The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 73

spanning length l of the clusters. l is de ned as the maximum separation of two sites or

two bonds in the simulated network:

l ≡ max j r i r j j

i ; j in cluster :

ð 3 : 7 Þ

The divergence of the correlation length or average spanning length is an essential property,
as we will see below. Such continuous phase transitions can be characterized by power laws
containing the critical exponents, and perhaps the most important of these is the exponent
describing the divergence of the correlation length when the transition is approached.
In Figure 3.6 , three curves are drawn: s av is related to the geometry of clusters of nite

size below the threshold, which diverge as p c is approached, whereas above p c the cluster
size is in nite. The other two curves, P(p) and σ / σ 0 , vanish below p c , whereas above p c

they have finite values. The critical region is located near the threshold, with the

condition that |p

p c | << 1. In this region the important parameters describing the
percolation or thermal phase transitions follow various power-law behaviours versus
the distance to the threshold. For instance, for

‒

1

S av ≈

Þ γ with p c

ð p Þ! 0 ; ð 3 : 8 Þ

ð

p c

p

the exponent is γ . For the average cluster size

1

l av ≈ with ð p c p Þ! 0 ; ð 3 : 9 Þ

Þ ν ð p c p

the exponent is .

For the other parameters, P(p) and

ν

σ (p), the power laws are given by

p c Þ β and

t as

P ≈ ð p σ ≈ ð p p c Þ ð p - p c Þ! 0 : ð 3 : 10 Þ

The exponents and t are positive numbers that are not integers. For all simulations

they appear to be independent of the lattice geometry, but strongly dependent on the
space dimensionality d. The values of the exponents for various parameters deduced
from simulations are gathered in Table 3.2 .
One can see from Table 3.2 that the exponents are different for percolation and for
thermal phase transitions, meaning that they do not belong to the same

γ , ν , β

' universality

class . However, the general structure of the theory is the same; the exponents are not

independent of each other but are related by scaling laws (Stauffer et al., 1982 ). The
exponent

'

< 1, while t > 1, which explains the difference in the shape of the curves just
beyond the threshold: the slope of P(p)isin

β

is null. For large
clusters, at p = p c there is a relation between the size l and the cluster size s:

nite, while the slope of σ

l if d s ≈ ; s ! ∞; ð 3 : 11 Þ

where f d is the fractal dimension of the cluster at the percolation threshold. It is found in
simulations that if d = 1.9 if d = 2, and if d = 2.6 when d =3.