Worm-like chain

The worm-like chain (WLC) model in polymer physics is used to describe the behavior of polymers that are semi-flexible: fairly stiff with successive segments pointing in roughly the same direction, and with persistence length within a few orders of magnitude of the polymer length. The WLC model is the continuous version of the Kratky–Porod model.

The WLC model envisions a continuously flexible isotropic rod. This is in contrast to the freely-jointed chain model, which is only flexible between discrete freely hinged segments. The model is particularly suited for describing stiffer polymers, with successive segments displaying a sort of cooperativity: nearby segments are roughly aligned. At room temperature, the polymer adopts a smoothly curved conformation; at ${\displaystyle T=0}$ K, the polymer adopts a rigid rod conformation.

For a polymer of maximum length ${\displaystyle L_{0}}$, parametrize the path of the polymer as ${\displaystyle s\in (0,L_{0})}$. Allow ${\displaystyle {\hat {t}}(s)}$ to be the unit tangent vector to the chain at point ${\displaystyle s}$, and ${\displaystyle {\vec {r}}(s)}$ to be the position vector along the chain, as shown to the right. Then:

The energy associated with the bending of the polymer can be written as:

${\displaystyle E={\frac {1}{2}}k_{B}T\int _{0}^{L_{0}}P\cdot \left({\frac {\partial ^{2}{\vec {r}}(s)}{\partial s^{2}}}\right)^{2}ds}$

where ${\displaystyle P}$ is the polymer's characteristic persistence length, ${\displaystyle k_{B}}$ is the Boltzmann constant, and ${\displaystyle T}$ is the absolute temperature. At finite temperatures, the end-to end distance of the polymer will be significantly shorter than the maximum length ${\displaystyle L_{0}}$. This is caused by thermal fluctuations, which result in a coiled, random configuration of the undisturbed polymer.

The polymer's orientation correlation function can then be solved for, and it follows an exponential decay with decay constant 1/P:

${\displaystyle \langle {\hat {t}}(s)\cdot {\hat {t}}(0)\rangle =\langle \cos \;\theta (s)\rangle =e^{-s/P}\,}$