Kuhn length

{{Short description|Idealization in polymer thermodynamics}}

{{More citations needed|date=December 2009}}

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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 $N$ Kuhn segments each with a Kuhn length $b$. Each Kuhn segment can be thought of as if they are freely jointed with each other.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}}{{cite book|last1=Doi|first1=M.|last2=Edwards|first2=S. F.|title=The Theory of Polymer Dynamics|date=1988|publisher=Volume 73 of International series of monographs on physics. Oxford science publications|isbn=0198520336|pages=391}} 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.

{{citation | title=Physics 127a: Class Notes; Lecture 8: Polymers |publisher=California Institute of Technology |author=Michael Cross |date= October 2006

|url=http://www.pmaweb.caltech.edu/~mcc/Ph127/a/Lecture_8.pdf|accessdate=2013-02-20

}} 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 $\backslash langle\; R^2\backslash rangle\; =\; Nb^2$.

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,{{clarify|date=February 2012|reason=What is dihedral angle energy?}} the mean square end-to-end distance can be obtained as

:$\backslash langle\; R^2\; \backslash rangle\; =\; n\; l^2\; \backslash frac\{1+\backslash cos(\backslash theta)\}\{1-\backslash cos(\backslash theta)\}\; \backslash cdot\; \backslash frac\{1+\backslash langle\backslash cos(\backslash textstyle\backslash phi\backslash ,\backslash !)\backslash rangle\}\{1-\backslash langle\backslash cos\; (\backslash textstyle\backslash phi\backslash ,\backslash !)\backslash rangle\}$,

::where $\backslash langle\; \backslash cos(\backslash textstyle\backslash phi\backslash ,\backslash !)\; \backslash rangle$ is the average cosine of the dihedral angle.

The fully stretched length $L\; =\; nl\backslash ,\; \backslash cos(\backslash theta/2)$. By equating the two expressions for $\backslash langle\; R^2\; \backslash rangle$ 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.Gert R. Strobl (2007) The physics of polymers: concepts for understanding their structures and behavior, Springer, {{ISBN|3-540-25278-9}}