Biomedical Engineering Reference
In-Depth Information
Solvent
Unswollen
gel
Swoll
en
gel
Swelling and de-swelling of a gel cube.
Figure 4.3
since the gel is assumed to swell af
nely (that is, each side deforms proportional to the
l )
3
, so that the degree of swelling q is also equal to
3
. (For
macroscopic strain
ε
), V =(
ε
ε
very high degrees of swelling, this ratio could also be de
ned in terms of the mass of gel.)
The factors which affect swelling also need to be considered. The excluded-volume
effect determines that the average coil dimensions tend to increase, in order to maximize
the number of polymer segment
solvent interactions. For the bulk (unswollen) gel, the
chains will have approximately unperturbed dimensions, so when immersed in solvent
the polymer will swell by an amount depending on the polymer molecular mass M
w
:
-
R
g
i=h
R
g
i
0
≈
M
w
1
=
5
h
:
ð
4
:
3
Þ
In a poor solvent the polymer will appear to shrink, and eventually
R
g
i ≈
M
w
2
=
3
h
:
ð
4
:
4
Þ
The exponents 0.2 and 2/3 follow from excluded-volume theory (Yamakawa,
1971
).
Obviously the overall increase (or decrease) in dimensions will depend upon the original
dimensions; and the higher the molecular mass, the greater the effect. Indeed, for the gel,
the nominal overall weight average molecular mass M
w
is in
nite degree of
swelling would be predicted. This does not happen because the tendency to swell is
counteracted by the elastic restoring force from rubber elasticity theory.
The total swelling pressure
nite, so an in
Π
can be written more formally as
X
i
¼
1
;
4
π
i
;
¼
ð
4
:
5
Þ
where the individual pressure terms are
π
i
¼
D
F
i
1
ϕ
1
is the volume fraction of solvent. Here, for example,
Δ
and
F
1
is the free energy for
π
1
is the excluded-volume (or mixing) term from Flory
-
swelling and
Huggins theory,
with an associated polymer
-
solvent
'
goodness
'
parameter
χ
.
π
2
of opposite sign re
ects
the change in con
gurational free energy with swelling (the rubber elasticity term).
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