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

28 1 Self-consistent field theory and its applications
Before moving on, we present a useful trick for evaluating f e that avoids the difficult step
of calculating g(z 0 ). It derives from the fact that E ⊥ [z α ;0, 1] = 0, which implies that for each
and every individual chain, the stretching energy equals minus the field energy. Thus,
f e (z 0 )
k B T = − 1 ∫ L
dz w(z)φ(z; z
aN 1/2 0 ) (1.122)
0
Substituting this into the integral of Eq. (1.121) gives
f e
k B T = − 1 L
∫ L
0
dz w(z)φ(z) =
3π2
8a 2 NL
∫ L
0
dz z 2 (1.123)
where we have made use of Eq. (1.118). The trivial integral of z 2 gives the previous result in
Eq. (1.121). An important advantage of this formula is that it easily extends to brushes grafted
to curved substrates, a fact that we will use in Section 1.8.4 for block copolymer morphologies.
1.6.2 Path-Integral Formalism for a Parabolic Potential
While the classical trajectory is the most probable, there are, nevertheless, nearby trajectories
that make a significant contribution to the partition function. To calculate their effect, we
expand about the lowest energy path in a Fourier sine series,
z α (s) =z 0 cos(πs/2) +
∞∑
z n sin(nπs) (1.124)
n=1
that constrains the two ends of the chain to z =0and z = z 0 . This allows the integrals to be
performed analytically, giving
where
E ⊥ [z α ;0, 1]
k B T
=
∞∑
k n zn 2 (1.125)
n=1
k n = 3π2 (4n 2 − 1)
16a 2 N
(1.126)
As required, k n > 0 for all n, implying that fluctuations about the classical path (i.e., z n ≠0)
increase the energy of the chain, more so for the higher harmonics.
Now we can evaluate the partition function, integrating over all possible paths, or equivalently
all possible amplitudes, z n , of each harmonic, n. The result is
∞∏
∫ ∞
∞
q(0,z 0 , 1) ∝ dz n exp(−k n zn 2 )= ∏
√ π
= constant (1.127)
k n
n=1
−∞
It might be disturbing that the product tends to zero as more factors are included, but there
is in reality a cutoff at n ≈ M, since sinusoidal fluctuations in the chain trajectory cease to
make any sense when the wavelength becomes smaller than the actual monomer size. What
n=1