General properties of polymer networks - Physical Gels from Biological and Synthetic Polymers - page 106

In Flory ' s original approach (Erman and Mark, 1997 ), the Flory - Huggins mixing

terms give

D F 1 ¼ RT n 1 ln ð 1 þ n 2 ln 2 þ χ 1 1 ν 2 Þ: ð 4 : 6 Þ

Here n 1 is number fraction of solvent molecules and ϕ 2 is the volume fraction of the

network, with − ϕ 2 ). The elastic term can be written in a number of ways, depend-

ing upon the network model, but is related to the number of elastically effective chains
and the volume deformation (the swelling ratio) (Dusek and Prins, 1969 ; Schroder and
Oppermann, 1996 ).
Further terms which may be added include

ϕ 1 =(1

π 3 , which is a measure of the difference in
osmotic pressure between the gel and the solution. This includes the Donnan contribution
from the mixing of ions with the solvent, where it is assumed that a polyelectrolyte gel
acts like an osmotic semi-permeable membrane. If the polymer chains have, say, a net
negative charge, there will be a net accumulation of mobile cations in the ' compartment '

containing the gel, which gives rise to this free energy contribution. Finally, a term π 4

comes from the free energy of electrostatic interactions. For neutral gels in non-polar
solvents, the original Flory

- Rehner case, only the first two of these terms are required.

The osmotic swelling force depends on the factor ecting solvent quality. As

already mentioned, the retractive force is in turn proportional to the number density of
cross-links (more formally, to the number density of elastically active network chains).
Thus for higher degrees of cross-linking, the net elastic force is higher (the shear modulus
is higher) so the equilibrium degree of swelling is lower; conversely, the lower the degree
of cross-linking, the greater the degree of swelling.
The previous comments assume that the polymer and solvent are not carrying any
charges. However, because of the polyelectrolyte effect, the need to reduce charge

χ − 0.5, re

-
charge interactions along the chain contour, a polyelectrolyte gel will swell to a large
extent in pure water, or in low ionic strength electrolyte solutions ( Chapter 5 ). At high
electrolyte concentrations the gel will tend to de-swell, because of the dependence of
chain dimensions on I − 1/2 , where I is the ionic strength. This effect can be ampli ed by

adjusting the mix of ionic species.
Overall, this means that the change in chemical potential due to swelling can be obtained
by evaluating the derivatives of the free energies with respect to n 1 . Swelling equilibrium is
then established for polyelectrolyte gels when the chemical potential of the solvent in the
gel and of the surrounding solvent are the same; the subscripts 1 to 4 refer respectively to
the mixing, elastic, osmotic (including Donnan) and electrostatic contributions:

Dμ 1 ;eq ¼ Dμ 1 ; 1 þ Dμ 1 ; 2 þ Dμ 1 ; 3 þ Dμ 1 ; 4 ; ð 4 : 7 Þ

where the mixing term is simply the classic Flory - Huggins contribution,

:

2
2

Dμ 1 ; 1 ¼ RT ln 1 ð 2 Þþ 2 þ χ ð 4 : 8 Þ

The other terms are more complicated to evaluate, and there are several alternate forms in
the literature (Erman and Mark, 1997 ). In terms of overall and readily measurable