The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 72

3.3 Percolation and phase transitions

The shape of the curve P(p) is similar to one known in critical phase transitions for the
'

order parameter ' , as explained above. At the threshold, the slope of P(p)isin nite and

the behaviour in the close vicinity of the threshold, when |p − p c | → 0, is controlled by a

' '

set of so-called . The reason why the probability P(p) increases so

sharply is related to the size distribution of the clusters near threshold, since it only needs,
in some cases, just a few extra bonds (or the

critical exponents

filling of a few sites) to convert several large

clusters into one of in nite size. However, conductivity does not increase so sharply, nor

at the same time, because the addition of new bonds from a large cluster increases the
number of loops in the network but not necessarily the connectivity of the whole lattice.
Many dead ends (or what we call in Chapter 4 ) are present near the

threshold in the percolation path, and only a small fraction of the bonds, called the
'

' network defects '

, contributes to the conductivity. This is why the increase in conductivity is
very small near the threshold. When the limit p

backbone '

1 is reached, all the bonds will

→

contribute to the conductivity.
Referring to the analogy between percolation and thermodynamic phase transitions,
we can examine Table 3.1 .
We have already discussed the analogy between temperature and the percolation
threshold. Spontaneous magnetization M, density ρ liq −ρ vap and percolation probability

P(p) are the respective order parameters for these transitions. All of them are zero in the
disordered state (unconnected bonds or high-temperature limit) and they increase very
sharply as the percolation threshold or the critical temperature is approached. The
magnetic susceptibility,

H, describes the ability of a ferromagnetic
material to increase its magnetization under the effect of a magnetic

χ de ned by χ = ∂ M/ ∂

field H. When the
Curie temperature is approached from above, T > T C , in the paramagnetic state (no
spontaneous magnetization) the susceptibility diverges. It can be shown that the analogue
of the magnetic susceptibility is formally the isothermal compressibility in a

fluid or the
mean cluster size in the percolation model. These parameters diverge when the phase
transition is approached from the high-temperature side or, in percolation, from below the
threshold.
The correlation length in thermal phase transitions is the mean cluster size below the
threshold. This mean cluster size l av or mean spanning length is the average of the

Table 3.1 Analogy between percolation and thermodynamic phase transitions.

Percolation Ferromagnet Liquid - vapour critical point

Percolation threshold p c Critical temperature T c Critical temperature T c

Percolation probability Spontaneous magnetization M ρ liq − ρ vap

Mean cluster size s av Magnetic susceptibility χ Isothermal compressibility κ

Mean cluster spanning length l av Correlation length ξ Correlation length ξ