The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 75

critical domain is the range of values of |p − p c |/p c where the critical exponents can be

observed.
Away from the threshold,

fluctuations decrease in amplitude, and in this case the
expected exponents are independent of the space dimensionality and are characterized by
the so-called mean field values. These values are listed in the last column of Table 3.2 .It

was shown by de Gennes ( 1977 ) and Daoud ( 1979 ) that the exponents expected in the
random cross-linking of high molecular mass polymers in the molten state are mean

eld
exponents, and this case is known as the vulcanization limit, by analogy with the cross-
linking of a rubber in a dense medium. In semi-dilute solutions, the width of the critical
domain is a function of the ratio c/c*, where c* is the overlap concentration; for instance,
for c =10c*, the width could be nearly 0.4.

3.4 Percolation and gelation

The connection between percolation and gelation was first suggested by Frisch and

Hammersley ( 1963 ) and was analysed in greater depth in independent papers by de
Gennes ( 1976 ) and Stauffer ( 1976 ) in 1976. The best analogy recognized is between
permanent cross-link formation in a chemical reaction and bond formation in a numerical
simulation. The extent of the reaction is the probability of bond formation; clusters are
branched macromolecules, while f is the functionality in a step growth process, as
discussed in Section 3.1 .
Table 3.3 gives the equivalence between the simulated process and a real cross-linking
process.
The equivalence between electrical conductivity and shear modulus appears at the
bottom of the table. Kirchhoff

s current law states that the sum of the electric currents at
each node of a network is zero:

'

X

g ij ð v i v j Þ¼ 0 ; ð 3 : 15 Þ

j

Table 3.3 Analogy between percolation and gelation.

Percolation Chemical gelation

Percolation threshold p c Gel point

Connected bond Reacted functional group

Fraction of connected bonds p Extent of reaction p

Lattice for p < p c Sol state

Lattice for p > p c Gel state

Mean cluster size s av (p), p < p c Average molecular mass M w (p)(p < p c )

Infinite cluster Gel network

Percolation probability P(p), p > p c Gel fraction

Coordination number z Functionality f

Network conductivity σ Equilibrium elastic modulus