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

30 1 Self-consistent field theory and its applications
1.2
φ(z;z 0
)
0.8
0.4
1
2
z 0
/aN 1/2 = 4
0.0
0 1 2 3 4
z/aN 1/2
Figure 1.6: Segment distributions, φ(z; z 0), of polymers extended to various z = z 0 in the
parabolic potential, Eq. (1.113). The dashed curves denote the distributions, Eq. (1.117), of the
classical paths.
The strong-stretching approximation is inaccurate until z 0 /aN 1/2 1, and even then it is
rather poor for z ≈ z 0 . Of course this should not be surprising, as there is relatively little
tension at the free end of the chain.
Now we evaluate the average elastic energy, f e (z 0 ), of a fluctuating chain extended to
z = z 0 . This is obtained by taking the free energy of a chain in the parabolic potential and
subtracting off the average field energy. In mathematical terms,
∫
f e (z 0 )
k B T
= − ln q(0,z 0, 1) − dz w(z)φ(z; z 0 ) (1.135)
Although there is no analytical formula for φ(z; z 0 ), Eq. (1.116) allows the field energy to be
expressed as
∫ 1
∫ L
ds dz w(z)ρ(z; z 0 ,s)=− 3π2 z0
2
0 0
16a 2 N − 1 (1.136)
4
which can be evaluated by integrating first over z and then s. Therefore, the entropic free
energy of the chain is
f e (z 0 )
k B T = 3π2 z0
2
16a 2 + constant (1.137)
N
which is precisely the same as predicted by SST, Eq. (1.120), apart from the additive constant.
The unknown constant appears because we are unable to specify the proportionality constant
in Eq. (1.127), which in turn is because the coarse-graining procedure prevents us from knowing
the absolute entropy of the chain; the best we can do is calculate relative changes in
entropy.