The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 68

Water inlet

Coffee

Vapour

Water outlet

Percolation transition in a porous medium, some passages being blocked. The connectivity
between the two sides of the porous medium is achieved by a single path via which water
can ow.

Figure 3.3

a system near a phase transition depend only on a small number of features, such as
dimensionality and symmetry, and are insensitive to the underlying microscopic proper-
ties of the system. This presentation may seem obscure for non-specialists in phase
transitions, but we shall try to show that this is the origin of a new and uni ed description

of gelation in terms of what is now known as percolation.
History tells us that the term

was given by the mathematician J. M.
Hammersley ( 1957 ) to a statistical geometric model which reminded him of the passage
of a

' percolation '

fluid through a network of channels in which of the channels, randomly distributed,
were blocked. A sketch of the percolation transitions through a porous medium, in this
case using a traditional Italian espresso pot, is shown in Figure 3.3 .
The percolation path allows the liquid to

flow through the medium. Another analogy is
provided by the electric circuit represented by a square lattice of interconnections, where
electric current circulates between two electrodes connected to a voltage source; see
Figure 3.4 . The interconnections are cut randomly. The current decreases until a critical
fraction of bonds is cut and then the current vanishes. Of course, when the current stops
circulating not all bonds have been cut, but there is no connected path between the two
electrodes below a critical fraction of bonds. This critical fraction depends on the type of
network. The transition from conducting to non-conducting network can be extremely
sharp when the lattice is very large, i.e. when the L, the distance between the electrodes, is
very large compared to a segment a; the mathematical limit is L/a >> 1.
In the limit of an in nitely large system, the critical value of the fraction of bonds is

also sharply de , the threshold is distributed around

the critical value. When the analogy between percolation and a physical phenomenon is

ned. For small lattices, of ' 'finite size '