Biomedical Engineering Reference
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where G
0
is the shear modulus at this MW in bulk, c is the concentration and
ρ
is the
polymer density. The master curve represents the modulus
temperature relationship for
gels of 900 kg mol
−
1
aPS with no solvent. The plateau modulus of 3 × 10
6
Pa is close to
the 10
5
-
10
6
Pa range for aPS in the rubbery plateau region and is signi
-
cantly below the
10
9
Pa modulus of aPS in the glassy state.
In the Flory
Rehner model for an ideal rubber, the modulus should depend only on the
molecular mass of the chain between junctions; for gels, this corresponds to the molec-
ular mass M
a
between association points. According to this, the simplest form of rubber
elasticity theory,
-
cRT
M
a
2M
a
M
G
¼
1
:
ð
8
:
2
Þ
If (
8.2
) is obeyed, M
a
should be independent of M.The experiments on aPS show that
M
a
is indeed largely invariant with M for a reduced temperature value corresponding to
the plateau region. However, the average value of M
a
is considerably smaller than the
calculated entanglement molecular mass of various concentrations, so it is apparent that
the physical associations that form the gel network are not simply entanglements, in
agreement with structural investigations by IR and neutron scattering.
The calculation of M
a
also suggests a lower molecular mass limit for gelation of aPS,
and this was subsequently con
rmed by observation. A strong temperature dependence
of M
a
was established. For all but the lowest molecular masses, M
a
increases rapidly with
T,re
ecting the rapidly decreasing modulus. In terms of rubber theory, this corresponds
to the gradual melting out of associations. Neutron scattering measurements by Izumi
et al.(
1995
) con
rmed that the formation of physical associations is favoured by low-
ering the temperature. The authors proposed a re
ned model for the molecular mass
dependence of the plateau modulus, using a statistical approach taking account of the
amount of polymer not forming part of the gel network. The apparent molecular masses
M
a
0
are larger than the M
a
found using the average approach of (
8.2
). The discrepancy
between M
a
and M
a
0
may be a result of the simplifying assumption that the gel network
structure is the same as in cross-linked rubbers; the structure of gels is not necessarily
homogeneous since they contain multi-chain associations.
The results of Koltisko et al.(
1986
) therefore encourage further measurements on
thermoreversible aPS in the transparent gel region, along the lines adopted for conven-
tional rubbers, including speci
c physical associations. The lifetime of the associations is
probably suf
ciently long that the end of the rubbery plateau (relaxation of the shear
modulus) is not reported in currently published experiments. All of these theories also
assume that, as in ideal rubber theory, elasticity is purely entropic, which is unlikely to be
justi
ed.
Additional rheological data is available from the work of Xie et al.(
1990
), who
investigated gelation time as a function of temperature, polymer concentration and M
w
of aPS in CS
2
. A master curve was found for the gelation time
gelation temperature
curves at various polymer concentrations, by shifting the curves along the temperature
axis. However, the gelation point, where solutions lost their
-
fluidity, was determined by
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