General properties of polymer networks - Physical Gels from Biological and Synthetic Polymers - page 104

for the classical networks above) by assuming that the gel consists of a network of
Langevin chains. In this so-called

model, both ends of a Langevin chain are
bound into a cross-link (or junction zone) region, but can be progressively released on
increasing the temperature by

' reel-chain '

, as if chains were being pulled off a reel. Using
this model, Nishinari and co-workers were able to calculate the temperature dependence of
the modulus (in particular the static Young

' sublimation '

smodulusE) and compare results with their
own data for PVA gel networks. They found that the temperature dependence of E was a
sensitive function of a constraint release parameter, de

'

ned as the ratio of the number of
chains released to the (RMS) end-to-end distance of the chain. They also found that E had a
maximum value as a function of temperature, first increasing and then decreasing.

Such reel-chain models have been published by others, including Higgs and Ball
( 1989 ), who drew on the NKO model (Nishinari et al., 1985 ) and also investigated
so-called rod-chain models, which apply before the unwinding or

occurs.
Using their model, they concluded that, rather than approaching its extension limit
and paying a high free-energy cost due to reduction in entropy, a Langevin chain will
tend to pull out another chain segment from the junction zone. Consequently, as long as a
Langevin chain does not approach its extension limit, it behaves in the same way as a
Gaussian chain, so the effects of limited extensibility are somewhat outweighed by the
unwinding of the junction zone.

' sublimation '

4.3 Swelling of gels

While the various theories of rubber elasticity allow us to calculate the retractive force on
a polymer network, the swelling of a polymer network or gel is assumed to be governed
by a solvent-induced expansion force (or, more formally, free energy) involving osmotic
and, for polyelectrolyte systems, ionic contributions. The net result is that a gel will tend
to swell to an equilibrium degree which depends upon its nature, the amount of cross-
linking, the nature of the solvent, the temperature and other factors. For a given network,
these factors allow the equilibrium degree of swelling to be altered, but the essential point
is clear: to achieve a high degree of swelling, the retractive force must be small, so the
degree of cross-linking must itself be not much more than the critical amount.
Since absolute degrees of swelling are, in the context of the present volume, of less
signi cance than the changes that can be wrought by changes of temperature, solvent

etc., we will discuss swelling below in terms of the simplest such model, that of Flory and
Rehner (Flory, 1953 ). This is now very ancient but, like the Flory Stockmayer theory,

can be regarded as a useful and easily comprehensible zeroth-order approximation, upon
which we can hang future developments and improvements.
As stated above, when a chemically cross-linked gel is immersed in an excess of liquid
and allowed to equilibrate, the size of the sample may increase, decrease or remain
constant; see Figure 4.3 . This effect depends strongly upon the

-

'
of the liquid for the gel and the degree of cross-linking. If a hypothetical cube of gel (with
sides of length l ) is immersed in a good solvent the volume will increase from an initial
value of V 0 to V. The swelling ratio q is de

'

thermodynamic quality

ned as V/V 0 . For a cube, V 0 is equal to l 3 ,and