General properties of polymer networks - Physical Gels from Biological and Synthetic Polymers - page 118

The zero-shear viscosity is expected to increase strongly at c m ~ c min . In weakly
compressed gels (c m > c min ), the authors assume that the deformation of micelles is
achieved through a hopping mechanism similar to the one adopted in microscopic

' free

volume theories of simple liquids. The creation of a void is necessary for a micelle to

jump into a new position. The process of hole formation implies an effective energy of
deformation of the micelles, where some of the bridges between micelles are stretched
while the micelle near the hole is compressed. The deformation barrier U associated with
a hole, for a weakly compressed gel, can be calculated ignoring the bridges. It is a
function of the degree of compaction X and of N agg . The activation energy of this process
is governed by the stress relaxation time

'

*.
For a strongly compressed gel (c m >>c min ), the mechanism of the elementary jump of a
micelle is different. The energy of vacancy is very high, but there is no need to create a
hole because the micelle can creep to another position without changing its volume, by
a motion which requires an (albeit considerable) deformation in shape of the given
micelle and of its neighbours. The corresponding free energy is (for shear amplitudes
of the order of 1)

τ

U ~GR m 3

: ð 4 : 19 Þ

Here G is again the elastic shear modulus and R m is the micellar size.
The stress relaxation time depends crucially on whether the deformation barrier U is
larger than the bridge exchange barrier B.IfU> B, the activation state is controlled by U
(micelles without bridges, with micelles and neighbours changing shape):

U ~

D F b þ U ; ð 4 : 20 Þ

Δ

where F b , the free energy of de-bridging of a given micelle from its neighbours, is

proportional to N 0 , the number of bridges per micelle.
On the other hand, when U< B,

U ~

D F b þ B : ð 4 : 21 Þ

Finally, the stress relaxation time τ

* is estimated as

τ ~ τ 0 exp U = ð k B T Þ: ð 4 : 22 Þ

The zero-shear viscosity ( 4.10 ) of the solution increases, and the most important con-
tribution to the viscosity comes from the energy barrier U* and therefore the change in
the characteristic relaxation time *.

Overall, the SJK model provides a more detailed analysis of the mechanism respon-
sible for the elasticity and the large viscosity increase in micellar gels of telechelic
polymers, via attractive interactions from bridging chains. Micelles are compared to
brushes that can slightly interpenetrate in the gel state. However, the predictions of the
model, such as the power-law dependence of the moduli with the aggregation number or
the polymer length, are dif

τ

cult to validate experimentally.