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

1.3 Strong-Stretching Theory (SST): The Classical Path 15
This equation provides the practical means of calculating the average segment concentration,
but there will also be instances where we need to identify it by the functional derivative,
φ α (r) =− N ρ 0
D ln(Q[w])
Dw(r)
≡− N ρ 0
lim
ɛ→0
ln(Q[w + ɛδ]) − ln(Q[w])
ɛ
(1.32)
The equivalence of this functional derivative to Eq. (1.31) follows almost immediately from
the alternative expression,
∫
Q[w] ∝
(
Dr α exp − 3
2a 2 N
∫ 1
0
ds|r ′ α(s)| 2 − ρ ∫
0
N
)
dr 1 w(r 1 ) ˆφ α (r 1 )
(1.33)
for the partition function. Note that Q[w + ɛδ] simply refers to the partition function evaluated
with w(r 1 ) replaced by w(r 1 )+ɛδ(r 1 − r). (See the Appendix for further information on
functional differentiation.)
Lastly, we calculate the entropy, S, or equivalently the entropic free energy, f e ≡−TS,
which, for reasons that will become apparent, is also called the elastic free energy of the
polymer. In accord with standard statistical mechanics, the entropic energy is
( )
f e Q[w]
k B T = − ln − ρ ∫
0
dr w(r)φ α (r) (1.34)
V N
The logarithm of Q[w] gives the total free energy of the polymer in the field, while the integral
removes the average internal energy acquired from the field, leaving behind the entropic energy.
The extra factor of V in the logarithm simply adds a constant to f e such that the entropic
energy is measured relative to the homogeneous state; it is easily confirmed that Eq. (1.34)
gives f e =0for the case of a uniform field.
Since the force exerted on the segments is given by the gradient of the field, an additive
constant to the field should have no effect on any physical quantities. Indeed, this is the case.
If a constant, w 0 , is added to the field such that w(r) ⇒ w(r) +w 0 , the partition functions
transform as
q(r,s) ⇒ q(r,s)exp(−sw 0 ) (1.35)
q † (r,s) ⇒ q † (r,s)exp(−(1 − s)w 0 ) (1.36)
Q[w] ⇒ Q[w]exp(−w 0 ) (1.37)
When these substitutions are entered into Eq. (1.31), all the exponential factors cancel, leaving
φ α (r) unaffected. Likewise, w 0 cancels out of Eq. (1.34) for f e , using the fact that the average
segment concentration is ¯φ α = N/ρ 0 V. This invariance will generally be used to set the
spatial average of the field, ¯w, to zero.
1.3 Strong-Stretching Theory (SST): The Classical Path
In circumstances where the field energy becomes large relative to the thermal energy, k B T ,
the polymer is effectively restricted to those configurations close to the ground state, referred
to as the classical path for reasons that will become apparent in the following section. This