Gelation in Polymerization

The classical theories of Carothers, Flory and Stockmayer^1-5 describe the conversion of growing polymers to insoluble gels which occurs through crosslinking. All three theories use a statistical approach and assume that polymers approach infinite size at the gel point, i.e. they  become one gigantic molecule. Many other theories have been developed over the last 50 years or so. Some give a more accurate estimate of the gel point, but the majority of these models are much more complicated and thus, in practice, are rarely used.

The molecular weight distribution of a linear condensation or addition polymer can be easily calculated if we assume that each functional group has an equal chance of reacting with other groups regardless of the size of the oligomers.⁶ According to Carother, the number-average degree of polymerization for such a reaction is given by

DP = 2 / (2 - p f_avg)

which can be rearranged to

p  = 2 / f_avg  -  2 / (DP · f_avg)

where p is the fraction of functional groups which have reacted and f_avg is the average number of functional groups per monomer molecule of the blend of monomers. The latter is defined by

f_avg = ∑_i N_i f_i / ∑_i N_i

Flory and others have shown that any cross-linkable polymer system possesses a sharply defined gel point at a certain critical extent of reaction p_c which should be independent of temperature, type and amount of catalyst, etc. When polymers gel, a sudden and dramatic increase in viscosity is observed and the polymer mass stops to flow and wraps around the mixer used to homogenize the reaction mixture. At this point, the molecular weight becomes  infinite, DP → ∞ . Thus, the second term of the Carother equation vanishes and the critical extent of the reaction p_c is given by

p_c  = 2 / f_avg

Carothers equation is applicable only to stoichiometric mixtures of addition monomers or if only one type of functional group undergoes polymerization as it is the case with chain growth. In the case of non-stoichiometric ratios of two or more different functional groups, Flory's and Stockmayer's statistical approaches give more accurate results.

Flory¹ consider first a system in which there is only one type of functional group, i.e. a  step-growth reaction of di- and tri-functional groups. This type of polymerization can be represented by

Flory assumed that both monomers ant their functional groups are equally reactive, and that a reacted group does not affect the reactivity of other unreacted groups of the same unit. Then the probability, p, that an A group has reacted is equal to the ratio of reacted A's to all A groups. Let ρ be the ratio of A's belonging to a branch unit to the total number of A's. Then the probability that one A group of a branch unit is connected to a sequence of n bifunctional units A-A followed by a branch unit is

pⁿ (1- ρ)ⁿ p ρ

The probability α that such a sequence ends in a branch regardless of the number n of bifunctional repeat units is given by the sum over all possible n:

α = ∑_n pⁿ (1- ρ)ⁿ p ρ =

For very large sequences, n_max → ∞, the sum has the solution

α = pρ / [1 - p(1- ρ)]

According to Flory¹, the criterion for gelation of a stoichiometric mixture containing branching units of functionality f is given by

α_c = 1 / (f - 1)

Thus for f = 3, an infinite network is only possible if α_c = 1/2, which means that at least one functionality of a branch unit at the end of chain will react with another unit.
If ρ = 1, i.e. if all monomers are trifunctional units, the limiting condition for gel formation is p_c =  α_c.

Another important case that Flory and others investigated is the reaction between two different types of monomers which is the common case in step-growth polymerization. For example, for the case of trifunctional and bifunctional A units reacting with difunctional B units, the step-growth reaction may be represented by

The probability of obtaining such a sequence is given by p_Aρ[p_A (1 - ρ) p_B]ⁿp_B. Then the probability α that a chain ends in a branch unit regardless of the number of linear repeat units inbetween the two branched ones is given by¹

α = p_A ρ p_B ∑_n [p_A (1 - ρ) p_B]ⁿ = p_A p_B ρ / [1 - p_A (1 - ρ) p_B]

where ρ is the ratio of A's that are part of a branch unit to the total number of A's and p_A and p_B are the extend of the reactions of A and B functional groups. Either p_A or p_B can be eliminated by using the ratio r of all A groups to all B groups which yields¹

α = p_B²ρ / [r - p_B²(1 - ρ)]

Combining this equation with α = (f - 1)^-1 yields following expression for the extent of reaction of the A groups at the gel point:

p_c = 1 / {r [1 + ρ (f - 2)]}^½

and for the special case r = 1, i.e. p_A = p_B = 1:

p_c = 1 / [1 + ρ (f - 2)]^½

Some experimental values observed by Flory^1,7 together with theoretical values calculated from Carother's and Flory's equation are listed below. It can be seen that Carother's equation gives too high values while Flory's method underestimates the extent of reaction at the gel point. This discrepancy has been explained by the fact that the functional groups have unequal reactivities and that some chains form ring structures through intramolecular reactions.⁸

 

It can also be seen that the average of both expressions deviates very little from the experimental value. For this reason, Pizzi suggested to combine the two expressions. For the simpler case ρ = 1, i. e. no A-A units are present, this expression reads⁹

p_c = 1/ f_avg + (α_c / r)^1/2 / 2

References and Notes

1. P.J. Flory, J. Amer. Chem. Soc., 63, pp 3083-3090 (1941)
2. P.J. Flory, J. Phys. Chem., Vol. 46, No. 1, pp 132-140 (1942)
3. W. H. Carothers, Trans. Faraday Soc., Vol. 32, pp 39-49 (1936)
4. W. H. Carothers, Collected Papers, Interscience, New York, 1940
5. W. H. Stockmayer, J. Chem. Phys., 12, 125 (1944)
6. This assumption is called Flory's equal reactivity principle. It states that the probability p_A that a given functional group A has reacted is equal to the ratio of reacted A's to total A's.
7. Paul L. Flory, Principles of Polymer Chemistry, Ithaca, New york, 1953
8. Z. Ahmed and R.F.T. Stepto, Poly. J., Vol. 14, No. 10, pp 767-772 (1982)
9. A. Pizzi, J. Appl. Polym. Sci. 63, 603 (1997) & 71, 517 (1999)
10. A. W. Fogiel and C. W. Stewart, J. Poly. Sci.: Part A-2, Vol. 7, 1116 (1969)