page_title: Worm-like chain
section_title: Model elements
hierarchical_section_title: Worm-like chain / Model elements
caption_attribution_description: English: Diagram of the wormlike chain model, an approximation of a flexible polymer.
context_page_description: The worm-like chain 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.
context_section_description: 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  K, the polymer adopts a rigid rod conformation.
  For a polymer of maximum length , parametrize the path of the polymer as . Allow  to be the unit tangent vector to the chain at point , and  to be the position vector along the chain, as shown to the right. Then:
   and the end-to-end distance  .
  The energy associated with the bending of the polymer can be written as:
  where  is the polymer's characteristic persistence length,  is the Boltzmann constant, and  is the absolute temperature. At finite temperatures, the end-to end distance of the polymer will be significantly shorter than the maximum length . 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:
  A useful value is the mean square end-to-end distance of the polymer:
  Note that in the limit of , then . This can be used to show that a Kuhn segment is equal to twice the persistence length of a worm-like chain. In the limit of , then , and the polymer displays rigid rod behavior. The figure to the right shows the crossover from flexible to stiff behavior as the persistence length increases.
caption: An illustration of the WLC model, with position and unit tangent vectors as shown.
exif: {}