Infinite branched polymer

These three assumptions were used to calculate the expected neutron scattering from trifunctional gels labeled around the crosslinks by assuming that the gels were made up of trifunctionally connected triblock copolymers. For comparison, the expected scattering from the following infinite branched polymers, all made up of identical triblock copolymers, was also calculated. Each triblock copolymer was assumed to have deuterated ends of equal length. 

Results of the calculations of the structure factors of the various infinite branched polymers and gels are given below, together with the structure factor of the triblock copolymer given in the same terms. In the equations below, A = (Q Rg) where Rg Is the root-raean-square radius of gyration of the triblock copolymer, Xa = exp (-Au/2) except in the case of infinite branched polymer 1 where Xq = exp (-Au/3), xb = exp (-Au) except in the case of infinite branched polymer 1 where xb = exp (-At /2), X2a — b fXaXb refers to the product of Xa and and so on. 

Figure 1 shows the right-hand side of Eq. 1 versus (Q Rg) i where Rg is the root-mean-square radius of gyration of the triblock copolymer subunit, for a 10% deuterium-labeled triblock copolymer, and of both branched polymer 2 and branched polymer 3. The branched polymers in Fig. 1 are made up of the triblock copolymer also shown in Fig. 1. As it happens, all the gel approximations give virtually the same curve that is shown for branched polymer 3 in Fig. 1. The infinite branched polymers shown in Fig, 1 have a more pronounced maximum in their scattering intensity curves than their constituent triblock copolymer, but the position of the maximum with respect to Q is hardly shifted at all. Figure 2 shows the same comparisons as Fig. 1, but now the triblock copolymers are 50% deuterated. 

The molecular distributions for polymers formed by condensations involving polyfunctional units of the type R—A/ resemble those for the branched polymers mentioned above, except for the important modification introduced by the incidence of gelation. The generation of an infinite network commences abruptly at the gel point, and the a-mount of this gel component increases progressively with further condensation. Meanwhile, the larger, more complex, species of the sol are selectively combined with the gel fraction, with the result that the sol fraction decreases in average molecular complexity as well as in amount. It is important to observe that the distinction between soluble finite species on the one hand and infinite network on the other invariably is sharp and by no means arbitrary. 

So far, we have only considered linear polymerization reactions, as in condensation reactions involving bifunctional monomers (A-B or A-A/B-B pairs). Obviously, incorporating multifunctional monomers into this type of polymerization results in the synthesis of highly branched polymers and can lead to the formation of very large interconnected molecules. These can have macroscopic dimensions and are considered to be infinite networks . The formation of such gels does not follow automatically from the incorporation of multifunctional monomers into the reaction pot, however. So, keeping in mind that an A can only react with a B and vice-versa, which of the reactions in Figure 5-11 do you think would lead to network formation ... 

By convention, the term branched implies that the polymer molecules are discrete, which is to say that their sizes can be measured by at least some of the usual analytical methods described in Chapter 3. A network polymer is an interconnected branch polymer. The molecular weight of such polymers is infinite, in the sense that it is too high to be measured by standard techniques. If the average functionality of a mixture of monomers is greater than 2, reaction to sufficiently high conversion yields network structures (p. 174). 

Network polymers can also be made by chemically linking linear or branched polymers. The process whereby such a preformed polymer is converted to a network structure is called cross-linking. Vulcanization is an equivalent term that is used mainly for rubbers. The rubber in a tire is cross-linked to form a network. The molecular weight of the polymer is not really infinite even if all the rubber in the tire is part of a single molecule (this is possible, at least in theory), since the size of the tire is finite. Its molecular weight is infinite, however, on the scale applied in polymer measurements, which require the sample to be soluble in a solvent. 

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. 

Below the gel point (forp < p ) there are only finite-size branched polymers, while above the gel point (for p > p ) there is also at least one infinite polymer (the gel) in addition to many finite-size branched polymers. The... 

A network polymer [Fig. 1.7(d)], on the other hand, can be described as an interconnected branched polymer. For example, a three-dimensional or space network structure will develop, instead of the branched structure (X), if styrene is copolymerized with higher concentrations of divinyl benzene. In a network structure, all polymer chains are linked to form one giant molecule and the molecular weight is infinite in the sense that it is too high to be measured by standard techniques (see Problem 1.5). Because of their network structure such polymers cannot be dissolved in solvents and cannot be melted by heat strong heating only causes decomposition. 

Polymerizations lead to linear macromolecules only if the monomers are strictly bifunctional. Monofunctional molecules were shown in Fig. 3.13 to limit the molecular length of step-growth polymers. Adding monomers with a higher functionality than, f = 2 to bifunctional monomers leads to network formation with stractures as discussed in Sect. 1.2. A sudden gel formation is illustrated in Fig 3.48. Gel formation is marked by the appearance of a macroscopic network. The network has a practically infinite molar mass. In polymerizations from the pure monomer the network is swollen by monomer, oligomers, and linear and branched polymer molecule. In polymerization from solution, the solvent makes up a large number of the small molecules. As the polymerization continues, most or all of the polymerizable material joins the network. A gel can hold a large number of small... 

In general, however, identification of the crystal cell is only part of the problem of characterizing the structure of crystalline polymers. Crystals are never perfect and the units cells do not infinitely duplicate through space even when they are grown very carefully from dilute solution using low molecular mass materials. As with the organic crystals considered in Chapter 3, a variety of defects can be observed and are associated with chain ends, kinks in the chain and jogs (defects where the chains do not lie exactly parallel). The presence of molecular (point) defects in polymer crystals is indicated by an expansion of the unit cell as has been shown by comparison of branched and linear chain polyethylene. The c parameter remains constant, but the a and b directions are expanded for the branched polymer crystals. Both methyl and... 

The randomly branched polymers grow as the total concentration c of Af units increases an infinite cluster (network) is formed above a certain critical concentration c. The statistics of branched polymers is analyzed below for an ideal system of Af units (no interactions apart from bond formation) which... 

Eqrration [86] essentially rmderestimates tbe average dimensions of randomly branched polymers and predicts an infinite increase in tbe intramolecular concentration of tbe monomer rmits as a function of N. Tbe effect of tbe intramolecular short-range (excluded-volume) repulsions on the conformations of the randomly branched polymers can be taken into account within the Flory mean-field approach by balancing the conformational entropy losses in the uniformly swollen randomly branched polymer ... 

Gelation links macromolecular chains to result in the formation of a branched polymer structure with a solubility that depends on the chemical nature of the starting materials (Lan et al., 2015 Wu and Morbidelli, 2014). The mixture containing water and the soluble branched polymer is called a sol. The solubility of the polymer decreases with increasing dimension of the structure. This infinite polymer is called the gel or network, and it is composed of several finite branched polymers (Li et al., 2012 Cravotto and Cintas, 2009). The transition from a system with finite branched polymer to inhnite molecules is called sol—gel transition or gelation, and the critical point where gel first appears is called the gel point. The gelation process is depicted in Fig. 10.6. 

The methods available for the preparation of star-branched polymers are described. After presentation of theoretical predictions of the effect of this type of branching on the properties of polymers, available experimental data are compared and critically discussed. Thermodynamic and visco-elastic properties at infinite dilution are first presented followed by visco-elastic properties in concentrated solution and the melt. FituMy the effect of star-branching on T of low molecular we t polynia s is described. 

Figure 3 shows the right-hand side of Eq. 1 versus (Q Rg) for branched polymer 1 as compared with the triblock copolymer and branched polymer 2. Branched polymer 1 is the only one of the calculated polymers that is not composed strictly of triblock copolymers. Branched polymer 1 may be considered as an infinite polymer made up of tetrablock copolymers or of triblock copolymers in the backbone with diblock copolymers as branches. The maximum in the scattering curve for branched polymer 1 is at larger values of (Q Rgn than that of the other polymers, partly because the Rg of this polymer pertains to the tetrablock copolymer subunit. Branched polymer 1 was included in the calculation because synthetic methods for this polymer are easier to find than for the other branched polymers. 

A conventional branched polymer results when a dimethacrylate is copolymerized in a random fashion with monofunctional monomers. However it is necessary to keep the dimethacrylate/initiator ratio less than 1/1 in order to prevent gelation. When the dimethacrylate/lnitiator ratio equals or exceeds a 1/1 ratio, there should be an infinite number of branches with each chain connected to at least two other chains, resulting in a gel. Table,1 shows that gels result when a 2/1 dimethacrylate/initiator ratio is used and the dimethacrylate is randomly copolymerized with a monofunctional monomer. This is true whether the polymerization is free radical or GTP. 

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