The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 67

with an encouraging proposal that the sol '

similar to those established for these other critical points, at least within the narrow range
of cross-linking parameters governing network formation close to the transition. In
thermodynamics, this parameter is temperature. In the case of gelation one has to
determine the microscopic parameter which enables the network formation for any
system.
One suggestion was to classify gelation as a

- gel transition should obey universal ' laws

fifth-order transition in the Ehrenfest sense
(Gordon and Ross-Murphy, 1975 ). This hypothesis is, of course, purely theoretical, as it
seems very unlikely that any experimental evidence could ever establish its truth or
otherwise. In any case, subtle effects such as the non-equilibrium states observed in many
physical gels, and reported throughout this topic, make the real situation still more
complicated.
In any second-order phase transition, the Landau theory de

'
which determines this approach to the critical temperature, and this parameter character-
izes the onset of order at the phase transition. It is a measure of the degree of order in a
system, with its extreme values being zero for total disorder, usually above the critical
point, and non-zero and approaching unity for complete order. In such a second-order
phase transition, the order parameter decreases continuously, following

nes an ' order parameter

'
when the measurements are performed very close to the critical temperature. In magnetic
transitions, the spontaneous magnetization M is the

' universal laws

' '

. At the critical
temperature, called the Curie temperature T C , when temperature decreases a spontaneous
magnetization appears and increases:

order parameter

M ≠ 0 for T 5 T C ;

M ¼ 0 for T > T C ;

5

M ! 0 when T ! T C with T T C :

At the critical point of the vapour - liquid phase transition, the ' order parameter ' is the

difference between liquid and vapour densities, ρ liq - ρ vap , a difference which decreases

smoothly towards zero, when temperature approaches the critical point. The universal
laws are in general power laws such as, for instance, for spontaneous magnetization:

Þ β ð

T C

for T 5 T C : ð 3 : 4 Þ

M ~ T

The exponent β is the same for spontaneous magnetization near the Curie point and near

the vapour - liquid critical point, β ≈ 0.32, as has been determined experimentally for such

diverse fluids as helium, xenon and water (Stauffer et al., 1982 ). This same exponent,

when derived from van der Waals equations, however, gives a different value, close to
β ≈

0.50.
The difference between these two values de

nes a different ' universality class ' . The

latter is that from the classical theory, where fluctuations are neglected near the phase

transition, while the former is deduced from, for example, renormalization group theory,
which takes into account the local

fluctuations of parameters, such as magnetization,

which actually diverge towards an in nite value at the approach of the critical point. The

theory of critical phenomena (Stanley, 1998 ) states that the thermodynamic properties of