Branching coefficient

The probability that a chain segment is capped at both ends by a branch unit is described by the branching coefficient a. The branching coefficient is central to the discussion of gelation, since whether gelation occurs or not depends on what happens after capping a section of chain with a potential branch point. 

Since the branching coefficient gives the probability of a chain segment being capped by potential branch points, the above development describes this situation ... 

Our interest from the outset has been in the possibility of crosslinking which accompanies inclusion of multifunctional monomers in a polymerizing system. Note that this does not occur when the groups enclosed in boxes in Table 5.6 react however, any reaction beyond this for the terminal A groups will result in a cascade of branches being formed. Therefore a critical (subscript c) value for the branching coefficient occurs at... 

As an example of the quantitative testing of Eq. (5.47), consider the polymerization of diethylene glycol (BB) with adipic acid (AA) in the presence of 1,2,3-propane tricarboxylic acid (A3). The critical value of the branching coefficient is 0.50 for this system by Eq. (5.46). For an experiment in which r = 0.800 and p = 0.375, p = 0.953 by Eq. (5.47). The critical extent of reaction, determined by titration, in the polymerizing mixture at the point where bubbles fail to rise through it was found experimentally to be 0.9907. Calculating back from Eq. (5.45), the experimental value of p, is consistent with the value =0.578. 

The kinetics and mechanism of vacuum decomposition of AgMn04 at 378—393 K [466] are believed to differ from the behaviour of KMn04 in that the effective chain branching coefficient diminishes with time and this leads (Chap 3, Sect. 3.2) to the modified form of the Prout—Tompkins equation... 

First of all it is necessary to determine the branching coefficient a, w hich is defined as the probability that a given functional group of a branch unit leads via a chain of bifunctional units to another branch unit. In a polymer of the type shown in Fig. 61, a is the probability that an A group selected at random from one of the trifunctional units is connected to a chain the far end of which connects to another trifunctional unit. As will be shown later, both the location of the gel point and the course of the subsequent conversion of sol to gel are directly related to a. 

If rings are formed, the branching coefficient must be redefined. 

As mentioned previously, the behavior of systems containing bifunctional as well as trifunctional reactants is also governed by the equations developed above. The variation of wx for the polymerization of bifunctional monomers, where the branching coefficient a is varied by using appropriate amounts of a trifunctional monomer, is similar to that observed for the polymerization of trifunctional reactants alone. The distribution broadens with increasing extent of reaction. The effect of unequal reactivity of functional groups and intramolecular... 

Such reactions have been used to explain the three limits found in some oxidation reactions, such as those of hydrogen or of carbon monoxide with oxygen, with an "explosion peninsula between the lower and the second limit. However, the phenomenon of the explosion limit itself is not a criterion for a choice between the critical reaction rate of the thermal theory and the critical chain-branching coefficient of the isothermal-chain-reaction theory (See Ref). For exothermic reactions, the temperature rise of the reacting system due to the heat evolved accelerates the reaction rate. In view of the subsequent modification of the Arrhenius factor during the development of the reaction, the evolution of the system is quite similar to that of the branched-chain reactions, even if the system obeys a simple kinetic law. It is necessary in each individual case to determine the reaction mechanism from the whole... 

The branching coefficient, of star polymers is defined as the ratio of the radius of gyration of branched and linear polymers having the same molecular weight [67],i.e.,... 

The understanding of the macromolecular properties of lignins requires information on number- and weight-average molecular weights (Mn, Mw) and their distributions (MWD). These physico-chemical parameters are very useful in the study of the hydrodynamic behavior of macromolecules in solution, as well as of their conformation and size (1). They also help in the determination of some important structural properties such as functionality, average number of multifunctional monomer units per molecule (2, 3), branching coefficients and crosslink density (4,5). 

The gel point is directly connected with the branching coefficient ab, defined as the probability that a given branch (emanating from a branch point) leads to a branch point. 

The branching coefficient can be calculated from the probabilities that the functionalities have reacted these probabilities are the mole fractions of reacted functionalities. For example, when RAf units react with R AS units (A reacting with A), r being the fraction of A groups belonging to RAf andp the probability that the A unit has reacted, the probability that the chain leads via i R Az units to a branch unit is... 

Cool flame behaviour is the result of highly limited degenerate branching, where the branching coefficient, a, is small, e.g. is of the order of 1.1, compared with being of the order of two or three in normal branching. 

The point in the reaction at which gelation occurs has been deduced by Flory. 4 The gel point is developed in terms of the branching coefficient, a, which is the probability that a given functional group on the multifunctional monomers leads, via a chain that can contain any number of bifunctional units, to another multifunctional monomer. The critical value of the branching coefficient, denoted by ac, at which gelation occurs is... 

The large branching coefficient, 7, and the fact that k3 > kx leads to a very sharp transition from a slow preacceleration reaction to a rapid accelerated phase. 

For a system containing only monomers with / > 2 the branching coefficient a is simply equal to p, the extent of reaction [see Problem 5.24(d)], or the probability of reaction for any given functional group all a s in Eqs. (5.149)-(5.151) can thus be replaced by p. These equations... 

Since only monomer A/ (/ > 2) is present, the branching coefficient (a), defined earlier, is simply p and the critical condition for gel formation is thus given by... 

A branching coefficient, a, is defined as the probability that a given functionality group on a branch unit is connected to another branch unit a thus equals p, . An expression for the branching cOeflficient is obtained... 

The probability that a functional group A on a branch point (see Fig. 5.12) leads to another branch point, which defines the branching coefficient a), is evidently equal to the probabihty that A has reacted, i.e., pA- Since B is the limiting group, replacing pB with the conversion p, we thus obtain from Eq.(5.175) an expression for the branching coefficient as... 

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