The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 70

(a)

p = 0.25

s = 3

(b)

p = 0.50

s = 8

(c)

p = 0.75

s = ∞

Site percolation on a square lattice for three values of the fraction p of filled sites.

Figure 3.5

Then, the number of sites in clusters of size s is proportional to sn(s), and the site-
weighted average cluster size is av for each value of the parameter p is given by

∞

s 2 n

ð s Þ

s ¼ 1

s av ¼ : ð 3 : 6 Þ

∞

sn ð s Þ

s ¼ 1

The sum in the denominator is equal to p. is av starts at 1 for p = 0 and increases as p
increases. In Figure 3.5b , where p = 0.50, there are fewer singlet sites and more clusters of
larger size than in Figure 3.5a .In Figure 3.5c , p = 0.75. Here there is a major difference
compared to the two previous frames: there is now a very large cluster which connects the
four edges of the frame. As the lattice grows in .

Note that there are still independent clusters which do not belong to the in

nitely large, s av → ∞

nite cluster,

and also empty sites (at least when p < 1). The in nite cluster is called the percolation