Reptation

Short description: Movement of entangled polymer chains

A peculiarity of thermal motion of very long linear macromolecules in entangled polymer melts or concentrated polymer solutions is reptation.[1] Derived from the word reptile, reptation suggests the movement of entangled polymer chains as being analogous to snakes slithering through one another.[2] Pierre-Gilles de Gennes introduced (and named) the concept of reptation into polymer physics in 1971 to explain the dependence of the mobility of a macromolecule on its length. Reptation is used as a mechanism to explain viscous flow in an amorphous polymer.[3][4] Sir Sam Edwards and Masao Doi later refined reptation theory.[5][6] Similar phenomena also occur in proteins.[7]

Two closely related concepts are reptons and entanglement. A repton is a mobile point residing in the cells of a lattice, connected by bonds.Cite error: Closing `</ref>` missing for `<ref>` tag this relaxation time has nothing to do with the reptation relaxation time.

## Models

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The blob model, explaining the entanglement of long polymer chains.

The tube model, explaining the basically one-dimensional mobility of long polymer chains.

Entangled polymers are characterized with effective internal scale, commonly known as the length of macromolecule between adjacent entanglements [math]\displaystyle{ M_{e} }[/math].

Entanglements with other polymer chains restrict polymer chain motion to a thin virtual tube passing through the restrictions.[8] Without breaking polymer chains to allow the restricted chain to pass through it, the chain must be pulled or flow through the restrictions. The mechanism for movement of the chain through these restrictions is called reptation.

In the blob model,[9] the polymer chain is made up of [math]\displaystyle{ n }[/math] Kuhn lengths of individual length [math]\displaystyle{ l }[/math]. The chain is assumed to form blobs between each entanglement, containing [math]\displaystyle{ n_{e} }[/math] Kuhn length segments in each. The mathematics of random walks can show that the average end-to-end distance of a section of a polymer chain, made up of [math]\displaystyle{ n_{e} }[/math] Kuhn lengths is [math]\displaystyle{ d=l \sqrt{n_{e}} }[/math]. Therefore if there are [math]\displaystyle{ n }[/math] total Kuhn lengths, and [math]\displaystyle{ A }[/math] blobs on a particular chain:

[math]\displaystyle{ A= \dfrac{n}{n_{e}} }[/math]

The total end-to-end length of the restricted chain [math]\displaystyle{ L }[/math] is then:

[math]\displaystyle{ L=Ad = \dfrac{nl\sqrt{n_{e}}}{n_{e}} = \dfrac{nl}{\sqrt{n_{e}}} }[/math]

This is the average length a polymer molecule must diffuse to escape from its particular tube, and so the characteristic time for this to happen can be calculated using diffusive equations. A classical derivation gives the reptation time [math]\displaystyle{ t }[/math]:

[math]\displaystyle{ t=\dfrac{l^2 n^3 \mu}{n_{e} k T} }[/math]

where [math]\displaystyle{ \mu }[/math] is the coefficient of friction on a particular polymer chain, [math]\displaystyle{ k }[/math] is Boltzmann's constant, and [math]\displaystyle{ T }[/math] is the absolute temperature.

The linear macromolecules reptate if the length of macromolecule [math]\displaystyle{ M }[/math] is bigger than the critical entanglement molecular weight [math]\displaystyle{ M_{c} }[/math]. [math]\displaystyle{ M_{c} }[/math] is 1.4 to 3.5 times [math]\displaystyle{ M_{e} }[/math].[10] There is no reptation motion for polymers with [math]\displaystyle{ M\lt M_{c} }[/math], so that the point [math]\displaystyle{ M_{c} }[/math] is a point of dynamic phase transition.

Due to the reptation motion the coefficient of self-diffusion and conformational relaxation times of macromolecules depend on the length of macromolecule as [math]\displaystyle{ M^{-2} }[/math] and [math]\displaystyle{ M^3 }[/math], correspondingly.[11][12] The conditions of existence of reptation in the thermal motion of macromolecules of complex architecture (macromolecules in the form of branch, star, comb and others) have not been established yet.

The dynamics of shorter chains or of long chains at short times is usually described by the Rouse model.

## See also

* Important publications in polymer physics
* Polymer characterization
* Polymer physics
* Protein dynamics
* Soft matter

## References

1. ↑ Pokrovskii, V. N. (2010). The Mesoscopic Theory of Polymer Dynamics. Springer Series in Chemical Physics. 95. doi:10.1007/978-90-481-2231-8. ISBN 978-90-481-2230-1. Bibcode: 2010mtpd.book.....P.
2. ↑ Rubinstein, Michael (March 2008). "Dynamics of Entangled Polymers". Pierre-Gilles de Gennes Symposium. New Orleans, LA: American Physical Society. http://www.aps.org/units/dpoly/resources/degennes.cfm. Retrieved 6 April 2015.
3. ↑ De Gennes, P. G. (1983). "Entangled polymers". Physics Today 36 (6): 33. doi:10.1063/1.2915700. Bibcode: 1983PhT....36f..33D. "A theory based on the snake-like motion by which chains of monomers move in the melt is enhancing our understanding of rheology, diffusion, polymer-polymer welding, chemical kinetics and biotechnology".
4. ↑ De Gennes, P. G. (1971). "Reptation of a Polymer Chain in the Presence of Fixed Obstacles". The Journal of Chemical Physics 55 (2): 572. doi:10.1063/1.1675789. Bibcode: 1971JChPh..55..572D.
5. ↑ Samuel Edwards: Boltzmann Medallist 1995, IUPAP Commission on Statistical Physics, archived from the original on 2013-10-17, https://web.archive.org/web/20131017061732/http://iupap.cii.fc.ul.pt/Boltz_Award/BA1995.html, retrieved 2013-02-20
6. ↑ Doi, M.; Edwards, S. F. (1978). "Dynamics of concentrated polymer systems. Part 1.?Brownian motion in the equilibrium state". Journal of the Chemical Society, Faraday Transactions 2 74: 1789–1801. doi:10.1039/f29787401789.
7. ↑ Bu, Z; Cook, J; Callaway, D. J. (2001). "Dynamic regimes and correlated structural dynamics in native and denatured alpha-lactalbumin". Journal of Molecular Biology 312 (4): 865–73. doi:10.1006/jmbi.2001.5006. PMID 11575938.
8. ↑ Edwards, S. F. (1967). "The statistical mechanics of polymerized material". Proceedings of the Physical Society 92 (1): 9–16. doi:10.1088/0370-1328/92/1/303. Bibcode: 1967PPS....92....9E.
9. ↑ Duhamel, J.; Yekta, A.; Winnik, M. A.; Jao, T. C.; Mishra, M. K.; Rubin, I. D. (1993). "A blob model to study polymer chain dynamics in solution". The Journal of Physical Chemistry 97 (51): 13708. doi:10.1021/j100153a046.
10. ↑ Fetters, LJ; Lohse, DJ; Colby, RH (2007). "25.3". in Mark, James E. Chain Dimensions and Entanglement Spacings in "Physical properties of polymers handbook" (2nd ed.). New York: Springer New York. p. 448. ISBN 978-0-387-69002-5.
11. ↑ Pokrovskii, V. N. (2006). "A justification of the reptation-tube dynamics of a linear macromolecule in the mesoscopic approach". Physica A: Statistical Mechanics and Its Applications 366: 88–106. doi:10.1016/j.physa.2005.10.028. Bibcode: 2006PhyA..366...88P.
12. ↑ Pokrovskii, V. N. (2008). "Reptation and diffusive modes of motion of linear macromolecules". Journal of Experimental and Theoretical Physics 106 (3): 604–607. doi:10.1134/S1063776108030205. Bibcode: 2008JETP..106..604P.

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