# Kuhn length

The **Kuhn length** is a theoretical treatment, developed by Werner Kuhn, in which a real polymer chain is considered as a collection of **Kuhn segments** each with a Kuhn length . Each Kuhn segment can be thought of as if they are freely jointed with each other.[1][2][3][4] 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 bonds and with fixed bond angles, torsion angles, and bond lengths, Kuhn considered an equivalent ideal chain with connected segments, now called Kuhn segments, that can orient in any random direction.

The length of a fully stretched chain is for the Kuhn segment chain.[5] 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 .

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 and bond angle θ with a dihedral angle energy potential, the average end-to-end distance can be obtained as

- ,

- where is the average cosine of the dihedral angle.

The fully stretched length . By equating the two expressions for and the two expressions for from the actual chain and the equivalent chain with Kuhn segments, the number of Kuhn segments and the Kuhn segment length can be obtained.

For worm-like chain, Kuhn length equals two times the persistence length.[6]

## References

- Flory, P.J. (1953)
*Principles of Polymer Chemistry*, Cornell Univ. Press, ISBN 0-8014-0134-8 - Flory, P.J. (1969)
*Statistical Mechanics of Chain Molecules*, Wiley, ISBN 0-470-26495-0; reissued 1989, ISBN 1-56990-019-1 - Rubinstein, M., Colby, R. H. (2003)
*Polymer Physics*, Oxford University Press, ISBN 0-19-852059-X - Doi, M.; Edwards, S. F. (1988).
*The Theory of Polymer Dynamics*. Volume 73 of International series of monographs on physics. Oxford science publications. p. 391. ISBN 0198520336. -
Michael Cross (October 2006),
*Physics 127a: Class Notes; Lecture 8: Polymers*(PDF), California Institute of Technology, retrieved 2013-02-20 - Gert R. Strobl (2007)
*The physics of polymers: concepts for understanding their structures and behavior*, Springer, ISBN 3-540-25278-9