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Journal of Statistical Physics

, Volume 102, Issue 5–6, pp 1177–1209 | Cite as

Trees at an Interface

  • E. J. Janse van Rensburg
Article
  • 27 Downloads

Abstract

A lattice tree at an interface between two solvents of different quality is examined as a model of a branched polymer at an interface. Existence of the free energy is shown, and the existence of critical lines in its phase diagram is proven. In particular, there is a line of first order transitions separating a positive phase from a negative phase (the tree being predominantly on either side of the interface in these phases), and a curve of localization–delocalization transitions which separate the delocalized positive and negative phases from a phase where the tree is localized at the interface. This model is generalized to a branched copolymer which is examined in a certain averaged quenched ensemble. Existence of a thermodynamic limit is shown for this model, and it is also shown that the model is self-averaging. Lastly, a model of branched polymers interacting with one another through a membrane is considered. The existence of a limiting free energy is shown, and it is demonstrated that if the interaction is strong enough, then the two branched polymers will adsorb on one another.

trees phase transition adsorbing localization–delocalization branched copolymer self-averaging 

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REFERENCES

  1. 1.
    E. Eisenriegler, K. Kremer, and K. Binder, Adsorption of polymer chains at surfaces, scaling and Monte Carlo analysis, J. Chem. Phys. 77:6296 (1982).Google Scholar
  2. 2.
    J. M. Hammersley, G. M. Torrie, and S. G. Whittington, Self-avoiding walks interacting with a surface, J. Phys. A: Math. Gen. 15:539 (1982).Google Scholar
  3. 3.
    K. De'Belland and T. Lockman, Surface phase transitions in polymer systems, Rev. Mod. Phys. 65:87 (1993).Google Scholar
  4. 4.
    C. Vanderzande, On knots in a model for the adsorption of ring polymers, J. Phys. A: Math. Gen. 28:3681 (1995).Google Scholar
  5. 5.
    T. Vrbová and S. G. Whittington, Adsorption and collapse of self-avoiding walks and polygons in three dimensions, J. Phys. A: Math. Gen. 29:6253 (1996).Google Scholar
  6. 6.
    T. Vrbová and S. G. Whittington, Adsorption and collapse of self-avoiding walks in three dimensions, J. Phys. A: Math. Gen. 31:3989 (1998).Google Scholar
  7. 7.
    T. Vrbova and S. G. Whittington, Adsorption and collapse of self-avoiding walks at a defect plane, J. Phys. A: Math. Gen. 31:7031 (1998).Google Scholar
  8. 8.
    E. J. Janse van Rensburg, Collapsing and adsorbing polygons, J. Phys. A: Math. Gen. 31:8295 (1998).Google Scholar
  9. 9.
    E. J. Janse van Rensburg and S. You, Adsorbing and collapsing trees, J. Phys. A: Math. Gen. 31:8635 (1998).Google Scholar
  10. 10.
    E. Eisenriegler, Polymers Near Surfaces (World Scientific, Singapore, 1993).Google Scholar
  11. 11.
    S. G. Whittington, A self-avoiding walk model of copolymer adsorption, J. Phys. A: Math. Gen. 31:3769 (1998).Google Scholar
  12. 12.
    S. You and E. J. Janse van Rensburg, A lattice tree model of branched copolymer adsorption, J. Phys. A: Math. Gen. 33:1171 (2000).Google Scholar
  13. 13.
    D. J. Klein, Rigorous results for branched polymer models with excluded volume, J. Chem. Phys. 75:5186 (1981).Google Scholar
  14. 14.
    E. J. Janse van Rensburg, The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles, Oxford Lecture Series in Mathematics and its Applications, Vol. 18 (Oxford University Press, Oxford, 2000).Google Scholar
  15. 15.
    E. J. Janse van Rensburg, Models of composite polygons, J. Phys. A: Math. Gen. 32:4351 (1999).Google Scholar
  16. 16.
    J. M. Hammersley, Percolation processes II: The connective constant, Math. Proc. Camb. Phil. Soc. 53:642 (1957).Google Scholar
  17. 17.
    J. M. Hammersley, The number of polygons on a lattice, Math. Proc. Camb. Phil. Soc. 57:516 (1961).Google Scholar
  18. 18.
    E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, AMS Colloquim Publications, Vol. 31 (American Mathematical Society, 1957).Google Scholar
  19. 19.
    J. M. Hammersley, Generalization of the fundamental theorem on sub-additive functions, Math. Proc. Camb. Phil. Soc. 58:235 (1962).Google Scholar
  20. 20.
    G. Grimmett and D. J. A. Welsh, Probability, An Introduction (Oxford Science Publications, 1991).Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • E. J. Janse van Rensburg
    • 1
  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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