backbone segments, other than the segments ej, and ‘k’, are defined equal to unity.
The recursive function I1 for the backbone segments are redefined as, ∕ι,ι(r) = 1 and
ʃi,j(r) = ʃ <⅛'∕jj-ι(r') exp[Dj-ι(r')]βrj∙-1(r')Δ^-1,∙7∖r, r'). Reasons for defining the
branch factors for backbone segments which do not have a branch and the I1 function
this way, help to generalize the derivation to chains with multiple branches.
Now substituting for (from eqn. 7.15) and Xjc (from eqn. 7.16) in the Euler-
Lagrange equation for backbone segment ‘j’ gives,
j bjr∏j
= eχp(β ∑ exp(0 Σ Mi)-Brj∙(rj) exp[Γj(rj)]∕ιj(rj). (7.17)
i=l i=bjl
Hence, for a backbone segment la, after segment ζj,,
1 ɑ-ɪ
—— = exp(β∑*μi)I1,a(ra). (7.18)
λbW i=1
where, ∑*Γ=ι1 ɪɛ ɑɪθ sum 0ver θɪɪ ɑɪθ backbone as well as the branch segments from
(backbone) segment T to (backbone) segment ‘a — Γ. Physically, this is depicted
in fig. 7.2a. This specifies the bonding sites ‘B’ on all the backbone segments of
the chain. Evidently, Xβ,s relate the chemical potential of the segment la, to the
environment experienced by segments connected to ia, through the bonding site lB,.
Such sharing of information along a molecule is essential to modeling the structure
of (heteronuclear) chain molecules with different segment types such as copolymers.
This also holds true for the other bonding sites on the segments. Note that we can
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