Randomly branched molecules

From Eq. (4) it can be seen that as the conversion is driven towards completion, i.e.,p is close to unity, the molecular weight distribution increases dramatically. Theoretically, polycondensation of A2B monomers should form an infinite molecule at extremely high conversions, though in practice this is seldom observed. Flory concluded that condensation of A B monomers would give randomly branched molecules without network formation [1]. However, the occurrence of unwanted reactions (an A group reacts with an A group, for instance) will eventually give rise to an infinite network. Therefore, side-reactions have to... 

Hyperscaling relates the fractal dimension of randomly branched molecules in the gelation reaction with the Fisher exponent and the space dimension (see Table 6.4) ... 

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. 

The molecules used in the study described in Fig. 2.15 were model compounds characterized by a high degree of uniformity. When branching is encountered, it is generally in a far less uniform way. As a matter of fact, traces of impurities or random chain transfer during polymer preparation may result in a small amount of unsuspected branching in samples of ostensibly linear molecules. Such adventitious branched molecules can have an effect on viscosity which far exceeds their numerical abundance. It is quite possible that anomalous experimental results may be due to such effects. 

All polymer molecules have unique features of one sort or another at the level of individual repeat units. Occasional head-to-head or tail-to-tail orientations, random branching, and the distinctiveness of chain ends are all examples of such details. In this chapter we shall focus attention on two other situations which introduce variation in structure into polymers at the level of the repeat unit the presence of two different monomers or the regulation of configuration of successive repeat units. In the former case copolymers are produced, and in the latter polymers with differences in tacticity. Although the products are quite different materials, their microstructure can be discussed in very similar terms. Hence it is convenient to discuss the two topics in the same chapter. 

Low density polyethylene produced by a high-pressure high-temperature reaction process. This creates a molecule with a high degree of random branching. Thus crystallinity and hence density are low. 

Theoretically, if each molecule in a polymer sample were to be linked to two of its neighbors, a single highly branched molecule would form that would encompass the whole sample. In practice, due to the statistical distribution of chain lengths and the random incorporation of crosslinks, the situation is far more complex. 

Figure 11.7 illustrates the three different subtypes of dendritically branched molecules that have been identified within the major architectural class of dendritic polymers. Random hyperbranched polymers, not only exhibit polydispersity in molecular mass between individual molecules, it should also be noted... 

This approximation is equivalent to assuming that the differences in internal densities and, consequently, in solvent draining, between a branched chain and the homologous linear chain, when included in their corresponding mean sizes, can describe both the friction coefficient and the viscosity. Besides these theoretical considerations, an empirical correlation in terms of a log-log fit of h vs f was employed by Roovers et al. [51]. Kurata and Fukatsu [48] and Ptitsyn [82] performed a more general Kirwood evaluation of the friction coefficient for different types of ideal branched molecules (uniform and randomly distributed stars, combs and random-branched structures). Their results for different structures are included within the limits l[Pg.60]

The other extreme to star molecules are the randomly branched macromolecules. Here the branching process has an immense influence on the shape of the distribution. It was first derived by Stockmayer [ 14] and later reproduced byFlo-ry [13]. The exact and asymptotic distributions are given by the following two equations ... 

Fig. 23. The intrinsic viscosity of several end-linked PS star molecules as a function ofM [95]. In the limit of low and high molar masses asymptotic power law behavior may be derived. That at low molar masses is widely controlled by the presence of non-reacted star molecules, that at high molar masses is expected from theory for randomly branched macromolecules. The exponents of the two asymptotic lines are a =0.49 0.08 for M <0.8x10 g/mol and a =0.18 0.05 for M >2.0xl0 g/mol. Reprinted with permission from [95]. Copyright [1997] American Society...

Fig. 28. The ratio A2M [rj] at large for star molecules (symbols) and randomly branched structures [25,26,108,130,131]. The shaded area indicates the range of the experimental findings with randomly and hyperbranched samples [144]. The line was drawn to guide the eye...

Randomly Branched Polystyrene. Branched molecules in solution are more compact than linear molecules and therefore the overall size of a branched polymer molecule in solution is smaller than the... 

We have used the uncharged polysaccharide dextran as a model describing the behaviour of water-soluble polymers. The dextrans used in this study have about 95 % oc-(l - 6) linkages within the main chain and side chains the 5 % non-a-(l -> 6) linkages are starting points of branched chains of which most are only stubs of about two glucose units 9). Therefore, while there is some branching in dextran, albeit low, its solution behaviour is that of a linear, random-coil molecule l0,ll). 

While these methods result in random branching with very little control over placement or extent of branching in any particular molecule, the products can be quite useful commercially. 

If interactions between parts of the molecule separated by many links (the excluded volume effect ) is absent, so that the chains obey random-flight statistics, takes its unperturbed value, (s ). Theoretical calculations of the dimensions of branched molecules usually assume random flight chains, and values of the mean-square radius so obtained are estimates of . 

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