Self-Consistent Field Theory and Its Applications by M. W. Matsen
Self-Consistent Field Theory and Its Applications by M. W. Matsen
Self-Consistent Field Theory and Its Applications by M. W. Matsen
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
60 1 <strong>Self</strong>-consistent field theory <strong>and</strong> its applications<br />
w A (r) <strong>and</strong> w B (r) are sufficiently small such that q(r,f) is well described <strong>by</strong> Eq. (1.99) with<br />
the field w A (r) <strong>and</strong> similarly such that q † (r,f) is given <strong>by</strong> Eq. (1.100) with the field w B (r).<br />
These approximations are then inserted into Eq. (1.274) to give<br />
Q[w A ,w B ]<br />
V<br />
∫<br />
1<br />
≈ 1+<br />
2(2π) 3 dk[S 11 w A (−k)w A (k)+<br />
V<br />
S 22 w B (−k)w B (k)+2S 12 w A (−k)w B (k)] (1.276)<br />
where S 11 ≡ g(x, f), S 22 ≡ g(x, 1 − f), <strong>and</strong> S 12 ≡ h(x, f)h(x, 1 − f). Differentiating<br />
Q[w A ,w B ] with respect to the Fourier transforms of the fields, as we did in Section 1.7.3,<br />
gives the Fourier transforms of the concentrations,<br />
φ A (k) = −S 11 w A (k) − S 12 w B (k) (1.277)<br />
φ B (k) = −S 22 w B (k) − S 12 w A (k) (1.278)<br />
for all k ≠0. These equations are then inverted to obtain the expressions,<br />
w A (k) = −S 22φ A (k)+S 12 φ B (k)<br />
det(S)<br />
(1.279)<br />
w B (k) = S 12φ A (k) − S 11 φ B (k)<br />
det(S)<br />
(1.280)<br />
for the fields, where det(S) ≡ S 11 S 22 − S12 2 . At this point, the incompressibility condition<br />
can be used to set φ B (k) =−φ A (k), for all k ≠0. Substituting the expressions for the two<br />
fields into that for Q[w A ,w B ], <strong>and</strong> then inserting that into the logarithm of Eq. (1.273) gives<br />
F<br />
nk B T<br />
= χNf(1 − f)+<br />
N<br />
to second order in |φ A (k)|, where<br />
S −1 (k) =<br />
2(2π) 3 V<br />
∫<br />
k≠0<br />
dk S −1 (k)φ A (−k)φ A (k) (1.281)<br />
g(x, 1)<br />
− 2χ (1.282)<br />
Ndet(S)<br />
is the inverse of the scattering function for a disordered melt. In a somewhat analogous but<br />
considerably more complicated procedure, the scattering functions can also be evaluated for<br />
periodically ordered phases (Yeung et al. 1996; Shi et al. 1996). It involves a very elegant<br />
method akin to b<strong>and</strong>-structure calculations in solid-state physics, providing yet another<br />
example of where SCFT draws upon existing techniques from quantum mechanics.<br />
Figure 1.16 shows the disordered-state S(k) for diblock copolymers of symmetric composition,<br />
f =0.5, plotted for a series of χN values. As the interaction strength increases, S(k)<br />
develops a peak over a sphere of wavevectors at radius, kaN 1/2 =4.77, which corresponds<br />
to composition fluctuations of wavelength, D/aN 1/2 =1.318. The peak eventually diverges<br />
at χN =10.495, at which point the disordered state becomes unstable <strong>and</strong> switches <strong>by</strong> a<br />
continuous transition to the ordered lamellar phase. With the exception of f =0.5, the spinodal<br />
point is preempted <strong>by</strong> a discontinuous transition to the ordered phase, but an approximate<br />
treatment of this requires, at the very least, fourth-order terms in the free energy expansion of<br />
Eq. (1.281) (Leibler 1980). Better yet, the next section will show how to calculate the exact<br />
mean-field phase boundaries.