Skip to main content
SpringerLink
Log in

Fun with Hard Spheres

  • Conference paper
  • First Online:
Statistical Physics and Spatial Statistics

Part of the book series: Lecture Notes in Physics ((LNP,volume 554))

Abstract

Thermostatistical properties of hard sphere and hard disk systems are discussed. In particular we focus on phase transitions such as freezing in the thermodynamic limit. Results based on theory and computer simulations are given. It is emphasized that suspensions of sterically-stabilized colloids represent excellent realizations of the hard sphere model. Finally a survey of current research activities for hard sphere systems is presented and some recent results are summarized.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
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. Allen, M.P., D.J. Tildesley (1989): Computer Simulation of Liquids (Clarendon Press, Oxford)

    Google Scholar 

  2. Allen, M.P., G.T. Evans, D. Frenkel, B. M. Mulder (1993): ‘Hard Convex Body Fluids’, Advances in Chemical Physics, Vol. LXXXVI, pp. 1–166

    Article  Google Scholar 

  3. Barrat, J.L., J.P. Hansen (1986): ‘On the stability of polydispersed colloidal crystals’, J. Physique 47, Paris, pp. 1547–1553

    Article  Google Scholar 

  4. Bartlett, P., R.H. Ottewill, P.N. Pusey (1992): ‘Superlattice Formation in Binary Mixtures of Hard-Sphere Colloids’, Phys. Rev. Letters 68, pp. 3801–3804

    Article  ADS  Google Scholar 

  5. Bartlett, P. (1997): ‘A geometrically-based mean-field theory of polydisperse hardsphere mixture’, J. Chem. Phys. 107, pp. 188–196

    Article  ADS  Google Scholar 

  6. Bartlett, P. (1998): ‘Fractionated crystallization in a polydisperse mixture of hard spheres’, J. Chem. Phys. 109, pp. 10970–10975

    Article  ADS  Google Scholar 

  7. Bartlett, P., P.B. Warren (1999): ‘Reentrant Melting in Polydispersed Hard Spheres’, Phys. Rev. Letters 82, pp. 1979–1982

    Article  ADS  Google Scholar 

  8. Barker, J.A., D. Henderson (1976): ‘What is a “liquid”? Understanding the states of matter’, Rev. Mod. Phys. 48, pp. 587–676

    Article  ADS  MathSciNet  Google Scholar 

  9. Baus, M. (1987): ‘Statistical Mechanical Theories of Freezing: An Overview’, J. Stat. Phys. 48, pp. 1129–1146

    Article  ADS  Google Scholar 

  10. Biben T., J.P. Hansen (1991): ‘Phase Seperation of Asymmetric Binary Hard-Sphere Fluids’, Phys. Rev. Letters 66, pp. 2215–2218

    Article  ADS  Google Scholar 

  11. Blum, L. and G. Stell (1979): ‘Polydisperse systems. I. Scattering function for polydisperse fluids of hard or permeable spheres’, J. Chem. Phys. 71, pp. 42–46

    Article  ADS  Google Scholar 

  12. Bolhuis, P.G., D.A. Kofke (1996a): ‘Numerical study of freezing in polydisperse colloidal suspensions’, J. Phys. Condensed Matter 8, pp. 9627–9631

    Article  ADS  Google Scholar 

  13. Bolhuis, P.G., D.A. Kofke (1996b): ‘Monte Carlo study of freezing of polydispersed hard spheres’, Phys. Rev. E 54, pp. 634–643

    ADS  Google Scholar 

  14. Bolhuis, P.G., D. Frenkel, S.-C. Mau, D.A. Huse (1997): ‘Fatigue, alcohol and performance impairment’, Nature 388, London, pp. 235–238

    Article  ADS  Google Scholar 

  15. Bolhuis, P.G., D. Frenkel (1997): ‘Tracing the phase boundaries of hard spherocylinders’, J. Chem. Phys. 106, pp. 666–687

    Article  ADS  Google Scholar 

  16. Bosse, J., J.S. Thakur (1987): ‘Delocalization of Small Particles in a Glassy Matrix’, Phys. Rev. Letters 59, pp. 998–1001

    Article  ADS  Google Scholar 

  17. Bruce, A.D., N.B. Wilding, G.J. Auckland: ‘Free Energy of Crystalline Solids: A Lattice-Switch Monte Carlo Method’, Phys. Rev. Letters 79, pp. 3002–3005

    Google Scholar 

  18. Callen, H.B. (1960): Thermodynamics (Wiley, New York)

    MATH  Google Scholar 

  19. Cipra, B. (1998): ‘Packing Challenge Mastered At Last’, Science 281, p. 1267

    Article  Google Scholar 

  20. Courtemanche, D.J., F. van Swol (1992): ‘Wetting State of a Crystal Fluid System of Hard Spheres’, Phys. Rev. Letters 69, pp. 2078–2081

    Article  ADS  Google Scholar 

  21. Cuesta, J.A. (1999): ‘Demixing in a single-peak distributed polydisperse mixture of hard spheres’, Europhysics Letters 46, pp. 197–203

    Article  ADS  Google Scholar 

  22. Denton, A.R., N.W. Ashcroft (1990): ‘Weighted-density-functional theory of nonuniform fluid mixtures: Application to freezing of binary hard-sphere mixtures’, Phys. Rev. A 42, pp. 7312–7329

    ADS  Google Scholar 

  23. Denton, A.R., N.W. Ashcroft (1991): “Vegard’s law”, Phys. Rev. A 43, pp. 3161–3164

    ADS  Google Scholar 

  24. Dietrich, S. (1988):‘Wetting Phenomena’. In: Phase transitions and Critical Phenomena, ed. by C. Domb, J.L. Lebowitz, Vol. 12 (Academic Press, London), pp. 1–128

    Google Scholar 

  25. Dijkstra, M., R. van Roij, R. Evans (1999a): ‘Direct Simulation of the Phase Behaviour of Binary Hard-Sphere Mixtures: Test of the Depletion Potential Description’, Phys. Rev. Letters 82, pp. 117–120

    Article  ADS  Google Scholar 

  26. Dijkstra, M., R. van Roij, R. Evans (1999b): ‘Phase diagramm of highly asymmetric binary hard-sphere mixtures’, Phys. Rev. E 59, pp. 5744–5771

    ADS  Google Scholar 

  27. Doliwa, B., A. Heuer (1998): ‘Cage Effect, Local Anisotropies and Dynamic Heterogeneities at the Glass Transiton: A Computer Study of Hard Spheres’, Phys. Rev. Letters 80, pp. 4915–4918

    Article  ADS  Google Scholar 

  28. Eldridge, M.D., P.A. Madden, D. Frenkel (1993): ‘The stability of the AB 13 crystal in a binary hard sphere system’, Mol. Phys. 79, pp. 105–120

    Article  ADS  Google Scholar 

  29. Evans, R. (1992): ‘Density Functionals in the Theory of Nonuniform Fluids’. In: Fundamentals of Inhomogeneous Fluids, ed. by D. Henderson (Marcel Dekker, New York), pp. 85–176

    Google Scholar 

  30. Finken, R., M. Schmidt, H. Löwen: to be published

    Google Scholar 

  31. Frenkel, D., B. Smit (1996): Understanding Molecular Simulation (Academic, San Diego)

    MATH  Google Scholar 

  32. Frisch, H.L., N. Rivier and D. Wyler (1985): ‘Classical Hard-Sphere Fluid in Infinitely Many Dimensions’, Phys. Rev. Letters 54, pp. 2061–2063

    Article  ADS  MathSciNet  Google Scholar 

  33. Frisch, H.L., J.K. Percus (1987): ‘Nonuniform classical fluid at high dimensionality’, Phys. Rev. A 35, pp. 4696–4702

    ADS  MathSciNet  Google Scholar 

  34. Frisch, H.L., J.K. Percus (1999): ‘High dimensionality as an organizing device for classical fluids’, Phys. Rev. E 60, pp. 2942–2948

    ADS  MathSciNet  Google Scholar 

  35. Fröhlich, J., C. Pfister (1981): ‘On the Absence of Spontaneous Symmetry Breaking and of Crystalline Ordering in Two-Dimensional Systems’, Commun. Math. Phys. 81, pp. 277–298

    Article  ADS  Google Scholar 

  36. Garzo, V., J.W. Dufty (1999): ‘Dense fluid transport for inelastic hard spheres’, Phys. Rev. E 59, pp. 5895–5911

    ADS  Google Scholar 

  37. Götzelmann, B., A. Haase, S. Dietrich (1996): Structure factor of hard spheres near a wall’, Phys. Rev. E 53, pp. 3456–3467

    ADS  Google Scholar 

  38. Götzelmann, B., R. Evans, S. Dietrich (1998): ‘Depletion forces in fluids’, Phys. Rev. E 57, pp. 6785–6800

    ADS  Google Scholar 

  39. Graf, H., H. Löwen, M. Schmidt (1997): ‘Cell theory for the phase diagram of hard spherocylinders’, Prog. Colloid Polymer Science 104, pp. 177–179

    Article  Google Scholar 

  40. Graf, H., H. Löwen (1999): ‘Density functional theory for hard spherocylinders: phase transitions in the bulk in the presence of external fields’, J. Phys. Condensed Matter 11, pp. 1435–1452

    Article  ADS  Google Scholar 

  41. Hansen, J.P., I.R. McDonald (1986): Theory of Simple Liquids, 2nd ed. (Academic Press, London)

    Google Scholar 

  42. Henderson, J.R., F. van Swol (1984): ‘On the interface between fluid and a planar wall: Theory and simulations of a hard sphere fluid at a hard wall’, Mol. Phys. 51, pp. 991–1010

    Article  ADS  Google Scholar 

  43. Heni, M., H. Löwen (1999): ‘Interfacial free energy of hard sphere fluids and solids near a hard wall’, Phys. Rev. E (in press)

    Google Scholar 

  44. Holyst, R., A. Poniewierski (1989): ‘Nematic-smectic-A transition for perfectly aligned hard spherocylinders: Application of the smoothed-density approximation’, Phys. Rev. A 39, pp. 2742–2744

    ADS  Google Scholar 

  45. Hoover, W.G., F.H. Ree (1968): ‘Melting Transitions and Communal Entropy for Hard Spheres’, J. Chem. Phys. 49, pp. 3609–3617

    Article  ADS  Google Scholar 

  46. Imhof, A., J.K.G. Dhont (1995): ‘Long-time self-Diffusion in binary colloidal hardsphere dispersions’, Phys. Rev. E 52, pp. 6344–6357

    ADS  Google Scholar 

  47. Imhof, A., J.K.G. Dhont (1997): ‘Phase behaviour and long-time self-diffusion in a binary hard sphere dispersion’, Colloids and Surfaces 122, pp. 53–61

    Article  Google Scholar 

  48. Jaster, A. (1998): ‘Orientational order of the two-dimensional hard-disk system’, Europhys. Letters 42, pp. 277–282

    Article  ADS  Google Scholar 

  49. Jaster, A. (1999): ‘Computer simulations of the two-dimensional melting transition using hard disks’, Phys. Rev. E 59, pp. 2594–2602

    ADS  Google Scholar 

  50. Kirkpatrick, T.R. (1986): ‘Ordering in the parallel hard hypercube gas’, J. Chem. Phys. 85, pp. 3515–3519

    Article  ADS  Google Scholar 

  51. Kofke, D.A., P.G. Bolhuis (1999): ‘Freezing of polydisperse hard spheres’, Phys. Rev. E 59, pp. 618–622

    ADS  Google Scholar 

  52. Kranendonk, W.G.T., D. Frenkel (1991): ‘Thermodynamic properties of binary hard sphere mixtures’, Mol. Phys. 72, pp. 715–733

    Article  ADS  Google Scholar 

  53. Lebowitz, J.L., O. Penrose (1964): ‘Convergence of virial expansions’, J. Math. Phys. 5, pp. 841–847

    Article  MathSciNet  ADS  Google Scholar 

  54. Lekkerkerker, H.N.W., P. Buining, J. Buitenhuis, G.J. Vroege, A. Stroobants (1995): ‘Liquid Crystal Phase Transition in Dispersions of Rodlike Colloidal Particles’. In: Observation, Prediction and Simulation of Phase Transitions in Complex Fluids, ed. by M. Baus et al. (Kluwer, Holland), pp. 53–112

    Google Scholar 

  55. Leppmeier, M. (1997): Kugelpackungen: von Kepler bis heute (Vieweg, Wiesbaden)

    MATH  Google Scholar 

  56. Luding, S. (1995): ‘Granular materials under vibration: simulations of rotating spheres’, Phys. Rev. E 52, pp. 4442–4457

    ADS  Google Scholar 

  57. Löwen, H. (1994): ‘Melting, Freezing and Colloidal Suspensions’, Phys. Rep. 237, pp. 249–324

    Article  ADS  Google Scholar 

  58. Mau, S.-C., D.A. Huse (1999): ‘Stacking entropy of hard-sphere crystals’, Phys. Rev. E 59, pp. 4396–4401

    ADS  Google Scholar 

  59. McCarley, J.S., N.W. Ashcroft (1994): ‘Hard-Sphere quasicrystals’, Phys. Rev. B 49, pp. 15600–15606

    ADS  Google Scholar 

  60. Mitus A.C., H. Weber, D. Marx (1997): ‘Local structure analysis of the hard-disk fluid near melting’, Phys. Rev. E 55, pp. 6855–6857

    ADS  Google Scholar 

  61. Münster, A. (1974): Statistical Thermodynamics. Vol. II (Springer-Verlag, Berlin), pp. 337–346

    Google Scholar 

  62. Murray, C.A., W.O. Sprenger, R.A. Wenk (1990): ‘Comparison of melting in three and two dimensions: Microscopy of colloidal spheres’, Phys. Rev. B 42, pp. 688–703

    ADS  Google Scholar 

  63. Murray, C.A. (1992): ‘Experimental Studies of Melting and Hexatic Order in Two-Dimensional Colloidal Suspensions’. In: Bond-orientational Order in Condensed Matter Systems, ed. by K.J. Strandburg (Springer, New York), pp. 137–215

    Google Scholar 

  64. Nägele, G., J.K.G. Dhont (1998): ‘Tracer-diffusion in colloidal mixtures: A modecoupling scheme with hydrodynamic interactions’, J. Chem. Phys. 108, pp. 9566–9576

    Article  ADS  Google Scholar 

  65. Nägele, G., J. Bergenholtz (1998): ‘Linear viscoelasticity of colloidal mixtures’, J. Chem. Phys. 108, pp. 9893–9904

    Article  ADS  Google Scholar 

  66. Németh, Z.T., H. Löwen (1998): ‘Freezing in finite systems: hard discs in circular cavities’, J. Phys. Condensed Matter 10, pp. 6189–6203

    Article  ADS  Google Scholar 

  67. Németh, Z.T., H. Löwen (1999): ‘Freezing and glass transition of hard spheres in cavities’, Phys. Rev E 59, pp. 6824–6829

    ADS  Google Scholar 

  68. Neser, S., C. Bechinger, P. Leiderer, T. Palberg (1997a): ‘Finite-Size Effects on the Closest Packing of Hard Spheres’, Phys. Rev. Letters. 79, pp. 2348–2351

    Article  ADS  Google Scholar 

  69. Neser, S.T. Palberg, C. Bechinger, P. Leiderer (1997b): ‘Direct observation of a buckling transition during the formation of thin colloidal crystals’, Progr Colloid Polymer Science 104, pp. 194–197

    Article  Google Scholar 

  70. Ohnesorge, R., H. Löwen, H. Wagner (1991): ‘Density-funcional theory of surface melting’, Phys. Rev. A 43, pp. 2870–2878

    ADS  Google Scholar 

  71. Pansu, B., P. Pieranski (1984): ‘Structures of thin layers of hard spheres: high pressure limit’, J. Physique 45, pp. 331–339

    Article  Google Scholar 

  72. Phan, S.-E., W.B. Russel, J. Zhu, P.M. Chaikin (1999): ‘Effects of polydispersity on hard sphere crystals’, J. Chem. Phys. 108, pp. 9789–9795

    Article  ADS  Google Scholar 

  73. Pieranski, P., L. Strzelecki, B. Pansu (1983): ‘Thin Colloidal Crystals’, Phys. Rev. Lett. 50, pp. 900–902

    Article  ADS  Google Scholar 

  74. Pronk, S., D. Frenkel (1999): ‘Can stacking faults in hard-sphere crystals anneal out spontaneously?’ J. Chem. Phys. 110, pp. 4589–4592

    Article  ADS  Google Scholar 

  75. Pusey, P.N. (1991): ‘Colloidal Suspensions’. In: Liquids, Freezing and the Glass Transition, ed. by J.P. Hansen, D. Levesque, J. Zinn-Justin (North Holland, Amsterdam), pp. 763–942

    Google Scholar 

  76. Reiss, H., H.L. Frisch, E. Helfand, J.L. Lebowitz (1960): ‘Aspects of the Statistical Thermodynamics of Real Fluids’, J. Chem. Phys. 32, pp. 119–124

    Article  MathSciNet  ADS  Google Scholar 

  77. Rosenfeld, Y. (1996): ‘Close-packed configurations,’ symmetry breaking’, and the freezing transition in density functional theory’, J. Phys. Condensed Matter 8, pp. L795–L801

    Article  ADS  Google Scholar 

  78. Rosenfeld, Y., M. Schmidt, H. Löwen, P. Tarazona (1997): ‘Fundamental-measure free-energy density functional for hard spheres: Dimensional crossover and freezing’, Physical Review E 55, pp. 4245–4263

    ADS  Google Scholar 

  79. Rosenfeld, Y. (1998): ‘Self-consistent density functional theory and the equation of state for simple liquids’, Mol. Phys. 94, pp. 929–936

    Article  ADS  Google Scholar 

  80. Schmidt, M., H. Löwen (1996): ‘Freezing between Two and Three Dimensions’,Phys. Rev. Letters 76, pp. 4552–4555

    Article  ADS  Google Scholar 

  81. Schmidt, M., H. Löwen (1997): ‘Phase diagram of hard spheres confined between two parallel plates’, Phys. Rev. E 55, pp. 7228–7241

    ADS  Google Scholar 

  82. Sear, R.P. (1998): ‘Phase seperation and crystallisation of polydisperse hard spheres’, Europhys. Letters 44, pp. 531–535

    Article  ADS  Google Scholar 

  83. Tonks, L. (1936): ‘The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres’, Phys. Rev. 50, pp. 955–963

    Article  MATH  ADS  Google Scholar 

  84. Truskett, T.M., S. Torquato, S. Sastry, P.G. Debenedetti, F.H. Stillinger (1998): ‘Structural precursor to freezing in the hard-disk and hard-sphere system’, Phys. Rev. E 58, pp. 3083–3088

    ADS  Google Scholar 

  85. Van Blaaderen, A. and P. Wiltzius (1995): ‘Real-Space Structure of Colloidal Hard-Sphere Glasses’, Science 270, pp. 1177–1179

    Article  ADS  Google Scholar 

  86. Van Winkle, D.H., C.A. Murray (1986): ‘Layering transitions in colloidal crystals as observed by diffraction and direct lattice imaging’, Phys. Rev. A 34 (1986), pp. 562–573

    ADS  Google Scholar 

  87. Warren, P.B. (1999): ‘Fluid-fluid phase seperation in hard spheres with a bimodal size distribution’, Europhys. Letters 46, 295–300

    Article  ADS  Google Scholar 

  88. Weiss, J., D.W. Oxtoby, D.G. Grier, C.A. Murray (1995): ‘Martensitic transition in a confined colloidal suspension’, J. Chem. Phys. 103, pp. 1180–1190

    Article  ADS  Google Scholar 

  89. Wyler, D., N. Rivier, H. Frisch (1987): ‘Hard-Sphere fluid in infinite dimensions’, Phys. Rev. A 36, pp. 2422–2431

    ADS  MathSciNet  Google Scholar 

  90. Xu, H., M. Baus (1992): ‘A density functional study of superlattice formation in binary hard-sphere mixtures’, J. Phys. Condensed Matter 4, pp. L663–L668

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Theoretische Physik II, Heinrich-Heine-Universität Düsseldorf, Universitätsstrase 1, D-40225, Düsseldorf, Germany

    Hartmut Löwen

Authors
  1. Hartmut Löwen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Fachbereich Physik, Bergische Universität Wuppertal, 42097, Wuppertal, Germany

    Klaus R. Mecke

  2. Institut für Stochastik, TU Bergakademie Freiberg, 09596, Freiberg, Germany

    Dietrich Stoyan

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Löwen, H. (2000). Fun with Hard Spheres. In: Mecke, K.R., Stoyan, D. (eds) Statistical Physics and Spatial Statistics. Lecture Notes in Physics, vol 554. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45043-2_11

Download citation

  • DOI: https://doi.org/10.1007/3-540-45043-2_11

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67750-5

  • Online ISBN: 978-3-540-45043-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation