The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 79

Three cases can be envisaged:

G 00 5 G 0 n 5 ½ ;

G 00 4 G 0 n 4 ½ ;

G 00 ¼ G 0 : n ¼ ½ ;

In the particular case where n =½,G 0 = G 00 , the gel point is easily located by measure-
ments at a single frequency. This was the case when they investigated samples with
appropriately balanced stoichiometry.
For n = m, the relaxation modulus G(t relax ) of the solution exhibits time dependence,
with a power law of slope - n:

t n
relax

G ð t Þ ≈ ; ð 3 : 26 Þ

where t relax is the elapsed time in the relaxation spectrum. The authors exclude the
high-frequency limit, where the material behaves as a glassy system, from their
analysis. The steady-state viscosity

η 0 can be calculated from the time dependence
of the relaxation modulus G(t relax ): it can be shown that, for any exponent n < 1, the
viscosity diverges to in

nite time decays to
zero for n > 0. The solution is very viscous, but not yet elastic at the point de

nity and the equilibrium modulus at in

ned
by Equations ( 3.23 ) and ( 3.24 ). The authors commented that a system which
exhibits power-law behaviour exhibits the ' classical attributes ' of the gel point.

nition of the gel point are therefore
based on two different approaches. Most percolation models are essentially static
theories. They do not take into account the mobility of the molecules. They predict the
relations between the structure of the solution schematically represented by lattices
and the fraction of bonds created at random. During gelation, larger and larger clusters
modifying the viscosity of the solution are formed, until an in

These two theoretical approaches to the de

nite cluster is created

and the solution starts to exhibit a finite time of

observation). The predictions for the average molecular masses of the clusters or the
gel fraction are derived straightforwardly from simulations. The predictions for viscosity
or elasticity are still debated and depend on the particular context of the analysis
(Monte Carlo simulations, analogy with conductance of static network, particular
model for network connectivity etc.). There is still an important debate about the
universality classes for these parameters, where identical critical exponents can be
predicted with various hypotheses.
The Winter

finite elasticity at zero frequency (or in

first of all on the experimental
observations these authors made by following an end-linking reaction of primary chains
in the absence of any physical association (crystallization, phase separation etc.), where the
precursor polymers were not entangled. At one particular moment in the reaction, both
viscoelastic moduli G 0 and G 00 measured in the linear regime exhibit power-law depend-
ences on the frequency. This indicates that an in

- Chambon criteria are based

nite viscosity at zero frequency is also
expected from the percolation model. The authors suggest that both approaches

- perco-

lation and mechanical spectrum - lead to the same determination. They also veri ed that, at

the gel point, the large cross-linked cluster became insoluble.