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.


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  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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