Theory of Swelling

A suitable expression for AFm may be obtained from Eq. (XII-22), bearing in mind that the number U2 of polymer molecules is to be equated to zero owing to the absence of individual polymer molecules in the network structure. Thus 

Evaluating the other two derivatives occurring in Eq. (36) by differentiating Eqs. (34) and (35) and expressing Ve in moles, we obtain  

The first three terms occurring in the right-hand member of Eq. (38), represent dAFu/ ni] they correspond to mi Mi according to Eq. (XII-26) for a polymer of infinite molecular weight (i.e., a = 00). The last member introduces the modification of the chemical potential due to the elastic reaction of the network structure. The activity ai 

The treatment given here, like that of rubber elasticity in Chapter XI, is developed for a network in which the ends of the chains are united tetrafunctionally, 

If the chemical potential difference ah Mi calculated according to Eq. (38) is plotted against it will be found that, owing to the positive contribution of the elastic term (with j e 0), the chemical potential Ml exceeds mi for the pure solvent for all concentrations below a certain polymer concentration In other words, the activity a would 

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In this section, we will describe the theory of swelling of polyelectrolyte networks. The simplest problem of this type concerns a network sample swelling freely in an infinite solvent. The solvent may contain some low-molecular-weight salt. This problem will be considered in Sect. 2.1. 

The first quantitative theory of the reentrant collapse was developed in Ref. [49], The theory explained the phenomenon of the simple reentrant collapse which was observed in Refs. [14, 41]. A more general theory of swelling and collapse of charged networks in the binary solvent was developed in Ref. [31] and described in Sect. 2.4.1. We have seen that one of the most essential features of the swelling behavior in mixed solvents is a redistribution of solvent molecules within the network giving a different solvent composition in the gel and the external solution. This redistribution is more pronounced for the collapsed gel, because the probability of contacts of the molecules of the solvent with polymer links in the collapsed gel is higher than in the swollen gel. 

In certain cases as for instance by treatment of cellulose with cuprammonium, swelling virtually dissolves the cellulose. (The theory of swelling and solution of nitrocellulose is developed in more detail on p. 244). During swelling a characteristic increase in the diameter of the fibre occurs, without any increase in its length. 

The classical theory of swelling developed by Flory and Rehner in 1953 (1) provides the following expression for swelling equilibrium ... 

In a more recent theory of swelling equilibrium developed by Flory in 1979 (19), the extent to which the swelling equihbrium deformation is non-affine is taken into account. The nonaffine behavior depends on the looseness with which the cross-links are embedded in the network and consequently is related to the network structure and the degree of swelling at equilibrium. The final expression is... 

The factor characterizes the extension by which the deformation approaches the affine limit. This theory is more difficult to apply than that corresponding to Eq. (3.46) because the factor contains some parameters that are not present in the classical theory of swelling. 

Neuburger NA, Eichinger BE. Critical experimental test of the Flory-Rehner theory of swelling. Macromolecules 1988 21 3060-3070. 

It was observed, that a gel swollen to equilibrium in the liquid solvent shrinks as soon as it is transferred to the vapour phase of the same solvent. This phenomenon is known as the paradox of Schroeder (Freundlich 1932). For a theory of swelling with solvents in various phases, see (Borchard Steinbrecht 1991). 

In the 1940s, Flory and Rehner (Flory and Rehner 1943a, b Flory 1953) were the first to formulate the theory of swelling of network structures . Here the change of ambient conditions can be represented by a change of the Gibb s free energy AF (see also Chap. 3). 

Murad, M.A. Cushman, J.H. 1997. A multiscale theory of swelling porous media II. dual porosity models for consolidation of clays incorporating physical chemical effects. Trans. Por. Media. 28(l) pp.69-108. 

The general theory of swelling assumes that the free energy of mixing and the elastic free energy in a swollen network are additive. The chemical potential difference between gel and solvent is given by the equation ... 

Below, the simplest theory of swelling of polyelectrolyte gels is described, mainly to illustrate that the translational entropy of counterions for this case is much more important than the Coulomb interactions. 

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