Monodisperse Melts

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. 

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. 

The simple replacement of P(t) by P as shown in Eq. 6.35 (which does not account for primitive path fluctuations) does not have much effect on the relaxation modulus of monodisperse linear chains, as mentioned above. This is because the relaxation spectrum for reptation is 

---

Monodisperse melts appear to exhibit a plateau region in the stress vs shear rate flow curve [51,62,65]. The capillary flow behavior actually closely resembles the oscillatory shear behavior in the sense that the flow curve essentially overlaps on the absolute value of complex modulus G vs the oscillation frequency (0 [62]. Thus.it appears that the transition-like capillary flow behavior of highly entangled monodisperse melts reflects constitutive bulk properties of the melts and is not interfacial in origin. It remains to be explored whether this plateau indeed manifests a real constitutive instability, i.e., whether it is double-valued. 

The storage and loss moduli, G and G", are obtained from the relaxation spectrum in the usual way—that is, using G = Gi[co rl/(l + co zl)] G — G,[mT /(l -P The longest relaxation mode of the relaxation modulus in Eq. (3-67) is the dominant one it accounts for 96% of the zero-shear viscosity. Thus, the reptation model predicts that for a nearly monodisperse melt, the relaxation spectrum is dominated by a single relaxation time, T = Ta. This is in reasonable accord with experimental data at low and moderate frequencies (see the dashed line in Fig. 3-29). As the frequency increases, however, there... 

The uniaxial extensional viscosity rj(s) and the viscometric functions rj(y) and ki(y), predicted by the Doi-Edwards model for monodisperse melts, are shown in Fig. 3-32. The Doi-Edwards model predicts extreme thinning in these functions the high-shear-rate asymptotes scale as 17 oc oc y , and4 i oc The second normal... 

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... 

An important consideration in the existence of a spinodal is the prescribed experimental conditions. In a monodisperse melt, liquid liquid coexistence can only occur along a line in the pressure-temperature/>—T plane. Hence, liquid liquid phase separation under isobaric conditions can only be transient, before the entire phase reverts to the dense liquid. On the other hand, an isochoric quench would be expected to yield true spinodal-like behaviour. The true system is probably something between the two extremes, with volume leaving the system on some timescale. Based on estimates of thermal diffusivity in melts, the time to shrink is of order 10 s (based on a 1 m sample thickness). If... 

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... 

Consider a monodisperse melt of randomly branched polymers with N Kuhn monomers of length b. Randomly branched polymers in an ideal state (in the absence of excluded volume interactions) have fractal dimension D = 4. Do these randomly branched polymers overlap in a three-dimensional monodisperse melt ... 

Consider a steady shear flow (with shear rate 7) of a monodisperse melt of unentangled N-mers with monomeric friction coefficient C ... 

The results of models that include tube length fluctuation modes [Fig. 9.23(b)] are in much better agreement with the experimentally measured loss modulus G" (ic) of monodisperse melts than the prediction of the Doi Edwards reptation model [Eq. (9.83)]. Tube length fluctuation corrections predict that the loss peak broadens with decreasing molar mass because the fraction of the stress released by fluctuations is larger for shorter chains. 

Consider an entangled monodisperse melt of H-polymers with Abb Kuhn monomers in the central backbone and monomers in each of the four arms, with Ng monomers between entanglements. 

The conformation of clusters under these conditions was determined using the approach proposed [80] for studying the conformation of a branched polymer in a monodisperse melt, i.e., in a solution of coils with identical sizes. In the case of minimisation of the free energy with respect to R ... 

And at last, for the collapsed state the model of branched polymer in monodisperse melt was offered, within the framework of which the following identity was obtained [6] ... 

M/Me- Thus, the line shape of the relaxation modulus G t) changes with concentration. As showm below, the viscoelastic behavior of the entangled component in the concentrated blend solution is well described by Eq. (9.19) with the entanglement molecular weight Me replaced by M, which is given by Eq. (11.4). This means that G t) is universal among monodisperse melt systems and blend solution systems, as all the G t) curves are described by... 

At this stage, however, the reader has to be warned that the fine structure of the behaviour of real polymers is by no means resolved. The contents of this book will produce answers to many questions, and these answers will have qualitatively the right forms. But they may not be correct in detail. For example, the viscosity of a monodisperse melt oi flexible polymers is shown to be proportional to tP whereas experimentally it is known to be Possible explanations of the discrepancy are discussed, but the problem is not entirely resolved. Thus the authors believe they are surveying real progress, but by no means the whole... 

To understand the dynamics of one chain in a melt, it is convenient to start from a slightly different problem. We consider one test chain of Ni monomers, embedded in a monodisperse melt of the same chemical species, with a number N of monomers per chain. We consider three types of motion for the test chain reptation, tube renewal, and Stokes-Einstein friction. We first describe tube renewal and show that this is probably negligible for most practical purposes. Then we discuss competition between reptation and Stokes-Einstein friction. 

The plots in Figure 3 show the adsorbed amount 0 versus H (the distance between the surfaces) as calculated through (28). The curves are for various values of P, the length of the nonfunctionalized chains in a monodisperse melt. The length of the function zed chains is given by N = 75 and the volume fraction of these chains in... 

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...

As the arm length increases above M q, we expect the onset of entanglement to cause marked deviations from this relationship. However, for star polymers it is observed that continues to increase linearly with M, in accord with the Rouse-Bueche model for Hnear, unentangled polymers. This is in contrast to the behavior of entangled, linear, monodisperse melts, for which 7s° is independent of M at large M as shown by Eq. 5.10. Figure 5.21 shows data of Graessley and Roovers for four and six arm polystyrenes [90]. The horizontal line is based on... 

In Eq. 6.43, is the normalized relaxation modulus for a monodisperse melt composed... 

As mentioned in Section 6.4.4.1 the double reptation theory contains no explicit treatment of primitive path fluctuations, and if the theoretical expression Eq. 6.28 is used for the reptation time, it will predict that a monodisperse melt will have a longest relaxation time proportional to the third power of molecular weight, in disagreement with the observed 3.4 power-law dependence. A simple way of dealing with this is to use the empirical formula, (Eq. 6.40), for the longest relaxation time. More sophisticated ways are available for dealing with this limitation [29], but here we confine ourselves to this simple fix, which is adequate for many commercial polymers with broad (but not too broad) molecular weight distributions. 

---