of very long chains in a matrix of much shorter chains and can be neglected in monodisperse solutions and melts. The effect of constraint release on stress relaxation is much more important than on the diffusion and cannot be neglected even for monodisperse systems. Constraint release can be described by Rouse motion of the tube. The stress relaxation modulus for the Rouse model decays as the reciprocal square root of time [Eq. (8.47)] ... 

Thus, a finite fraction of the stress relaxes by constraint release at time scales of the order of the constraint lifetime in the Rouse model of constraint release. This is also the time scale at which the stress relaxes by reptation in monodisperse entangled solutions and melts. Both processes simultaneously contribute to the relaxation of stress. Therefore, constraint release has to be taken into account for a quantitative description of stress relaxation even in monodisperse systems. The contribution of constraint -release to stress relaxation in pefydispcrse solutions and melts is even more important as will be discussed below. 

Reptation and tube length fluctuations of surrounding chains release some of the entanglement constraints they impose on a given chain and lead to Rouse-like motion of its tube, called constraint release. Constraint release modes are important for stress relaxation, especially in polydisperse entangled solutions and melts. 

In panels (a)-(c), comparison of the blend moduli calculated from the model (curves Equations 3.69 through 3.73) with the moduli data (symbols) for various high-M polyisoprene/ poly(p-tert butyl styrene) (PI/PtBS) blends as indicated. The sample code nmnbers of the blends indicate 10 M of the components. The model considers the cooperative Rouse equilibration and successive constraint release (CR)/reptation relaxation of the component chains, and the model parameters summarized in Table 3.1 were determined experimentally. (Redrawn, with permission, from Watanabe, H., Q. Chen, Y. Kawasaki, Y. Matsumiya, T. Inoue, and O. Urakawa. 2011. Entanglement dynamics in miscible polyisoprene/poly(p-fert-butylstyrene) blends. Macromolecules 44 1570-1584). 

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