The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 89

provided p p c ; and the critical concentration itself can be shown to be

≳

M w
K s f 2 ;

c 0 ¼ ð 3 : 31 Þ

The assumption of high f is a very arti cial limitation, even for the cross-linking of long

primary chains, so Clark and Ross-Murphy ( 1985 , 1987 ) derived a more general form
valid for any functionality, which gives

M w f ð 1 Þ

c 0 ¼

2 ;

ð 3 : 32 Þ

K s ff ð 2 Þ

in which case

p
p c ≠

c
c 0 :

On the basis of this approximation, and applying a polymer network branching model,
the general relationship between gel modulus and concentration was also derived. In
carrying out such a manoeuvre, there is no assumption that networks should be ideal
rubbers, since the so-called rubber elasticity ' front factor ' need not be close to unity, as it

is for rubber networks ( Chapter 4 ).
In this way the underlying network connectivity (percolation) can be factored out of
the contribution to the modulus from each cross-link, which in turn will depend on the
chain species,

flexibility etc. It is via the system-independent connectivity part of the
treatment that such essential features as critical concentration and the initial steepness of
the concentration dependence are derived. The nal - high - concentration modulus

tends to re ect the modulus per cross-link terms. Clark ( 1993 ) showed subsequently that

the form of the equations above still holds even if the system, rather than being in
equilibrium, is under kinetic control.
The method has been quite successful, not only for cold-set gels such as gelatin
( Chapter 7 ) but also for heat-set particulate and fibrillar gels such as those formed from

β -lactoglobulin ( Chapter 9 ). Not least, it has also served as a useful interpolation method

in evaluating the modulus of mixed-gel systems ( Chapter 10 ).
For thermoreversible gels, the critical concentration will obviously depend on temper-
ature because, above the melting temperature, the critical concentration is nominally
in nite. This means that there must be a maximum gelation temperature, and this relates

to the induction time originally de

ned by te Nijenhuis, now more commonly referred to
as the critical gelation time t c , which also tends to in

nity at this maximum gelation
temperature. Various empirical models for the temperature dependence of c 0 and t c are
discussed by te Nijenhuis in his review (te Nijenhuis, 1997 ).
For some workers, c 0 is related to the overlap concentration c* in solution, and some
even treat the two as the same. That said, it seems that for physical gels there is little
correlation between c*andc 0. Indeed it has even been argued (Clark and Ross-