# 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

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

• ${\ displaystyle K}$ [-] of the correlation function, with (usually: )${\ displaystyle -1 \ leq K \ leq 1}$ ${\ displaystyle 0 • ${\ displaystyle l}$ [-] 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

## Individual evidence

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3. persistence length. Retrieved June 21, 2018 .
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6. a b en: Persistence length
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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 .).
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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 .