Kratky-Porod or Worm-like Chain Model

The worm-like chain model was first suggested by Kratky and Porod1 (1949) and is often called the Kratky-Porod model. It is particularly suited for describing the conformations of stiff(er) polymer chains. The successive segments of a stiff chain all point roughly in the same direction; thus, the polymer adopts conformations that resemble a smoothly curved path and at very low temperatures the polymer adopts a rigid rod-like conformation.

The projection of the end-to-end vector of a polymer chain, R, on the direction of the first bond vector l1 for the freely rotating chain is given by

where θ is the angle between two successive bonds. In the limit Nl → ∞ this series converges to

This quantity is often called persistence length. It is the length along the chain backbone that one must travel before the next statistical unit no longer correlates with the end-to-end vector. For example, the persistence length of a freely jointed chain is one segment length and that of a rod-like chain is equal to the end-to-end distance.

 The mean square end-to-end vector of a freely rotating chain is

When we combine the two expression above, we obtain

For small bond angles, cosθ can be expressed by a Taylor series.

where the expansion has been truncated after the second term. Then we can make the approximations

The wormlike chain model is the continuous curvature limit of the freely rotating chain, such that the bond length l goes to zero and the number of bonds N goes to infinity, but the contour length of the chain L = Nl and the persistance length lp are kept constant:

In this limit the expression for the mean square end-to-end vector for the freely rotating chain converges to

When the contour length L of the chain is much larger than the persistance length, L >> lp, the stiffness of the chain is negligble, and the wormlike chain reduces to a freely-rotating chain, <R2> / L → 2l.

References
  1. O. Kratky and G. Porod, Rec. trav. chim. 68, 1108 (1949)
  • Summary

    Worm-like Chain

    For very stiff chains, the worm-like chain model is applied. It is the continuous curvature limit of the freely rotating chain.

  • The persistence length is a measure of chain stiffness (or flexibility). It is the average sum of the projections of all bonds on the first bond in an infinitley long chain.

  • When the contour length of the chain is much larger than the persistance length, the stiffness of the chain is negligble, and the wormlike chain reduces to a freely-rotating chain.

Kuhn & Persistance
Length

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