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Kuhn length - Wikipedia

  • ️Wed Feb 20 2013

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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 {\displaystyle N} Kuhn segments each with a Kuhn length {\displaystyle b}. 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 {\displaystyle n} bonds and with fixed bond angles, torsion angles, and bond lengths, Kuhn considered an equivalent ideal chain with {\displaystyle N} connected segments, now called Kuhn segments, that can orient in any random direction.

The length of a fully stretched chain is {\displaystyle L=Nb} 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 mean square end-to-end distance for a chain satisfying the random walk model is {\displaystyle \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 {\displaystyle l} and bond angle θ with a dihedral angle energy potential,[clarification needed] the mean square end-to-end distance can be obtained as

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

The fully stretched length {\displaystyle L=nl\,\cos(\theta /2)}. By equating the two expressions for {\displaystyle \langle R^{2}\rangle } and the two expressions for {\displaystyle L} from the actual chain and the equivalent chain with Kuhn segments, the number of Kuhn segments {\displaystyle N} and the Kuhn segment length {\displaystyle b} can be obtained.

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

  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. ^ 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.
  5. ^ Michael Cross (October 2006), Physics 127a: Class Notes; Lecture 8: Polymers (PDF), California Institute of Technology, retrieved 2013-02-20
  6. ^ Gert R. Strobl (2007) The physics of polymers: concepts for understanding their structures and behavior, Springer, ISBN 3-540-25278-9