less than the sum of two van der Waals radii of the carbon atoms, which means that
the carbon ends of the pentane segment will overlap with each other. Therefore, due
to their volume-exclusion interactions upon overlapping, such kinds of
conformations cannot be accepted.
2.2.4 Characterization of Static Semi-Flexibility of Polymers
The static flexibility of a semi-flexible polymer chain is related not only to the
potential energy differenceDe, but also to the temperatureT. The following
quantities are often used to characterize the conformational states of semi-flexible
polymer chains.
- Persistence length
The persistence lengthis theoretically defined by the projection of the chain end
along the direction of the first bond vector (Flory 1969 )as
bpb 0 expDe
kT(2.10)
whereb 0 is the projection of each backbone bond on the direction of chain exten-
sion. The persistence length represents the correlation length of the backbone-bond
orientations along the polymer chain. It is also used to describe the chain rigidity
from other sources besides hindered internal rotation, such as the charge interactions
along the polyelectrolyte chain, the double helix formation of DNA, microtubules,
and the conjugated covalent bonds in the liquid crystal polymers or conductive
polymers. This quantity originates from the worm-like-chain model describing the
semi-rigid polymer chains (Kratky and Porod 1949 ).
- Length of Kuhn segment
The Kuhn segmentis defined by the minimum freely jointed unit along the chain
(Kuhn 1936 ). Assuming that a chain containsnbackbone bonds, and each bond
contributesb 0 projection length, the projection length of the whole chain is thus
L¼nb 0 ¼nKbK (2.11)wherenKandbKare the number and the sequence length of Kuhn segments,
respectively. They make the mean-square end-to-end distance
<R^2 >¼nKb^2 K (2.12)The polymer chain formed bynKandbKis also called the equivalently freely
jointed chain. Therefore, the Kuhn segment lengthbKis
20 2 Structure–Property Relationships