The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 81

exponents in agreement with, for example, percolation predictions. They also measured
the viscosity of the sol and found the exponent k = 1.36 ± 0.09 and the exponent for
elasticity t = 2.71 ± 0.30. At

first sight, the results seem convincing, but the authors
themselves point out, in connection with analyses of others ' work, that ' without an

independent measure of the critical point ... the determination of critical exponents ... is

extremely imprecise and subjective .

Other experiments by the same authors (Lusignan et al., 1999 ) were designed to
explore the vulcanization limit for the cross-linking of a melt of long linear chains of N
monomers, where mean

'

field (classical) exponents are predicted (Ginzburg, 1960 ;
Daoud, 1979 ). Here it is expected that the critical (percolation) domain is limited to the
range |p

p c |/p c <N − 1/3 , which is very much reduced when the chains are long enough, so
that classical exponents are expected to be observed experimentally. The analysis turned
out to be more complex: the exponent for viscosity s was found to increase with the ratio
N/N e , where N e is the entanglement length. The values for s are constant, s = 1.33 while
N < N e , and rose to s = 6 or 7 for N=30N e . The authors concluded that, for long chains,
the exponents change systematically with the number of entanglements N/N e per chain.
The exponent for elasticity was found to be t = 3.2 ± 0.2.
Other investigations on epoxy resins (Martin et al., 1988 ) and silica particles
(Gauthier-Manuel et al., 1987 ) tried to derive the critical exponent of viscosity by
assuming that the relative distance to the gel point

−

ε −

=|p p c |/p c could well be replaced

by the distance in reaction time t reac , t c )/t c . However, for any reaction the extent

p is not linear with reaction time and, even if one explores a narrow domain of the cross-
linking reaction, there is no

ε =(t reac −

fixed proportionality between these parameters (Ross-
Murphy, 2005 ). Therefore such approximations

should not be considered as

corroborating critical exponent determinations.
In summary, in this experimental work it appears that only the melt polymerization of
branched molecules (polyesters) with N = 2 monomers between junction points gives a
good illustration of the percolation critical exponents. For longer chains the scaling
behaviour seems to depend strongly on topological constraints, and no universal behav-
iour can be observed. The direct proof of the percolation models is best established by
direct measurement of the cluster characteristics, where there is a clear prediction for
exponents such as . It appears, as Stauffer himself pointed out some years ago

(Stauffer, 1998 ), that such experiments are very testing to carry out, and he even goes so
far as to suggest a

β , γ or ν

!
When it comes to rheological measurements, there is no consensus either from the
theoretical or experimental viewpoint about what is expected for viscosity or elasticity.
Percolation requires evaluating the extent of reaction using an independent method (such
as spectroscopy or calorimetry). Very careful experiments are needed for an experimental
determination of the critical exponents and of the gelation threshold. When dealing with
kinetics of gelation that cannot be stopped at any moment, rheological measurements are
limited by the time evolution of the system, and the limit of t relax → ∞

' failure of cooperation between physics and chemistry '

ω →

0 is not
accessible to experiments. In dynamic oscillatory measurements, the Newtonian behav-
iour of viscosity is de

or

ned by