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

12 1 Self-consistent field theory and its applications
The energy of a polymer configuration, r α (s), turns out to be a local property, meaning that
any given interval of the chain, s 1 ≤ s ≤ s 2 , can be assigned a definite quantity of energy,
E[r α ,s 1 ,s 2 ], independent of the rest of the chain. This will become a crucial property in the
derivation that follows. The formula for the energy,
E[r α ; s 1 ,s 2 ]
k B T
=
∫ s2
s 1
ds
( 3
2a 2 N |r′ α(s)| 2 + w(r α (s))
)
(1.14)
involves one term that accounts for the Gaussian probability from Eq. (1.11) and another term
for the energy of the field. Note that square brackets are used for E[r α ; s 1 ,s 2 ] to denote that
it is a functional (i.e., a function of a function).
The two primary quantities of 〈 the polymer that need to be calculated are the ensembleaveraged
concentration, φ α (r) ≡ ˆφα (r)〉
, and the configurational entropy, S. In order to
evaluate these, some partition functions are required. To begin, consider the first sN segments
(0 ≤ s ≤ 1) of the chain constraining the two ends at r α (0) = r 0 and r α (s) =r. The energy
of this fragment is E[r α ;0,s] and thus its partition function is
∫ (
q(r, r 0 ,s) ∝ Dr α exp − E[r α;0,s]
k B T
)
δ(r α (0) − r 0 )δ(r α (s) − r) (1.15)
This is a functional integral over all configurations, r α (t), for 0