Self-Consistent Field Theory and Its Applications by M. W. Matsen

64 1 Self-consistent field theory and its applications
narrows as the segregation increases. Plots 1.17(b) and 1.17(c) show these trends on logarithmic
scales so as to highlight the well-known scaling behavior that emerges for stronglysegregated
microstructures. The dashed line in Fig. 1.17(b) denotes the domain scaling of
D ∝ aχ 1/6 N 2/3 , where the proportionality constant will be derived in the next Section using
SST. At intermediate segregations, the rate of increase is somewhat more rapid, consistent
with the fact that the experimentally measured exponent for N tends to be ∼ 0.8 (Almdal et al.
1990). The dashed and dotted lines in Fig. 1.17(c) show SST predictions for the interfacial
width, which will be discussed in the next Section.
L C S G PL
Figure 1.18: Minority domains from the periodically ordered phases observed in diblock
copolymer melts. The first three, lamellar (L), cylindrical (C), and spherical (S), are referred
to as the classical phases, and the latter two, gyroid (G) and perforated-lamellar (PL), are denoted
as the complex phases. Although the PL phase is often observed, our best experimental
evidence indicates that it is only ever metastable (Hajduk et al. 1997).
The lamellar phase is only observed for f ≈ 0.5. When the diblock copolymer becomes
sufficiently asymmetric, the domains transform into other periodic geometries as depicted in
Fig. 1.18. In the cylindrical (C) phase, the shorter minority blocks form cylindrical domains
aligned in a hexagonal packing with the longer majority blocks filling the intervening space.
In the spherical (S) phase, the minority blocks form spheres that generally pack in a bodycentered
cubic (bcc) arrangement. Note, however, that there are occasions where the spherical
(S cp ) phase prefers close-packed (e.g., fcc or hcp) arrangements. In addition to these classical
phases, two complex phases, gyroid (G) and perforated-lamellar (PL), have also been
observed. The minority domains of the G phase (Hajduk et al. 1994; Schulz et al. 1994)
form two interweaving networks composed of three-fold coordinated junctions. The PL phase
(Hamley et al. 1993) is much like the classical L phase, but the minority-component lamellae
develop perforations through which the majority-component layers become connected. The
perforations tend to be aligned hexagonally within the layers and staggered between neighboring
layers.
Fortunately, the only part of the spectral method that changes when considering the nonlamellar
morphologies are the basis functions, and these changes are entirely contained within
the values of λ i and Γ ijk . Take the C phase for example. Its hexagonal unit cell is shown
in Fig. 1.19 with a coordinated system defined such that the cylinders are aligned in the z