Gelation classical theories

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. 

Theoretical and experimental treatments of gels go hand-in-hand. The former are covered first because they will help us understand gel point and other concepts. Two main theories have been used to interpret results of experimental studies on gels the classical theory based on branching models developed developed by Floiy and Stockmayer, and the percolation model credited to de Gennes. Gelation theories predict a critical point at which an infinite cluster first appears. As with other critical points, the sol-gel transition can be in general characterized in terms of a set of generally applicable (universal) critical exponents. 

Experimental detection of the gel point is not always easy since the equilibrium shear modulus is technically zero at the gel point and any applied stress will eventually relax, but only at infinite time. From the classical theory, the attributes of the gel point are an infinite steady-shear viscosity and a zero equilibrium modulus at zero frequency limit (Figure 6-3) (Flory, 1953). These criteria have been widely employed to detect the gel point of chemical gels. However, because continuous shearing affects gel formation, accurate information from viscosity measurement is not possible in the close vicinity of the gel point. Further, information regarding the transition itself could only be obtained by extrapolation, thereby introducing uncertainties in the determination of the gelation moment. 

Statistical network models were first developed by Flory (Flory and Rehner, 1943, Flory, 1953) and Stockmayer (1943, 1944), who developed a gelation theory (sometimes referred to as mean-field theory of network formation) that is used to determine the gel-point conversions in systems with relatively low crosslink densities, by the use of probability to determine network parameters. They developed their classical theory of network development by considering the build-up of thermoset networks following this random, percolation theory. 

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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