The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 76

where g ij is the bond conductance and vi i is the equilibrium voltage at each node (or site) i.
The formal analogy between Kirchhoff

s current law and the equilibrium between forces
acting upon a monomer appears in ( 3.15 ). A monomer i connected to bonds j with spring
constants k ij is in equilibrium when the sum of the forces exerted upon it by the j
neighbours is zero:

'

X

k ij ð u i u j Þ¼ 0 ; ð 3 : 16 Þ

j

where u i is the displacement of monomer i.
In the same way that the conductivity of a random network of resistors depends on the
fraction of conducting bonds, the shear modulus should depend on the fraction of cross-
links. The shear modulus is the measure of the elastic stiffness which opposes deforma-
tion, when volume is conserved. In the sol state, the shear modulus is zero. It is expected
by analogy with conductivity that the shear modulus should vary with the fraction of
bonds with the same exponent t ( Table 3.2 ). The critical exponent of the conductivity for
a resistor

insulator mixture is t = 1.6.
Before proceeding further, it is important to estimate the extent of the Ginzburg critical
domain for the gelation case. In general, this is very dif

-

cult to do; evaluation of the
extent of the critical domain remains the realm of the theorist, but the best estimate we
have, due to Stauffer, Coniglio and Adam (Stauffer et al., 1982 ), and a useful and
practical guide, is that the upper limit is |p/p c − 10 - 1 . In practice this can be very

1| ≤

dif cult to achieve, particularly for physical gels and networks. Indeed, it suggests

that, in order to test theory and experiment, really good data needs to be collected
when p/p c > 0.99 or better: a very testing requirement.
The increase of the cluster size near the threshold in the sol state induces signi cant

changes in the viscosity of the solution as the system approaches p c , corresponding to the
formation of branched species of larger size. Percolation on a lattice does not include the
presence of solvent. How can we predict the changes of the sol viscosity from numerical
simulations on lattices? The first attempt (Stauffer et al., 1982 ) was to express the viscosity

contribution of each cluster size in terms of the cluster radius and to refer to a monodisperse
suspension of solid spheres in solution, occupying a volume fraction

ϕ .TheEinstein

equation for dilute suspensions at very small volume fractions states that

5
2 þ…

¼ 0 1 þ ; ð 3 : 17 Þ

where η 0 is the solvent viscosity. In the case of gelation, when the distribution of polymer

sizes in solution is highly polydisperse, the volume fraction of the clusters of size s is ϕ s ≈

n s R s 3 . If one neglects the cluster - cluster interactions and considers the clusters as solid

spheres, Einstein ' s approximation leads to

!

X

n s R s þ…: ¼ 0 1 þ const : : ð 3 : 18 Þ

s