Biomedical Engineering Reference
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composition in the solution will be a superposition of sine waves of
fixed wavelength
q = q
m
, but random in orientation, phase and amplitude. A periodic distribution of
higher and lower concentrations increases, with a characteristic periodicity length
Λ
m
=2
/q
m
predicted.
Overall, then, the Cahn
π
-
Hilliard linearized model suggests:
(a) The logarithm of the scattered intensity plotted against time should increase linearly.
(b) A plot of
ω
q
/q
2
versus q
2
(Cahn
-
Hilliard plot) is a straight line if the effective
cient D
eff
is independent of q. From the slope and the intercept, D
eff
and q
m
can be calculated.
(c) In order to compare the coarsening of phase separating systems at various temper-
atures and test for universal behaviour, the processes are considered on a reduced
time scale, which is the time it takes to build up an inhomogeneity of characteristic
size 1/q
m
. The characteristic time
diffusion coef
τ
for the early stages of SD can be de
ned by
1
D
eff
q
2
m
:
τ ¼
ð
10
:
5
Þ
10.3.4
Spinodal decomposition in polymer mixtures
A speci
c model was developed for polymer mixtures by de Gennes (
1980
) for incom-
pressible binary liquids composed of macromolecules, which he called the
'
metallurgy of
polymer blends
'
. de Gennes discussed the qualitative features of the concentration
fluctuations of binary polymer melts of two species A and B, with individual molecular
masses (or weights) N
A
and N
B
, starting with a description of the free energy density of
homogeneous mixtures using the Flory
Huggins formula.
The interaction parameter or Flory parameter
-
is often positive in polymer
blends and favours segregation. If the two species are not too different,
χ
χ
is small.
Therefore the de Gennes model mainly concerns the case of small positive
. Provided
that both N
A
and N
B
are large, the static properties of the blends are supposed to be well
described by the mean
χ
field approach. For instance, in the symmetric case (N
A
= N
B
= N)
there is a critical value
χ
s
(spinodal point or critical point of the polymer blend)
de
ned by
1
χ
s
¼
2Nc
ð
1
c
Þ;
ð
10
:
6
Þ
where c is the concentration of polymer A. The
χ
parameter is temperature-dependent.
For
χ
s
phase separation may occur.
As far as the early stages of demixing kinetics are concerned, by comparison with
other systems, demixing in polymer mixtures proceeds very slowly, so that the initial
stages should be studied more easily. Among others, Hashimoto et al.(
1983
)showed
by time-resolved light scattering studies that binary polymer mixtures of poly(styrene)
(PS) and poly(vinylmethylether) (PVME) obeyed de Gennes
χ
<
χ
s
the system is entirely miscible, while for
χ
>
'
s predictions for the early
stage of SD.
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