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

1.8 Block Copolymer Melts 69
Second of all, the symmetry requires that the strongly-stretched chains follow straight trajectories
in the radial direction, for which Eq. (1.123) is easily generalized. The stretching
energy of the γ-type blocks (γ =A or B) becomes
f γ,e
k B T = 3π2
8a 2 × 1
N γ V γ
∫
dr z 2 d (1.305)
which involves an average of z 2 d over the volume, V γ, of the γ domain, where z d is the radial
distance to the interface and N γ is the number of segments in the γ-type block. For a C phase
with A-type cylinders and a B-type matrix, the stretching energy of the A blocks becomes
f A,e
k B T = 3π 2 ∫ RI
4a 2 Nf 2 R 2 dρ ρ(ρ − R I ) 2 = π2 R 2
0
16a 2 (1.306)
N
and that of the B blocks is
f B,e
k B T = 3π 2 ∫ R
4a 2 N(1 − f) 2 R 2 dρ ρ(ρ − R I ) 2 = π2 R 2
R I
16a 2 N α B (1.307)
where
α B ≡ (1 − √ f) 3 (3 + √ f)
(1 − f) 2 (1.308)
Combining the three contributions, the total free energy of the C phase is written as
√ ( ) −1 ( ) 2
F C χNf R
nk B T =2 + π2 (1 + α B ) R
(1.309)
6 aN 1/2 16 aN 1/2
Again, minimization leads to a domain size that scales as R ∼ aχ 1/6 N 2/3 , which when
inserted back into Eq. (1.309), gives
F C
nk B T = 1 4 (18π2 (1 + α B )χN) 1/3 (1.310)
Comparing F L and F C , we find that the C phase has a lower free energy than the L phase for
f0.7009. The analogous calculation for the spherical (S)
phase (Matsen and Bates 1997b) predicts C/S boundaries at f =0.1172 and f =0.8828.
There is one problem that has been overlooked. Equation (1.305) provided the stretching
energies while avoiding the need to work out the distribution of chain ends. However, had
we evaluated the concentration of B ends, we would have encountered a negative distribution
along side the interface. Of course a negative concentration is unphysical. The proper solution
(Ball et al. 1991) has instead a narrow exclusion zone next to the interface that is free of
chain ends, and as a consequence the SST field, w B (r), deviates slightly from the parabolic
potential. (The classical-mechanical analogy with a simple harmonic oscillator used in Section
1.6.1 only holds if the chain ends have a finite population over the entire domain.) Fortunately,
the effect is exceptionally small and can be safely ignored (Matsen and Whitmore 1996).