Biomedical Engineering Reference
In-Depth Information
Although other theories were developed in the intervening years, that presented here
(and referred to as the TE model) was proposed in a series of papers in 1992 by Tanaka
and Edwards (
1992a
,
1992b
,
1992c
). This model deals with
flexible polymers with a
uniform molecular mass M, each carrying associative groups at both ends, as represented
schematically in
Figure 4.6
, and deals with the un-entangled regime M<<M
e
.
The junctions of the network are created by the
functional groups attached at
the ends of the chains. There are two kinds of chains in the network: elastically active
chains and dangling chains, the latter being chains with only one end attached to a
junction, shown by open circles in
Figure 4.6
. The TE model introduces a potential
barrier W for the dissociation of an end reactive group from a junction. One end can
dissociate from a junction either by its own thermal motion or by being pulled by the chain
attached to it. The probability for an isolated end group (with no chain connected to it) to
dissociate is exp[
'
sticky
'
−
(W/k
B
T)] and hence the dissociation rate of the reactive group is given by
β
0
¼ ω
0
exp
ð
W
=
k
B
T
Þ;
ð
4
:
11
Þ
where
ω
0
is a characteristic frequency of thermal vibration of the reactive group,
estimated to be of the order of 10
8
10
9
s
−
1
.
Because the reactive group is attached to a chain, an additional force works in the
direction of the other end of the chain, and the potential barrier for the attached end to
dissociate is reduced, owing to the elasticity of the chain. The TE model introduces a
disentanglement rate
-
β
(r) which depends on the end-to-end distance of the attached
chain:
3r
Na
s
;
βð
r
Þ¼β
0
exp
ð
4
:
12
Þ
where N is the number of monomers (residues) in the chain and a
s
is the monomer size.
This correction accounts for the non-Newtonian behaviour in steady-state shear, because
the chains can stretch under shear.
As well as this, a dangling chain can also capture one of the junctions in its neighbour-
hood by its reactive end, with a probability per unit time which is the recombination rate
p
r
. In the TE model, p
r
is given by
p
r
≈ β
0
exp
ð
B
=
k
B
T
Þ;
ð
4
:
13
Þ
where B is the binding energy.
When n is the total number of chains per unit volume, the equilibrium number of active
chains
ν
0
under no external force is given by
ν
0
n
≈
exp
ð
B
=
k
B
T
Þ
Þ
:
ð
4
:
14
Þ
1
þ
exp
ð
B
=
k
B
T
The TE model predicts the linear viscoelastic response of the transient networks as a
function of the angular frequency at various temperatures. It shows that the modulus
-
frequency curves at any temperature T can be superposed on the curve at the reference
Search WWH ::
Custom Search