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

1.7 Polymer Blends 49
where S(k), referred to as the scattering function, is defined by
S −1 (k) =
1
Nφ(1 − φ)g(k 2 a 2 − 2χ (1.218)
N/6, 1)
The scattering function for a 50-50 blend is plotted in Fig. 1.11 at several values of χN
within the one-phase region of the phase diagram in Fig. 1.10(b). The inverse of S(k) determines
the free energy cost of producing a compositional fluctuation of wavelength, D =2π/k.
In the one-phase region, S −1 (k) > 0 for all k, implying that all fluctuations cost energy and
are thus suppressed. However, as the two-phase region is entered, S −1 (k) becomes negative,
starting with the longest wavelengths. For small k,
S −1 (k) ≈ 1+ 1 18 k2 a 2 N
Nφ(1 − φ)
− 2χ (1.219)
which implies that the blend becomes unstable to long-wavelength fluctuations at
χN =
1
2φ(1 − φ)
(1.220)
This is precisely the spinodal line, Eq. (1.215), calculated in the preceding section.
incident ray (wavelength = λ)
A(k)
θ
scattered rays (wavelength = λ)
D
a
θ/2
b
c
Figure 1.12: Diagram showing scattered rays of wavelength, λ, due to a composition fluctuation,
φ A(k), of period, D ≡ 2π/k. The sketch at the bottom shows the condition for a
first-order Bragg reflection, where the path difference, ab − ac, equals one wavelength, λ.
The quantity, S(k), is called the scattering function because of its direct relevance to
the pattern, I(θ), produced by the scattering of radiation. First of all, there is a one-to-one