Branched polymers reptation

Figure 16 Tube model for reptation of a branched polymer molecule from the work of Blackwell et al. [124]. Reproduced with permission from Blackwell et al. [124]. Copyright 2000, The Society of Rheology, Inc.

The pom-pom polymer reptation model was developed by McLeish and Larson (60) to represent long chain-branched LDPE chains, which exhibit pronounced strain hardening in elongational flows. This idealized pom-pom molecule has a single backbone confined in a reptation tube, with multiple arms and branches protruding from each tube end, as shown in Fig. 3.12(a). Mb is the molecular weight of the backbone and Ma, that of the arms. 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

Complex Architectures. Perhaps the most significant recent advances in molecular understanding of polymer melts have emerged from the study of branched polymer architectures. We have noted above how a tube theory for star-polymers provided the means to treat fluctuations in entangled path length in linear polymers (see Figure lb). This is simply due to the complete suppression of reptation in star polymers without fluctuation there is no stress-relaxation at all ... 

Unfortunately, simple reptation moves cannot be used to simulate branched polymers or block copolymers, because the geometry of the molecule would be altered in the process. Furthermore, for a long molecule a significant amount of time is spent moving the chain ends back and forth, while leaving its center portions unchanged. For this reason, reptation moves are usually supplemented by other moves. 

The contour length fluctuation plays an essential role in the dynamics of branched polymers. Consider for example the star-shaped polymer shown in Fig. 6.11. Obviously simple reptation is not possible, but the polymer can change its conformation by utilizing the contour length fluctuation. 

As discussed in Section 6.4.5, reptation is severely suppressed if the polymer has long branches. Indeed it has been observed that the dynamical properties of branched polymers are quite distinct from those of linear polymers. So far studies have been done for branched polymers of the simplest type, the star-shaped polymer in which / chains are connected to a centre. The observed phenomena are ... 

Perhaps the most important problem is the tube reorganization. We have seen that the tube reorganization is important in branched polymers and in linear polymers with polydispersity. It will also be important in a nonuniform system sudi as polymer mixtures. So far the reptation theory is based on the assumption that there is a tube which is characterized by a single parameter a, the step length of the tube. Though the outcome of this simple assumption is quite fruitfiil, one could ask to what extent is this picture correct ... 

This theoretical conclusion has not been confirmed by direct reptation experiments, but it has some implications. Mechanical measurements on strongly entangled, high molecular weight chains may be completely dominated by the presence of a few branch points. If exponential laws such as eq. (VIII.23) are involved, we need only a small fraction of branch points, and such fractions cannot be detected by standard physicochemical methods. We conclude that mechanical measurements in long chain systems can be extremely sensitive to certain chemical defects. Unfortunately, we do not have reptation data on controlled branched polymers. We do have data on mechanical properties of branched melts,but the melt problem is much more complex than the reptation problem, as shown in next section. 

An alternative picture called tube dilution was employed to describe CR during relaxation of branched polymers. If one end of the chain is fixed to a branch point, reptation is suppressed and the whole relaxation must proceed by CLFs. However, large fluauations are exponentially suppressed, resulting in a large separation between the characteristic relaxation times of the tube segments 5 and s-tl. Ball and McLeish su ested that because of this separation, one can assume that the effective tube diameter also depends on s since on the timescale r(s) constraints created by all segments closer to the chain end than s had disappeared many times and can be discarded. Thus, the... 

From the considerations presented in the previous section, it is evident that it is desirable to choose MC moves X X such that the relaxation time resulting from a Markov chain of such moves for the configurations of the polymer chains is as small as possible. This is particularly important for dense melts of long (and hence mutually entangled ) polymer chains, where the reptation concepts imply an asymptotic scaling rocN , that is, the relaxation is distinctly slower than for isolated chains (cf. eqns [13]-[16]).The situation would be even worse for dense melts of star polymers (or other branched polymers) where even an exponential scaling (lnr°=N) may result. ... 

Doi [50] has since proposed a tube renewal concept adapted frx)m the melt characteristics of star branched polymers which assumes the motion of rings to be a result of the rearrangement of the tube rather than by the reptation of the polymer down the tube. 

In branched polymers, the plateau compliance is often not well defined even for narrow molecular weight distribution, there is no obvious plateau. - This is probably related to the suppression of reptation, which forces each molecule to relax by different mechanisms with a broad spectrum of relaxation times. 

The dynamics of linear or branched polymers is known to follow the reptation mechanism, in which the chain ends of polymer molecules play a critically important role. On the contrary, ring polymers are unique in the absence of the chain ends and therefore their non-reptation dynamics has been a fundamental issue in both theory and practice in polymer materials science. 

Another case where tube contraction is important is the relaxation of branched polymers/as epitomized by star-polymers. Because the branch point is highly immobile, reptation by Brownian motion of an arm as a whole is strongly inhibited. Escape from the tube can only occur by contraction of the primitive path. Any significant contraction has a high free energy (discussed in the exercise above). The time for a fractional contraction i.e. A is... 

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. 

The constraint-release models discussed above have been tested by comparing their predictions to experimental data, as shown in Figures 7.9 and 7.10. For linear polymers for which the molecular weight distribution is unimodal, and not too broad, dynamic dilution is not very important, and theories that account for constraint release without assuming any tube dilation are adequate. Such is the case with the version of the Milner-McLeish theory for linear polymers used to make the predictions shown in Fig. 6.13. The double reptation theory also neglects tube dilation. The dual constraint theory mentioned in Chapter 6 does include dynamic dilution, although its effect is not very important for narrowly dispersed linear polymers. As described above, dynamic dilution becomes important for some bimodal blends, and is certainly extremely important for branched polymers, as discussed in Chapter 9. 

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... 

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]. 

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. 

Figure 9.20 Hierarchical algorithm for computing the linear relaxation of the arbitrary comb-branched polymer illustrated in (a). The molecule consists of arms and backbone segments. Initially, after a step strain, only the arms can move, by primitive path fluctuationsy from the arm tips Inward. When an arm fully relaxes, it is pruned away, and replaced by a bead at the branch point to represent the frictional drag contributed by that arm see (b). Continued arm relaxation converts the molecule into a star (c), and finally a linear chain (d).The linear chain can complete its relaxation by reptation. From Park and Larson [49].

At late times, reptation takes over. For a linear molecule, the reptation is of the ordinary kind, but for branched polymers, the reptation is a compound process each step of which requires... 

---