Branching finite probability

The purpose of the following treatment is to define the conditions under which indefinitely large chemical structures, or infinite networks, will occur. To this end we seek the answer to the question Under what conditions is there a finite probability that an element of the structure selected at random occurs as part of an infinite network In order to simplify the problem, any given molecule such as the one shown in Fig. 61 may be regarded as an assemblage of chains connected together through polyfunctional, or branch, units (trifunctional in... 

To make predictions about the onset of gelation, Flory considered the conditions under which there will be a finite probability that any chain element, picked at random from the reaction mixture, occurs as part of an infinite network. He also introduced the branching coefficient, a, defined as the probability that a particular functional group belonging to a branch unit leads (via bifiinctional imits) to another branch unit. 

Now conduct a thought experiment, where we begin at A, and walk along the chain to the branch point at the other end. When we arrive at the new branch point, several alternatives are possible we may find two new chains that both lead to additional branch points we may find one chain that leads to an additional branch point and one that leads to a terminal end or we may find two chains that lead to terminal ends. If we tend to find branch points that lead to additional branch points, more often than to terminal ends, which happens if a > 0.5, there is a finite probability that we are walking on an infinite gel molecule, so some very large gel molecules will be present in the reacting mixture. 

Most of the E-SBR contains about 24% by weight of styrene and it is a random copolymer with butadiene. Some specific grades contain as much as 40-46% styrene, and are much stiffen The polymerisation is by free radical initiator and there is a finite probability of chain-transfer reaction, which generates long branches. Because the chain-transfer targets... 

It is important to note, however, that beyond a = 1/2 by no means all of the material will be combined into infinite molecules. For example, in spite of the favorable probability of branching, a chain selected at random may be terminated at both ends by unreacted functional groups. Or it may possess a branch at only one end, and both of the succeeding two chains may lead to unreacted dead ends. These and other finite species will coexist with infinite networks as long as l/2[Pg.353]

Choose any point in the system at random, and ask what is the expected number N 1) of /-neighbours (sites connected to this point by a path of / bonds in chemical space). Obviously N 1) = N(l — l)ps where p is the bond probability and s the branching ratio (s -f 1 is the maximum functionality at a junction). The critical point is when N oo) first becomes finite, so that pcS = 1 at this point, N 1) is a constant. But N 1) may also be written as a sum over clusters ... 

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