Persistence length

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In polymer physics , the persistence length is a length that is a measure of the rigidity of a polymer chain . It is defined as the length over which the directional correlation of the segments in the polymer chain is lost.

In a polymer melt or solution , a polymer takes on a random coil shape . Depending on the type of chemical bond between the monomers , a polymer chain is more "stiff" or more flexible, which results in a more or less loose coil conformation.

Mathematical definition

Definition via the correlation function

If one generalizes this to uneven lengths, this means: If a linear coordinate along the contour of the chain and a unit vector , which indicates the local orientation of the chain, is called the correlation function


  • [-] of the correlation function, with (usually: )
  • [-] the relative length with , with
  • [-] the unit vector in tangential direction with
  • [m] the running coordinate along the axis of the chain
  • [m] the total length must be significantly greater than , and can be anything.

for great strive towards zero. The more compact the polymer ball, the faster the correlation function drops.

The persistence length P as a measure of the compactness of the coil structure is defined as the integral over the correlation function:


  • [m] the persistence length
  • [-] the correlation function, with

Alternative version

Generalization of the Kuhn length

In the case of an infinitely long polymer chain, whose binding vectors ΔL are all the same length and correspond to the length ΔL i , the persistence length of any binding vector from the chain is defined as the sum of the projections of all binding vectors ΔL with L> L i in the direction of ΔL i .

That means that:

Here, ΔL i is the bond length and Θ i, j is the angle between the bond vectors ΔL i and ΔL in a current conformation. The product is equal to the length of the projection of the binding vector ΔL at the point L_j in the direction of ΔL i . This means that the mean value over all conformations is the projection of ΔL onto ΔL i . For all bond vectors .DELTA.L i and .DELTA.L is true as well , but is valid for sufficiently distant bond vectors: .

In practice, the persistence length is a measure of the internal flexibility of a polymer chain. For a “stiff” polymer molecule with severely restricted rotation, P is “large” and for a “statistical coil” “small”.

Exponential Definitions

Stiffness definition


  • G. Strobl: The Physics of Polymers. Springer, Berlin (inter alia) 1996, ISBN 3-540-60768-4

See also

Individual evidence

  1. ^ Peter W. Atkins, Julio De Paula: Physical chemistry . John Wiley & Sons, 2013, ISBN 978-3-527-33247-2 , pp. 703 ( limited preview in Google Book search).
  2. Atsushi Ikai: Introduction to Nanobiomechanics Imaging and Measurement by Atomic Force Microscopy . John Wiley & Sons, 2012, ISBN 3-527-63295-6 ( limited preview in Google Book Search).
  3. persistence length. Retrieved June 21, 2018 .
  4. a b c M. D. Lechner, Klaus Gehrke, Eckhard H. Nordmeier: Macromolecular Chemistry A textbook for chemists, physicists, materials scientists and process engineers . Springer-Verlag, 2014, ISBN 978-3-642-41769-6 , pp. 50 ( limited preview in Google Book search).
  5. ^ A b John A. Schellman: The flexibility of DNA: II. Spontaneous and ligand induced distortions . In: Elsevier (Ed.): Biophysical Chemistry . tape 11 , no. 3 , June 1980, ISSN  0301-4622 , p. 329-337 , doi : 10.1016 / 0301-4622 (80) 87005-0 .
  6. a b en: Persistence length
  7. Ott, A. and Magnasco, M. and Simon, A. and Libchaber, A .: Measurement of the persistence length of polymerized actin using fluorescence microscopy . In: American Physical Society (Ed.): Phys. Rev. E . tape 48 , no. 3 , September 1993, p. R1642-R1645 , doi : 10.1103 / PhysRevE.48.R1642 .
  8. C. Bouchiat and MD Wang and J.-F. Allemand and T. Strick and SM Block and V. Croquette: Estimating the Persistence Length of a Worm-Like Chain Molecule from Force-Extension Measurements . In: Biophysical Journal . tape 76 , no. 1 , 1999, ISSN  0006-3495 , pp. 409-413 , doi : 10.1016 / S0006-3495 (99) 77207-3 .
  9. ^ Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz: Statistical physics Part1 . Pergamon, Oxford 1958, Chapter XII Fluctuations: §127 Fluctuations in the curvature of long molecules, p. §127 (English, [PDF], Russian: Курс теоретической физики Ландау и Лифшица .).
  10. Lev Davidovich Landau, and Evgenii Mikhailovich Lifshitz: Textbook of Theoretical Physics Volume V: Statistical Physics Part 1 [Statistical Physics Part 1] . In: Textbook of theoretical physics . Akademie Verlag, Berlin 1966, Chapter XII Fluctuations: §127 Fluctuations in the bending of long molecules, p. §127 ( [PDF] Russian: Курс теоретической физики Ландау и Лифшица .).
  11. Jeffrey Skolnick and Marshall Fixman: Electrostatic persistence length of a wormlike polyelectrolyte . In: ACS Publications (Ed.): Macromolecules . tape 5 , September 1977, pp. 944-948 , doi : 10.1021 / ma60059a011 .
  12. Barkley, Mary D and Zimm, Bruno H: Theory of twisting and bending of chain macromolecules; analysis of the fluorescence depolarization of {DNA} . In: AIP Publishing (Ed.): The Journal of Chemical Physics . tape 70 , no. 6 , March 1979, p. 2991-3007 , doi : 10.1063 / 1.437838 .