The Kuhn length is a theoretical treatment, developed by Werner Kuhn, in which a real polymer chain is considered as a collection of
N
Kuhn segments each with a Kuhn length
b
. Each Kuhn segment can be thought of as if they are freely jointed with each other. Each segment in a freely jointed chain can randomly orient in any direction without the influence of any forces, independent of the directions taken by other segments. Instead of considering a real chain consisting of
n
bonds and with fixed bond angles, torsion angles, and bond lengths, Kuhn considered an equivalent ideal chain with
N
connected segments, now called Kuhn segments, that can orient in any random direction.
The length of a fully stretched chain is
L
=
N
b
for the Kuhn segment chain. In the simplest treatment, such a chain follows the random walk model, where each step taken in a random direction is independent of the directions taken in the previous steps, forming a random coil. The average end-to-end distance for a chain satisfying the random walk model is
⟨
R
2
⟩
=
N
b
2
.
Since the space occupied by a segment in the polymer chain cannot be taken by another segment, a self-avoiding random walk model can also be used. The Kuhn segment construction is useful in that it allows complicated polymers to be treated with simplified models as either a random walk or a self-avoiding walk, which can simplify the treatment considerably.
For an actual homopolymer chain (consists of the same repeat units) with bond length
l
and bond angle θ with a dihedral angle energy potential, the average end-to-end distance can be obtained as
⟨
R
2
⟩
=
n
l
2
1
+
cos
(
θ
)
1
−
cos
(
θ
)
⋅
1
+
⟨
cos
(
ϕ
)
⟩
1
−
⟨
cos
(
ϕ
)
⟩
,
The fully stretched length
L
=
n
l
cos
(
θ
/
2
)
. By equating
⟨
R
2
⟩
and
L
for the actual chain and the equivalent chain with Kuhn segments, the number of Kuhn segments
N
and the Kuhn segment length
b
can be obtained.
For worm-like chain, Kuhn length equals two times the persistence length.
