The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 77

In percolation theory, the sum diverges logarithmically at the gel point:

≈ log ð p p c Þ; ð 3 : 19 Þ

while in the original Einstein theory it remains constant, so using this equation for
describing gelation is unrealistic. It is necessary to include other terms with quadratic
contributions from the volume fractions of virial expansion type:

" 1 #

X

s s 2

2 X

5

¼ 0 þ

s s þ

: þ… : ð 3 : 20 Þ

const

At the gel point, the interaction between the clusters becomes too strong and this
approximation also becomes unreliable: the third term in ( 3.20 ) becomes larger than
the second term and the Taylor expansion in ϕ s is no longer valid.

A completely different approach was proposed by de Gennes ( 1979 ) based on the
analogy between viscosity and elasticity. The viscosity below p c is the counterpart of the
elasticity above p c . To predict the behaviour of the viscosity in the sol state, he suggested
simulating a random mixture of superconductors with fraction p and normal conductors
with fraction 1

− '

s critical exponent equals the
critical conductivity exponent which, for d = 3, is given by k = 0.7:

p. According to this analogy, the viscosity

Þ k ≈ ð

p c

;

p 5 p c :

ð 3 : 21 Þ

p

Results based on molecular dynamics and Monte Carlo simulations show good agree-
ment with this prediction (Farago and Kantor, 2000 ; Vernon et al., 2001 ).
Another approach is the so-called Rouse approximation (Stauffer et al., 1982 ), in
which the contribution of a cluster s to the viscosity is proportional to sR s 2 : near the
threshold, it states that the viscosity varies as
¼ 0 1

X

n s sR s þ … p 5 p c : þ const : ; ð 3 : 22 Þ

s

The exponent k for viscosity is related to the exponents , which gives

k = 1.3 for d = 3. This approximation neglects hydrodynamic interactions and the
excluded-volume interactions between monomers, and so may not be valid near the gel
point. Subsequently, however, the Bound Fluctuation method (Carmesin and Kremer,
1988 ), based on an algorithm which takes into account excluded-volume interactions,
supported the prediction that, for d =3,k = 1.3.
Despite these attempts, there is no de

β and ν by k =2 ν − β

nite answer as to the value of the exponent k for
the divergence of the viscosity near the threshold. A review by de Arcangelis ( 2003 )
suggests that there could be two distinct universality classes for the viscoelastic critical
behaviour, with a crossover between different dynamic regimes. As we have already
mentioned, the problem is very complex, since the distribution of cluster sizes is very
(arguably tending to in nitely) polydisperse as the gel point is approached, and the

relationship between size and M w is complicated even for linear polymers.