Skip to main content
SpringerLink
Log in

Strictly two-dimensional self-avoiding walks: Density crossover scaling

  • Published:
Polymer Science Series C Aims and scope Submit manuscript

Abstract

The density crossover scaling of thermodynamic and conformational properties of solutions and melts of self-avoiding and highly flexible polymer chains without chain intersections confined to strictly two dimensions (d = 2) is investigated by means of molecular dynamics and Monte Carlo simulations of a standard coarse grained bead-spring model. We focus on properties related to the contact exponent set by the intrachain subchain size distribution. With R ∼ N ν being the size of chains of length N and ρ the monomer density, the interaction energy e int between monomers from different chains and the corresponding number n int of interchain contacts per monomer are found to scale as

with ν = 3/4 and θ2 = 19/12 for dilute solutions and ν = 1/d and θ2 = 3/4 for N≫ g(ρ) ≈ 1/ρ2. Irrespective of ρ, long chains thus become compact packings of blobs of contour length

with d p = d − θ2 = 5/4 being the fractal line dimension. Due to the generalized Porod scattering of the compact chains, the Kratky representation of the intramolecular form factor F(q) reveals a non-monotonous behavior approaching with increasing chain length and density a power-law slope \(F(q)q^d /\rho \approx 1/(qR)^{\theta _2 } \) in the intermediate regime of the wavevector q. The specific intermolecular contact probability is argued to imply an enhanced compatibility for polymer blends confined to ultrathin films. We comment briefly on finite persistence length effects.

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. P. G. De Gennes, Scaling Concepts in Polymer Physics (Cornell Univ. Press, Ithaca, 1979).

    Google Scholar 

  2. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon, Oxford, 1986).

    Google Scholar 

  3. J. Des Cloizeaux and G. Jannink, Polymers in Solution: Their Modeling and Structure (Clarendon, Oxford, 1990).

    Google Scholar 

  4. M. Rubinstein and R. Colby, Polymer Physics (Oxford Univ. Press, Oxford, 2003).

    Google Scholar 

  5. L. Schafer, Excluded Volume Effects in Polymer Solutions (Springer, New York, 1999).

    Book  Google Scholar 

  6. K. Binder, Adv. Polym. Sci. 112, 181 (1994).

    Article  CAS  Google Scholar 

  7. J. Wittmer, A. Cavallo, H. Xu, J. Zabel, P. Poliñska, N. Schulmann, H. Meyer, J. Farago, A. Johner, S. Obukhov, and J. Baschnagel, J. Stat. Phys. 145, 1017 (2011).

    Article  CAS  Google Scholar 

  8. A. Semenov and I. Nyrkova, in Comprehensive Polymer Science, Ed. by A. Khoklov and F. Kremer (Elsevier, Rotterdam, 1997), p. 303.

  9. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, San Francisco, 1982).

    Google Scholar 

  10. D. Morse and J. Chung, J. Chem. Phys. 130, 224901 (2009).

    Article  Google Scholar 

  11. E. Meyer, R. Overney, K. Dransfeld, and T. Gyalog, Nanoscience: Friction and Rheology on the Nanometer Scale (World Sci., Singapore, 1998).

    Book  Google Scholar 

  12. B. Frank, A. Gast, T. Russel, H. Brown, and C. Hawker, Macromolecules 29, 6531 (1996).

    Article  CAS  Google Scholar 

  13. R. Jones, S. Kumar, D. Ho, R. Briber, and T. Russel, Nature 400, 146 (1999).

    Article  CAS  Google Scholar 

  14. B. Maier and J. O. Rädler, Phys. Rev. Lett. 82, 1911 (1999).

    Article  CAS  Google Scholar 

  15. B. Maier and J. O. Rädler, Macromolecules 33, 7185 (2000).

    Article  CAS  Google Scholar 

  16. S. Granick, S. Kumar, and E. Amis, J. Polym. Sci., Part B: Polym. Phys. 41, 2755 (2003).

    Article  CAS  Google Scholar 

  17. X. Wang and V. J. Foltz, J. Chem. Phys. 121, 8158 (2004).

    Article  CAS  Google Scholar 

  18. K. Shin, H. Xiang, S. Moon, T. Kim, T. MacCarthy, and T. Russel, Science 306, 76 (2004).

    Article  CAS  Google Scholar 

  19. P. O’Connell and G. McKenna, Science 307, 1760 (2005).

    Article  Google Scholar 

  20. G. T. Gavranovic, J. M. Deutsch, and G. G. Fuller, Macromolecules 38, 6672 (2005).

    Article  CAS  Google Scholar 

  21. W. Kuhlman, E. Olivetti, L. Griffith, and A. Mayes, Macromolecules 39, 5122 (2006).

    Article  CAS  Google Scholar 

  22. S. Sheiko, F. Sun, A. Randall, D. Shirvanyants, M. Rubinstein, H. Lee, and K. Matyjaszewski, Nature 440, 191 (2006).

    Article  CAS  Google Scholar 

  23. F. Sun, A. Dobrynin, D. Shirvanyants, H. Lee, K. Matyjaszewski, G. Rubinstein, M. Rubinstein, and S. Sheiko, Phys. Rev. Lett. 99, 137801 (2007).

    Article  Google Scholar 

  24. M. Gallyamov, B. Tartsch, I. Potemkin, H. Borner, K. Matyjaszewski, A. Khokhlov, and M. Moller, Eur. Phys. J. E 29, 73 (2009).

    Article  CAS  Google Scholar 

  25. K. Shin, S. Obukhov, J.-T. Chen, J. Huh, Y. Hwang, S. Mok, P. Dobriyal, P. Thiyagarjan, and T. Russell, Nat. Mater. 6, 961 (2007).

    Article  CAS  Google Scholar 

  26. F. Monroy, F. Ortega, R. G. Rubio, H. Ritacco, and D. Langevin, Phys. Rev. Lett. 95, 056103 (2005).

    Article  Google Scholar 

  27. F. Monroy, F. Ortega, R. G. Rubio, and M. G. Velarde, Adv. Colloid Interface Sci. 134–135, 175 (2007).

    Article  Google Scholar 

  28. A. Maestro, H. M. Hilles, F. Ortega, R. G. Rubio, D. Langevin, and F. Monroy, Soft Matter 6, 4407 (2010).

    Article  CAS  Google Scholar 

  29. L. R. Arriaga, F. Monroy, and D. Langevin, Soft Matter 7, 7754 (2011).

    Article  CAS  Google Scholar 

  30. K. Sugihara and J. Kumaki, J. Phys. Chem. B 116, 6561 (2012).

    Article  CAS  Google Scholar 

  31. F. Brochard and P.-G. De Gennes, J. Phys., Lett. 40, L399 (1979).

    Article  Google Scholar 

  32. E. Nikomarov and S. Obukhov, JETP 53, 328 (1981).

    Google Scholar 

  33. B. Duplantier, Phys. Rev. B: Condens. Matter 35, 5290 (1987).

    Article  Google Scholar 

  34. B. Duplantier, J. Stat. Phys. 54, 581 (1989).

    Article  Google Scholar 

  35. A. N. Semenov and A. Johner, Eur. Phys. J. E 12, 469 (2003).

    Article  CAS  Google Scholar 

  36. N. Lee, J. Farago, H. Meyer, J. Wittmer, J. Baschnagel, S. Obukhov, and A. Johner, Europhys. Lett. 96, 48002 (2011).

    Article  Google Scholar 

  37. E. Eisenriegler, Polymers Near Surfaces (World Sci., Singapore, 1993).

    Google Scholar 

  38. J. L. Jacobsen, N. Read, and H. Saleur, Phys. Rev. Lett. 90, 090601 (2003).

    Article  CAS  Google Scholar 

  39. Y. Ikhlef, J. Jacobsen, and H. Saleur, J. Stat. Mech.-Theory Exp. 05, P05005 (2007).

    Article  Google Scholar 

  40. A. Baumgärtner, Polymer 23, 334 (1982).

    Article  Google Scholar 

  41. I. Carmesin and K. Kremer, J. Phys. 51, 915 (1990).

    Article  CAS  Google Scholar 

  42. P. H. Nelson, T. A. Hatton, and G. Rutledge, J. Chem. Phys. 107, 1269 (1997).

    Article  CAS  Google Scholar 

  43. B. Ostrovsky, M. Smith, and Y. Bar-Yam, Int. J. Mod. Phys. C 8, 931 (1997).

    Article  Google Scholar 

  44. P. Polanowski and T. Pakula, J. Chem. Phys. 117, 4022 (2002).

    Article  CAS  Google Scholar 

  45. N. K. Balabaev, A. A. Darinskii, I. M. Neelov, N. V. Lukasheva, and I. Emri, Polymer Sci., Ser. A 44, 781 (2002).

    Google Scholar 

  46. A. Yethiraj, Macromolecules 36, 5854 (2003).

    Article  CAS  Google Scholar 

  47. A. Cavallo, M. Muller, and K. Binder, Europhys. Lett. 61, 214 (2003).

    Article  CAS  Google Scholar 

  48. A. Cavallo, M. Müller, and K. Binder, J. Phys. Chem. B 109, 6544 (2005).

    Article  CAS  Google Scholar 

  49. A. Cavallo, M. Müller, J. P. Wittmer, A. Johner, and K. Binder, J. Phys.: Condens. Matter 17, S1697 (2005).

    Article  CAS  Google Scholar 

  50. H. Meyer, T. Kreer, A. Cavallo, J. P. Wittmer, and J. Baschnagel, Eur. Phys. J., Spec. Top. 141, 167 (2007).

    Article  Google Scholar 

  51. H. Meyer, T. Kreer, M. Aichele, A. Cavallo, A. Johner, J. Baschnagel, and J. P. Wittmer, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 79, 050802(R) (2009).

    Article  Google Scholar 

  52. H. Meyer, J. P. Wittmer, T. Kreer, A. Johner, and J. Baschnagel, J. Chem. Phys. 132, 184904 (2010).

    Article  Google Scholar 

  53. H. Meyer, N. Schulmann, J. E. Zabel, and J. P. Wittmer, Comput. Phys. Commun. 182, 1949 (2011).

    Article  CAS  Google Scholar 

  54. N. Schulmann, H. Meyer, J. P. Wittmer, A. Johner, and J. Baschnagel, Macromolecules 45, 1646 (2012).

    Article  CAS  Google Scholar 

  55. J. P. Wittmer, H. Meyer, A. Johner, T. Kreer, and J. Baschnagel, Phys. Rev. Lett. 105, 037802 (2010).

    Article  CAS  Google Scholar 

  56. N. Schulmann, H. Xu, H. Meyer, P. Poliñska, J. Baschnagel, and J. P. Wittmer, Eur. Phys. J. E 35, 93 (2012b).

    Article  CAS  Google Scholar 

  57. M. Allen and D. Tildesley, Computer Simulation of Liquids (Oxford Univ. Press, Oxford, 1994).

    Google Scholar 

  58. D. Frenkel and B. Smit, Understanding Molecular Simulation-From Algorithms to Applications (Academic, San Diego, 2002).

    Google Scholar 

  59. D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge Univ. Press, Cambridge, 2000).

    Google Scholar 

  60. G. S. Grest and K. Kremer, Phys. Rev. A 33, 3628 (1986).

    Article  CAS  Google Scholar 

  61. K. Kremer and G. Grest, J. Chem. Phys. 92, 5057 (1990).

    Article  CAS  Google Scholar 

  62. S. J. Plimpton, J. Comput. Phys. 117, 1 (1995).

    Article  CAS  Google Scholar 

  63. J. Baschnagel, J. P. Wittmer, and H. Meyer, in Computational Soft Matter: From Synthetic Polymers to Proteins, Ed. by N. Attig (NIC Series, Julich, 2004), Vol. 23, p. 83.

  64. J. P. Wittmer, N. Schulmann, P. Poliñska, and J. Baschnagel, J. Chem. Phys. 135, 186101 (2011).

    Article  CAS  Google Scholar 

  65. J. S. Rowlinson, Liquids and Liquid Mixtures (Butterworths Sci., London, 1959).

    Google Scholar 

  66. P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge Univ. Press, Cambridge, 1995).

    Book  Google Scholar 

  67. A. Kopf, B. Dünweg, and W. Paul, J. Chem. Phys. 107, 6945 (1997).

    Article  CAS  Google Scholar 

  68. A. Milchev, J. P. Wittmer, P. Van der Schoot, and D. P. Landau, Europhys. Lett. 54, 58 (2001).

    Article  CAS  Google Scholar 

  69. B. Schnell, H. Meyer, C. Fond, J. Wittmer, and J. Baschnagel, Eur. Phys. J. E 34, 97 (2011).

    Article  CAS  Google Scholar 

  70. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

    Google Scholar 

  71. R. Everaers, I. Graham, and M. Zuckermann, J. Phys. A 28, 1271 (1995).

    Article  Google Scholar 

  72. S. Caracciolo, A. Pelissetto, and A. Sokal, J. Phys. A: Math. Gen. 23, L969 (1990).

    Article  Google Scholar 

  73. L. Schäfer and K. Elsner, Eur. Phys. J. E 13, 225 (2004).

    Article  Google Scholar 

  74. M. Müller, K. Binder, and L. Schäfer, Macromolecules 33, 4568 (2000).

    Article  Google Scholar 

  75. J. Higgins and H. Benoĭt, Polymers and Neutron Scattering (Oxford Univ. Press, Oxford, 1996).

    Google Scholar 

  76. H. D. Bale and P. W. Schmidt, Phys. Rev. Lett. 53, 596 (1984).

    Article  CAS  Google Scholar 

  77. P. Z. Wong and A. J. Bray, Phys. Rev. Lett. 60, 1344 (1988).

    Article  Google Scholar 

  78. G. B. West, J. H. Brown, and B. J. Enquist, Science 284, 1677 (1999).

    Article  CAS  Google Scholar 

  79. M. Cates and J. Deutsch, J. Phys. 47, 2121 (1986).

    Article  CAS  Google Scholar 

  80. S. Obukhov, M. Rubinstein, and T. Duke, Phys. Rev. Lett. 73, 1263 (1994).

    Article  CAS  Google Scholar 

  81. M. Müller, J. P. Wittmer, and M. E. Cates, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 61, 4078 (2000).

    Article  Google Scholar 

  82. T. Vettorel, A. Grosberg, and K. Kremer, Phys. Biol. 6, 025013 (2009).

    Article  Google Scholar 

  83. J. D. Halverson, W. Lee, G. Grest, A. Grosberg, and K. Kremer, J. Chem. Phys. 134, 204904 (2011).

    Article  Google Scholar 

  84. J. Suzuki, A. Takano, T. Deguchi, and Y. Matsushita, J. Chem. Phys. 131, 144902 (2009).

    Article  Google Scholar 

  85. H.-P. Hsu, W. Nadler, and P. Grassberger, Macromolecules 37, 4658 (2004).

    Article  CAS  Google Scholar 

  86. A. Baumgärtner and D. Yoon, J. Chem. Phys. 79, 521 (1983).

    Article  Google Scholar 

  87. A. Yethiraj, B. J. Sung, and F. Lado, J. Chem. Phys. 122, 094910 (2005).

    Article  Google Scholar 

  88. H.-P. Hsu, W. Paul, and K. Binder, Europhys. Lett. 95, 68004 (2011).

    Article  Google Scholar 

  89. J. Jacobsen and J. Kondev, Phys. Rev. Lett. 92, 210601 (2004a).

    Article  Google Scholar 

  90. J. Jacobsen and J. Kondev, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 69, 066108 (2004b).

    Article  Google Scholar 

  91. I. Potemkin, A. Khokhlov, S. Prokhorova, S. Sheiko, M. Möller, K. Beers, and K. Matyjaszewski, Macromolecules 37, 3918 (2004).

    Article  CAS  Google Scholar 

  92. I. Potemkin, Macromolecules 39, 7178 (2006).

    Article  CAS  Google Scholar 

  93. I. Potemkin and K. Popov, J. Chem. Phys. 129, 124901 (2008).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Charles Sadron, Université de Strasbourg and CNRS, 23 rue du Loess, P 84047, 67034, Strasbourg Cedex 2, France

    N. Schulmann, H. Meyer, A. Johner, J. Baschnagel & J. P. Wittmer

  2. Leibniz-Institut für Polymerforschung Dresden e.V., Postfach 120 411, 01005, Dresden, Germany

    T. Kreer

  3. Dirtimento di Fisica, Universita degli Studi di Salerno, via Ponte don Melillo, I-84084, Fisciano, Italy

    A. Cavallo

Authors
  1. N. Schulmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Meyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. Kreer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Cavallo
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. Johner
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. J. Baschnagel
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. J. P. Wittmer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. P. Wittmer.

Additional information

The article is published in the original.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schulmann, N., Meyer, H., Kreer, T. et al. Strictly two-dimensional self-avoiding walks: Density crossover scaling. Polym. Sci. Ser. C 55, 181–211 (2013). https://doi.org/10.1134/S1811238213070072

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1811238213070072

Keywords

Search

Navigation