The Kuhn length is a theoretical treatment, developed by Hans Kuhn, in which a real

polymer A polymer (; Greek ''poly-'', "many" + '' -mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic and ...

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 In polymer chemistry, a random coil is a conformation of polymers where the monomer subunits are oriented randomly while still being bonded to adjacent units. It is not one specific shape, but a statistical distribution of shapes for all the ch ...

. 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 In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb ...

or a

self-avoiding walk In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon (S ...

, 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 A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...

energy potential, the average end-to-end distance can be obtained as :

\langle R^2 \rangle = n l^2 \frac \cdot \frac

, ::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.Gert R. Strobl (2007) ''The physics of polymers: concepts for understanding their structures and behavior'', Springer,

References

{{DEFAULTSORT:Kuhn Length Polymer chemistry Polymer physics