Chemistry: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 [math]\displaystyle{ N }[/math] Kuhn segments each with a Kuhn length [math]\displaystyle{ b }[/math]. 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 [math]\displaystyle{ n }[/math] bonds and with fixed bond angles, torsion angles, and bond lengths, Kuhn considered an equivalent ideal chain with [math]\displaystyle{ N }[/math] connected segments, now called Kuhn segments, that can orient in any random direction.
The length of a fully stretched chain is [math]\displaystyle{ L=Nb }[/math] 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 [math]\displaystyle{ \langle R^2\rangle = Nb^2 }[/math].
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 [math]\displaystyle{ l }[/math] and bond angle θ with a dihedral angle energy potential,[clarification needed] the average end-to-end distance can be obtained as
- [math]\displaystyle{ \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} }[/math],
- where [math]\displaystyle{ \langle \cos(\textstyle\phi\,\!) \rangle }[/math] is the average cosine of the dihedral angle.
The fully stretched length [math]\displaystyle{ L = nl\, \cos(\theta/2) }[/math]. By equating the two expressions for [math]\displaystyle{ \langle R^2 \rangle }[/math] and the two expressions for [math]\displaystyle{ L }[/math] from the actual chain and the equivalent chain with Kuhn segments, the number of Kuhn segments [math]\displaystyle{ N }[/math] and the Kuhn segment length [math]\displaystyle{ b }[/math] 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. pp. 391. ISBN 0198520336.
- ↑ Michael Cross (October 2006), Physics 127a: Class Notes; Lecture 8: Polymers, California Institute of Technology, http://www.pmaweb.caltech.edu/~mcc/Ph127/a/Lecture_8.pdf, 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
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