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

58 1 Self-consistent field theory and its applications
In this section, we consider a melt of n identical diblock copolymer molecules, occupying
a fixed volume of V = nN/ρ 0 . This time, there is only a single molecular species, and thus
one set of functions, r α (s) with α =1,2,...,n, is sufficient to specify the configuration of the
system. In terms of these trajectories, the dimensionless A-segment concentration is expressed
as
ˆφ A (r) = N ρ 0
∑
n ∫ f
α=1
0
ds δ(r − r α (s)) (1.265)
and that of the B segments is the same, but with the integration extending from s = f to s =1.
As before with the homopolymer blend, the segment interactions are described by the same
U[ ˆφ A , ˆφ B ] defined in Eq. (1.192).
1.8.1 SCFT for a Diblock Copolymer Melt
The SCFT for diblock copolymers (Helfand 1975) is remarkably similar to that for the homopolymer
blend considered in Section 1.7.1. The partition function for a block copolymer
melt,
Z ∝ 1 ∫ n
(
∏
˜Dr α exp − U[ ˆφ A , ˆφ
)
B ]
δ[1 −
n!
k B T
ˆφ A − ˆφ B ] (1.266)
α=1
is virtually the same, except that there are now only functional integrals over one molecular
type. Proceeding as before, the partition function is converted to the same form as Eq. (1.198)
for the polymer blend, but with
(
F
nk B T = − ln Q[WA ,W B ]
V
)
+ 1 ∫
V
dr[χNΦ A (1 − Φ A ) −
W A Φ A − W B (1 − Φ A )] (1.267)
where
∫
Q[W A ,W B ] ∝
(
˜Dr α exp −
∫ f
ds W A (r α (s)) −
∫ 1
0
f
ds W B (r α (s))
)
(1.268)
is identified as the partition function for a single diblock copolymer with its A and B blocks
subjected to the fields, W A (r) and W B (r), respectively. Note that an irrelevant constant of
one has been dropped from Eq. (1.267).
As always in SCFT, the free energy of the melt is approximated by F [φ A ,w A ,w B ], where
the functions, φ A , w A , and w B , correspond to a saddle point obtained by equating the functional
derivatives of Eq. (1.267) to zero. The derivative of F [Φ A ,W A ,W B ] with respect to
W A leads to the condition,
φ A (r) =−V D ln(Q[w A,w B ])
Dw A (r)
(1.269)