Constraint Release - Double Reptation

Consider an isolated long probe P-mer entangled in a melt of shorter Wmers. Tube dilation assumes that as soon as short chains relax, stress in the long P-mer drops to zero. In particular, a version of tube dilation called double reptation imposes an exact symmetry between single chains in a tube and multi-chain processes. As one chain reptates away, stress at a common entanglement (stress point) is relaxed completely. In constraint release models, this stress relaxes only partially due to connectivity of the P-mer. 

M. Rubinstein (Eastman Kodak Company) In the des Cloizeaux double reptation model which is similar to the Marrucci Viovy model, it is assumed that a release of constraint chain A imposes on chain B when chain A reptates away completely relaxes the stress in that region for both chains. This would imply that for a homopolymer binary blend of long and short chains would be completely relaxed after each of these K entanglements is released only once. But if an entanglement is released, another one is formed nearby. I believe that to completely relax this section one needs disentanglement events and that the Verdier-Stockmayer flip-bond model or the Rouse model is needed to describe the motion and relaxation of the primitive path due to the constraint release process, as was proposed by Prof, de Gennes, J. Klein, Daoud, G. de Bennes and Graessley and used recently by many other scientists. The fact that double reptation is an oversimplification of the constraint release process has been confirmed by experiments. 

Tj represents some relaxation time, of component i, which in terms of the tube model, is related to the idealised Doi-Edwards relaxation time for component i in a matrix of fixed obstacles, tde. by, Xi = (1/2)tde. Hence, in the double reptation model, the effect of constraint release is to half the relaxation time (if single exponential decay is assumed), from that predicted for a polymer in a fixed matrix. In the heterogeneous blends considered here, the tj are the tube survival times for chains of species i in an idealised environment, in which the chemical heterogeneity matches that of the blend, but all chains share the same relaxation time. That is, double reptation accounts for mutual effects in topological stress relaxation, but not for direct effects of local composition on the monomeric friction factors. The parameters of the double reptation model should be treated as phenomenological, to be determined from independent linear rheology experiments in the one phase region (see for example reference [61]). 

The process of constraint release is in general very complex, and a completely general, rigorous, theory has not yet been developed. Nevertheless, there is a simple description of constraint release called double reptation that is reasonably accurate for many cases of practical importance. More rigorous, general theories of constraint release are presented in Chapter 7. 

Here Tj l and are the reptation times of the long and short chains, respectively. If we neglect constraint release, the relaxation modulus G(f) is just P t) times the plateau modulus G. To calculate the stress relaxation modulus, including the effects of constraint release using the double reptation theory, we merely square P t) as follows ... 

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. 

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...

Thus, up to now, in our discussion of constraint release, we have assumed that for non-dilute concentrations of long chains, the reptation time of the long-chain component of a bidisperse melt is unaffected by relaxation of the other components (except for the factor of two correction predicted by double reptation). This is not always true. To take an extreme example, if a monodisperse polymer melt is diluted with a small-molecule solvent, entanglements will become less dense, and the plateau modulus will drop, thus increasing the tube diameter a as indicated by to Eq. 6.23 ... 

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 upper limit on the summation is r, the degree of polymerization, which is M/Mq. In a polydisperse system, this approach must be modified. Montfort etal. [48] account for the effects of polydispersity in two ways. First, they use the double reptation concept with the Doi-Edwards kernel function to account for constraint release, but they also let the relaxation times depend on the molecular weight distribution, a concept originally proposed by Graessley [49]. Specifically, they represent the terminal relaxation time in a polydisperse system as the harmonic average of the reptation time and a tube renewal time, Tp which depends on the molecular weight distribution. 

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. 

Our understanding of how the MWD of a linear polymer is reflected in its rheological behavior is now sufficiently advanced that it is possible to use rheological data to infer the MWD except when is it is very narrow or very broad. The earliest methods made use of the viscosity data and required no assumption regarding the shape of the distribution. More recent methods are based on the tube model. If reptation is the only relaxation method taken into account, use of the double-reptation scheme to account for constraint release makes it possible to infer the MWD from storage modulus data, but the omission of other relaxation mechanisms limits the applicability of this method. The most elaborate methods take into account all possible relaxation mechanisms, but their use requires the assumption of an equation to describe the distribution. For reliable results, the data must be very accurate and precise. 

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