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Entanglement effects in model polymer networks

  • R. Everaers
  • K. Kremer
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 519)

Abstract

The influence of topological constraints on the local dynamics in cross-linked polymer melts and their contribution to the elastic properties of rubber elastic systems are a long standing problem in statistical mechanics. Polymer networks with diamond lattice connectivity (Everaers and Kremer 1995, Everaers and Kremer 1996a) are idealized model systems which isolate the effect of topology conservation from other sources of quenched disorder. We study their behavior in molecular dynamics simulations under elongational strain. In our analysis we compare the measured, purely entropic shear moduli G to the predictions of statistical mechanical models of rubber elasticity, making extensive use of the microscopic structural and topological information available in computer simulations. We find (Everaers and Kremer 1995) that the classical models of rubber elasticity underestimate the true change in entropy in a deformed network significantly, because they neglect the tension along the contour of the strands which cannot relax due to entanglements (Everaers and Kremer (in preparation)). This contribution and the fluctuations in strained systems seem to be well described by the constrained mode model (Everaers 1998) which allows to treat the crossover from classical rubber elasticity to the tube model for polymer networks with increasing strand length within one transparant formalism. While this is important for the description of the effects we try to do a first quantitative step towards their explanation by topological considerations. We show (Everaers and Kremer 1996a) that for the comparatively short strand lengths of our diamond networks the topology contribution to the shear modulus is proportional to the density of entangled mesh pairs with non-zero Gauss linking number. Moreover, the prefactor can be estimated consistently within a rather simple model developed by Vologodskii et al. and by Graessley and Pearson, which is based on the definition of an entropic interaction between the centers of mass of two loops in a conserved topological state.

Polymer networks are the basic structural element of systems as different as tire rubber and gels. They are not only technically important but also commonly found in biological systems such as the cytoskeleton. Networks of flexible macromolecules display an elastic and thermoelastic behaviour quite different from ordinary solids. (Treloar 1975) Crystals, metals, ceramics, or glasses can be stretched only minimally. Small deformations of the sample extend down to atomic scales and lead to an increase of the internal energy. Rubber-like materials reversibly sustain elongations of up to 1000% with small strain elastic moduli that are four or five orders of magnitude smaller than for other solids. Most importantly, the tension induced by a deformation is almost exclusively due to a decrease in entropy. As a consequence, the underlying mechanism has to be different from the case of conventional solids.

Keywords

Tube Model Rubber Elasticity Entanglement Effect Strand Length Unstrained State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Barsky, S. J., Plischke, M., Zhou, Z. and Joós, B. (1996): Phys. Rev. E, 54, 5370CrossRefADSGoogle Scholar
  2. Barsky, S. J. and Plischke, M. (1996): Phys. Rev. E, 53, 871CrossRefADSGoogle Scholar
  3. Bird, R.B., Armstrong, R. C. and Hassager, O. (1977): Dynamics of Polymeric Liquids, volume 1. Wiley, New YorkGoogle Scholar
  4. Curro, J. G. and Pincus, P. A. (1983): Macromolecules, 16, 559CrossRefADSGoogle Scholar
  5. Curro, J. G., Pearson, D. S. and Helfand, E. (1985): Macromolecules, 18, 1157CrossRefADSGoogle Scholar
  6. Curro, J. G., Schweizer, K. S., Grest, G. S. and Kremer, K. (1991): J. of Chem. Phys., 91, 1357CrossRefADSGoogle Scholar
  7. Deam, R. T. and Edwards, S. F. (1976): Phil. Trans. R. Soc. A, 280, 317CrossRefADSMathSciNetGoogle Scholar
  8. Doi, M. and Edwards, S. F. (1986): The Theory of Polymer Dynamics. Claredon Press, OxfordGoogle Scholar
  9. Duering, E. R., Kremer, K. and Grest, G. S. (1991): Phys. Rev. Lett., 67, 3531CrossRefADSGoogle Scholar
  10. Duering, E. R., Kremer, K. and Grest, G. S. (1992): Progr. Colloid Polym. Sci., 90, 13CrossRefGoogle Scholar
  11. Duering, E. R., Kremer, K. and Grest, G. S. (1993): Macromolecules, 26, 3241CrossRefADSGoogle Scholar
  12. Duering, E. R., Kremer, K. and Grest, G. S. (1994): J. of Chem. Phys., 101, 8169CrossRefADSGoogle Scholar
  13. Edwards, S. F. (1967a): Proc. Phys. Soc., 92, 9CrossRefADSGoogle Scholar
  14. Edwards, S. F. (1967b): Proc. Phys. Soc., 91, 513zbMATHCrossRefADSGoogle Scholar
  15. Edwards S. F. (1968): J. of Phys. A, 1, 15CrossRefADSGoogle Scholar
  16. Edwards, S. F. and Vilgis, T. A. (1986): Polymer, 27, 483CrossRefGoogle Scholar
  17. Edwards, S. F. and Vilgis, T. A. (1988): Rep. Progr. Phys., 51, 243CrossRefADSMathSciNetGoogle Scholar
  18. Erman, B. and Flory, P. J. (1978): J. of Chem. Phys., 68, 5363CrossRefADSGoogle Scholar
  19. Erman, B. and Flory, P. J. (1982): Macromolecules, 15, 806CrossRefADSGoogle Scholar
  20. Everaers, R. and Kremer, K. in preparationGoogle Scholar
  21. Everaers, R. and Kremer, K. (1994): Comp. Phys. Comm., 81, 19CrossRefADSGoogle Scholar
  22. Everaers, R. Kremer, K. and Grest, G. S. (1995): Macromol. Symposia, 93, 53Google Scholar
  23. Everaers, R. and Kremer, K. (1995): Macromolecules, 28, 7291CrossRefADSGoogle Scholar
  24. Everaers, R. and Kremer, K. (1996a): Phys. Rev. E, 53, R37Google Scholar
  25. Everaers, R. and Kremer, K. (1996b): J. Mol. Mod., 2, 293CrossRefGoogle Scholar
  26. Everaers, R. (1998): Eur. J. Phys. B, 4, 341CrossRefADSGoogle Scholar
  27. Flory, P. J. and Rehner, J. (1943): J. of Chem. Phys., 11, 512CrossRefADSGoogle Scholar
  28. Flory, P. J. (1976): Proc. Royal Soc. London Ser. A., 351, 351ADSCrossRefGoogle Scholar
  29. Flory, P.J. (1977): J. of Chem. Phys., 66, 5720CrossRefADSGoogle Scholar
  30. Flory, P. J. and Erman, B. (1982): Macromolecules, 15, 800CrossRefADSGoogle Scholar
  31. Frank-Kamenetskii, M. D. Lukashin, A. V. and Vologodskii, A. V. (1975): Nature, 258, 398CrossRefADSGoogle Scholar
  32. Gao, J. and Weiner, J. H. (1995): J. of Chem. Phys., 103, 1614CrossRefADSGoogle Scholar
  33. de Gennes, P. G. (1971): J. of Chem. Phys., 55, 572CrossRefADSGoogle Scholar
  34. Gottlieb, M., Macosko, C. W., Benjamin, G. S., Meyers, K. O. and Merrill, E. W. (1981): Macromolecules, 14, 1039CrossRefADSGoogle Scholar
  35. Graessley, W. W. and Pearson, D. S. (1977): J. of Chem. Phys., 66, 3363CrossRefADSGoogle Scholar
  36. Graessley, W. W. (1982): Adv. Pol. Sci., 47, 67CrossRefGoogle Scholar
  37. Grest, G. S. and Kremer, K. (1990a): J. de Physique (France), 51, 2829CrossRefGoogle Scholar
  38. Grest, G. S. and Kremer, K. (1990b): Macromolecules, 23, 4994CrossRefADSGoogle Scholar
  39. Grest, G. S., Kremer, K. and Duering, E. R. (1992): Europhysics Lett., 19, 195CrossRefADSGoogle Scholar
  40. Grest, G. S., Kremer, K. and Duering, E. R. (1993): Physica A, 194, 330CrossRefADSGoogle Scholar
  41. Heinrich, G. Straube, E. and Helmis, G. (1988): Adv. Pol. Sci., 85, 34Google Scholar
  42. Helfand, E. and Tonelli, A. E. (1974): Macromolecules, 7, 832CrossRefADSGoogle Scholar
  43. Herrmann, H. J., Hong, D. C. and Stanley, H. E. (1984): J. of Phys. A, 17, L261Google Scholar
  44. Iwata, K. (1982): J. of Chem. Phys., 76, 6363CrossRefADSMathSciNetGoogle Scholar
  45. Iwata, K. (1985): J. of Chem. Phys., 83, 1969CrossRefADSMathSciNetGoogle Scholar
  46. James, H. (1947): J. of Chem. Phys., 15, 651CrossRefADSGoogle Scholar
  47. James, H. and Guth, E. (1947): J. of Chem. Phys., 15, 669CrossRefADSGoogle Scholar
  48. Kästner, S. (1981): Colloid Polym. Sci., 259, 499 and 508CrossRefGoogle Scholar
  49. Kremer, K. and Grest, G. S. (1990): J. of Chem. Phys., 92, 5057CrossRefADSGoogle Scholar
  50. Kremer, K. and Grest, G. S. (1991): J. of Chem. Phys., 94, 4103CrossRefADSGoogle Scholar
  51. Kremer, K. and Grest, G. S. (1995): In K. Binder, editor, Monte Carlo and Molecular Dynamics Simulations in Polymer Science. Oxford University Press, New York and OxfordGoogle Scholar
  52. Larsson, I. and Kramer, O. (1993): Makromol. Chem., Macromol. Symp., 76, 117Google Scholar
  53. Lazár, M., Rado, R. and Rychly, J. (1990): Adv. Polym. Sci., 95, 149Google Scholar
  54. Leung, Y. K. and Eichinger, B. E. (1984): J. of Chem. Phys., 80, 3877 and 3885CrossRefADSGoogle Scholar
  55. Mark, J.E. (1982): Adv. Pol. Sci., 44, 1CrossRefGoogle Scholar
  56. Oeser, R., Ewen, B., Richter, D. and Farago, B. (1988): Phys. Rev. Lett., 260, 1041CrossRefADSGoogle Scholar
  57. Opperman, W. and Rennar, N. (1987): Prog. Colloid Polym. Sci., 75, 49CrossRefGoogle Scholar
  58. Patel, S. K. Malone, S., Cohen, C. and Gillmor, J. R (1992): Macromolecules, 25, 5241CrossRefADSGoogle Scholar
  59. Pearson, D. S. and Graessley, W. W. (1980): Macromolecules, 13, 1001CrossRefADSGoogle Scholar
  60. Queslel, J. P. and Mark, J. E. (1984): Adv. Pol. Sci., 65, 135CrossRefGoogle Scholar
  61. Ronca, G. and Allegra, G. (1975): J. of Chem. Phys., 63, 4990CrossRefADSGoogle Scholar
  62. Schweizer, K. S. and Curro, J. G. (1994): Adv. Polym. Sci., 116, 319CrossRefGoogle Scholar
  63. Shy, L. Y. and Eichinger, B. E. (1986): Macromolecules, 19, 2787CrossRefADSGoogle Scholar
  64. Straube, E., Urban, V., Pyckhout-Hintzen, W. Richter, D. and Glinka, C. J. (1995): Phys. Rev. Lett., 74, 4464CrossRefADSGoogle Scholar
  65. Tonelli, A. E. and Helfand, E. (1974): Macromolecules, 7, 59CrossRefADSGoogle Scholar
  66. Treloar, L.R.G. (1975): The Physics of Rubber Elasticity. Clarendon Press, OxfordGoogle Scholar
  67. Wiegel, F. W. (1986): Introduction in Path Integral Methods in Physics and Polymers Science. World Scientific, PhiladelphiaGoogle Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • R. Everaers
    • 1
    • 2
  • K. Kremer
    • 1
  1. 1.Max-Planck-Institut für PolymerforschungMainzGermany
  2. 2.Section de rechercheInstitut CurieParis Cedex 05France

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