The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 88

' values of the melting enthalpy, and these are discussed in the te Nijenhuis review

(te Nijenhuis, 1997 ).
In 1972, Takahashi ( 1972 ) derived a different equation, based on the Flory copolymer
melting treatment, to address the same problem. Rather than plotting ln c versus 1/T m ,it
uses Flory

best '

Huggins coordinates, so that 1/T m is now the abscissa and ln v 2 N the ordinate,
with v 2 the polymer volume fraction and N the degree of polymerization. While criticisms
similar to those mentioned above for EF could be made about this treatment, it does
appear to be quite successful in allowing the superposition of samples of differing
molecular mass.
Yet another variation on the EF method, based on a more rigorous underlying
theoretical model, is due to Tanaka and Nishinari ( 1996 ). They suggested that a plot of
ln c versus 1000/T m +lnM would also allow the apparent multiplicity of junction zones
to be determined. Here the factor 1000 simply helps to represent the two plot components
successfully, analogous to the scaling of the classical Zimm plot for light scattering.
The two slopes of this rectilinear Zimm type plot can be related respectively to the
multiplicity and to the usual melting enthalpy. They describe the application of the
method to a number of synthetic thermoreversible gel systems, including poly(styrene)
in carbon disulphide (which gives a maximum in the EF plot) and PVA in water. In the
latter case s was calculated to be in the range 2

-

-

3, with higher values, perhaps as

expected, for the high-melting samples.

3.7 Critical gel concentration

One aspect of the model mentioned above is the critical gel concentration, here denoted
c 0 . That this exists follows from the existence of a percolation threshold, as described
above and derived from both classical and non-classical formulations. Apart from this, it
is all but self-evident that there has to be a certain (

) concentration of polymer
present before a gel can be formed, and only a few approaches (Bremer et al., 1993 ) have
made the bold assumption that this concentration is effectively zero.
Using the equilibrium assumption of Eldridge and Ferry, Hermans ( 1965 ) assumed that
there was a monomer

' critical '

two free sites) equilibrium with a site dissoci-
ation constant K s . This could then be written in the form of an Ostwald dilution law,

dimer (or link

⇔ ⇔

h i N 0 f

K s

2

¼ð 1 p Þ = p ;

where N 0 is the number of primary chains per unit volume, p is the percolation site
occupancy and f is the functionality. In the limit of weak binding (where K s

1) and
high functionality, f can be shown (Clark and Ross-Murphy, 1987 ) to give a straight
proportionality, equivalent in our terms to

≫

p
p c ¼

c
c 0 ;