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

1.8 Block Copolymer Melts 67
a function of composition, f, and segregation, χN. This theoretical diagram is in excellent
agreement with experiment (Matsen 2002a) apart from a few minor differences. Experimental
phase diagrams (Bates et al. 1994) have varying degrees of asymmetry about f =0.5, but
this is well accounted for by conformational asymmetry (Matsen and Bates 1997a) occuring
due to unequal statistical lengths of the A and B segments. Aside from that, the small remaining
differences with experiment are generally attributed to fluctuation effects, which will
be discussed later in Section 1.10. According to the theory, the only complex microstructure
present in the equilibrium phase diagram is G, and based on extrapolations (Matsen and Bates
1996a) its region of stability pinches off at χN ≈ 60. Although the PL phase never, at any
point, possesses the lowest free energy, it is very close to being stable in the G region. This
integrates well with recent experiments (Hajduk et al. 1997) demonstrating that occurances
of PL eventually convert to G, given sufficient time. In addition to predicting a phase diagram
in agreement with experiment, the theory also provides powerful intuitive explanations
for the behavior in terms of spontaneous interfacial curvature and packing frustration (Matsen
2002a).
Many, particularly older, block copolymer calculations employ a unit-cell approximation
(UCA), where the hexagonal cell is replaced by a circular cell of equal area as depicted in
Fig. 1.19. This creates a rotational symmetry that transforms the two-dimensional diffusion
equation into a simpler one-dimensional version. An equivalent approximation is also available
for the spherical phase. The diblock copolymer phase diagram calculated (Vavasour and
Whitmore 1992) with this UCA is very similar to that in Fig. 1.20, except that it leaves out
the G phase as there are no UCA’s for the complex phases. Nevertheless, the complex phase
region is a small part of the diblock copolymer phase diagram, and thus its omission is not
particularly serious. While there is no longer any need to invoke this approximation for simple
diblock copolymer melts, there are many other systems with dauntingly large parameter
spaces (e.g., blends) that are still computationally demanding. For those that are dominated
by the classical phases, the omission of the complex phases is a small price to pay for the
tremendous computational advantage gained by implementing the UCA. The spectral method
can still be used exactly as described above, but with the expansion performed in terms of
Bessel functions (Matsen 2003a).
1.8.4 SST for the Ordered Phases
The free energy of a well-segregated diblock-copolymer microstructure has three main contributions,
the tension of the internal interface and the stretching energies of the A and B blocks
(Matsen and Bates 1997b). In the large-χN regime, these energies can all be approximated
by simple expressions, producing a very useful analytical strong-segregation theory (SST)
(Semenov 1985). In fact, the interface of a block copolymer melt is the same as that of a homopolymer
blend, to a first-order approximation. The derivation in Section 1.7.5 just requires
sufficiently steep field gradients at the interface such that the ground-state dominance applies,
and this continues to be the case. Therefore, the tension, γ I , in Eq. (1.244) and the interfacial
width, w I , in Eq. (1.239) also apply to block copolymer melts. Indeed, the approximation
for the width, shown by the dashed line in Fig. 1.17(c), is reasonable, although not nearly as