Kuhn length

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title: "Kuhn length" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polymer-chemistry", "polymer-physics"] description: "Idealization in polymer thermodynamics" topic_path: "science/chemistry" source: "https://en.wikipedia.org/wiki/Kuhn_length" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Idealization in polymer thermodynamics ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/66/Molecule_1.jpg" caption="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. 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. 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 mean square 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, the mean square 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 the two expressions for \langle R^2 \rangle and the two expressions for L from 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.

References

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. (1988). "The Theory of Polymer Dynamics". Volume 73 of International series of monographs on physics. Oxford science publications.
5. Michael Cross. (October 2006). "Physics 127a: Class Notes; Lecture 8: Polymers". California Institute of Technology.
6. Gert R. Strobl (2007) ''The physics of polymers: concepts for understanding their structures and behavior'', Springer, {{ISBN. 3-540-25278-9

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