the derivation is presented for a pure fluid of branched chain molecules having m
backbone segments and two branches at backbone segments ‘j’ and lk,, as shown in
fig. 7.la. The branch at ‘j’ has mj- segments enumerated as bj-l, bj2, ..., bjmj. This
notation is used to distinguish the branch segments from the backbone segments. The
generalization to chains with multiple branches will follow.
(a)
Figure 7.1: (a) Model of a branched chain molecule. The backbone has ‘m’ segments and
the two branches at backbone segment ‘j’ and ‘k’ are ‘b/ and ζb∕c,, respectively, bj has mj
segments while bʌ, has n⅛ segments. The total number of segments in the chain, N = m
+ mj∙ + rn⅛. (b) Schematic of the formation of a branched polymer chain of N segments
from N associating spheres. For a middle segment a , site ‘A’ associates with site ‘B’ on
the segment a' = a + 1 and site ‘B’ associates with site ‘A’ on the segment a' = a - 1.
The branch points on the backbone T and ‘k’ have additional associating site ‘C’ which
associates with the branch segments, 1bj m3, and ζb⅛ m⅛,, respectively. End segments 1, m,
tb7 1’ and ‘bfc 1’ have only one associating site, ‘A’ or ‘B’.
The grand free energy (Ω) of the chain fluid at fixed volume ( V), temperature ( T),
and chemical potential (μ) in the presence of an external field ( Vearf(R)) is related to
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