Similar expressions can be derived for the segments in the branch ζbfc,.
This derivation can be generalized to chains with multiple branches. The deriva-
tion can also be extend to cases where the backbone segments have multiple branches.
In that case,
n
Bra(r) = ɪɪ Brα,i(r) (7.27)
i=l
where the product is over all the ‘n’ branches on the backbone segment ‘a’ and Brctil
is the branch factor from the branch T of (backbone) segment la,. /ɪ and I2 functions
for the backbone segments and the I1 for the individual branches remains same as
defined earlier. Only the initiator to the I2 function for branch segments changes to
account for the presence of multiple branches.
l2,b*i1(r) = ∕⅛'exp[Dα(r')] ɪɪ Brc^(r‰a(r')I2, ɛjr')ʌ^'^ŋ(r,r'), (7.28)
k=l,k≠i
where brctl denotes the branch 4i, at backbone segment a with ‘n’ branches.
We can also extend this derivation to cases where the branch segment themselves
have branches. In that case, we have to define the branch factors for all the segments
on the chain such that the branch factor is unity if the segment has no branches. The
functions Ii and I2 have to be defined for each of the linear branches and their parent
backbone segments. Using these functions the segment density profiles of the chain
molecule can be obtained.
Once the segment density profiles are obtained, the equilibrium grand free energy
194
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