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

50 1 Self-consistent field theory and its applications
correspondence between k and θ due to the fact that a periodic compositional fluctuation of
wavevector, k, predominantly scatters radiation by a particular angle, θ, determined by the
Bragg condition outlined in Fig. 1.12. First-order constructive interference requires that the
path difference, ab − ac, equals one wavelength, λ, of the incident radiation. By simple
trigonometry, the path-length between points a and b is ab = D/ sin(θ/2), where D =2π/k
is the period of the composition fluctuation. The alternative path-length between a and c is
ac = ab cos(θ) =ab[1 − 2sin 2 (θ/2)]. Thus, the Bragg condition relating k and θ is
( ) λk
θ =2sin −1 (1.221)
4π
Second of all, the intensity of the scattered radiation, I(θ), is proportional to the ensemble average
of |φ A (k)| 2 , where the orientation of k fulfils the Bragg condition. Since the probability
of a harmonic fluctuation is given by the Boltzmann factor, exp(−cS −1 (k)|φ A (k)| 2 ), where
c is independent of k and φ A (k), it follows that 〈 |φ A (k)| 2〉 ∝ S(k). This, in turn, implies
I(θ) ∝ S(k), and hence the name, scattering function.
1.7.4 SCFT for a Homopolymer Interface
When a blend macrophase separates, there is inevitably an interface between the two phases
as illustrated in Fig 1.3(b). Here, we calculate its properties and determine the resulting excess
free energy per area, referred to as the interfacial tension, γ I . Since the interface is unfavorable,
it tends to be flat so as to minimize its total area, A. Therefore, we set up a Cartesian
coordinate system with the z axis perpendicular to the interface, and we consider a volume,
V = LA, where L is sufficiently large that the system attains the properties of the bulk A- and
B-rich phases at z = −L/2 and z = L/2, respectively. Because of the translational symmetry
parallel to the interface, there will be no spatial variation in the x and y directions. These facts
allow us to impose reflecting boundary conditions at the edges of V.
As for the brush in Section 1.6.6, we again solve this problem using the spectral method.
For the symmetry of our current problem, the appropriate basis functions are
{ 1 , if i =0
f i (z) = √ (1.222)
2cos(iπ(1 + 2z/L)) , if i>0
This is, in fact, the same set of functions used for the polymer brush in Eq. (1.166), but with
β =0. In this case, the eigenvalues of the Laplacian become
λ i =4π 2 i 2 (1.223)
and the symmetric tensor simplifies to
Γ ijk =(δ i+j,k + δ |i−j|,k )/ √ 2 (1.224)
when all the indices are nonzero. If one or more of the indices are zero, then it reduces to a
single delta function (e.g. Γ ij0 = δ ij when k =0).
The SCFT calculation for this system proceeds remarkably like that of the polymer brush,
with only a couple of key differences. The first difference occurs because there are now two