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 $$N$$ Kuhn segments each with a Kuhn length $$b$$. Each Kuhn segment can be thought of as if they are freely jointed with each other. 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. 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 $$\langle R^2\rangle = Nb^2$$.

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 $$l$$ and bond angle θ with a dihedral angle energy potential, the average end-to-end distance can be obtained as


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


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

The fully stretched length $$L = nl\, \cos(\theta/2)$$. By equating the two expressions for $$\langle R^2 \rangle$$ 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.