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

16 1 Self-consistent field theory and its applications
opens up the possibility of low-temperature series approximations, but one first needs to locate
the ground-state trajectory by minimizing
where
E[r α ;0, 1] =
∫ 1
0
dsL(r α (s), r ′ α(s)) (1.38)
L(r α (s), r ′ α(s)) ≡ 3
2a 2 N |r′ α(s)| 2 + w(r α (s)) (1.39)
This is a standard calculus of variations problem, which, as demonstrated in the Appendix, is
equivalent to solving the Euler-Lagrange equation,
d
ds
( ∂L
∂r ′ α
)
− ∂L
∂r α
=0 (1.40)
This is a slight generalization of Eq. (1.331), due to the fact that r α (s) is a vector rather than
a scalar function. Nevertheless, the differentiation of L in Eq. (1.40) is easily performed and
leads to the differential equation,
3
a 2 N r′′ α (s) −∇w(r α(s)) = 0 (1.41)
Solving this vector equation subject to possible boundary conditions for r α (0) and r α (1)
provides the classical path.
Although Eq. (1.41) is sufficient, we can derive an alternative equation that provides a
useful relation between the local chain extension, |r ′ α(s)|, and the field energy, w(r α (s)).
This is done by dotting Eq. (1.41) with r ′ α (s) to give
3
a 2 N r′ α (s) · r′′ α (s) − r′ α (s) ·∇w(r α(s)) = 0 (1.42)
which, in turn, can be rewritten as
(
d 3|r
′
α (s)| 2 )
ds 2a 2 − d N ds w(r α(s)) = 0 (1.43)
Performing an integration reduces this to the first-order differential equation,
3|r ′ α (s)|2
2a 2 N − w(r α(s)) = constant (1.44)
In cases of appropriate symmetry, the classical trajectory can be derived from this scalar equation
alone.
Once the classical path is known, the average segment concentration is given by
φ α (r) = N ∫ 1
ds δ(r − r α (s)) (1.45)
ρ 0
0
and the entropic free energy is
f e
k B T = 3 ∫ 1
2a 2 ds |r ′
N
α(s)| 2 (1.46)
0
This entropic energy is often referred to as the elastic or stretching energy, because it has the
equivalent form to the stretching energy of a thin, elastic thread.