The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 78

The elasticity of the gel near the threshold can also be predicted by the scaling
relation t = d

, which in three dimensions gives t = 2.6 (Daoud and Coniglio, 1981 ).
The model assumes that elasticity is dominated by the entropic term. When there is a
bond-bending contribution to the elastic energy of the network, as expected for
particulate gels, the predictions for the exponent of elasticity are higher: t =3.7 for
d = 3 (Feng and Sen, 1984 ;Fenget al., 1984 ; Kantor and Webman, 1984 ). More
recently, a direct investigation of viscoelastic properties was performed by introducing
bond

ν

fluctuation dynamics into the percolation model, taking into account the con-
formational changes of the polymer and the excluded-volume interactions.
Simulations have shown a critical exponent t =2.5ford =3(DelGadoet al., 2002 ),
in agreement with the prediction t = d for entropic networks. Here again the numerical

work suggests the possibility that there are two distinct universality classes, one
characterized by an exponent t = d

ν

ν ≈ 2.64 and the other, based on the electrical

analogy, by t 2.

Before examining experimental results on gelation, we present another approach
widely used to determine the

≈ 1.6 -

' gel point ' , known as the Winter - Chambon criteria.

3.4.1 Winter - Chambon criteria

Winter and Chambon ( 1986 ) investigated the end-linking reaction of polydimethylsilox-
ane (PDMS) chains, to provide a good model system. Divinyl terminated PDMS was
linked with a tetrasilane (f = 4, functional) moiety, and the molecular mass of the
prepolymer was below the entanglement limit. Using the time

temperature superposition
procedure, they were able to reconstruct mechanical spectra over a very wide range of
frequencies, far from the glass transition temperature. They were also able to stop the
chemical reaction at intermediate times of conversion. They observed that the spectra of
the shear storage modulus G 0 and loss modulus G 00 of their system exhibit, at some stage
of the conversion, a power-law behaviour in the entire measurable radial frequency
domain.
They proposed to de

-

ne the gel point by the following properties:

G 0 ðωÞ ≈ ω

n

0 5ω5∞; ð 3 : 23 Þ

G 00 ðωÞ ≈ ω

m

0 5ω5∞: ð 3 : 24 Þ

Kronig relation requires the two exponents n and m to be equal, so the
complex moduli are then given by

The Kramers -

G 00 ðωÞ
tan n 2 ¼

G 00 ðωÞ
tan

G 0 ðωÞ¼

n 5 1 ; ð 3 : 25 Þ

δðωÞ

and, the relation being independent of the frequency, the loss angle δ ( ω ) is independent of

the frequency.