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# Persistence length

The persistence length is a basic mechanical property quantifying the stiffness of a long polymer.

Informally, for pieces of the polymer that are shorter than the persistence length, the molecule behaves rather like a flexible elastic rod, while for pieces of the polymer that are much longer than the persistence length, the properties can only be described statistically, like a three-dimensional random walk.

Formally, the persistence length is defined as the length over which correlations in the direction of the tangent are lost. Let us define the angle Θ between a vector that is tangent to the polymer at position 0 (zero) and a tangent vector at a distance L away from position 0. It can be shown that the expectation value of the cosine of the angle falls off exponentially with distance,

$<\cos{\theta}> = e^{-(L/P)} \,$

where P is the persistence length and < > denotes the average over all starting positions.

A piece of cooked spaghetti has a persistence length on the order of 10 cm. Double-helical DNA has a persistence length of about 50 nanometers.

In polymer science, persistence length is one half of the Kuhn length, the length of hypothetical segments that the chain can be considered as freely joined. The persistence length equals the average projection of the end-to-end vector on the tangent to the chain contour at a chain end in the limit of infinite chain length. [1]

## References

1. ^ IUPAC Compendium of Chemical Terminology