Persistence length

The persistence length is a basic mechanical property quantifying the bending stiffness of a polymer. The molecule behaves like a flexible elastic rod/beam (beam theory). Informally, for pieces of the polymer that are shorter than the persistence length, the molecule behaves like a rigid 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, 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]}

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

where ${\displaystyle k_{B}}$ is the Boltzmann constant and T is the temperature.

${\displaystyle B_{s}=EI\,}$

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

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

where a is the radius.

For charged polymers the persistence length depends on the surrounding salt concentration due to electrostatic screening. The persistence length of a charged polymer is described by the OSF (Odijk, Skolnick and Fixman) model.^[8]