Critical molar mass

In most cases polymer solutions are not ideally dilute. In fact they exhibit pronounced intermolecular interactions. First approaches dealing with this phenomenon date back to Bueche [35]. Proceeding from the fundamental work of Debye [36] he was able to show that below a critical molar mass Mw the zero-shear viscosity is directly proportional to Mw whereas above this critical value r 0 is found to be proportional to (Mw3,4) [37,38]. This enhanced drag has been attributed to intermolecular couplings. Ferry and co-workers [39] reported that the dynamic behaviour of polymeric liquids is strongly influenced by coupling points. 

For semi-dilute solutions, two regimes with different slopes are similarly obtained the powers of M, however, can be lower than 1.0 and 3.4. Furthermore, the transition region from the lower to the higher slope is broadened. The critical molar mass, Mc, for polymer solutions is found to be dependent on concentration (decreasing as c increases), although in some cases the variation appears to be very small [60,63]. 

Let us consider a homopolymer of initial molar mass Mn0 (molar mass of the monomer unit m0) and a critical molar mass (entanglement threshold... 

Crystallisation of most polymers is accompanied by the separation of different molecular species, a process referred to as molecular fractionation. Bank and Krimm [147] provided the first direct evidence of molecular fractionation in polyethylene. The first extensive study performed by Wunderlich and Mehta [148] indicated that, at each crystallisation temperature, there exists a critical molar mass (MCIjt) such that the molecules of molar mass greater than Mcrjt, are able to crystallise at this temperature, whereas... 

Fig. 26 Critical molar mass of melt-crystallised linear polyethylene as a function of crystallisation temperature. Filled circles data for a broad molar mass sample with Mn=8500 g mor1, Mw=153,000 g mor1 of Mehta and Wunderlich [149], Open circles data for a sample with Mn=12,900 g mor1, Mw=108,000 g mol-1 from Gedde et al. [152]. From [91] with permission from Kluwer, Doordrecht, Netherlands...

The critical molar mass, above which segregation occurs, Mcr, is 2.5 to 5 times greater then for equivalent blends. This means a better miscibility of block copolymers, as a result of the entropy restrictions in the domain structure. 

Thus, if scheme A is applicable, the embrittlement critical molar mass would be 70 30 kg.mol, which would lead according to Fig. 4 to an embrittlement lifetime of about 350 hours versus 300 hours as the experimental value for the PE under study. This difference would be acceptable in practice, but two questions remain ... 

Within this framework, one can imagine a two dimensional ductile-brittle transition criterion which would be composed of two perpendicular boundaries in the graph la-Mw (Fig. 7). The vertical boundary would correspond to a critical molar mass M c- In polymers of low crystallinity or fully amorphous, M c would be sharply linked to the entanglement molar mass M c 5Me. In polymers of medium to high crystallinity, M c would be the molar mass below which it is impossible to have values higher than lac- The horizontal boundary would correspond to U = lac-... 

It has been tried to put the result of Kennedy et al (//) in a la-Mw graph (Fig. 7). Two samples families are distinguished quenched samples and slowly crystallized samples. In both families, the ductile-brittle transition occurs at lac = 6 nm. The corresponding critical molar mass defined in scheme A is higher for slowly crystallized samples (M c 100 kg.mof ) than for quenched ones (M c 60 kg.mof ). This result can be considered as a proof that scheme B is more pertinent than scheme A. 

In these graphs (Figs. 7 and 8), the quasi vertical part of the curve would correspond to annealing, the rest would be due to chemicrystallization. In this case, all the trajectories would converge towards an embrittlement point not very far from the intersection of both boundaries Uc and M c and the concept of a critical molar mass (scheme A) would apparently be valid. 

It is well known that below a critical molar mass, the properties of polymers are very sensitive to changes in molar mass. Very little, however, is known about the molar mass dependence of the properties of LCPs. This is partly due to the difficulties in controlling molar mass in the reactions usually used to synthesize LCPs and partly due to the lack of good molar mass data on most LCPs, which are often soluble only in foirly harsh solvents. The chapter by Kim and Blumstein describes the preparation of series of main-chain LCPs varying in molar mass and the effect of molar mass on the thermal properties. 

Equation (8.136) is tested in Fig. 8.17 (solid curves) and found to describe the molar mass dependence of constant friction coefficient viscosity data for all three of these linear polymers. The critical molar mass Me for entanglement effects in viscosity is always a factor of 2-4 larger than the entanglement molar mass Mg. that was defined in Eq. (7.47). 

Molar mass dependence of viscosity for polymer melts reduced by their critical molar mass. Open circles are polyisobutylene with M = MnnOgmor, from T. G. Fox... 

Extension of these studies to the polymer in solution and the measurement of the attendant decrease in molar mass showed that it was the chain length which was important, not the solid-state properties. In particular the solid-state studies showed that there was a critical molar mass before mechanochemical effects could be observed (for polyethylene (PE), a degree of polymerization of about 100). In solution a similar effect was observed since the degradation of poly(methyl methacrylate) (PMMA) could not continue when the chain length reached about 4000 repeat units (Sohma, 1989a, 1989b). This was shown by a slow decrease in molar mass to the asymptotic value as shown in Figure 1 -39. 

Orrah DJ, Semiyen JA, Rossmurphy SB (1988) Studies of cyclic and linear poly(dimethyl-siloxanes). 27. Bulk viscosities above the critical molar mass for entanglement. Polymer 29 1452-1454... 

A critical molar mass is analogously found for melts. Below the critical molar mass, the melt viscosity at zero shear gradient is directly proportional to the molar mass ... 

In contrast, the viscosity above the critical molar mass is given by (see Fig. 7-8) ... 

The tensile strength of polymers at first increases strongly, and then weakly with increasing molar mass (Figure 11-23). Below a certain critical molar mass, the tensile strength is practically zero. The transition from sharp to weak molar mass dependence is probably due to entanglements. At very... 

The critical molar mass (Me) parameter [18,19] denotes the transition in the melt viscosity/molar mass relation as the exponents change from 1 to 3.4. Table 25.5 presents the polymers for which the Me values are known while Table 25.6 lists the polymers for which, seemingly, Mc/Mg is one. The ratio of Me/Mg was long taken to be 2 [18] and thus to be species independent. However, a recent empirical compilation [19] has shown that the ratio is p dependent and varies from 3.5 (PE p = 1.69) to 1.4 (a-PCHE p = 5.59). Based upon the data of Table 25.5 this ratio is empirically expressed as ... 

Eayolle B, Audouin L, Verdu J. A critical molar mass separating the ductile and brittle regimes as revealed by thermal oxidation in polypropylene. Polymer 2004 45 4323. 

Fig. 3.7. Equilibrium critical molar mass of PE melt in the crystallization...

Orrah, D. J. Semiyen, J. A. Ross-Murphy, S. B., Studies of Cyclic and Linear Poly(dimethylsiloxanes) 27. Bulk Viscosities Above the Critical Molar Mass for Entanglement. Polymer 1988, 29,1452-1454. 

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