The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 71

500 1. 0

400 0.8

P ( p )

300 0.6

s av P , σ

200 0.4

s av ( p )

σ / σ 0

10 0 0.2

0 0

0 0.2 0.4 0.6 0.8 1.0

p c

p

Behaviour of various quantities as a function of the fraction p of filled bonds, in a bond percolation
simulation on a two-dimensional lattice.

Figure 3.6

cluster or the percolation path. The percolation path for an in nitely large cluster, as

observed in Monte Carlo simulations, appears at the percolation threshold p c .The
percolation threshold for site percolation on a square lattice is found at p c =0.59.
Below the threshold there is no percolating path; above the threshold there is one path.
When the simulation on a square lattice takes into account the number of bonds created
instead of the number of sites occupied, the so-called bond percolation on a square
lattice has a threshold of p c =0.5.
Another important parameter is the probability that a site belongs to the in nite

cluster when p > p c .Referringtothebond percolation simulation on a square lattice, in
Figure 3.6 one can see the probability P(p) of a bond belonging to the in

nite
cluster. This probability is exactly zero below the threshold, and then increases very
steeply at the threshold. As p approaches 1, P(p) p. Other important parameters

are seen in Figure 3.6 : in particular, the increase in s av as p approaches p c from
below. This simulation also represents the increase of the conductance of a random
resistor network,

≈

σ ο versus p, after the threshold, as seen in Figure 3.6 .Thereisa
small but increasing curvature at and just after p c , and then a monotonic but essentially
linear increase up to a value of 1 at full occupancy (p =1).
It is important, at this stage, to clarify that the FS and later theories are not qualitatively
different from those based on percolation

σ /

-

in fact the FS theory corresponds exactly to
percolation on a so-called Bethe lattice or Cayley tree, a cycle-free lattice in which there
is no overall connectivity until p > p c and then the increase is monotonic. The difference
is simply that this lattice is totally

, i.e. it is not embedded in any dimensional
space. Instead, the difference comes from quantitative predictions of the various critical
exponents.

' oppy '