# Persistence length

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

Informally, for pieces of the polymer that are shorter than the persistence length, the molecule behaves rather like a flexible elastic rod/beam (beamtheory), 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, P, is defined as the length over which correlations in the direction of the tangent are lost. In a more chemical based manner it can also be defined as the average sum of the projections of all bonds j ≥ i on bond i in an infinitely long chain.[1]

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, along the contour of the chain. It can be shown that the expectation value of the cosine of the angle falls off exponentially with distance,[2][3]

${\displaystyle \langle \cos {\theta }\rangle =e^{-(L/P)}\,}$

where P is the persistence length and the angled brackets denote the average over all starting positions.

The persistence length is considered to be 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.[4]

The persistence length can be also expressed using the bending stiffness ${\displaystyle B_{s}}$, the Young's modulus E and knowing the section of the polymer chain. [2] [5] [6] [7]

${\displaystyle P={\frac {B_{s}}{k_{B}T}}\,}$
${\displaystyle B_{s}=EI\,}$

In the case of a rigid and uniform rod I can be expressed as:

${\displaystyle I={\frac {\pi a^{4}}{4}}\,}$

For example a piece of uncooked spaghetti has a persistence length on the order of ${\displaystyle 10^{18}}$ m (taking in consideration a Young modulus of 5 GPa and a radius of 1 mm).[8] Double-helical DNA has a persistence length of about 500 Angstroms. Such large persistent length for spaghetti does not mean that it is not flexible. It just means that its stiffness is such that it needs ${\displaystyle 10^{18}}$ m of length for thermal fluctuations at 300K to bend it.

Another example:[9]
Imagine a long cord that is slightly flexible. At short distance scales, the cord will basically be rigid. If you look at the direction the cord is pointing at two points that are very close together, the cord will likely be pointing in the same direction at those two points (i.e. the angles of the tangent vectors are highly correlated). If you choose two points on this flexible cord (imagine a piece of cooked spaghetti that you've just tossed on your plate) that are very far apart, however, the tangent to the cords at those locations will likely be pointing in different directions (i.e. the angles will be uncorrelated). If you plot out how correlated the tangent angles at two different points are as a function of the distance between the two points, you'll get a plot that starts out at 1 (perfect correlation) at a distance of zero and drops exponentially as distance increases. The persistence length is the characteristic length scale of that exponential decay. For the case of a single molecule of DNA the persistence length can be measured using optical tweezers and atomic force microscopy.[10][11]

## References

1. ^ Flory, Paul J. (1969). Statistical Mechanics of Chain Molecules. New York: Interscience Publishers. ISBN 0-470-26495-0.
2. ^ a b Landau, Lev Davidovič; Lifšic/Lifshitz/Lifshits, Evgenii Mikhailovich (between 1958 and 1981). Statistical Physics. Oxford [and other publisher] : Pergamon Press. p. §127. Check date values in: |date= (help)
Landau, Lev Davidovič; Lifshitz, Evgenii Mikhailovich; Lenk, Richard (translated Russian to German) (1979). Lehrbuch der Theoretischen Physik: Statistische Physik: Teil 1 (5.Auflage) (in German). Berlin: Akadmie-Verlag. p. §127.
3. ^ Doi, M.; Edwards, S.F. (1986). The Theory of Polymer Dynamics. Clarendon, Oxford. p. 317.
4. ^
5. ^ Gittes, Frederick; Mickey, Brian; Nettleton, Jilda; Howard, Jonathon (1993). "Flexural rigidity of microtubules and actin filaments measured from thermal fluctuations in shape.". The Journal of cell biology. 120 (4). Rockefeller Univ Press. pp. 923–934.
6. ^ Baumann, Christoph G.; Smith, Steven B.; Bloomfield, Victor A.; Bustamante, Carlos (1997). "Ionic effects on the elasticity of single DNA molecules.". 94 (12). Proceedings of the National Academy of Sciences. pp. 6185–6190.
7. ^ Mofrad, Mohammad R.K.; Kamm, Roger D. (2006). Cytoskeletal mechanics: models and measurements. Cambridge University Press.
8. ^ Guinea, G. V. "Brittle failure of dry spaghetti". Engineering Failure Analysis.