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 ... 

There is growing evidence that t-T superposition is not valid even in miscible blends well above the glass transition temperature. For example, Cavaille et al. [1987] reported lack of superposition for the classical miscible blends — PS/PVME. The deviation was particularly evident in the loss tangent vs. frequency plot. Lack of t-T superposition was also observed in PI/PB systems [Roovers and Toporowski, 1992]. By contrast, mixtures of entangled, nearly mono-dispersed blends of poly(ethylene-a/f-propylene) with head-to-head PP were evaluated at constant distance from the glass transition temperature of each system, homopolymer or blend [Gell et al, 1997]. The viscoelastic properties were best described by the double reptation model , viz. Eq 7.82. The data were found to obey the time-temperature superposition principle. 

The double reptation model was used to evaluate viscoelastic behavior of metallocene-catalyzed polyethylene and low-density polyethylene blends by Peon et al. (2003). They compared their results with those obtained for HDPE/BPE blends prepared under similar conditions. Since this model assumes miscibility between the mixed species, the experimental viscosity of HDPE/BPE blends showed only small deviation compared to that expected according to the reputation miscible model. However, the model underestimated the compositional dependence of the zero-shear viscosity for mPE/LDPE blends, especially at intermediate levels. The enhanced zero-shear viscosity in immiscible blends such as PETG/EVA, PP/EVA, or EVA/PE blends was found to be more abrupt than it is for mPE/LDPE blends (Lacroix et al. 1996, 1997 Peon et al. 2003). 

Yu et al. (2011) studied rheology and phase separation of polymer blends with weak dynamic asymmetry ((poly(Me methacrylate)/poly(styrene-co-maleic anhydride)). They showed that the failure of methods, such as the time-temperature superposition principle in isothermal experiments or the deviation of the storage modulus from the apparent extrapolation of modulus in the miscible regime in non-isothermal tests, to predict the binodal temperature is not always applicable in systems with weak dynamic asymmetry. Therefore, they proposed a rheological model, which is an integration of the double reptation model and the selfconcentration model to describe the linear viscoelasticity of miscible blends. Then, the deviatirMi of experimental data from the model predictions for miscible... 

The rheological properties of miscible blends tmder different temperatures can be obtained from some theoretical models. One such model is the double reptation self-concentration. The DRSC (double reptation self-concentration) model actually includes the temperature dependency and concentration dependency through a complex mixing mle given by the double reptation model and self-concentration model, which helps to exclude the complex contribution from miscible components under different temperatures in the experimental data and only illustrate the effect of the concentration fluctuation and interface formation. This model is applied to study PMMA/SMA (Wei 2011). 

The Doi and Edwards [67] reptation model provides simple mixing rules for miscible systems without the thermodynamic interactions. For athermal systems Tsenoglou [16, 191] proposed the double reptation model ... 

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]). 

Garcia-Franco and Mead [142] proposed the use of the parameters of Eq. 5.65 to describe the behavior of polyethylenes prepared by means of anionic polymerization, gas-phase metallocene catalysis, and Ziegler-Natta catalysis. They reported that Eqs. 5.67 gave a good fit of their data and suggested that it is valid for all linear, flexible polymers with monomodal molecular weight distributions except in the terminal zone. They found that except in the terminal zone it provides a representation of the data that is similar to that given by the double-reptation model. They... 

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... 

Figure 6.15 Comparison of the predictions of the double reptation model (lines) to experimental data (symbols) for (a) the storage modulus G, and (b) the loss modulus G", for bidisperse polystyrenes (MW = 160,000 and 670,000) at 160 °C [35]. The volume fractions of the high molecular weight component ((j ) from right to left are 0.0,0.05,0.1,0.2, and 0.5, and 1.0, respectively.The parameter values are G 5 = 2 10 Pa and /f = 4.6 10 s/(mol) " The latter value, obtained by a best fit to the data for monodisperse samples, is almost identical to the value (K = 4.55 -10" s/(mol) ) obtained using Eq. 7.3 with =0.00375. Adapted from Pattamaprom and Larson [19].

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 ... 

To illustrate this, in Figs. 6.17a and b, we compare the predictions of the double reptation model against data for both a monodisperse and a polydisperse polystyrene of nearly the same molecular weight at 150 °C [19]. The sample of Fig. 6.17a is a nearly monodisperse polystyrene, with M = 363,000, and M /M = 1.03. The linear rheology for this sample has been fitted by the double reptation theory in the terminal region, yielding a value of K = 2.275 10 at 150 °C for the double reptation constant. Using this value of K, we make the a priori predictions shown in Fig. 6.17b (at 150 °C) of the linear moduli of the polydisperse sample = 2.3) with = 357,000, which is very close to the molecular... 

Figure 6.17 (a) Comparison of the predictions of the dual constraint model (solid lines) and the double reptation model (broken lines) to experimental data (symbols) for the storage modulus, G, and the loss modulus, G", for monodisperse linear polystyrene (M = 363,000) at 150 °C. The parameter values are G 5 = 2 -10 Pa.and = 0.05 s,the latter value being obtained as a best fit. From this value of Tg,after multiplying it by the correction factor ofO.375 in footnote (g) of Table 7.1, the value K = 2.275 10" s/(mol) for the double reptation model is obtained from Eq. 7.3 (from Pattamaprom and Larson [19]).(b)The same as (a), except the sample is a polydisperse polystyrene M = 357,000 = 2.3) constructed from 11... 

Wasserman and Graessley [30] showed that if two high-molecular-weight components with molecular weights of around 3.8 and 4.5 million are added at a total concentration of only 1% to the mixture of 11 components described above, then the terminal region of the G curve is measurably altered compare G for the 11-component mixture Ml with the 13-component mixture M2 in Fig. 6.18. Very importantly, the G curves predicted for both Ml and M2 by the dual constraint theory match the experimental data very well. Similar sensitivity to high molecular weight components is shown by the double reptation model [27]. 

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... 

Wasserman [29] also developed a method for calculating MWD that is based on the double reptation model. However, whereas Mead [23] chose to use the integral form of the equation and employ various mathematical transforms to manipulate it, Wasserman used discrete variables and numerical techniques. Thus, he writes the double reptation relationship as ... 

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. 

Another method that makes use of the double reptation model and the assumption of a Gex MWD is that of Guzman et al. [37]. They also accoimt for the effect of unentangled chains. Their method avoids the use of a regularization technique to infer G t) from dynamic data, and their analysis leads to an estimate of the reliability of the prediction. 

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 ° °... 

Apart from the above approach relying on the double-reptation model, Wood-Adams and Dealy proposed a simpler method of evaluating MWD that is based on an empirical mixing mle for the zero-shear viscosity, = S, Wi til/f, where t]o,i is the zero-shear viscosity of component i in its bidk state and a = 3.4 (cf. eqn [59c]). (Since M,-,... 

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