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Effect of Confinement: Polygons in Strips, Slabs and Rectangles

  • Anthony J Guttmann
  • Iwan Jensen
Part of the Lecture Notes in Physics book series (LNP, volume 775)

In this chapter we will be considering the effect of confining polygons to lie in a bounded geometry. This has already been briefly discussed in Chapters 2 and 3, but here we give many more results. The simplest, non-trivial case is that of SAP on the two-dimensional square lattice Z2, confined between two parallel lines, say x = 0 and x = w. This problem is essentially 1-dimensional, and as such is in principle solvable. As we shall show, the solution becomes increasingly unwieldy as the distance w between the parallel lines increases. Stepping up a dimension to the situation in which polygons in the simple-cubic lattice Z3 are confined between two parallel planes, that is essentially a two-dimensional problem, and as such is not amenable to exact solution.

Self-avoiding walks in slits were first treated theoretically by Daoud and de Gennes [4] in 1977, and numerically by Wall et al. [14] the same year. Wall et al. studied SAW on Z2, in particular the mean-square end-to-end distance. For a slit of width one they obtained exact results, and also obtained asymptotic results for a slit of width two. Around the same time, Wall and co-workers [13, 15] used Monte Carlo methods to study the width dependence of the growth constant for walks confined to strips of width w. In 1980 Klein [9] calculated the behaviour of SAW and SAP confined to strips in Z2 of width up to six, based on a transfer matrix formulation.

Keywords

Double Layer Transfer Matrix Bottom Wall Growth Constant Transfer Matrix Method 
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. 1.
    Alm S E and Janson S, (1990) Random self-avoiding walks on one dimensional lattices Com-mun Statist-Stochastic Models, 6 (2), 189–212MathSciNetGoogle Scholar
  2. 2.
    Alvarez J, Janse van Rensburg E J, Soteros E G and Whittington, S G (2008) Self avoiding polygons and walks in slits J. Phys A: Math. Gen, 41 Google Scholar
  3. 3.
    Bousquet-Mélou, M, Guttmann, A J and Jensen, I, (2005) Self-avoiding walks crossing a square J. Phys A: Math. Gen, 38 9159–9181.MATHCrossRefADSGoogle Scholar
  4. 4.
    Daoud, M and de Gennes P G (1977) Statistics of macromolecular solutions trapped in small pores J. de Physique 38 85–93CrossRefGoogle Scholar
  5. 5.
    deGennes P G (1979) Scaling concepts in Polymer Physics (Ithaca: Cornell University Press)Google Scholar
  6. 6.
    Di Marzio E A and Rubin R J (1971) Adsorption of a chain polymer between two plates J. Chem. Phys, 55 4318–36.CrossRefADSGoogle Scholar
  7. 7.
    Hammersley J M and Whittington S G (1985) Self-avoiding walks in wedges J. Phys A: Math. Gen, 18 101–11CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Janse van Rensburg E J, Orlandini E and Whittington, S G (2006) Self-avoiding walks in a slab: rigorous results J. Phys A: Math. Gen, 39 13869–13902.MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Klein, D (1980) Asymptotic distributions for self-avoiding walks constrained to strips, cylin ders and tubes J. Stat. Phys 23 561–86.CrossRefADSGoogle Scholar
  10. 10.
    Martin, R, Orlandini E, Owczarek A L, Rechnitzer A and Whittington, S G Exact enumera tions and Monte Carlo results for self avoiding walks in a slab, (2007) J. Phys A: Math. Gen, 40 7509–21.MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Soteros C E and Whittington S G (1988) Polygons and stars in a slit geometry J. Phys A: Math. Gen, 21 L857–861.CrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Stilck J F and Machado K D Tension of polymers in a strip (1998) Eur. Phys. J. B 5 899–904.CrossRefADSGoogle Scholar
  13. 13.
    Wall, F T, Mandel, F and Chin J C (1976) Self-avoiding random walks subject to external spatial constraints J. Chem. Phys, 65 2231–2234.CrossRefADSGoogle Scholar
  14. 14.
    Wall, F T, Seitz W A and Chin J C (1977) Self-avoiding walks subject to boundary constraints J. Chem. Phys, 67 434–8.CrossRefADSGoogle Scholar
  15. 15.
    Wall, F T, Seitz W A Chin J C and deGennes P G (1978) Statistics of self-avoiding walks confined to strips and capillaries Proc. Nat. Acad. Sci 75 no 5 2069–70.CrossRefADSGoogle Scholar

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© Canopus Academic Publishing Limited 2009

Authors and Affiliations

  • Anthony J Guttmann
    • 1
  • Iwan Jensen
    • 1
  1. 1.Department of Mathematics and StatisticsThe University of MelbourneVictoriaAustralia

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