Biomedical Engineering Reference
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separated by interfaces much narrower than the lamellar period
λ
— so-called “strong segregation.” Assuming that the chains are
uniformly stretched, an approximate expression for the free energy
per chain for the lamellar phase (
F
) is given by:
LAM
F
3(
λ
/2)
2
γA
k
LAM
k
= +
(2.4)
T
2
Na
2
T
B
B
(a)
(b)
λ
Figure 2.3
Microphase separation in diblock copolymers
.
The shaded
regions represent domains rich in A or B monomers. (a)
f
=
f
B
= 1/2,
A
> 1/2,
spontaneous curvature of domains toward the minority phase in gyroidal,
cylindrical, or spherical morphologies. The characteristic domain spacing
lamellar phase separation with flat domain interfaces. (b)
f
A
λ
is typically in the range 10-100 nm.
The first term in Eq. 2.4 is the entropic stretching penalty
for a Gaussian chain of
, extended to a
distance equal to half the lamellar domain period (
N
monomers, each of size
a
). The second
term represents the repulsive energetic interactions confined to
the (sharp) A-B interface, as a product of the contact area per
chain (
λ
). A simple treatment of the
polymer-polymer interface for highly immiscible polymers
A
) and the interfacial tension (
γ
gives us
1
). The volume-
filling condition (assuming incompressibility) constrains the contact
area per chain,
an estimate for the interfacial tension
γ
= (
k
T
/
a
)√(
χ
2
B
Aλ
/2=
Na
. Now, minimizing Eq. 2.4 with respect to
λ
,
3
we obtain an expression for the equilibrium lamellar period (
λ
0
):
λ
/2 ~
aχ
1/6
N
2/3
(2.5)
0
1
See, for example, reference [3].
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