Polydisperse Melts

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  

If we were to naively apply this theory to a polydisperse polymer, we would simply sum the weighted relaxation functions for the various molecular weights present and write  

However, this fails to account for the fact that the shorter molecules reptate faster than the longer ones, so the tube cannot be assumed to be fixed during the entire period of relaxation of a long molecule. This is where the double reptation theory can be employed to great 

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 ( o) curves rather well for such melts, better than it does for monodisperse or 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  

is the normalized relaxation modulus for a monodisperse melt composed 

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The transition from the Newtonian plateau to the Power Law region is sharp for monodispersed polymer melts and broad for polydispersed melts (see Fig. 3.5). 

The Doi-Edwards equation predicts an overshoot in shear stress as a function of time after inception of steady shearing, but no overshoot in the first normal stress difference (Doi and Edwards 1978a). Typical overshoots in these quantities for a polydisperse melt are shown in Fig. 1-10. For monodisperse melts, the Doi-Edwards model predicts that the shear-stress maximum should occur at a shear strain yt = Yp, of about 2, roughly independently of... 

On the other hand, in a polydisperse melt a spinodal instability would be expected eventually to reach biphasic equilibrum, in which case standard Cahn-Hilliard kinetics would be expected. Phase separation could be due to several effects, including chemical polydispersity, coupling of mass density to the density of polymer ends, and conformation-density-orientation coupling, which has a natural prediliction for inducing phase separation (longer chains are, per unit monomer, more nematic than their short chain cousins ). There are evidently several effects to sort out, and close collaboration between theory and experiment will help to attack the various mechanisms. 

If an A chain is smaller than the thermal blob (TVa < A g), its conformation is almost ideal. In a monodisperse melt with Aa = Ab, or in a weakly polydisperse melt, all chains have ideal statistics. On the other hand, strongly asymmetric binary blends of dilute long chains in a melt of short chains with have swollen long chains. The size of these swollen... 

Demonstrate that the excluded volume in a polydisperse melt is v = b jN, where N, is the weight-average molar mass of the melt. 

Nanocomposites based on polymer matrices have been studied extensively (Paul and Robeson 2008). In the majority of these studies, the focus has been the enhancements in properties through the addition of relatively small quantities of nanoparticles. In this chapter, we centre our attention on the effects, if any, of the nanoparticles on the behaviour of the polymer matrix and in particular the nucle-ation and growth of crystal phases in the case of crystallizable polymers. We first consider the influence of macroscopic particles on this behaviour by examining the influence of polymer fibres on the matrix behaviour. We then consider the situation in polydisperse melts where extended objects can be formed fi om the high molecular weight fraction in the melt, and then finally we review work reported in this field in the context of nanoscale fillers. 

In general, polymer samples are polydisperse and are expected to exhibit a distribution of diffusion coefficients. Previous studies on polydisperse melts have shown the influence of the molecular weight distribution on the echo attenuation. " > In the model proposed by von Meerwall, Eq. (7) is extended to a multicomponent system considering additivity of the NMR signals of the N components ... 

Figure 5.2 Relaxation moduli of three samples of a linear polymer A) an unentangled molten sample, B) an entangled,monodisperse molten sample,C) an entangled, polydisperse molten sample, and D) acrosslinked sample. At short times,all the samples relax first by a glassy mechanism and then by Rouse relaxation involving only very short segments of the chain (log scales). The unentangled melt then flows in the terminal zone.The entangled, monodisperse melt has a plateau modulus followed by terminal relaxation, while in the polydisperse melt the plateau zone of the longest molecules overlaps with the terminal zones of the shorter molecules.

Figure 5.S Storage moduli of same materials as in Fig. 5.2 A) unentangied poiymer, B) entangied, monodisperse melt, C) entangled, polydisperse melt, D) crossiinked polymer (logarithmic scales).The plateau modulus is G 5,G isthe glassy modulus,and is the equilibrium modulus...

Although reptation and primitive path fluctuations together provide a nearly quantitative prediction of the linear viscoelasticity of monodisperse melts of linear chains, for polydisperse melts it is clear that these are not the only important relaxation mechanisms. To develop quantitative, or even qualitative, theories for polydisperse melts, constraint release must be taken into accoimt. 

A problem with inclusion of the CCR term in the equation for stretch (Eq. 11.34) is that in a polydisperse melt, there are chains/ whose reptation time Tj j might be faster than the retraction time Tsi of some other chains i. In this case, the constraint-release term in Eq. 11.34 would cause chains i to retract by constraint release at a rate faster than their own Rouse time, which is the relaxation time those chains would have in the absence of entanglements. Since the release of entanglements cannot allow relaxation to be faster than it would be without entanglements, when this occurs one should replace in Eq. 11.34 by Ts j, or implement some other fix . 

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