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

70 1 Self-consistent field theory and its applications
The only significant inaccuracy comes from the unit-cell approximation (UCA) (Matsen
and Whitmore 1996). An improved SST-based calculation (Likhtman and Semenov 1994)
with the proper unit cell of the C phase, but still assuming a circular interface and radial trajectories,
predicts a L/C phase boundary at f =0.293. Although a relatively minor problem
in this instance, the assumption of straight trajectories (in general, the shortest distance to an
interface (Likhtman and Semenov 1994)) minimizes the stretching energy without enforcing
the connectivity of the A and B blocks. This approximation can, in fact, lead to highly erroneous
predictions (Matsen 2003b). One way of enforcing the connectivity is to subdivide the
A and B domains into wedges along which the diblocks follow straight paths with a kink at
the interface (Olmsted and Milner 1998). Furthermore, the proper interface is not a perfect
circle, but is slightly perturbed towards the six corners of the hexagonal unit cell (Matsen
and Bates 1997b). Likhtman and Semenov (1997) have recently presented a method for performing
SST calculations that minimize the free energy with respect to the interfacial shape
while maintaining the connectivity of the blocks (the trajectory of each block is still straight
and the exclusion zones are still ignored), but the method is highly computational. With this
more accurate SST algorithm, they showed that the gyroid (G) phase becomes unstable in
the strong-segregation limit, consistent with the SCFT extrapolations by Matsen and Bates
(1996a), but unfortunately they did not calculate accurate values for the remaining L/C and
C/S boundaries. At present, our best SST estimates for these boundaries are f =0.294 and
f =0.109, respectively, obtained by adjusting the UCA predictions according to SCFT-based
corrections calculated by Matsen and Whitmore (1996). These values are denoted by arrows
at the top of Fig. 1.20, and indeed they correspond reasonably well with the SCFT boundaries
at finite χN.
1.9 Current Track Record and Future Outlook for SCFT
Although SCFT has only been demonstrated here for three relatively simple systems, it is a
fantastically versatile theory that can be applied to multi-component mixtures with any number
of species (Hong and Noolandi 1981; Matsen 1995a) and to polymer architectures of virtually
any complexity (Matsen and Schick 1994b; Matsen and Schick 1994c). Solvent molecules can
also be included in the theory (Naughton and Matsen 2002). Interactions are easily generalized
(Matsen 2002a) beyond the simple Flory-Huggins form in Eq. (1.192), and the constraint,
ˆφ A (r)+ˆφ B (r) =1, can be relaxed to allow for some degree of compressibility (Yeung et al.
1994). To include liquid-crystalline interactions (Netz and Schick 1996) or to account for
chain stiffness (Morse and Fredrickson 1994; Matsen 1996), the Gaussian model for flexible
polymers can be substituted by a worm-like chain model (Takahashi and Yunoki 1967), in
which the flexibility can be adjusted. The possibilities are truly limitless.
Not only is SCFT highly versatile, it also has a track record to rival any and all theories
in soft condensed matter physics. On the topic of polymer brushes, SCFT has recently
resolved (Matsen and Gardiner 2001) a discrepancy between SST (Leibler et al. 1994) and
experiment (Reiter and Khanna 2000) on the subtle effect of autophobic dewetting, where
a small interfacial tension of entropic origins causes a homopolymer film to dewet from a
chemically identical brush. However, to date, the majority of its triumphs have involved the
subtleties of block copolymer phase behavior. Most notably, SCFT correctly predicted the