Randomly-branched polymers

The zero shear viscosities of these randomly branched polystyrenes were measured and compared with those of linear polystyrenes and it was found that t]0 for all of the branched polymers were far lower than that of linear homologues of the same overall molecular weight. In addition, a scaling of fJo was observed for the first two generations of each branched series of 

For the subsequent generation of arborescent graft polystyrenes, a dramatic increase in rj0 was observed by Hempenius et al. [43] for each of the three series included in their study. However, despite this increase in viscosity, the rj0 for each of these is still lower than that of the linear homologue polystyrenes of the same overall molecular weight. This jump in viscosity is due to an increase in branch density which in turn results in increase in chain extension similar to that observed by Roovers [31] for highly branched star polymers. 

A similar reduction in viscosity to that found by Masuda and co-workers 

Random branching always leads to a broad distribution of structures, making it difficult to distinguish between the effects of branching and polydispersity. In fact, Wood-Adams and Dealy [109] have shovm how it is possible, in principle, to prescribe the molecular weight distribution of a linear polymer that would have the same complex viscosity as any given branched polymer. 

In order to accoimt for the fact that the molecular size of branched polymers is affected by both the molecular weight and the branching structure, Mendelson etal. [114] multiplied by the branching factor g, the mean square radius of gyration ratio defined by Eq. 2.16, to 

Note that this describes a line that is different from that for linear polyethylene, ie., if gis set equal to unity, we do not recover the line for HDPE. Thus, LDPE is a distinctly different material than HDPE, and one cannot think of decreasing the level of branching in LDPE so that it approaches HDPE in a continuous manner. 

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Another definition, taking into account polymerization conversion, has been more recently proposed.192 Perfect dendrimers present only terminal- and dendritic-type units and therefore have DB = 1, while linear polymers have DB = 0. Linear units do not contribute to branching and can be considered as structural defects present in hyperbranched polymers but not in dendrimers. For most hyperbranched polymers, nuclear magnetic resonance (NMR) spectroscopy determinations lead to DB values close to 0.5, that is, close to the theoretical value for randomly branched polymers. Slow monomer addition193 194 or polycondensations with nonequal reactivity of functional groups195 have been reported to yield polymers with higher DBs (0.6-0.66 range). 

In the 1940s and 1950s, random branching in polymers and its effect on their properties was studied by Stockmayer, Flory, Zimm and many others. Their work remains a milestone on the subject to this day. Flory dedicated several chapters of his Principles of Polymer Chemistry to non-linear polymers. Especially important at that time was the view that randomly branched polymers are intermediates to polymeric networks. Further developments in randomly branched polymers came from the introduction of percolation theory. The modern aspects of this topic are elaborated here in the chapter by W. Burchard. 

At low monomer conversion, the polymerization leads to fairly small molecules, and the branching process can proceed largely unimpeded by the finite volume. Thus, the common behavior of randomly branched polymers is observed. At larger conversion of monomer, the polymer has grown in size, such that the largest species have already reached dimensions of the latex sphere. On further branching, the molecular dimensions... 

A simple algorithm (Kuchanov et al., 1988) enables one to determine the probability of any fragments of macromolecules of the Gordonian polymers. Their comparison with the NMR spectroscopy data permits estimating the adequacy of the chosen kinetic model of the process of synthesis of a particular polymer specimen. These probabilities also enter in the expressions for the glass transition temperature and some structure-additive properties of randomly branched polymers (Chompff, 1971). 

Another important feature controlling the properties of polymeric systems is polymer architecture. Types of polymer architectures include linear, ring, star-branched, H-branched, comb, ladder, dendrimer, or randomly branched as sketched in Fig. 1.5. Random branching that leads to structures like Fig. 1.5(h) has particular industrial importance, for example in bottles and film for packaging. A high degree of crosslinking can lead to a macroscopic molecule, called a polymer network, sketched in Fig. 1.6. Randomly branched polymers and th formation of network polymers will be discussed in Chapter 6. The properties of networks that make them useful as soft solids (erasers, tires) will be discussed in Chapter 7. 

An ideal randomly branched polymer is a fractal object with fractal dimension V — 4. In Chapter 6, we will learn how this polymer can fit into three-dimensional space. What is the ratio of molar masses M1/M2 of two ideal randomly branched polymers if the ratio of their sizes is R1/R2 = 3 ... 

The Kramers theorem effectively cuts a randomly branched polymer with N monomers into two parts, with N andN- Ni monomers. 

Substituting this average [Eq. (2.66)] into the Kramers theorem [Eq. (2.65)] recovers the classical result for the radius of gyration of an ideal linear chain [Eq. (2.54)]. In Section 6.4.6, we apply the Kramers theorem [Eq. (2.65)] to ideal randomly branched polymers. In this case the average is not only over different ways of dividing a molecule into two parts, but also over different branched molecules with the same degree of polymerization N. 

Consider a randomly branched polymer in a dilute solution. Let us assume that the radius of gyration for this polymer in an ideal state (in the absence of... 

Use a Flory theory to determine the size R of this randomly branched polymer in a good solvent with excluded volume y. What is the size of a randomly branched polymer with N— 1000, b — 3A, v = 21.6A Compare this size to the size of a linear chain with the same degree of polymerization in the same good solvent and in -solvent. 

Consider a randomly branched polymer with N monomers of length b. The polymer is restricted to the air-water interface and thus assumes a two-... 

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 ... 

Hint What would be the iV-dependence of density if monodisperse randomly branched polymers overlapped in the melt ... 

The regular lattice constructed in this way is called a Bethe lattice (see Fig. 6.13). The mean-field model of gelation corresponds to percolation on a Bethe lattice (see Section 6.4). The infinite Bethe lattice does not fit into the space of any finite dimension. Construction of progressively larger randomly branched polymers on such a lattice would eventually lead to a congestion crisis in three-dimensional space similar to the one encountered here for dendrimers. 

This number density distribution function is more convenient than the previously used number fraction, when dealing with systems like A/ condensation that can form gels. The reason is that number fraction applies o the randomly branched polymers present butnot to the gel, which makes this quantity awkwardly normalized beyond the gel point. The number density n(p, N) avoids this complication since the total number of monomers is independent of extent of reaction. 

Here we calculate the size of ideal randomly branched polymers, ignoring excluded volume interactions and allowing each molgcule to achieve the state of maximum entropy (recall the discussion of ideal chains in Chapter 2). Since branched molecules have many ends, the mean-square end-to-end distance used to characterize the size of linear chains is not appropriate for them. The simplest quantity describing the size of branched molecules is their mean-square radius of gyration j g [see Eq. (2.44) fonthe definition]. 

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