Force-Extension Curve of an Entangled Polymer Chain: A Superspace Approach

Huan Gong; Jian-Feng Li; Hong-Dong Zhang; An-Chang Shi

doi:10.1007/s10118-021-2623-y

Your Location：

Home >

Browse articles >

Force-Extension Curve of an Entangled Polymer Chain: A Superspace Approach

RAPID COMMUNICATION | Updated：2021-07-23

- Force-Extension Curve of an Entangled Polymer Chain: A Superspace Approach
- Chinese Journal of Polymer Science Vol. 39, Issue 11, Pages: 1345-1350(2021)
- Affiliations：
  
  a.The State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
  b.Department of Physics and Astronomy, McMaster University, Hamilton, Ontario Canada L8S 4M1
  c.School of Computer Science and Technology, Fudan University, Shanghai 200433, China
- Author bio：
  
  lijf@fudan.edu.cn (J.F.L.)
  shi@mcmaster.ca (A.C.S.)
- Funds：
- DOI：10.1007/s10118-021-2623-y
  CLC：
Scan for full text
Huan Gong, Jian-Feng Li, Hong-Dong Zhang, et al. Force-Extension Curve of an Entangled Polymer Chain: A Superspace Approach. [J]. Chinese Journal of Polymer Science 39(11):1345-1350(2021)
DOI：

Huan Gong, Jian-Feng Li, Hong-Dong Zhang, et al. Force-Extension Curve of an Entangled Polymer Chain: A Superspace Approach. [J]. Chinese Journal of Polymer Science 39(11):1345-1350(2021) DOI： 10.1007/s10118-021-2623-y.

摘要

Abstract

The statistical mechanics of an ideal polymer chain entangled with static topological constraints is studied using a superspace approach, in which the probability distribution of the polymer is obtained as solutions of the Fokker-Planck equation in a superspace with an inner structure characterized by the ,n,-generator free group. The theory predicts that the force-extension curve of the polymer under the topological constraints has the generic form ,F,=,kl,+,Z,/,l, where ,l, is an effective extension. Aside from the elastic term that is linear in ,l, the force-extension curve contains a universal term of the form ,Z,/,l,. The magnitude of this topological term is determined by the topological charge number ,Z, which characterizes the topological nature of the static constraints. The theoretical results are further verified by a scaling analysis based on a blob model of the chain conformations.

关键词

Keywords

Topological constraintsProbability distributionPolymer entanglement

references

Spitzer, F. . Some theorems concerning 2-dimensional Brownian motionTrans . Am. Math. Soc. , 1958 . 87 187 .

Edwards, S. F. . The statistical mechanics of polymers with excluded volume . Proc. Phys. Soc. , 1965 . 85 613 DOI:10.1088/0370-1328/85/4/301http://doi.org/10.1088/0370-1328/85/4/301 .

Edwards, S. F. . Statistical mechanics with topological constraints . Proc. Phys. Soc. , 1967 . 91 513 DOI:10.1088/0370-1328/91/3/301http://doi.org/10.1088/0370-1328/91/3/301 .

Prager, S.; Frisch, H.L. . Statistical mechanics of a simple entanglement . J. Chem. Phys. , 1967 . 46 1475 DOI:10.1063/1.1840877http://doi.org/10.1063/1.1840877 .

Rudnick, J.; Hu, Y. . Winding angle of a self-avoiding random walk . Phys. Rev. Lett. , 1988 . 60 712 DOI:10.1103/PhysRevLett.60.712http://doi.org/10.1103/PhysRevLett.60.712 .

Grosberg, A.; Frisch, H. . Winding angle distribution for planar random walk, polymer ring entangled with an obstacle, and all that: Spitzer–Edwards–Prager–Frisch model revisited . J. Phys. A: Math. Gen. , 2003 . 36 8955 DOI:10.1088/0305-4470/36/34/303http://doi.org/10.1088/0305-4470/36/34/303 .

It should be noticed that the case of a closed chain in an array of obstacles has been discussed in by the work: Khokhlov, A. R.; Nechaev, S. K. Polymer chain in an array of obstacles. Phys. Lett. A 1985, 112, 156. Different from the current work, Khokhlov and Nechaev’s work mainly focuses on the overall size of the polymer chain.

Nechaev, S.; Voituriez, R. . Random walks on three-strand braids and on related hyperbolic groups . J. Phys. A: Math. Gen. , 2003 . 36 43 DOI:10.1088/0305-4470/36/1/304http://doi.org/10.1088/0305-4470/36/1/304 .

Lalley, S. P. . Finite range random walk on free groups and homogeneous trees . Ann. Probab. , 1993 . 4 2087 .

Grillet, P. A. Abstract Algebra. Springer: New York, 2007.

A Gaussian perturbation refers to the perturbation Δx that causes the Hamiltonian to vary to the order Δx2 and can be, thus, eliminated by the Gauss integral during the evaluation of partition function.

Thouless, D. J. Topological quantum numbers in nonrelativistic physics. World Scientific: Singapore, 1998.

Fredrickson, G. H. The equilibrium theory of inhomogeneous polymers. Clarendon Press: Oxford, 2006.

Helfand, E. J. . Theory of inhomogeneous polymers: Fundamentals of the Gaussian random-walk model . J. Chem. Phys. , 1975 . 62 999 DOI:10.1063/1.430517http://doi.org/10.1063/1.430517 .

As shown in Figure S3, nine subspaces are sufficient to obtain accurate solutions of the propagator, and it is impossible and unnecessary to consider infinite number of subspaces.

de Gennes, P. G. Scaling concepts in polymer physics. Cornell University Press: Ithaca and London, 1979.

Kalathi, J. T.; Yamamoto, U.; Schweizer, K. S.; Grest, G. S.; Kumar, S. K. . Nanoparticle Diffusion in Polymer Nanocomposites . Phys. Rev. Lett. , 2014 . 112 108301 DOI:10.1103/PhysRevLett.112.108301http://doi.org/10.1103/PhysRevLett.112.108301 .

Cai, L. H.; Panyukov, S.; Rubinstein, M. . Hopping Diffusion of Nanoparticles in Polymer Matrices . Macromolecules , 2015 . 48 847 DOI:10.1021/ma501608xhttp://doi.org/10.1021/ma501608x .

Views

Downloads

CSCD

Alert me when the article has been cited

Submit

Tools

Publicity Resources

No data

Related Author

No data

Related Institution

No data

⁰