Cavity averages for hard spheres in the presence of polydispersity and incomplete data

  • Michael Schindler
  • A. C. Maggs
Regular Article


We develop a cavity-based method which allows to extract thermodynamic properties from position information in hard-sphere/disk systems. So far, there are available-volume and free-volume methods. We add a third one, which we call available volume after take-out, and which is shown to be mathematically equivalent to the others. In applications, where data sets are finite, all three methods show limitations, and they do this in different parameter ranges. We illustrate the principal equivalence and the limitations on data from molecular dynamics: In particular, we test robustness against missing data. We have in mind experimental limitations where there is a small polydispersity, say 4% in the particle radii, but individual radii cannot be determined. We observe that, depending on the used method, the errors in such a situation are easily 100% for the pressure and 10kT for the chemical potentials. Our work is meant as guideline to the experimentalists for choosing the right one of the three methods, in order to keep the outcome of experimental data analysis meaningful.

Graphical abstract


Soft Matter: Colloids and Nanoparticles 


  1. 1.
    V.J. Anderson, H.N.W. Lekkerkerker, Nature 416, 811 (2002).CrossRefADSGoogle Scholar
  2. 2.
    A. van Blaaderen, P. Wiltzius, Science 270, 1177 (1995).CrossRefADSGoogle Scholar
  3. 3.
    W.K. Kegel, A. van Blaaderen, Science 14, 290 (2000).CrossRefADSGoogle Scholar
  4. 4.
    C.P. Royall, W.C.K. Poon, E.R. Weeks, Soft Matter 9, 17 (2013).CrossRefADSGoogle Scholar
  5. 5.
    P.J. Yunker, K. Chen, M.D. Gratale, M.A. Lohr, T. Still, A.G. Yodh, Rep. Prog. Phys. 77, 056601 (2014).CrossRefADSGoogle Scholar
  6. 6.
    K. Zahn, A. Wille, G. Maret, S. Sengupta, P. Nielaba, Phys. Rev. Lett. 90, 155506 (2003).CrossRefADSGoogle Scholar
  7. 7.
    T. Still, C.P. Goodrich, K. Chen, P.J. Yunker, S. Schoenholz, A.J. Liu, A.G. Yodh, Phys. Rev. E 89, 012301 (2014).CrossRefADSGoogle Scholar
  8. 8.
    B. Liu, T.H. Besseling, M. Hermes, A.F. Demirörs, A. Imhof, A. van Blaaderen, Nat. Commun. 5, 3092 (2014).ADSGoogle Scholar
  9. 9.
    J. Taffs, S.R. Williams, H. Tanaka, C.P. Royall, Soft Matter 9, 297 (2013).CrossRefADSGoogle Scholar
  10. 10.
    A. Ghosh, R. Mari, V.K. Chikkadi, P. Schall, A.C. Maggs, D. Bonn, Physica A 390, 3061 (2011).CrossRefADSGoogle Scholar
  11. 11.
    J. Baumgartl, R.P.A. Dullens, M. Dijkstra, R. Roth, C. Bechinger, Phys. Rev. Lett. 98, 198303 (2007).CrossRefADSGoogle Scholar
  12. 12.
    R. Dreyfus, Y. Xu, T. Still, L.A. Hough, A.G. Yodh, S. Torquato, Phys. Rev. E 91, 012302 (2015).CrossRefADSGoogle Scholar
  13. 13.
    R. Zargar, J. Russo, P. Schall, H. Tanaka, D. Bonn, EPL 108, 38002 (2014).CrossRefADSGoogle Scholar
  14. 14.
    R. Zargar, B. Nienhuis, P. Schall, D. Bonn, Phys. Rev. Lett. 110, 258301 (2013).CrossRefADSGoogle Scholar
  15. 15.
    C.L. Klix, F. Ebert, F. Weysser, M. Fuchs, G. Maret, P. Keim, Phys. Rev. Lett. 109, 178301 (2012).CrossRefADSGoogle Scholar
  16. 16.
    V. Chikkadi, P. Schall, Phys. Rev. E 85, 031402 (2012).CrossRefADSGoogle Scholar
  17. 17.
    A. Ghosh, V. Chikkadi, P. Schall, D. Bonn, Phys. Rev. Lett. 107, 188303 (2011).CrossRefADSGoogle Scholar
  18. 18.
    H.L. Schöpe, O. Marnette, W. van Megen, G. Bryant, Langmuir 23, 11534 (2007).CrossRefGoogle Scholar
  19. 19.
    W.C.K. Poon, E.R. Weeks, C.P. Royall, Soft Matter 8, 21 (2012).CrossRefADSGoogle Scholar
  20. 20.
    B. Widom, J. Chem. Phys. 39, 2808 (1963).CrossRefADSGoogle Scholar
  21. 21.
    R.J. Speedy, J. Chem. Soc., Faraday Trans. 2 76, 693 (1980).CrossRefGoogle Scholar
  22. 22.
    W.G. Hoover, W.T. Ashurst, R. Grover, J. Chem. Phys. 57, 1259 (1972).CrossRefADSGoogle Scholar
  23. 23.
    J.G. Kirkwood, J. Chem. Phys. 18, 380 (1950).CrossRefADSGoogle Scholar
  24. 24.
    W.W. Wood, J. Chem. Phys. 20, 1334 (1952).CrossRefADSGoogle Scholar
  25. 25.
    R.J. Speedy, J. Chem. Soc., Faraday Trans. 2 77, 329 (1981).CrossRefGoogle Scholar
  26. 26.
    R.J. Speedy, J. Phys. Chem. 92, 2016 (1988).CrossRefGoogle Scholar
  27. 27.
    R.J. Speedy, H. Reiss, Mol. Phys. 72, 999 (1991).CrossRefADSGoogle Scholar
  28. 28.
    D.S. Corti, R.K. Bowles, Mol. Phys. 96, 1623 (1999).CrossRefADSGoogle Scholar
  29. 29.
    S. Sastry, D.S. Corti, P.G. Debenedetti, F.H. Stillinger, Phys. Rev. E 56, 5524 (1997).MathSciNetCrossRefADSGoogle Scholar
  30. 30.
    S. Sastry, T.M. Truskett, P.G. Debenedetti, S. Torquato, F.H. Stillinger, Mol. Phys. 95, 289 (1998).CrossRefADSGoogle Scholar
  31. 31.
    M. Maiti, A. Lakshminarayanan, S. Sastry, Eur. Phys. J. E 36, 5 (2013).CrossRefGoogle Scholar
  32. 32.
    M. Maiti, S. Sastry, J. Chem. Phys. 141, 044510 (2014).CrossRefADSGoogle Scholar
  33. 33.
    K. Chen, T. Still, S. Schoenholz, K.B. Aptowicz, M. Schindler, A.C. Maggs, A.J. Liu, A.G. Yodh, Phys. Rev. E 88, 022315 (2013).CrossRefADSGoogle Scholar
  34. 34.
    R.K. Bowles, R.J. Speedy, Mol. Phys. 83, 113 (1994).CrossRefADSGoogle Scholar
  35. 35.
    R.J. Speedy, H. Reiss, Mol. Phys. 72, 1015 (1991).CrossRefADSGoogle Scholar
  36. 36.
    D.C. Rapaport, The Art of Molecular Dynamics Simulation, 2nd edition (Cambridge University Press, Cambridge, UK, 2004).Google Scholar
  37. 37.
    D. Forster, Hydrodynamic fluctuations, broken symmetry, and correlation functions (Addison-Wesley, 1990).Google Scholar
  38. 38.
    M. Schindler, Chem. Phys. 375, 327 (2010).CrossRefADSGoogle Scholar
  39. 39.
  40. 40.
    N. Clisby, B.M. McCoy, J. Stat. Phys. 122, 15 (2006).zbMATHMathSciNetCrossRefADSGoogle Scholar
  41. 41.
    B.J. Gellatly, J.L. Finney, J. Non-Cryst. Solids 50, 313 (1982).CrossRefADSGoogle Scholar
  42. 42.
    M. Caroli, P.M.M. de Castro, S. Loriot, O. Rouiller, M. Teillaud, C. Wormser, Robust and Efficient Delaunay Triangulations of Points on or Close to a Sphere, in 9th International Symposium on Experimental Algorithms (2010), Vol. 6049 of Lecture Notes in Computer Science (Springer, Berlin, 2010) pp. 462--473.Google Scholar
  43. 43.
    CGAL 4.6, Computational Geometry Algorithms Library, (2015).
  44. 44.
    M. Caroli, M. Teillaud, Computing 3D Periodic Triangulations, in Proceedings 17th European Symposium on Algorithms (2009), Vol. 5757 of Lecture Notes in Computer Science (Springer, Berlin, 2009) pp. 59--70.Google Scholar
  45. 45.
    M. Yvinec, in CGAL User and Reference Manual (CGAL Editorial Board, 2015), 4.6 edition,
  46. 46.
    N. Kruithof, in CGAL User and Reference Manual (CGAL Editorial Board, 2015), 4.6 edition,
  47. 47.
    N.P. Dolbilin, D.H. Huson, Period. Math. Hung. 34, 57 (1997).zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
  49. 49.
    R.P.A. Dullens, W.K. Kegel, D.G.A.L. Aarts, Oil, Gas Sci. Technol. Rev. IFP 3, 295 (2008).CrossRefGoogle Scholar
  50. 50.
    R. Zargar, D. Bonn, private communication.Google Scholar
  51. 51.
    K. Chen, G. Ellenbroek, Z. Zhang, D. Chen, P. Yunker, S. Henkes, C. Brito, O. Dauchot, W. Sarloos, A. Liu et al., Phys. Rev. Lett. 105, 025501 (2010).CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.UMR Gulliver 7083 CNRS, ESPCI ParisTechPSL Research UniversityParisFrance

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