(Helfand1975a). The theory was further developed by Hong and Noolandi ( 1981 ).
Helfand also applied this theoretical methodto the study of microdomain structures in
block copolymer systems (Helfand1975b), and tried to make precise computations of
the phase diagrams by numerical methods (Helfand and Wasserman 1976 ).
The microphase separation exhibits quite different behaviors at regions ofwr
between near and far away (r!1) from the order-disorder transition. The former is
normally regarded asweak segregation, while the latter is calledstrong segregation.
Leibler proposed the weak segregation theory by using the expansion of free energy
near the homogeneous phase (Leibler 1980 ). Semenov proposed the strong segregation
theory by separating the free energy directly into the interface contribution and the
stretching contribution (Semenov 1985 ). Since the gyroid phase structures of diblock
copolymers contain quite a lot of curved interfaces, unfavorable to the lowering of the
total free energy in the system, they could not be stable in the strong-segregation region.
Furthermore, at the lower end of the gyroid phase region, the stable Fddd orthorhombic
network phase has been discovered (Tyler and Morse 2005 ).
In order to obtain the precise computational results of phase diagrams for various
geometric features of microdomain structures, Matsen and Schick proposed the
reciprocal-space method for the numericalsolutions of the self-consistent mean-field
equations by the use of the crystal symmetry of the ordered phases (Matsen and Schick
1994 ). The results are shown in Fig.9.14, which have been well identified by experi-
mental observations. However, this method can only be applied to the stability study of
the known symmetric structures, and cannot predict the microdomain structures with
un-known symmetries. Therefore, Drolet and Fredrickson proposed the so-called real-
space method that starts from the randominitial field to obtain all the possible
symmetric structures of block copolymers via self-consistent iterations (Drolet and
Fredrickson 1999 ). Bohbot-Raviv and Wang proposed the similar but more efficient
method (Bohbot-Raviv and Wang 2000 ). But this approach cannot guarantee to obtain
the ordered phases with the minimum free energy, and the computation precision is not
superior to the reciprocal-space method. The more efficient method might be first to
obtain the ordered phase structures via the real-space method and then to evaluate their
thermodynamic stability via the reciprocal-space method according to their
Fig. 9.13 Illustration of the
disordered state and the
ordered state separated by the
critical segregation strength
182 9 Polymer Phase Separation