Persistence length

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. ${\displaystyle P}$

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

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There are different definitions in literature - Johannes Kalliauer 17:22, Apr 8, 2017 (CEST)

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 ${\displaystyle l}$ ${\displaystyle {\vec {e}}(l)}$

${\displaystyle K(\Delta L)=\langle {\vec {e}}(l)\cdot {\vec {e}}(l+\Delta l)\rangle ={\frac {1}{L_{ges}-\Delta L}}\cdot \int _{0}^{L_{ges}-\Delta L}{\vec {e}}({\frac {L}{L_{ges}-\Delta L}})\;{\vec {e}}({\frac {L+\Delta L}{L_{ges}-\Delta L}})\;{\mbox{d}}L=\int _{0}^{1}{\vec {e}}(l)\;{\vec {e}}(l+{\frac {\Delta L}{L_{ges}-\Delta L}})\;{\mbox{d}}l}$

With

• ${\displaystyle K}$[-] of the correlation function, with (usually: ) ${\displaystyle -1\leq K\leq 1}$ ${\displaystyle 0<K<1}$
• ${\displaystyle l}$[-] the relative length with , with ${\displaystyle l={\frac {L}{L_{ges}-\Delta L}}}$ ${\displaystyle 0\leq l\leq 1}$
• ${\displaystyle {\vec {e}}}$ [-] the unit vector in tangential direction with ${\displaystyle ||{\vec {e}}||=1}$
• ${\displaystyle L}$ [m] the running coordinate along the axis of the chain
• ${\displaystyle \Delta L\ll L_{ges}\approx \infty }$[m] the total length must be significantly greater than , and can be anything. ${\displaystyle L_{ges}}$ ${\displaystyle \Delta L}$ ${\displaystyle \Delta L}$

for great strive towards zero. The more compact the polymer ball, the faster the correlation function drops. ${\displaystyle {\frac {\Delta L}{P}}}$

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

${\displaystyle P=\int _{0}^{L_{ges}\approx \infty }K(\Delta L)\;{\mbox{d}}\,\Delta L}$

With

• ${\displaystyle P}$ [m] the persistence length
• ${\displaystyle K}$ [-] the correlation function, with ${\displaystyle -1\leq K\leq 1}$

Alternative version

${\displaystyle \int _{0}^{P}K(\Delta l)\mathrm {d} \Delta l=\int _{P}^{\infty }K(\Delta l)\mathrm {d} \Delta l}$

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:

${\displaystyle P\equiv \lim _{\Delta L\to 0}\left(\Delta L\cdot \sum _{j=i+1}^{j\to \infty }\langle \cos \Theta _{i,j}\rangle \right)=\lim _{\Delta L\to 0}\left(\Delta L\cdot \sum _{j=i+1}^{j\to \infty }\langle {\vec {e}}(L_{i})\cdot {\vec {e}}(L_{i}+\Delta L)\rangle \right)=\lim _{\Delta L\to 0}\left(\Delta L\cdot \sum _{j=i+1}^{j\to \infty }K(\Delta L)\right)=\int _{L=L_{i}}^{L\to \infty }\left(\mathrm {d} L\cdot K(\mathrm {d} L)\right)=\int _{L=L_{i}}^{L\to \infty }\left(K(\mathrm {d} L)\cdot \mathrm {d} L\right)}$

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: . ${\displaystyle \Delta L\cdot \langle \cos \Theta _{i,j}\rangle }$ ${\displaystyle \langle \cos \Theta _{i,j}\rangle }$ ${\displaystyle -1\leq \cos \Theta _{i,j}\leq 1}$ ${\displaystyle \langle \cos \Theta _{i,j}\rangle >0}$ ${\displaystyle \lim _{l_{i}\to \infty }\left(\langle \cos \Theta _{i,j}\rangle \right)=0}$

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”.

${\displaystyle P=P_{\infty }=\langle L_{1}\cdot \sum _{i=1}L_{i}\rangle /L_{0}}$

${\displaystyle P_{\alpha }=L_{0}\cdot (1+L_{2}\cdot L_{1}/{L_{0}^{2}}+L_{3}\cdot L_{1}/{L_{0}^{2}}+...)}$

Exponential Definitions

${\displaystyle \langle \cos {\theta }\rangle =e^{-(L/P)}}$

${\displaystyle \langle t(s)\cdot t(s+x)\rangle =e^{-x/2P}}$

Stiffness definition

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

literature

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

See also

• Contour length
• Kuhn length

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, fulviofrisone.com [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 ( fulviofrisone.com [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 .