# Kuhn length

Bond angle

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.[1][2][3] 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 $L=Nb$ for the Kuhn segment chain.[4] 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 $\langle R^2\rangle = Nb^2$.

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,[clarification needed] the average end-to-end distance can be obtained as

$\langle R^2 \rangle = n l^2 \frac{1+\cos(\theta)}{1-\cos(\theta)} \cdot \frac{1+\langle\cos(\textstyle\phi\,\!)\rangle}{1-\langle\cos (\textstyle\phi\,\!)\rangle}$,
where $\langle \cos(\textstyle\phi\,\!) \rangle$ is the average cosine of the dihedral angle.

The fully stretched length $L = nl\, \cos(\theta/2)$. By equating $\langle R^2 \rangle$ 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 semiflexible chain, Kuhn length equals two times the persistence length.[5]

## References

1. ^ Flory, P.J. (1953) Principles of Polymer Chemistry, Cornell Univ. Press, ISBN 0-8014-0134-8
2. ^ Flory, P.J. (1969) Statistical Mechanics of Chain Molecules, Wiley, ISBN 0-470-26495-0; reissued 1989, ISBN 1-56990-019-1
3. ^ Rubinstein, M., Colby, R. H. (2003)Polymer Physics, Oxford University Press, ISBN 0-19-852059-X
4. ^ Michael Cross (October 2006), Physics 127a: Class Notes; Lecture 8: Polymers, California Institute of Technology, retrieved 2013-02-20
5. ^ Gert R. Strobl (2007) The physics of polymers: concepts for understanding their structures and behavior, Springer, ISBN 3-540-25278-9