Branch points per molecule

As we conjectured in the introduction, the fundamental role of topology in this approach to entangled polymer dynamics would indicate that changes to the topology of the molecules themselves would radically affect the dynamic response of the melts. In fact rheological data on monodisperse star-branched polymers, in which a number of anionically-polymerised arms are coupled by a multifunctional core molecule, pre-dated the first application of tube theory in the presence of branching [22]. Just the addition of one branch point per molecule has a remarkable effect, as may be seen by comparing the dissipative moduli of comparable linear and star polymer melts in Fig. 5. 

Fig. 18. Reciprocal particle-scattering factors of star-molecules with polydisperse rays, where f denotes the number of rays per molecule. The same functions are obtained also for the ABC-type polycondensates, where nb denotes the number of branching points per molecule. The case f = 1 or nb = 0 is identical to linear chains obeying the most probable lengths distribution. It also represents the scattering behaviour of randomly branched f-functional polycondensates 1...

The extent of the latter reaction is a function not only of the nature and concentration of the catalyst used, but also of the temperature at which the melt (or solution) polyaddition reaction is carried out. Extensive branching can lead even at low conversions of the phenolic OH group to gelation. It is, therefore, of considerable practical importance to determine the extent of branching in such systems. Earlier NMR studies (l)on conventional epoxide resins (M = 1500-4000) has shown thaT the branching points per molecule varies between 0.09 to 0.6. 

NBP number branching points per molecule NITSE nonintermeshing twin-screw... 

Table 1. Determination of polymer size and number of branching points per molecule...

Amylose has a degree of polymerization (DP) between 100 and 10,000. One end of each macromolecule is reducing and the other is nonreducing. It holds around 2 to 8 branch points per molecule. DP of these branch chains ranges from 4 to 100 DP [45]. 

For randomly-branched molecules of uniform molecular weight, each with n branch points having a functionality off, Zimm and Stockmayer [8] made several simplifying assumptions to arrive at the expressions for gf(n) shown below as Eqs. 2.18 to 2.20 for one, two and three branch points per molecule (n) respectively. 

For heterogeneous polymers with larger, uniform numbers of randomly distributed branch points per molecule with a random distribution of branch lengths, they derived Eqs. 221 and 2.22 for tri- and tetra-functional branch points, respectively ... 

In these equations, is the weight average number of branch points per molecule. Lecacheux... 

The level of branching is often expressed simply in terms of A, the average number of branch points per thousand carbon atoms, and this is related to j8, the average number of branch points per molecule through Eq. 3.3 for the case of polyethylene. 

However, this single parameter contains no information about the distributions of branch points per molecule and the molecular weights of the various types of segment. For example, we note that many combinations of and / can give the same value of A. To determine the detailed structure, it is necessary to make a more detailed analysis of the reaction. Because this reaction system has been analyzed in great detail, it will be of interest to summarize the models used. Soares and Hamielec [87] developed a kinetic model based on the following reactions ... 

Figure 3.6 Average number of branch points per molecule p versus M for portions of simulated elute, having weight average molecular weights of M for the polyethylenes whose segment compositions are shown in Fig. 3.4.These were computed using the method of Costeux etal. [94] and the Zimm-Stockmeyer relationships (calculations carried out by S. Costeux).

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