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

54 1 Self-consistent field theory and its applications
for an arbitrary field, w(r). Applying the equivalent steps here to the Euler-Lagrange equation
leads to
a 2 N
6 [Θ′ (z)] 2 − χN sin 2 (Θ(z)) cos 2 (Θ(z)) = constant (1.237)
The boundary conditions on Θ(z) coupled with the fact that the profile must become flat in
the limit, z →±∞, implies that the constant of integration must be zero, and thus
where
w I Θ ′ (z) =2sin(Θ(z)) cos(Θ(z)) (1.238)
w I ≡
2a √ 6χ
(1.239)
Returning to the original function, φ A (z), Eq. (1.238) becomes
w I φ ′ A(z) =4φ A (z)[1 − φ A (z)] (1.240)
which can be rearranged as
dφ A
φ A (1 − φ A ) = 4dz
(1.241)
w I
Integrating this equation, using the method of partial fractions on the right-hand side, and
setting the constant of integration such that φ A (0) = 1/2 results in the predicted profile,
φ A (z) = 1 [ ( )] 2z
1+tanh
(1.242)
2
w I
Referring to the definition in Eq. (1.232), the constant, w I , can now be interpreted as the
interfacial width. This approximate width is plotted as a dashed line in Fig. 1.13(b) along side
the SCFT prediction, and indeed it is reasonably accurate for χN 10.
Inserting the calculated profile into the free energy expression, Eq. (1.234), gives
F
nk B T = aN L
√ χ
6
(1.243)
where the integral, ∫ dx cosh −2 (x) =2, has been used. Given that the free energy of the two
bulk phases is F s =0in the strong-segregation limit, it follows from Eq. (1.233) that the
interfacial tension is
γ I = k B Taρ 0
√ χ
6
(1.244)
Figure 1.14 demonstrates that this well-known estimate becomes reasonably good once χN
10.