Biomedical Engineering Reference
In-Depth Information
6
5
4
3
2
1
4
2
0
2
4
6
ω a T (rad s 1 )
log
Figure 4.14 Master curves for storage modulus for a series of poly(butadiene)s with various degrees of
modi
). Reduced temperature is 0°C;
molecular mass M n = 26 000 g mol 1 . Adapted from de Lucca Freitas and Stadler (1987) © 1987
American Chemical Society.
cation x (mol%): x =0(
), 0.5 (
), 2(
), 5 (
), 7.5 (
Let p 0 be the average number of stickers which are closed and
τ
the average lifetime of a
sticker in the associated state. For lifetimes longer than
is the
average time of a sticker in the associated state), the dynamics of the network changes as
stickers detach from one tie point and reassociate at another tie point.
The chains can diffuse in the reversible network and the stress can relax. The model is
based on the topology of the chain in a con
τ
(frequencies < 1/
τ
, where
τ
ning tube. The chains diffuse by a reptation
mechanism (Doi and Edwards, 1986 ). Three important relaxation times must be consid-
ered:
, the average time of a sticker in the
associated state; and T d , the time required for a chain to escape from its tube by curvi-
linear diffusion.
There are four time regimes that are important for the stress relaxation modulus of
reversible gels. At times shorter than
τ e , the Rouse time of an entanglement strand;
τ
τ e , relaxation is indistinguishable from that in the
polymer without stickers.
On time scales
τ e < t <
τ
there is a rubbery plateau analogous to the one observed in
permanently cross-linked networks or entangled melts, with a modulus which has
contributions from both cross-links and entanglements. The plateau modulus G 1 is
then the product of the number density of elastically active network strands and the
stored energy per strand, k B T. If one makes the assumption that the number densities of
entanglements and cross-links are additive, then G 1 can be written as
p 0
N S þ
1
N ent
G 1
cRT
;
ð
4
:
23
Þ
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