Classical theory of gelation

This chapter presents the definition of gels and gives some typical examples, followed by a description of their structures and fundamental properties. A statistical mechanical treatment of the chemical gels in the polycondensation reaction is developed to find the molecular weight distribution, average molecular weight, gel point, and the gel fraction. 

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Cations can be seen as acting as ionic crosslinks between polyanion chains. Although this may appear a naive concept, crosslinking can be seen as equivalent to attractions between polyions resulting from the fluctuation of the counterion distribution (Section 4.2.13). Moreover, it relates to the classical theory of gelation associated with Flory (1953). Divalent cations (Zn and Ca +) have the potential to link two polyanion chains. Of course, unlike covalent crosslinks, ionic links are easily broken and re-formed under stress there could therefore be chain slipping and this may explain the plastic nature of zinc polycarboxylate cement. 

Figure 9.6 Representation of a trifunctional monomer according to the classical theory of gelation. The monomer can hold four reactive states, from 0 to 3, which indicate the numher of functional groups that have been reacted, linking this unit with its neighbor. Source Adapted with permission from Dusek K, MacKnight WJ. Crosslinking and structure of polymer networks. In Labana SS, Dickie RA, Bauer RS, editors. CrossLinked Polymers. American Chemical Society 1988. p. 2 [88]. Copyright 1988 American Chemical Society.

We next consider the condensation reaction of polyfunctional molecules of the type R A/. The molecular weight distribution for the special case / = 3 was first studied by Flory [10], The result was later extended to the general case of / by Stockmayer [11] under the assumption of no intramolecular cycle formation. Their theories are called the classical theory of gelation reaction. 

To derive a specific form of the equilibrium constants bi, let us introduce a simple model for the internal structure of clusters. Clusters are assumed to take a tree structure with no internal loops (Cayley tree). Cycle formation within a cluster is neglected. This is a crude approximation on the basis of the classical theory of gelation presented in Section 3.2 [5,6,7,8], but in fact it is known to work very weU at least in the pregel regime. 

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