Biomedical Engineering Reference
In-Depth Information
Z
k
1
2
χ
=
χ
=
(
ε
− (
ε
+
ε
)),
(2.3)
AB
AB
AA
BB
T
B
where
is the number of nearest-neighbor contacts in a lattice
model of the polymer, and
Z
is the interaction energy per monomer
between A and B monomers. Polymer mixing is, therefore, controlled
by the product
ε
AB
χN
rather than
χ
as is the case for a simple monomer
blend. A critical value of
= 2 separates the case in which mixtures
of all compositions are stable (
χN
< 2) from the case in which
mixtures at some compositions will phase separate (
χN
χN
≥ 2). Since
N
(i.e.,
very weak repulsive monomer interactions) is required for the free
energy of mixing to favor phase separation over a single-phase
mixture of polymer chains.
In block copolymer melts, repulsive monomer interactions
are again a strong driving force for phase separation into A and B
rich domains. However, since each block is covalently tethered
to its reluctant partner, the scale of phase separation is limited
to macromolecular (mesoscopic) dimensions. The result is
spontaneous formation of a spectrum of periodic, ordered structures
on the 5-100 nm length scale, known as
can be extremely large, only a very small positive value of
χ
. At
sufficiently elevated temperatures, as with homopolymer blends,
entropic terms overwhelm the energetic interaction terms in the
free energy and result in a single disordered phase. The transition
from a homogeneous melt phase of copolymer chains to chemically
heterogeneous microdomains occurs on cooling at a critical value of
χN
microphase separation
known as the order-disorder transition (ODT). The morphology
of the microphase and the temperature of the ODT depend on the
composition of the copolymer, i.e., the volume fraction of one block
f
A
). The microphase morphology adopted must
balance minimizing unfavorable A-B interactions with the entropic
penalty of stretching the two blocks away from each other as the
chain adopts an extended configuration. For example, a symmetric
diblock (
(with
f
= 1 −
f
B
A
) will microphase separate into an alternating
lamellar morphology with a flat interface, as illustrated in Fig. 2.3a.
A simplified but illuminating treatment of this situation was given
by Bates and Fredrickson [2]. At low temperatures (large
f
=
f
A
B
), the
lamellar microdomains are nearly pure in A or B components,
χ
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