Relaxation Double reptation model

For entangled systems, the two first conditions are fulfilled in the framework of reptation theories a comprehensive expression of the monodisperse relaxation modulus G(M,t) is given by expression 3-24 and the double reptation model generalized to a continous molecular weight distribution provides the integral relation between the MWD function P(M) and the polydisperse experimental... 

What value of a corresponds to the double reptation model In Section 9.3, we have presented theoretical arguments and experimental data supporting the value a =4/3 in -solvents (and melts with ideal chain statistics). What is the expression of the stress relaxation modulus of tube dilation models corresponding to a = 4/3 ... 

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]). 

In general, there are multiple relaxation processes in polymers, many of which are much too complex to be described by simple rheological theories (such as the double reptation model presented below), and it is not our objective to describe all such processes in detail. The interested reader can find the details in the book by Doi and Edwards [ 1 ], and in the review article by Watanabe [2]. Nevertheless, in Chapter 9 we will present some advanced theories for polymer melts, including theories of McLeish, Milner, and coworkers, that include all the known important mechanisms of polymer relaxation, and in Chapter 11, we will combine... 

The sharpness of the predicted peaks is due, to a small degree, to the use of a single relaxation time for each component of the bidisperse melt. This deficiency can easily be fixed by including the full reptation relaxation spectrum for each component. That is, for P(f) we can generalize Eq. 6.25 for the double reptation model to include two components ... 

If this is now squared, using the double reptation formula, Eq. 6.3 5, we obtain many relaxation terms that correspond to the cross terms for each pair of terms in the above summations. This will broaden the spectrum of relaxation times compared to the single-relaxation-time approximation. Nevertheless, because the Doi-Edwards relaxation spectrum is so narrow (i.e., the modes higher than the first mode have very little weight), inclusion of these extra modes does not improve the predictions of the double reptation theory very much. The major reason the basic double reptation model does poorly in describing the shape of the peaks in is... 

In fact, it is the performance of the double reptation theory for broad molecular weight distributions that is of the greatest practical importance. The double reptation model predicts the shapes of the G (< ) and G"(bidisperse melts. The reason for this is that when the molecular weight distribution is broad, the peak in G"( < ) is smeared out, or entirely eliminated, and the omission of the fast fluctuation modes for a given molecular weight is masked by the longest-relaxationtime contributions of the other molecular weights. For polydisperse polymers, the double reptation formula for the relaxation modulus is written as ... 

We also note that similar predictions can be obtained for other polydisperse linear melts, including polyethylene see, for example. Figs. 7.13 and 9.5a. And other, related models appear to give predictions roughly equivalent to those of the dual constraint model. Of particular note is the work of Marin and coworkers [20,21 ], whose model is described in more detail in Chapter 8, and the double reptation model with a more complex kernel relaxation function F(t) [29]. 

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. 

Let us start by illustrating the conceptual limitations of the double reptation idea. Consider the case of a polymer of high molecular weight at a volume concentration in a matrix of a polymer of much lower molecular weight. This case was considered in Section 6.4.4.2, and we found that the double reptation model predicts two relaxation peaks in G", a peak at a high frequency roughly equal to the inverse of the reptation time, g, of the short chains, and a low-frequency peak, whose frequency is the inverse of half the reptation time, T(Jl/2, of the... 

Nobile and Cocchini [33] used the double reptation model to calculate the relaxation modulus, the zero-shear viscosity and the steady-state compliance for a given MWD. They compared three forms of the relaxation function for monodisperse systems the step function, the single integral, and the BSW. In the BSW model, they set the parameter j8 equal to 0.5, which gives /s° G equal to 1.8. The molecular weight data were fitted to a Gex function to facilitate the calculations (see Section 2.2.4 for a description of distribution functions). For the step function form of the relaxation function is given by Eq. 8.37. 

The double-reptation model, ° ° an approximate version of the ftill-DTD model explained earlier (cf. Figure 12b), is known to be valid for entangled polymers with a broad unimodal MWD because the CR-equilibration occurs rather rapidly in these polymers. The relaxation modulus deduced from this model can be cast in the fotm ° °... 

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. 

The enhanced viscoelastic functions are attributable to additional relaxation processes that occur at low frequencies associated with deformation of the dispersed phase. Therefore, for cases such as mPE/LDPE, where partial miscibility at high LDPE content and the extremely different relaxatimi times of the phases in the blends rich in mPE are observed, a hybrid model including the double reptation approach for the matrix and the linear Palieme approach for the whole system could successfully explain the viscoelastic response of these blends (Peon et al. 2003). 

Physically, double reptation accounts, in a simple way, for the fact that polymers do not reptate in fixed tubes, as assumed in the original tube model. The surrounding polymers, which form the tube, also relax, and so the constraints which form the tube decay with a characteristic time-scale. In other words, stress relaxation depends not only the dynamics of each individual polymer, but also on the dynamics of the surrounding polymers. It should be noted that more sophisticated and therefore complex rheological models for polymer mixtures have been proposed however, a particular appeal of this model is the simple relation between the stress relaxation fimction of the blend and the Doi-Edwards stress relaxation function for polymers in a fixed network. 

As for the form of the monodisperse relaxation function, several models have been proposed. First, since double reptation is a direct descendent of the Doi-Edwards reptation model, it seems appropriate to use the original D-E modulus, which is given by Eq. 8.32. 

B) Models based on double reptation without taking into account other relaxation mechanisms. (Mead, Wasserman)... 

The models based on double reptation that do not take other relaxation mechanisms into account have the decided advantage of being readily subject to mathematical manipulation. 

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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