Biomedical Engineering Reference
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inhomogeneities in composition: the solution undergoes local concentration
fluctuations
in concentration, so that in some regions it is c
0
+
δ
c and in others c
0
− δ
c. ATaylor series
expansion of the change in the free energy
G around the composition c
0
shows that it is
related to the second derivative of the free energy with respect to concentration. If this is
positive,
Δ
fluctuations will die out and
the phase is stable. Conversely, in the unstable region of the phase diagram, any small
Δ
G accompanying the
fluctuation is positive, so the
fluctuations
will tend to grow. Since there is no thermodynamic barrier to reaction inside the spinodal
region, decomposition is determined solely by diffusion. If diffusion takes place in a
stable solution,
fluctuations in concentration that occur spontaneously because of thermal
from the high- to the low-
concentration domain and the system will return to equilibrium with a uniform density
(single phase). However, if the system is unstable, the mutual diffusion coef
the molecules will diffuse
'
downhill
'
cient is
negative and material is transported from regions of low concentration to regions of high
concentration (
), a process called spinodal decomposition (SD). Such
phase separation processes have been observed, for instance, in the solidi
'
uphill diffusion
'
cation of metal
alloys and in block copolymer blends (Hashimoto et al.,
1984
; Sasaki and Hashimoto,
1984
), and they differ qualitatively from those in the nucleation and growth process. For
a range of compositions that would give approximately equal volume fractions, and close
to the critical point composition, the two phases form an interconnected, often bicontin-
uous structure.
The theoretical formulation of SD was
first proposed by Cahn and Hilliard in a series
of papers (Cahn and Hilliard,
1958
,
1959
; Cahn,
1965
). Below the spinodal, the free
energy can be lowered (
D
G
5
0
) if the amplitude of the
fluctuations is large enough, and
so these
fluctuations will tend to increase. Then the inhomogeneous phase is no longer
stable, and the so-called demixing with SD occurs. Consequently, if the system is
quenched rapidly at a composition close to the critical composition, the spinodal mech-
anism is likely to take place, while systems quenched slowly through the metastable
region will decompose by nucleation and growth steps, allowing critical nuclei to be
formed (Papon et al.,
2002
).
Consider an inhomogeneous solution whose composition everywhere differs only
slightly from the average composition, and with small composition gradients. The
difference between the free energy of inhomogeneous and homogeneous parts of the
system corresponds to the free energy of creation of the new phase, written
ð
"
1
2
2
#
dV
2
f
∂
∂
2
D
G
¼
ð
c
c
0
Þ
þ
K
jr
c
j
;
ð
10
:
1
Þ
c
2
c
0
where c
0
is the overall composition and f(c) is the free energy density of a mixture of a
homogeneous material with composition c. The additional term K
2
accounts for the
contribution to the free energy density of small concentration gradients.
In the development of the Cahn
jr
c
j
-
Hilliard theory, in order to predict the time and spatial
dependence of
fluctuations in the reciprocal space), it is convenient to
recast the theory in terms of Fourier components of the composition, rather than the
fluctuations (or the
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