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If one of the monomers in a copolymerization has a functionality greater than 2, a branched copolymer can be formed. It is also possible for the branches to react and create cross-links. In this way, “infinite” polymer networks called gels are made. the polymers in the system are multifunctional and highly branched but not yet cross-linked.
PA = probability that A reacts
PB = probability that B reacts
NAo = original number of A groups (@ t = 0)
NBo = original number of B groups (@ t = 0)
Assuming A can only react with B:
NAoPA = NBoPB and rPA = PB
For gelation to occur, q, the fraction of all monomer units in the sample that form cross-links, must be greater than qc, the critical value of q:
qc =1/(f-1) and q=(PAPBp)/(1-PAPB(1-p))
where f is the number of functional groups on the molecule with highest functionality. Additionally, only the highest functionality molecule reacts and causes branching, so another factor, p, must be considered.
p = (functionality of branched molecule * number of moles)/(sum(functionality * number of moles of all molecules of that type))
For example, if 0.5 moles of trifuctional A, 1 mole of difunctional A, and 2 moles of difunctional B molecules were reacted:
p = (3*0.5 moles)/(2*1 mole+3*0.5 moles) = 1.5/3.5 = 0.43
The limiting reactant in this situation is A because NAo = 3.5 mol < NBo = 4 mol. Solving for PA gives the fractional conversion of limiting reagent required to react for gelation to occur.
- Rudin, Alfred and Choi, Phillip. The Elements of Polymer Science and Engineering, 3rd Edition. 2012. Elsevier Science. p 410. ISBN 9780123821782