Skip to main content
SpringerLink
Log in

Geometric effects of cross sections on equilibrium of helical and twisted ribbon

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

In the framework of elastic rod model, the Euler-Lagrange equations characterizing the equilibrium configuration of the polymer chain are derived from a free energy functional associated with the curvature, torsion, twisting angle, and its derivative with respect to the arc-length. The configurations of the helical ribbons with different cross-sectional shapes are given. The effects of the elastic properties, the cross-sectional shapes, and the intrinsic twisting on the helical ribbons are discussed. The results show that the pitch angle of the helical ribbon decreases with the increase in the ratio of the twisting rigidity to the bending rigidity and approaches the intrinsic twisting. If the bending rigidity is much greater than the twisting rigidity, the bending and twisting of the helical ribbon always appear simultaneously.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chung, D. S., Benedek, G. B., Konikoff, F. M., and Donovan, J. M. Elastic free energy of anisotropic helical ribbons as metastable intermediates in the crystallization of cholesterol. The Proceedings of the National Academy of Sciences of the United States of America, 90, 11341–11345 (1993)

    Article  Google Scholar 

  2. Selinger, R. L. B., Selinger, J. V., Malanoski, A. P., and Schnur, J. M. Shape selection in chiral self-assembly. Physical Review Letters, 93, 158103 (2004)

    Article  Google Scholar 

  3. Kong, X. Y. and Wang, Z. L. Spontaneous polarization-induced nanohelixes, nanosprings, and nanorings of piezoelectric nanobelts. Nano Letters, 3, 1625–1631 (2003)

    Article  Google Scholar 

  4. Zhang, L., Deckhardt, E., Weber, A., Schönenberger, C., and Grützmacher, D. Controllable fabrication of SiGe/Si and SiGe/Si/Cr helical nanobelts. Nanotechnology, 16, 655–663 (2005)

    Article  Google Scholar 

  5. Cho, A. Pretty as you please, curling films turn themselves into nanodevices. Science, 313, 164–165 (2006)

    Article  Google Scholar 

  6. Srivastava, S., Santos, A., Critchley, K., Kim, K. S., Podsiadlo, P., Sun, K., Lee, J., Xu, C., Lilly, G. D., Glotzer, S. C., and Kotov, N. A. Light-controlled self-assembly of semiconductor nanoparticles into twisted ribbons. Science, 327, 1355–1359 (2010)

    Article  Google Scholar 

  7. Smith, S. B., Cui, Y. J., and Bustamante, C. Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules. Science, 271, 795–799 (1996)

    Article  Google Scholar 

  8. Bouchiat, C. and Mezard, M. Elasticity model of a supercoiled DNA molecule. Physical Review Letters, 80, 1556–1559 (1998)

    Article  Google Scholar 

  9. Tsuru, H. and Wadati, M. Elastic model of highly supercoiled DNA. Biopolymers, 25, 2083–2096 (1986)

    Article  MATH  Google Scholar 

  10. Goldstein, R. E. and Langer, S. A. Nonlinear dynamics of stiff polymers. Physical Review Letters, 75, 1094–1097 (1995)

    Article  Google Scholar 

  11. Benham, C. J. Geometry and mechanics of DNA superhelicity. Biopolymers, 22, 2477–2496 (1983)

    Article  Google Scholar 

  12. Tanaka, F. and Takahashi, H. Elastic theory of supercoiled DNA. The Journal of Chemical Physics, 83, 6017–6026 (1985)

    Article  Google Scholar 

  13. Bret, M. L. Twist and writhing in short circular DNAs according to first-order elasticity. Biopolymers, 23, 1835–1867 (1984)

    Article  Google Scholar 

  14. Cui, S. X., Yu, Y., and Zhang, B. L. Modeling single chain elasticity of single-stranded DNA: a comparison of three models. Polymer, 50, 930–935 (2009)

    Article  Google Scholar 

  15. Murayama, Y., Sakamaki, Y., and Sano, M. Elastic response of single DNA molecules exhibits a reentrant collapsing transition. Physical Review Letters, 90, 018102 (2003)

    Article  Google Scholar 

  16. Coleman, B. D., Swigon, D., and Tobias, I. Elastic stability of DNA configurations, II: supercoiled plasmids with self-contact. Physical Review E, 61, 759–770 (2000)

    Article  MathSciNet  Google Scholar 

  17. Tobias, I., Swigon, D., and Coleman, B. D. Elastic stability of DNA configurations, I: general theory. Physical Review E, 61, 747–758 (2000)

    Article  MathSciNet  Google Scholar 

  18. Fain, B., Rudnick, J., and Ostlund, S. Conformations of linear DNA. Physical Review E, 55, 7364–7368 (1997)

    Article  Google Scholar 

  19. Moroz, J. D. and Nelson, P. Torsional directed walks, entropic elasticity and DNA twist stiffness. The Proceedings of the National Academy of Sciences of the United States of America, 94, 14418–14422 (1997)

    Article  Google Scholar 

  20. Smith, B., Zastavker, Y. V., and Benedek, G. B. Tension-induced straightening transition of self-assembled helical ribbons. Physical Review Letters, 87, 278101 (2001)

    Article  Google Scholar 

  21. Zastavker, Y. V., Asherie, N., Lomakin, A., Pande, J., Donovan, J. M., Schnur, J. M., and Benedek, G. B. Self-assembly of helical ribbons. The Proceedings of the National Academy of Sciences of the United States of America, 96, 7883–7887 (1999)

    Article  Google Scholar 

  22. Thomas, B. N., Lindemann, C. M., and Clark, N. A. Left- and right-handed helical tubule intermediates from a pure chiral phospholipid. Physical Review E, 59, 3040–3047 (1999)

    Article  Google Scholar 

  23. Oda, R., Huc, I., Schmutz, M., Candau, S. J., and MacKintosh, F. C. Tuning bilayer twist using chiral counterions. nature, 399, 566–569 (1999)

    Article  Google Scholar 

  24. Schnur, J. M. Lipid tubules: a paradigm for molecularly engineered structures. Science, 262, 1669–1676 (1993)

    Article  Google Scholar 

  25. MacKintosh, F. C., Käs, J., and Janmey, P. A. Elasticity of semiflexible biopolymer networks. Physical Review Letters, 75, 4425–4428 (1995)

    Article  Google Scholar 

  26. Fygenson, D. K., Elbaum, M., Shraiman, B., and Libchaber, A. Microtubules and vesicles under controlled tension. Physical Review E, 55, 850–859 (1997)

    Article  Google Scholar 

  27. Hinner, B., Tempel, M., Sackmann, E., Kroy, K., and Frey, E. Entanglement, elasticity, and viscous relaxation of actin solutions. Physical Review Letters, 81, 2614–2617 (1998)

    Article  Google Scholar 

  28. Zhao, S. M., Zhang, S. L., Yao, Z. W., and Zhang, L. Equilibrium conformation of polymer chains with noncircular cross section. Physical Review E, 74, 032801 (2006)

    Article  Google Scholar 

  29. Selinger, J. V., MacKintosh, F. C., and Schnur, J. M. Theory of cylindrical tubules and helical ribbons of chiral lipid membranes. Physical Review E, 53, 3804–3818 (1996)

    Article  Google Scholar 

  30. Ghafouri, R. and Bruinsma, R. Helicoid to spiral ribbon transition. Physical Review Letters, 94, 138101 (2005)

    Article  Google Scholar 

  31. Yu, M. F., Dyer, M. J., Chen, J., Qian, D., Liu, W. K., and Ruoff, R. S. Locked twist in multiwalled carbon-nanotube ribbons. Physical Review B, 64, 241403 (2001)

    Article  Google Scholar 

  32. Feoli, A., Nesterenko, V. V., and Scarpetta, G. Functionals linear in curvature and statistics of helical proteins. Nuclear Physics B, 705, 577–592 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Nesterenko, V. V., Feoli, A., and Scarpetta, G. Dynamics of relativistic particles with Lagrangians dependent on acceleration. Journal of Mathematical Physics, 36, 5552–5564 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, S. L., Zuo, X. J., Xia, M. G., Zhao, S. M., and Zhang, E. H. General equilibrium shape equations of polymer chains. Physical Review E, 70, 148–168 (2004)

    Google Scholar 

  35. Thamwattana, N., Mccoy, J. A., and Hill, J. M. Energy density functions for protein structure. The Quarterly Journal of Mechanics and Applied Mathematics, 61, 431–451 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tu, Z. C. and Ouyang, Z. C. Elastic theory of low-dimensional continua and its applications in bioand nano-structures. Journal of Computational and Theoretical Nanoscience, 5, 422–448 (2008)

    Article  Google Scholar 

  37. Huang, Z. X. Modulating DNA configuration by interfacial traction: an elastic rod model to characterize DNA folding and unfolding. Journal of Biological Physics, 37, 79–90 (2011)

    Article  Google Scholar 

  38. Xiao, Y., Huang, Z. X., and Wang, S. N. An elastic rod model to evaluate effects of ionic concentration on equilibrium configuration of DNA in salt solution. Journal of Biological Physics, 40, 179–192 (2014)

    Article  Google Scholar 

  39. Xiao, Y., Huang, Z. X., Qiang, L., and Gao, J. Elastic response of DNA molecules under the action of interfacial traction and stretching: an elastic thin rod model. Modern Physics Letters B, 29, 1550193 (2015)

    Article  MathSciNet  Google Scholar 

  40. Xiao, Y. and Huang, Z. X. The influences of geometric shape of cross section on equilibrium configuration of DNA in elastic rod model. AIP Advances, 5, 117235 (2015)

    Article  Google Scholar 

  41. Chen, W. H. Differential Geometry (in Chinese), Beijing University Press, Beijing (2006)

    Google Scholar 

  42. Panyukov, S. and Rabin, Y. Fluctuating filaments: statistical mechanics of helices. Physical Review E, 62, 7135–7146 (2000)

    Article  Google Scholar 

  43. Kessler, D. A. and Rabin, Y. Stretching instability of helical springs. Physical Review Letters, 90, 024301 (2003)

    Article  Google Scholar 

  44. Marko, J. F. and Siggia, E. D. Bending and twisting elasticity of DNA. Macromolecules, 27, 981–988 (1994)

    Article  Google Scholar 

  45. Liu, Y. Z. Nonlinear Mechanics of Thin Elastic Rod: Theoretical Basis of Mechanical Model of DNA (in Chinese), Tsinghua Press, Beijing (2006)

    Google Scholar 

  46. Landau, L. D. and Lifshitz, E. M. Theory of Elasticity, Pergamon Press, Oxford (1986)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Ye Xiao & Zaixing Huang

Authors
  1. Ye Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zaixing Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zaixing Huang.

Additional information

Project supported by the National Natural Science Foundation of China (No. 11172130)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiao, Y., Huang, Z. Geometric effects of cross sections on equilibrium of helical and twisted ribbon. Appl. Math. Mech.-Engl. Ed. 38, 495–504 (2017). https://doi.org/10.1007/s10483-017-2182-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-017-2182-6

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation