Gelation in Polymerization

The classical theories of Carothers, Flory and Stockmayer1-5 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 favg)

which can be rearranged to

p  = 2 / favg  -  2 / (DP · favg)

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

favg = i Ni fi / i Ni

Flory and others have shown that any cross-linkable polymer system possesses a sharply defined gel point at a certain critical extent of reaction pc 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 pc is given by

pc  = 2 / favg

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, nmax → ∞, the sum has the solution

α = / [1 - p(1- ρ)]

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

αc = 1 / (f - 1)

Thus for f = 3, an infinite network is only possible if αc = 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 pcαc.

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 pAρ[pA (1 - ρ) pB]npB. 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 = pA ρ pB n [pA (1 - ρ) pB]n = pA pB ρ / [1 - pA (1 - ρ) pB]

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

α = pB2ρ / [r - pB2(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:

pc = 1 / {r [1 + ρ (f - 2)]}½

and for the special case r = 1, i.e. pA = pB = 1:

pc = 1 / [1 + ρ (f - 2)]½

Some experimental values observed by Flory1,7 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

Observed and Calculated Gel Points of Polyesters1

r =
[CO2H ] /[OH]

ρ

pc,Car a pc,Flory b pc,avg pc,obs c
1.000 0.293 0.951 0.879 0.915 0.911
1.000 0.194 0.968 0.916 0.942 0.939
1.002 0.404 0.933 0.843 0.888 0.894
0.800 0.375 N/A 0.953 N/A 0.991
aCarother's equation;   bFlory's equation;   cData from Flory (1941)1;

 

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

pc = 1/ favg + (αc / 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 pA 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)
  • Summary

    Polymer Networks

    A cross-linkable polymer system possesses a sharply defined gel point at a critical extent of reaction.

  • At this point, the crosslinked polymers approach infinite size, i.e. they become one gigantic molecule.

  • Gelation causes a sudden and dramatic increase in viscosity, so that the polymer mass stops to flow.

  • The basic statistical theories of chemical gelation were developed in the 1940s by Paul Flory and Walter Stockmayer.

  • Gelation occurs in a three dimensional polymerization long before most of the monomers have been bound together in one gigantic molecule.

  • According to Wallace Carothers, gelation can only happen when at least one bond per initial monomer has been formed.