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| [[File:Molecule 1.jpg|thumb| Bond angle]]
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| The '''Kuhn length''' is a theoretical treatment, developed by [[Werner Kuhn]], in which a real [[polymer]] chain is considered as a collection of <math>N</math> '''Kuhn segments''' each with a Kuhn length <math>b</math>. Each Kuhn segment can be thought of as if they are freely jointed with each other.<ref>Flory, P.J. (1953) ''Principles of Polymer Chemistry'', Cornell Univ. Press, ISBN 0-8014-0134-8</ref><ref>Flory, P.J. (1969) ''Statistical Mechanics of Chain Molecules'', Wiley, ISBN 0-470-26495-0; reissued 1989, ISBN 1-56990-019-1</ref><ref>Rubinstein, M., Colby, R. H. (2003)''Polymer Physics'', Oxford University Press, ISBN 0-19-852059-X</ref> 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 [[Polymer physics#Real Chains|real chain]] consisting of <math>n</math> bonds and with fixed bond angles, torsion angles, and bond lengths, Kuhn considered an equivalent [[Polymer physics#Ideal Chains|ideal chain]] with <math> N </math> connected segments, now called Kuhn segments, that can orient in any random direction. | |
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| The length of a fully stretched chain is <math>L=Nb</math> for the Kuhn segment chain.<ref>
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| {{citation | title=Physics 127a: Class Notes; Lecture 8: Polymers |publisher=California Institute of Technology |author=Michael Cross |date= October 2006
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| |url=http://www.pma.caltech.edu/~mcc/Ph127/a/Lecture_8.pdf|accessdate=2013-02-20
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| }}</ref> 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>\langle R^2\rangle = Nb^2</math>.
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| 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.
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| For an actual homopolymer chain (consists of the same repeat units) with bond length <math>l</math> and bond angle θ with a [[dihedral angle]] energy potential,{{clarify|date=February 2012|reason=What is dihedral angle energy?}} the average end-to-end distance can be obtained as
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| :<math>\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>, | |
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| ::where <math>\langle \cos(\textstyle\phi\,\!) \rangle</math> is the average cosine of the dihedral angle.
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| The fully stretched length <math>L = nl\, \cos(\theta/2)</math>. By equating <math>\langle R^2 \rangle</math> and <math>L</math> for the actual chain and the equivalent chain with Kuhn segments, the number of Kuhn segments <math>N</math> and the Kuhn segment length <math>b</math> can be obtained.
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| For [[worm-like chain|semiflexible chain]], Kuhn length equals two times the [[persistence length]].<ref>Gert R. Strobl (2007) ''The physics of polymers: concepts for understanding their structures and behavior'', Springer, ISBN 3-540-25278-9</ref>
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Kuhn Length}}
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| [[Category:Polymer chemistry]]
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| [[Category:Polymer physics]]
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