Skip to main content
SpringerLink
Log in

The Intermediate Phase and Self-organization in Network Glasses

  • Chapter
Phase Transitions and Self-Organization in Electronic and Molecular Networks

Part of the book series: Fundamental Materials Research ((FMRE))

  • 282 Accesses

Summary

We have discussed the rigidity of random and self-organized networks. We find that there is a single transition from floppy to rigid in random networks, but an intermediate phase intervenes in the self-organized networks. This intermediate phase is rigid but contains no redundant bonds and so is stress-free.

Some of this work, in the form of more extensive lecture notes has been presented at a NATO-ASI in Czech Republic in the summer of 2000, and will appear in the NATO-ASI series. We should like to thank D.J. Jacobs for his many contributions to the “pebble game” algorithm and to the US National Science Foundation for support under grant numbers DMR-0078361 and CHE-9903706.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Zachariasen, W.H. (1932) The atomic arrangement in glass, J. Am. Chem. Soc. 54, 3841–3851.

    Article  CAS  Google Scholar 

  2. Polk, D.E. (1971) Structural model for amorphous silicon and germanium, J. Non-Cryst. Solids 5, 365–376.

    Article  CAS  Google Scholar 

  3. Wooten, F., Winer, K., Weaire, D. (1985) Computer generation of structural models of amorphous Si and Ge, Phys. Rev. Lett. 54, 1392–1395.

    Article  CAS  Google Scholar 

  4. Kirkwood, J.G. (1939) Skeletal modes of vibration of long-chain molecules, J. Chem. Phys. 7, 506–509.

    CAS  Google Scholar 

  5. Keating, P.N. (1966) Theory of the third-order elastic constants of diamond-like crystals, Phys. Rev. 145, 674–678.

    Article  Google Scholar 

  6. Car, R. and Parrinello, M. (1985) Unified approach for molecular dynamics and density functional theory, Phys. Rev. Lett. 55, 2471–2474.

    Article  CAS  Google Scholar 

  7. Mousseau, N. and Barkema, G.T. (1998) Traveling through potential energy landscapes of disordered materials: The activation-relaxation technique, Phys. Rev. E57, 2419–2424; Barkema, G.T. and Mousseau, N. (1996) Event-based relaxation of continuous disordered systems, Phys. Rev. Lett. 77, 4358–4361.

    Google Scholar 

  8. Maxwell, J.C. (1864) On the calculation of the equilibrium and stiffness of frames, Philos. Mag. 27, 294–299.

    Google Scholar 

  9. Lagrange, J.L. (1788) Mécanique Analytique, Desaint, Paris.

    Google Scholar 

  10. Guyon, E., Roux, S., Hansen, A., Bideau, D., Trodec, J.-P. and Crapo, H. (1990) Nonlocal and nonlinear problems in the mechanics of disordered systems — application to granular media and rigidity problems, Rep. Progr. Phys. 53, 373–419.

    Google Scholar 

  11. Feng, S. and Sen, P. (1984) Percolation on elastic networks — new exponent and threshold, Phys. Rev. Lett. 52, 216–219.

    Article  Google Scholar 

  12. Jacobs, D.J. and Thorpe, M.F. (1995) Generic rigidity percolation: the pebble game, Phys. Rev. Lett. 75, 4051–4054.

    CAS  Google Scholar 

  13. Hendrickson, B. (1992) Conditions for unique graph realizations, SIAM J. Comput. 21, 65–84 and private communications.

    Article  Google Scholar 

  14. Laman, G. (1970) On graphs and rigidity of plane skeletal structures, J. Engrg. Math. 4, 331–340.

    Google Scholar 

  15. Lovasz, L. and Yemini, Y. (1982) On generic rigidity in the plane, SIAM J. Alg. Disc. Meth. 3, 91–98.

    Article  Google Scholar 

  16. Thorpe, M.F. (1983) Continuous deformations in random networks, J. Non-Cryst. Solids 57, 355–370.

    Article  CAS  Google Scholar 

  17. Cai, Y. and Thorpe, M.F. (1989) Floppy modes in network glasses, Phys. Rev. B40, 10535–10542.

    Google Scholar 

  18. Feng, S., Thorpe, M.F. and Garboczi, E.J. (1985) Effective-medium theory of percolation on central-force elastic networks, Phys. Rev. B31, 276–280.

    Google Scholar 

  19. Day, A.R., Tremblay, R.R. and Tremblay, A.-M.S. (1986) Rigid backbone: A new geometry for percolation, Phys. Rev. Lett. 56, 2501–2504.

    Article  Google Scholar 

  20. He, H. and Thorpe, M.F. (1985) Elastic properties of glasses, Phys. Rev. Lett. 54, 2107–2110.

    Article  CAS  Google Scholar 

  21. Moukarzel, C. and Duxbury, P.M. (1995) Stressed backbone and elasticity of random central-force systems, Phys. Rev. Lett. 75, 4055–4058.

    Article  CAS  Google Scholar 

  22. Phillips, J.C. (1979) Topology of covalent non-crystalline solids. 1. Short-range order in chalcogenide alloys, J. Non-Cryst. Solids 34, 153–181.

    Article  CAS  Google Scholar 

  23. Phillips, J.C. (1981) Topology of covalent non-crystalline solids. 2. Medium-range order in chalcogenide alloys and a-Si(Ge), J. Non-Cryst. Solids 43, 37–77.

    Article  CAS  Google Scholar 

  24. Boolchand, P. and Thorpe, M.F. (1994) Glass-forming tendency, percolation of rigidity, and one fold-coordinated atoms in covalent networks, Phys. Rev. B50, 10366–10368.

    Google Scholar 

  25. Boolchand, P., Zhang, M. and Goodman, B. (1996) Influence of one-fold-coordinated atoms on mechanical properties of covalent networks, Phys. Rev. B53, 11488–11494.

    Google Scholar 

  26. Döhler, G.H., Dandoloff, R. and Bilz, H. (1981) A topological-dynamical model of amorphycity, J. Non-Cryst. Solids 42, 87–96.

    Google Scholar 

  27. Jacobs, D.J. and Thorpe, M.F. (1996) Generic rigidity percolation in two dimensions, Phys. Rev. E53, 3682–3693.

    Google Scholar 

  28. Jacobs, D.J. (1998) Generic rigidity in three-dimensional bond-bending networks, J. Phys. A.: Math. Gen. 31, 6653–6668.

    CAS  Google Scholar 

  29. Jacobs, D.J., Hendrickson, B. (1997) An algorithm for two-dimensional rigidity percolation: The pebble game, J. Comput. Phys. 137, 346–365.

    Article  Google Scholar 

  30. Jacobs, D.J., Kuhn, L.A. and Thorpe, M.F. (1999) Flexible and rigid regions in proteins, in M.F. Thorpe and P.M. Duxbury (eds.), Rigidity Theory and Applications, Kluwer Academic / Plenum Publishers, New York, pp. 357–384.

    Google Scholar 

  31. Fortuin, C.M. and Kasteleyn, P.W. (1972) On the random-cluster model. 1. Introduction and relation to other models, Physica 57, 536–564; Kasteleyn, P.W. and Fortuin, C.M. (1969) Phase transitions in lattice systems with random local properties, J. Phys. Soc. Japan 26 sup, 11–14.

    Article  Google Scholar 

  32. Essam, J.W. (1980) Percolation theory, Rep. Prog. Phys. 43, 833–912.

    Article  CAS  Google Scholar 

  33. Duxbury, P.M., Jacobs, D.J., Thorpe, M.F. and Moukarzel, C. (1999) Floppy modes and the free energy: Rigidity and connectivity percolation on Bethe lattices, Phys. Rev. E59, 2084–2092.

    Google Scholar 

  34. Djordjevic, B.R., Thorpe, M.F. and Wooten, F. (1995) Computer model of tetrahedral amorphous diamond, Phys. Rev. B52, 5685–5689.

    Google Scholar 

  35. Thorpe, M.F., Jacobs, D.J., Chubynsky, N.V. and Rader, A.J. (1999) Generic rigidity of network glasses, in M.F. Thorpe and P.M. Duxbury (eds.), Rigidity Theory and Applications, Kluwer Academic/Plenum Publishers, New York, pp. 239–278.

    Google Scholar 

  36. Press, W.H., Teukolsky, S.A., Vetterling, V.T. and Flannery, B.P. (1992) Numerical Recipes in FORTRAN, 2nd ed., Cambridge University Press, New York.

    Google Scholar 

  37. Franzblau, D.S. and Tersoff, J. (1992) Elastic properties of a network model of glasses, Phys. Rev. Lett. 68, 2172–2175.

    Article  Google Scholar 

  38. Moukarzel, C. (1998) A fast algorithm for backbones, Int. J. Mod. Phys. C9, 887–895.

    Google Scholar 

  39. Straley, J.P. (1979) Phase transitions in treelike percolation, Phys. Rev. B19, 4845–4846.

    Google Scholar 

  40. Manna, S.S. and Subramanian, B. (1996) Quasirandom spanning tree model for the early river network, Phys. Rev. Lett. 76, 3460–3463.

    Article  CAS  Google Scholar 

  41. Kirchhoff, G. (1847) Über die Auflösung der Gleichungen, auf welche man bei der untersuchung der linearen verteilung galvanischer Ströme geführt wird, Ann. Phys. Chem. 72, 497–508.

    Google Scholar 

  42. See, e.g., Cieplak, M., Giacometti, A., Maritan, A., Rinaldo, A., Rodriguez-Iturbe I. and Banavar, J.R. (1998) Models of fractal river basins, J. Stat. Phys. 91, 1–15.

    Article  Google Scholar 

  43. Prim, R.C. (1957) Shortest connection networks and some generalizations, Bell Syst. Tech. J. 36, 1389–1401.

    Google Scholar 

  44. Cieplak, M., Maritan, A. and Banavar, J.R. (1994) Optimal paths and domain walls in the strong disorder limit, Phys. Rev. Lett. 72, 2320–2323; Cieplak, M., Maritan, A. and Banavar, J.R. (1996) Invasion percolation and Eden growth: Geometry and universality, Phys. Rev. Lett. 76, 3754–3757.

    Article  Google Scholar 

  45. Frank, D.J., Lobb, C.J. (1988) Highly efficient algorithm for percolative transport studies in two dimensions, Phys. Rev. B37, 302–307.

    Google Scholar 

  46. Stauffer, D. and Aharony, A. (1992) Introduction to Percolation Theory, 2nd ed., Taylor & Francis, London.

    Google Scholar 

  47. Braswell, W.D., Family, F., Straley, J.P. (1984) Treelike percolation in two dimensions, Phys. Rev. A29, 254–256.

    Google Scholar 

  48. Wu, F.Y. (1978) Absence of phase transitions in tree-like percolation in 2 dimensions, Phys. Rev. B18, 516–517.

    Google Scholar 

  49. Straley, J.P. (1990) Treelike percolation, Phys. Rev. A41, 1030–1033.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics and Astronomy, Michigan State University, East Lansing, MI, 48824

    M.F. Thorpe & M.V. Chubynsky

Authors
  1. M.F. Thorpe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M.V. Chubynsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Michigan State University, East Lansing, Michigan

    M. F. Thorpe

  2. Lucent Technologies, Bell Labs Innovations, Murray Hill, New Jersey

    J. C. Phillips

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Kluwer Academic Publishers

About this chapter

Cite this chapter

Thorpe, M., Chubynsky, M. (2002). The Intermediate Phase and Self-organization in Network Glasses. In: Thorpe, M.F., Phillips, J.C. (eds) Phase Transitions and Self-Organization in Electronic and Molecular Networks. Fundamental Materials Research. Springer, Boston, MA. https://doi.org/10.1007/0-306-47113-2_4

Download citation

  • DOI: https://doi.org/10.1007/0-306-47113-2_4

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-0-306-46568-0

  • Online ISBN: 978-0-306-47113-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation