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Relating thermodynamic quantities of convex-hard-body fluids to the body's shape

Thomas Franosch1,*, Cristiano De Michele2,†, and Rolf Schilling3,‡

  • *Contact author: thomas.franosch@uibk.ac.at
  • †Contact author: demichel@roma1.infn.it
  • ‡Contact author: rschill@uni-mainz.de
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Phys. Rev. Research 7, 023260 – Published 13 June, 2025

DOI: https://doi.org/10.1103/5shk-zjsc

Abstract

For a fluid of convex hard particles, characterized by a length scale σmin and an anisotropy parameter ε, we develop a formalism allowing one to relate thermodynamic quantities to the body's shape. In a first step, its thermodynamics is reduced to that of spherical particles. The latter have a hard core of diameter σmin and a soft shell with a thickness εσmin/2. Besides their hard core repulsion at σmin, they interact by effective entropic forces which will be calculated. Based on this mapping, a second step provides a perturbative method for the systematic calculation of thermodynamic quantities with the shape anisotropy ε as a smallness parameter. In leading order in ε, the equation of state is derived as a functional of the particle's shape. To illustrate these findings, they are applied to a one- and two-dimensional fluid of ellipses and compared with results from different analytical approaches and our computer simulations. The mapping to spherical particles also implies that any phase transition of spherical particles, e.g., the liquid-hexatic transition, also exists for the nonspherical ones, and shifts linearly with ε for weak shape anisotropy. This is supported by our Monte Carlo simulation.

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References (75)

  1. L. Onsager, The effects of shape on the interaction of colloidal particles, Ann. N.Y. Acad. Sci. 51, 627 (1949).
  2. J. P. Hansen and I. R. McDonald, Theory of Simple Liquids, 4th ed. (Academic Press, London, 2013).
  3. E. Priestly, Introduction to Liquid Crystals (Springer Science & Business Media, 2012).
  4. M. J. Stephen and J. P. Straley, Physics of liquid crystals, Rev. Mod. Phys. 46, 617 (1974).
  5. S. C. Glotzer and M. J. Solomon, Anisotropy of building blocks and their assembly into complex structures, Nat. Mater. 6, 557 (2007).
  6. J. Vieillard-Baron, Phase transitions of the classical hard-ellipse system, J. Chem. Phys. 56, 4729 (1972).
  7. D. Frenkel, B. M. Mulder, and J. P. McTague, Phase diagram of a system of hard ellipsoids, Phys. Rev. Lett. 52, 287 (1984).
  8. G. Odriozola, Revisiting the phase diagram of hard ellipsoids, J. Chem. Phys. 136, 134505 (2012).
  9. M. E. Foulaadvand and M. Yarifard, Two-dimensional system of hard ellipses: A molecular dynamics study, Phys. Rev. E 88, 052504 (2013).
  10. W.-S. Xu, Y.-W. Li, Z.-Y. Sun, and L.-J. An, Hard ellipses: Equation of state, structure, and self-diffusion, J. Chem. Phys. 139, 024501 (2013).
  11. G. Bautista-Carbajal and G. Odriozola, Phase diagram of two-dimensional hard ellipses, J. Chem. Phys. 140, 204502 (2014).
  12. I. Torres-Díaz, R. S. Hendley, A. Mishra, A. J. Yeh, and M. A. Bevan, Hard superellipse phases: particle shape anisotropy & curvature, Soft Matter 18, 1319 (2022).
  13. P. Marienhagen and J. Wagner, Reexamining equations of state of oblate hard ellipsoids of revolution: Numerical simulation utilizing a cluster Monte Carlo algorithm and comparison to virial theory, Phys. Rev. E 105, 014125 (2022).
  14. C. De Michele, R. Schilling, and F. Sciortino, Dynamics of uniaxial hard ellipsoids, Phys. Rev. Lett. 98, 265702 (2007).
  15. P. Pfleiderer, K. Milinkovic, and T. Schilling, Glassy dynamics in monodisperse hard ellipsoids, Europhys. Lett. 84, 16003 (2008).
  16. W.-S. Xu, Z.-Y. Sun, and L.-J. An, Relaxation dynamics in a binary hard-ellipse liquid, Soft Matter 11, 627 (2015).
  17. M. Alhissi, A. Zumbusch, and M. Fuchs, Observation of liquid glass in molecular dynamics simulations, J. Chem. Phys. 160, 164502 (2024).
  18. J. S. Rowlinson, Virial expansions in an inhomogeneous system, Proc. R. Soc. Lond. A 402, 67 (1985).
  19. A. Isihara, Determination of molecular shape by osmotic measurement, J. Chem. Phys. 18, 1446 (1950).
  20. H. Hadwiger, The kinetic radius of non-spherical molecules', Experientia 7, 395 (1951).
  21. T. Kihara, Virial coefficients and models of molecules in gases, Rev. Mod. Phys. 25, 831 (1953).
  22. B. Freasier and R. Bearman, Virial expansion for hard ellipsoids of revolution, Mol. Phys. 32, 551 (1976).
  23. I. Nezbeda, Virial expansion and an improved equation of state for the hard convex molecule system, Chem. Phys. Lett. 41, 55 (1976).
  24. M. Rigby, Hard ellipsoids of revolution, Mol. Phys. 66, 1261 (1989).
  25. S. R. G. Tarjus, P. Viot, and J. Talbot, New analytical and numerical results on virial coefficients for 2-D hard convex bodies, Mol. Phys. 73, 773 (1991).
  26. M. Rigby, Virial coefficients of hard convex molecules in two dimensions, Mol. Phys. 78, 21 (1993).
  27. X.-M. You, A. Y. Vlasov, and A. J. Masters, The equation of state of isotropic fluids of hard convex bodies from a high-level virial expansion, J. Chem. Phys. 123, 034510 (2005).
  28. A. J. Masters, Virial expansions, J. Phys.: Condens. Matter 20, 283102 (2008).
  29. E. Herold, R. Hellmann, and J. Wagner, Virial coefficients of anisotropic hard solids of revolution: The detailed influence of the particle geometry, J. Chem. Phys. 147, 204102 (2017).
  30. M. Kulossa, P. Marienhagen, and J. Wagner, Virial coefficients of hard hyperspherocylinders in R4: Influence of the aspect ratio, Phys. Rev. E 105, 064121 (2022).
  31. M. Kulossa and J. Wagner, Virial coefficients of hard, two-dimensional, convex particles up to the eighth order, Mol. Phys. 122, e2289699 (2023).
  32. M. Kulossa and J. Wagner, Geometric measures of uniaxial solids of revolution in higher-dimensional euclidean spaces and their relation to the second virial coefficient, Phys. Rev. E 111, 024112 (2025).
  33. C. G. Gray and K. E. Gubbins, Theory of Molecular Fluids I: Fundamentals (Clarendon Press, 1984).
  34. J. R. Solana, Perturbation Theories for the Thermodynamic Properties of Fluids and Solids (CRC Press, Boca Raton, FL, 2013).
  35. B. Barboy and W. M. Gelbart, Series representation of the equation of state for hard particle fluids, J. Chem. Phys. 71, 3053 (1979).
  36. B. Mulder and D. Frenkel, The hard ellipsoid-of-revolution fluid II. The y-expansion equation of state, Mol. Phys. 55, 1193 (1985).
  37. A. Bellemans, Free energy of an assembly of nonspherical molecules with a hard core, Phys. Rev. Lett. 21, 527 (1968).
  38. I. Nezbeda and T. W. L. Thomas, Conformal theory of hard non-spherical molecule fluids, J. Chem. Soc. Faraday Trans. 2 75, 193 (1979).
  39. J. K. Percus, Sphericalized molecular interactions, Ann. N.Y. Acad. Sci. 452, 67 (1985).
  40. H. Reiss, H. L. Frisch, and J. L. Lebowitz, Statistical mechanics of rigid spheres, J. Chem. Phys. 31, 369 (1959).
  41. R. Gibbons, The scaled particle theory for particles of arbitrary shape, Mol. Phys. 17, 81 (1969).
  42. T. Boublík, Statistical thermodynamics of convex molecule fluids, Mol. Phys. 27, 1415 (1974).
  43. T. Boublík, Equation of state of hard convex body fluids, Mol. Phys. 42, 209 (1981).
  44. Y. Rosenfeld, Density functional theory of molecular fluids: Free-energy model for the inhomogeneous hard-body fluid, Phys. Rev. E 50, R3318 (1994).
  45. Y. Rosenfeld, Chemical Applications of Density-Functional Theory (ACS Publications, 1996), p. 198.
  46. G. Cinacchi and F. Schmid, Density functional for anisotropic fluids, J. Phys.: Condens. Matter 14, 12223 (2002).
  47. R. Wittmann, M. Marechal, and K. Mecke, Fundamental measure theory for non-spherical hard particles: Predicting liquid crystal properties from the particle shape, J. Phys.: Condens. Matter 28, 244003 (2016).
  48. J. D. Parsons, Nematic ordering in a system of rods, Phys. Rev. A 19, 1225 (1979).
  49. S.-D. Lee, A numerical investigation of nematic ordering based on a simple hard-rod mode, J. Chem. Phys. 87, 4972 (1987).
  50. Y. Song and E. A. Mason, Equation of state for a fluid of hard convex bodies in any number of dimensions, Phys. Rev. A 41, 3121 (1990).
  51. C. Vega, Virial coefficients and equation of state of hard ellipsoids, Mol. Phys. 92, 651 (1997).
  52. J. R. Solana, Equations of state of hard-body fluids: a new proposal, Mol. Phys. 113, 1003 (2015).
  53. M. P. Allen, G. T. Evans, D. Frenkel, and B. Mulder, Hard convex body fluids, in Advances in Chemical Physics (John Wiley & Sons, Inc., Hoboken, NJ, 1993), Vol. 86, pp. 1–166.
  54. M. Dijkstra, Entropy-driven phase transitions in colloids: From spheres to anisotropic particles, Adv. Chem. Phys. 156, 35 (2014).
  55. K. R. Mecke, Theory of Simple Liquids: With Applications to Soft Matter (Springer, Berlin, 2000), pp. 111–184, 4th ed.
  56. S. Torquato and Y. Jiao, Exclusion volumes of convex bodies in high space dimensions: applications to virial coefficients and continuum percolation, J. Stat. Mech. (2022) 093404.
  57. T. Boublík, Two-dimensional convex particle liquid, Mol. Phys. 29, 421 (1975).
  58. K. J. Strandburg, Two-dimensional melting, Rev. Mod. Phys. 60, 161 (1988).
  59. J. W. Perram and M. Wertheim, Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function, J. Comput. Phys. 58, 409 (1985).
  60. S. Asakura and F. Oosawa, On interaction between two bodies immersed in a solution of macromolecules, J. Chem. Phys. 22, 1255 (1954).
  61. S. Asakura and F. Oosawa, Interaction between particles suspended in solutions of macromolecules, J. Polym. Sci. 33, 183 (1958).
  62. T. Franosch, S. Lang, and R. Schilling, Fluids in extreme confinement, Phys. Rev. Lett. 109, 240601 (2012); Erratum: 110, 059901(E) (2013); T. Franosch and R. Schilling, Erratum: Fluids in extreme confinement [Phys. Rev. Lett. 109, 240601 (2012)], 128, 209902(E) (2022).
  63. X. Zheng and P. Palffy-Muhoray, Distance of closest approach of two arbitrary hard ellipses in two dimensions, Phys. Rev. E 75, 061709 (2007).
  64. X. Zheng, W. Iglesias, and P. Palffy-Muhoray, Distance of closest approach of two arbitrary hard ellipsoids, Phys. Rev. E 79, 057702 (2009).
  65. J. Lebowitz, J. Percus, and J. Talbot, On the orientational properties of some one-dimensional model systems, J. Stat. Phys. 49, 1221 (1987).
  66. L. Tonks, The complete equation of state of one, two and three-dimensional gases of hard elastic spheres, Phys. Rev. 50, 955 (1936).
  67. Z. W. Salsburg, R. W. Zwanzig, and J. G. Kirkwood, Molecular distribution functions in a one-dimensional fluid, J. Chem. Phys. 21, 1098 (1953).
  68. J. Kolafa and M. Rottner, Simulation-based equation of state of the hard disk fluid and prediction of higher-order virial coefficients, Mol. Phys. 104, 3435 (2006).
  69. E. Helfand, H. L. Frisch, and J. L. Lebowitz, Theory of the two- and one-dimensional rigid sphere fluids, J. Chem. Phys. 34, 1037 (1961).
  70. NIST, Elliptic integrals, https://dlmf.nist.gov/19.2.
  71. L. F. Tóth, Some packing and covering theorems, Acta Sci. Math. Szeged 12, 62 (1950).
  72. E. P. Bernard and W. Krauth, Two-step melting in two dimensions: First-order liquid-hexatic transition, Phys. Rev. Lett. 107, 155704 (2011).
  73. S. C. Kapfer and W. Krauth, Two-dimensional melting: From liquid-hexatic coexistence to continuous transitions, Phys. Rev. Lett. 114, 035702 (2015).
  74. J. Talbot, D. Kivelson, M. P. Allen, G. T. Evans, and D. Frenkel, Structure of the hard ellipsoid fluid, J. Chem. Phys. 92, 3048 (1990).
  75. T. Franosch and C. DeMichele, Monte Carlo data for compressibility factors in 1D and 2D + equation of state [Data set], Universität Innsbruck (2025), https://researchdata.uibk.ac.at/records/avvca-fcx04.

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Phys. Rev. Research 7, 023260– Published 13 June, 2025
  • Received 17 January 2025
  • Accepted 13 May 2025
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