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