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

1.2 Gaussian Chain in an External Field 13
because Eq. (1.16) must hold for t → 0. Naturally q(r, r 0 , 0) = 0 when r ≠ r 0 , since s =0
implies that r and r 0 both correspond to the same point on the chain.
The recursion relation, Eq. (1.16), provides a means of building up the partition function
for a long chain fragment from those of very short fragments. This reduces the problem to
the easier one of finding q(r, r 0 ,ɛ), where ɛ is small enough that the field can be treated as a
constant for typical chain extensions of |r − r 0 | a(ɛN) 1/2 . In this case, the field simply
adds a constant energy of ɛw(r)k B T to each configuration, and thus does not affect the relative
probability of its configurations. Therefore, the partition function can be expressed as
( ) 3/2 3
q(r, r 0 ,ɛ) ≈ exp
(− 3|r − r 0| 2 )
2πɛ
2a 2 − ɛw(r)
(1.18)
Nɛ
which assumes the Gaussian distribution in Eq. (1.11) for a chain of ɛN segments in a homogeneous
environment, adjusted for the constant energy of the field and normalized so that
it reduces to Eq. (1.17) in the limit ɛ → 0. For extensions, |r − r 0 |, significantly beyond
a(ɛN) 1/2 , the partition function approaches zero regardless of whether or not the field energy
is well-approximated by ɛw(r)k B T . Hence, Eq. (1.18) remains accurate over all extensions,
provided that a(ɛN) 1/2 is sufficiently small relative to the spatial variations in w(r).
Now the partition function for large fragments can be calculated iteratively using
∫
1
q(r, r 0 ,s+ ɛ) =
dr
(a 2 N) 3/2 ɛ q(r, r + r ɛ ,ɛ)q(r + r ɛ , r 0 ,s) (1.19)
The first partition function, q(r, r + r ɛ ,ɛ), under the integral implies that we only need to
know the second one, q(r + r ɛ , r 0 ,ɛ), accurately for |r ɛ | a(ɛN) 1/2 . We therefore expand it
in the Taylor series,
q(r, r 0 ,s+ ɛ) =
∫
1
(a 2 N) 3/2
dr ɛ q(r, r + r ɛ ,ɛ) ×
[
1+r ɛ ·∇+ 1 ]
2 r ɛr ɛ : ∇∇ q(r, r 0 ,s) (1.20)
to second order in r ɛ . Here, we have used the dyadic notation (Goldstein 1980) for the secondorder
term. The expansion now makes it possible to explicitly integrate over r ɛ , giving
[
]
q(r, r 0 ,s+ ɛ) ≈ exp (−ɛw(r)) 1+ a2 N
6 ɛ∇2 q(r, r 0 ,s) (1.21)
[
]
≈ 1+ a2 N
6 ɛ∇2 − ɛw(r) q(r, r 0 ,s) (1.22)
to first order in ɛ. Alternatively, a Taylor-series expansion in s gives
[
q(r, r 0 ,s+ ɛ) ≈ 1+ɛ ∂ ]
q(r, r 0 ,s) (1.23)
∂s
By direct comparison, it immediately follows that
[
∂
a 2 ]
∂s q(r, r N
0,s)=
6 ∇2 − w(r) q(r, r 0 ,s) (1.24)