Order, Frustration, and Two-Dimensional Glass

  • D. R. Nelson
Conference paper
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 46)


Models of two-dimensional disordered materials, designed to illuminate the behavior of frustrated 2d liquids and solids, and provide insights into defect theories of the glass transition, are discussed. Important topological differences in the way particles pack in two, as opposed to three dimensions are emphasized. In d=2, one can introduce a controllable amount of disorder by varying the concentration of particles of the wrong size. Because of dislocation trapping by impurities, quenched analogues of the equilibrium hexatic phase appear at low temperatures. Large orientational correlation lengths characterize many nominally amorphous, translationally disordered particle configurations. Crystalline films with a quenched distribution of impurities become unstable at low temperatures, where thermally excited dislocation pairs are broken apart by the random impurity potential. Frustration can be introduced into arrays of identical particles by packing them on a manifold of constant negative curvature. The curvature can be chosen to mimic problems associated with dense random packing of tetrahedra in three dimensions. Some conjectures on the behavior of such arrays with decreasing temperature are presented.


Curvature Scale Metallic Glass Orientational Order Flat Space Average Coordination Number 
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  1. 1.
    See, e. g., Proceedings of the 1978 Les Houches Summer School: I11-Condensed Matter, edited by R. Balian, R. Maynard, and G. Toulouse ( North-Holland, New York, 1979 ).Google Scholar
  2. 2.
    D. R. Nelson, Phys. Rev. BT8, 2318 (1978).Google Scholar
  3. 3.
    B. I. Halperin and D. R. Nelson, Phys. Rev. Lett. 41 121; E41, 519 (1978); D.R. Nelson and B. I. Halperin, Phys. Rev. B19, 2457 (1979).CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    A. P. Young, Phys. Rev. B19, 1855 (1979).CrossRefADSGoogle Scholar
  5. 5.
    A. Zippelius, B. I. Halperin, and D. R. Nelson, Phys. Rev. B22, 2514 (1980).CrossRefADSGoogle Scholar
  6. 6.
    J. M. Kosterlitz and D. J. Thouless, J. Phys. C6, 1181 (1973)ADSGoogle Scholar
  7. 7.
    C. Domb and M. S. Green, Phase Transitions and Critical Phenomena, Vol. 6 ( Academic, New York 1976).Google Scholar
  8. 8.
    C. A. Angell, J. H. R. Clarke, and L. V. Woodcock, in Advances in Chemical Physics, edited by I. Prigogine and S. Rice (Wiley, New York, 1981 ) Vol. 48.Google Scholar
  9. 9.
    See, e. g., P. A. Heiney, R. J. Birgeneau, G. S. Brown, P. M. Horn, D. E. Moncton, and P. W. Stephens, Phys. Rev. Lett. 48, 104 (1982).CrossRefADSGoogle Scholar
  10. 10.
    R. Pindak, D. J. Bishop, and W. O. Springer, Phys. Rev. Lett. 44, 1461 (1980).CrossRefADSGoogle Scholar
  11. 11.
    R. N. Abraham, K. Miyano, and J. B. Ketterson, in Ordering in Two Dimensions, edited by S.K. Sinha ( North-Holland, Amsterdam, 1980 ).Google Scholar
  12. 12.
    P. S. Pershan, J. de Physique Colloque. 40, C3 (1979).Google Scholar
  13. 13.
    C. Y. Young, R. Pindak, N. A. Clark, and R. B. Meyer, Phys. Rev. Lett. 40, 773 (1978).CrossRefADSGoogle Scholar
  14. 14.
    P. Pieranski, Phys. Rev. Lett. 45, 569 (1980).CrossRefADSGoogle Scholar
  15. 15.
    J. P. McTague, M. Nielson, and L. Passell, in Ordering in Strongly Fluctuating Condensed Matter Systems, edited by T. Riste ( Plenum, New York, 1980 ).Google Scholar
  16. 16.
    See, e. g., F. Spaepen in Proceedings of the Les Houches Summer School: Physics of Defects, edited by J.P. Poirier and M. Kidman (Amsterdam, North-Holland).Google Scholar
  17. 17.
    F. Spaepen, J. Non-crystalline Solids 31, 207 (1978).CrossRefADSGoogle Scholar
  18. 18.
    J. P. McTague, D. Frenkel, and M. Allen, Proceedings of the Conference on Ordering in Two Dimensions, edited by S. Sinha ( Amsterdam, North-Holland, 1980 ).Google Scholar
  19. 19.
    C. S. Cargill, Ann. N.Y. Acad. Sci., 27, 208 (1976).CrossRefADSGoogle Scholar
  20. 20.
    M..R. Hoare, J. Non-Cryst. Solids 31, 157 (1978).CrossRefADSGoogle Scholar
  21. 21.
    See, e. g., F. Spaepen in Ref. 17.Google Scholar
  22. 22.
    M.H. Cohen and D. Turnbull, J. Chem. Phys. 31, 1164 (1959); D. Turnbull and M.H. Cohen, J. Chem. Phys. 34, 120 (1961).CrossRefADSGoogle Scholar
  23. 23.
    J. H. Gibbs and E. A. DiMarzio, J. Chem. Phys. 28, 373 (1958);G. Adam and J. H. Gibbs, J. Chem. Phys. 43, 139 (1965).CrossRefADSGoogle Scholar
  24. 24.
    M. Kliman and J. F. Sadoc, J. de Phys. Lett. 40, L569 (1979).CrossRefGoogle Scholar
  25. 25.
    D. R. Nelson, M. Rubinstein, and F. Spaepen, Phys. Mag. A46, 105 (1982).CrossRefADSGoogle Scholar
  26. 26.
    D. R. Nelson, Harvard University preprint.Google Scholar
  27. 27.
    D. Turnbull and R. L. Cormia, J. Appl. Phys. 31, 674 (1960).CrossRefADSGoogle Scholar
  28. 28.
    A. M. Kosevich in Dislocations in Solids, edited by F.R.N. Nabarro ( North-Holland, Amsterdam, 1979 ).Google Scholar
  29. 29.
    See, e. g., R. Collins, in Phase Transitions and Critical Phenomena, edited by C. Domb and M.S. Green (Academic, New York, 1972 ) Vol. II.Google Scholar
  30. 30.
    F. C. Frank, Proc. Roy. Soc. London 215A, 43 (1952).CrossRefADSGoogle Scholar
  31. 31.
    D. Turnbull, J. Chem. Phys. 20, 411(1952).CrossRefADSGoogle Scholar
  32. 32.
    P. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys. Rev. Lett. 47, 1297 (1981). and to be Published.CrossRefADSGoogle Scholar
  33. 33.
    For a lucid introduction to the differential geometry of surfaces, see H. S.M. Coxeter, Introduction to Geometry (Wiley and Sons, Inc., 1969), Chapters 19–21.Google Scholar
  34. 34.
    CRC Standard Mathematical Tables, edited by S.M. Selby ( The Chemical Rubber Co., Cleveland, 1970 ), pp. 15–16.Google Scholar
  35. 35.
    This picture is taken from a computer simulation of particles interacting with a repulsive 1/r potential by R. Morf.Google Scholar
  36. 36.
    For a discussion of boundary effects see J.P. Gaspard, R. Mosseri, and J.F. Sadoc, Saclay preprint.Google Scholar
  37. 37.
    D.S. Fisher, Phys. Rev. B2, 1190 (1980), and references therein.Google Scholar
  38. 38.
    M. Tinkham, Introduction to Superconductivity (McGraw-Hill, New York,1975).Google Scholar
  39. 39.
    M. Kliman, in Dislocation in Solids, edited by F. R. N. Nabarro ( North- Holland, New York, 1980 ) Vol. 5, p. 243.Google Scholar
  40. 40.
    J. P. Hansen, D. Levesque, and J. J. Weise, Phys. Rev. Lett. 43, 979 (1979).CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • D. R. Nelson
    • 1
  1. 1.Lyman Laboratory of PhysicsHarvard UniversityCambridgeUSA

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