Biomedical Engineering Reference
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where
F
is the free energy in the absence of the field,
ε
(
r
) is the
0
local dielectric constant, and
E
(
r
) the electric field [31]. We assume
a linear dielectric response.
Thurn-Albrecht
gave an informative evaluation of Eq. 2.7
for the simplified picture of an infinitely long, isolated cylinder
enclosed in an infinite matrix, with dielectric constants
et al.
ε
and
ε
,
1
2
∇⋅
respectively [32]. Solving Maxwell's equation (
)
to find the field distribution and substituting into Eq. 2.7, the
difference in free energy per unit volume between states with the
long axis parallel (
[
ε
(
r
)
E
(
r
)] =
ρ
free
F
) or perpendicular (
F
) to the field vector is
∥
⊥
1
2
(
ε
−
ε
)
2
1
2
Δ
F
=
F
−
F
= − |
E
|
2
ε
. (2.8)
∥
⊥
0
0
ε
+
ε
1
2
Since Δ
is always negative, the orientation of the cylinder axis
parallel to the applied field is always the lower free energy state,
whether
F
ε
>
ε
or
ε
<
ε
. In this orientation, all spatial gradients
1
2
1
2
of
dielectric
constant
(sharp
interfaces
in
this
picture)
are
perpendicular to the field vector; that is,
dielectric interfaces tend
to align parallel to the electric field
. The material heterogeneity of a
microphase-separated copolymer has an associated spatial variation
in the dielectric constant. As a consequence, anisotropic copolymer
phases such as L and C will favor an orientation with their long axes
parallel to the electric field vector. Admundson
calculated the
free-energy contribution for a more realistic description of arrays of
block copolymer microdomains by expanding the dielectric constant
in powers of the local composition amplitude (the volume fraction
of one component minus its mean value) [33, 34]. The essential
physics of electric field alignment for domains with sharp interfaces
is, however, captured by Eq. 2.8. Most important, the driving force
is proportional to both the square of the dielectric contrast (Δ
et al.
),
and the square of the electric field strength. In practice, Thurn-
Albrecht found that a cylinder-forming PS-
ε
-PMMA copolymer film
(with dielectric constants at elevated temperatures of
b
ε
= 6 and
PMMA
= 2.45) could be completely aligned on a gold substrate in a
40 V/μm field.
There are numerous theoretical [33-35] and experimental
[36-39] studies of the electric field alignment of block copolymers
using both in-plane and vertical fields. Wang
ε
PS
et al.
found that ion
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