Bidisperse Melts

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  

This relaxation modulus is different from the one obtained without constraint release, Eq. 6.37, in three ways. First, the factor of two appears in the exponents of the first and second terms of Eq.6. 38, just as it did for monodisperse polymers in Eq. 6.36. Second, the weighting of the contributions to the modulus from the long and short chains is proportional to the concen- 

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 

While the basic double reptation theory, with a single-exponential relaxation kernel function for each molecular weight component, does not predict very accurately the shapes of linear viscoelastic moduli, des Cloizeaux [28] has suggested a more sophisticated kernel function that provides much more accurate predictions for nearly monodisperse or bidisperse polymers. Still other kernel functions have also been suggested, which, when combined with the double reptation ansatz, have been recently shown to be quite successful in matching linear viscoelastic data, especially when the double reptation exponent is made slightly higher (around 2.25) than its canonical value of two [28]. In addition to the canonical value of two, a theoretical value for this exponent of around 7/3 could also be rationalized see Section 9.32. 

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Fig. 5.18. Self-diffusion constants for a bidisperse (i.e. two different chain lengths) PE melt with Mn = 20 coarse-grained monomers. Open triangles are for d = 2, filled diamonds for d = 4, open squares for d = 6 and filled circles for d = 8. There are always two symbols of the same kind shown in the figure, since the bidisperse melt contains two species of different chain length. The numbers quoted in the figure correspond to these chain lengths for a given polydisparsity d. For instance, d = 8 corresponds to Mi = 12 and M2 = 52. From [184].

These results on bidisperse melts are clearly rather preliminary but, nevertheless, encouraging, since they show that the problem is now within reach of computer simulation analysis. Again this is a line of research which will require more efforts in the future. 

J. A. Kornfield, G. G. Fuller, and D. S. Pearson, Third normal stress difference and component relaxation spectra for bidisperse melts under oscillatory shear, Macromolecules, 24, 5429 (1991). 

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"(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 ... 

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. 

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 ... 

Commercial melts are neither monodisperse nor bidisperse, but usually have a broad, continuous, distribution of relaxation times with polydispersity ratios, M /M, greater than or equal to 2.0. We will start our discussion of the effect of polydispersity by replacing the exponential in Eq. 6.36 with a dimensionless relaxation function P t,M) to obtain a general expression for the relaxation modulus (reptation only) of a monodisperse polymer ... 

Figure 7.9 Storage moduli of bidisperse linear polybutadiene melts whose components have molecular weights of 36800 and 168000 (Gr = MiMg/M =0.008), at long-chain volume fractions of, from right to left, 0.0,0.1,0.3,0.5,0.7,and 1.0 at 7 = 25 °C.The symbols are data of Struglinski and Graessley [15],andthe linesare predictions of the Milner-McLeish [19] theory as extended to binary blends by Park and Larson [17] with the long-chain reptation time given by td,L = 3 r zl (1 - for reptation in the undiluted tube.The factor 4i,L tbe fraction of the...

Figure 7.10 Storage moduli of bidisperse linear polybutadiene melts whose components have molecular weights of 2 10" and 5.5 10 (Gr = M M /M =0.167) at long-chain volume fractions,from right to left, of 0.0,0.01,0.05,0.1,0.2, and 1.0 at 7= 25 °C.The dotted linesare model predictions with the long-chainreptationtakentobeintheundilutedtube,T,jL = 3 TgZ f (1 and the solid lines are the model predictions with the long-chain reptation taken to be in the diluted tube, = 3 Tg Z (1 - (/y.The parameter values are the same as in Fig. 7.9. From Park and Larson [17].

Thus nonlinear constitutive equations that describe reptation, time-dependerrt chairr retractiorr, and constraint release by reptation and chain retraction, are in promisingly good agreemerrt with nonlinear data in shear flows of monodisperse or bidisperse entangled polymer solutiorrs. To be of use in practical polymer processing applications, however, such equations must be accurate for melts of polydisperse polymers. In steady shear flows of polymer melts, the steady-state viscosity as a function of shear rate T] y) is frequently found to be numerically roughly equal to the dynamic shear viscosity as a function of frequency tf (o) = [ tj io)) +, ... 

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