by
J^<7(r,s; [w]) = ^∙V2g(r,s; [w]) - w(γ)q(γ,s; [w]), (8.22)
with the initial condition, <γ(r, 0; [w]) = 1. The average segment density of the single
chain in presence of the external field is given by
p(r; [w]) = <ρ(r)>jw] =
ʃ Drp(r) exp (—Z3t70[r] - βUι[r, w])
ʃ Drexp (-βU0[r] - ∕3Dι[r,w])
1
VQ[w]
ds q(γ, N — s; [w])q(γ, s; [w]).
(8.23)
We can also derive the density of the segment located at the contour position, s,
^ɪɪ^(r, N-s- [w])9(r, s- [w]).
(8.24)
These results (eqs. 8.20 and 8.23) are for linear homopolymers. The derivation can
be generalized to a variety of chain structures, as shown in detail in ref. [269].
8.2 From particles to fields
The preliminary step in a field-based approach, such as SCFT, is to convert the
standard particle-based model to a statistical field theory. This particle-to-field trans-
formation technique for a homopolymer in an explicit solvent is discussed here in
detail. The intramolecular long-range interferences and inter-molecular interactions
between the segments of the polymer chains as well as the interactions with the sol-
215
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