Kuhn length

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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 ${\displaystyle N}$ 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 average 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 average 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]