The Kuhn length is a theoretical treatment, developed by Werner Kuhn, in which a real polymer chain is considered as a collection of N {\displaystyle N} Kuhn segments each with a Kuhn length b {\displaystyle b} . Each Kuhn segment can be thought of as if they are freely jointed with each other. 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 {\displaystyle n} bonds and with fixed bond angles, torsion angles, and bond lengths, Kuhn considered an equivalent ideal chain with N {\displaystyle N} connected segments, now called Kuhn segments, that can orient in any random direction. The length of a fully stretched chain is L = N b {\displaystyle L=Nb} for the Kuhn segment chain. 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 ⟨ R 2 ⟩ = N b 2 {\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 l {\displaystyle l} and bond angle θ with a dihedral angle energy potential, the mean square end-to-end distance can be obtained as
⟨ R 2 ⟩ = n l 2 1 + cos ( θ ) 1 − cos ( θ ) ⋅ 1 + ⟨ cos ( ϕ ) ⟩ 1 − ⟨ cos ( ϕ ) ⟩ {\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 ⟨ cos ( ϕ ) ⟩ {\displaystyle \langle \cos(\textstyle \phi \,\!)\rangle } is the average cosine of the dihedral angle. The fully stretched length L = n l cos ( θ / 2 ) {\displaystyle L=nl\,\cos(\theta /2)} . By equating the two expressions for ⟨ R 2 ⟩ {\displaystyle \langle R^{2}\rangle } and the two expressions for L {\displaystyle L} from the actual chain and the equivalent chain with Kuhn segments, the number of Kuhn segments N {\displaystyle N} and the Kuhn segment length b {\displaystyle b} can be obtained. For worm-like chain, Kuhn length equals two times the persistence length.
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