Biomedical Engineering Reference
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composition itself. If f (c) is the free energy density of a mixture of homogeneous
material, the approach shows that, in the unstable region, the term (
2
f
c
2
∂
=∂
Þ5
0
;
so
that
vectors) tend to increase the free energy
and therefore stabilize the homogeneous phase. On the other hand,
fluctuations with a short wavelength (large
q
fluctuations with large
wavelengths (small
vectors) stabilize the inhomogeneous phase and work in favour of
transformation into separate phases. K, the energy gradient coef
q
cient, is always positive
and a characteristic of the system, so demixing will take place when
Δ
G
q
< 0, and this
determines a critical wave vector q
c
:
"
#
c
0
þ
2Kq
2
∂
2
f
∂
c
2
D
G
q
~
ð
10
:
2
Þ
c
0
þ
2
f
∂
2Kq
2
q
2
q
c
;
5
0
when
5
∂
c
2
with
2
f
∂
∂
c
2
q
c
¼
c
0
:
ð
10
:
3
Þ
2K
Equation (
10.2
) states that
*
when q > q
c
the solution is stable with respect to concentration
fluctuations, so spinodal
fluctuations will decay
*
when q < q
c
the solution becomes inhomogeneous and spinodal decomposition occurs.
10.3.3
Kinetics of demixing
The difference in chemical potentials between the components is obtained from the
derivative of
Δ
G with respect to composition. The Fourier components of the concen-
tration
uctuation
δ
c(
q
,t) decay with a characteristic rate
ω
q
depending on the wave
vector
q
:
q
2
q
c
D
ef f
q
2
ω
q
¼
1
;
ð
10
:
4
Þ
where D
eff
is the effective diffusion coef
cient that changes sign at the spinodal line.
In the thermodynamically unstable region for wave vectors (q < q
c
) the characteristic
decay rates become negative (
fluctuations are not damp-
ened, so they grow with time. If the material is represented in one dimension
(
Figure 10.2
),
ω
q
< 0) and the concentration
ed with time and lead
to SD within the blend. This wavelength has a maximum at q
m
= q
c
/
fluctuations with large wavelengths are ampli
√
2 and it becomes
an
'
ampli
cation factor
'
. The theory predicts that one particular wavelength of
uctu-
ations in the composition q
m
grows most rapidly, leading to a well-de
ned peak in the
scattering pattern. Thus, at some time after phase separation starts, a description of the
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