Gel point

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.

Definitions:

P_A = probability that A reacts

P_B = probability that B reacts

N_Ao = original number of A groups (@ t = 0)

N_Bo = original number of B groups (@ t = 0)

r=N_Ao/N_Bo

Assuming A can only react with B:

N_AoP_A = N_BoP_B and rP_A = P_B

For gelation to occur, q, the fraction of all monomer units in the sample that form cross-links, must be greater than q_c, the critical value of q:

q_c =1/(f-1) and q=(P_AP_Bp)/(1-P_AP_B(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 N_Ao = 3.5 mol < N_Bo = 4 mol. Solving for P_A gives the fractional conversion of limiting reagent required to react for gelation to occur.