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

42 1 Self-consistent field theory and its applications
expressed as
ˆφ A (r) = N ρ 0
∑n A ∫ 1
α=1
0
ds δ(r − r A,α (s)) (1.188)
with an analogous expression specifying the B-segment concentration, ˆφB (r).
With the inclusion of two chemically distinct segments, it is no longer possible to sidestep
the issue of monomer-monomer interactions. For discussion purposes, assume that the
interaction between two monomers, i and j, obeys a simple Lennard-Jones potential,
[ ( ) 12 ( ) ] 6 σ
σ
u(r ij )=ɛ − 2
(1.189)
r ij r ij
where r ij is their separation. The incompressibility assumption
ˆφ A (r)+ˆφ B (r) =1 (1.190)
acts to maintain the spacing between neighboring monomers at the minimum of the potential
energy, r ij = σ. The detailed shape of the potential is therefore no longer needed; we can
simply assume that the interaction energy is −ɛ between neighboring monomers and zero
otherwise. In the brush system, all the interactions were the same and thus they added up
to a constant energy that could be ignored, assuming the total number of monomer contacts
remained more or less fixed. In the blend, however, we must account for the fact that the
depth of the potential will differ between the A-A, A-B, and B-B contacts. Noting that the
range of the monomer-monomer interactions, σ, is negligible on the coarse-grained scale, the
interactions can be treated as contact energies and thus the internal energy can be expressed
as
U ≡− N 2 ɛ AA
∑
∫
2ρ 2 ds dt δ(r A,α (s) − r A,β (t)) −
0 α,β
N 2 ɛ AB
∑
∫
ρ 2 ds dt δ(r A,α (s) − r B,β (t)) −
0 α,β
N 2 ɛ BB
∑
∫
2ρ 2 ds dt δ(r B,α (s) − r B,β (t)) (1.191)
0
α,β
Using the incompressibility assumption, ˆφ A (r)+ ˆφ B (r) =1, this simplifies to
U[ ˆφ A , ˆφ B ]
k B T
= χρ 0
∫
dr ˆφ A (r) ˆφ B (r) (1.192)
where the interaction strengths are grouped together into a single dimensionless parameter,
χ ≡ ɛ AA − 2ɛ AB + ɛ BB
2k B Tρ 0
(1.193)