Skip to main content
SpringerLink
Log in

Renormalization group study of the melting of a two-dimensional system of collapsing hard disks

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the melting of a two-dimensional system of collapsing hard disks (a system with a hard-disk potential to which a repulsive step is added) for different values of the repulsive-step width. We calculate the system phase diagram by the method of the density functional in crystallization theory using equations of the Berezinskii–Kosterlitz–Thouless–Halperin–Nelson–Young theory to determine the lines of stability with respect to the dissociation of dislocation pairs, which corresponds to the continuous transition from the solid to the hexatic phase. We show that the crystal phase can melt via a continuous transition at low densities (the transition to the hexatic phase) with a subsequent transition from the hexatic phase to the isotropic liquid and via a first-order transition. Using the solution of renormalization group equations with the presence of singular defects (dislocations) in the system taken into account, we consider the influence of the renormalization of the elastic moduli on the form of the phase diagram.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Alcoutlabi and G. B. McKenna, “Effects of confinement on material behaviour at the nanometre size scale,” J. Phys.: Condens. Matter, 17, R461–R524 (2005).

    ADS  Google Scholar 

  2. S. A. Rice, “Structure in confined colloid suspensions,” Chem. Phys. Lett., 479, 1–13 (2009).

    Article  ADS  Google Scholar 

  3. L. B. Krott and M. C. Barbosa, “Anomalies in a waterlike model confined between plates,” J. Chem. Phys., 138, 084505 (2013).

    Article  ADS  Google Scholar 

  4. A. M. Almudallal, S. V. Buldyrev, and I. Saika-Voivod, “Phase diagram of a two-dimensional system with anomalous liquid properties,” J. Chem. Phys., 137, 034507 (2012).

    Article  ADS  Google Scholar 

  5. L. B. Krott and J. R. Bordin, “Distinct dynamical and structural properties of a core-softened fluid when confined between fluctuating and fixed walls,” J. Chem. Phys., 139, 154502 (2013).

    Article  ADS  Google Scholar 

  6. L. B. Krott and M. C. Barbosa, “Model of waterlike fluid under confinement for hydrophobic and hydrophilic particle–plate interaction potentials,” Phys. Rev. E, 89, 012110 (2014).

    Article  ADS  Google Scholar 

  7. N. N. Bogoliubov, “Quasiaverage in problems of statistical mechanics [in Russian],” in: Collection of Scientific Works: Statistical Mechanics, Vol. 6, Equilibrium Statistical Mechanics: 1945–1986, Nauka, Moscow (2006), pp. 236–360.

    Google Scholar 

  8. N. D. Mermin, “Crystalline order in two dimensions,” Phys. Rev., 176, 250–254 (1968); Erratum, Phys. Rev. B, 20, 4762–4762 (1979); Erratum: Erratum, 74, 149902 (2006).

    Article  ADS  Google Scholar 

  9. M. Kosterlitz and D. J. Thouless, “Ordering, metastability, and phase transitions in two-dimensional systems,” J. Phys. C, 6, 1181–1203 (1973).

    Article  ADS  Google Scholar 

  10. B. I. Halperin and D. R. Nelson, “Theory of two-dimensional melting,” Phys. Rev. Lett., 41, 121–124 (1978); Erratum,, 41, 519 (1978).

    Article  ADS  MathSciNet  Google Scholar 

  11. D. R. Nelson and B. I. Halperin, “Dislocation-mediated melting in two dimensions,” Phys. Rev. B, 19, 2457–2484 (1979).

    Article  ADS  Google Scholar 

  12. A. P. Young, “Melting and the vector Coulomb gas in two dimensions,” Phys. Rev. B, 19, 1855–1866 (1979).

    Article  ADS  Google Scholar 

  13. U. Gasser, C. Eisenmann, G. Maret, and P. Keim, “Melting of crystals in two dimensions,” Chem. Phys. Chem., 11, 963–970 (2010).

    Article  Google Scholar 

  14. K. Zahn and G. Maret, “Dynamic criteria for melting in two dimensions,” Phys. Rev. Lett., 85, 3656–3659 (2000).

    Article  ADS  Google Scholar 

  15. P. Keim, G. Maret, and H. H. von Grünberg, “Frank’s constant in the hexatic phase,” Phys. Rev. E, 75, 031402 (2007).

    Article  ADS  Google Scholar 

  16. S. Deutschländer, T. Horn, H. Löwen, G. Maret, and P. Keim, “Two-dimensional melting under quenched disorder,” Phys. Rev. Lett., 111, 098301 (2013); Erratum, 111, 259901 (2013).

    Article  ADS  Google Scholar 

  17. T. Horn, S. Deutschländer, H. Löwen, G. Maret, and P. Keim, “Fluctuations of orientational order and clustering in a two-dimensional colloidal system under quenched disorder,” Phys. Rev. E, 88, 062305 (2013).

    Article  ADS  Google Scholar 

  18. S. T. Chui, “Grain-boundary theory of melting in two dimensions,” Phys. Rev. B, 28, 178–194 (1983).

    Article  ADS  Google Scholar 

  19. W. Janke and H. Kleinert, “Monte Carlo study of two-step defect melting,” Phys. Rev. B, 41, 6848–6863 (1990).

    Article  ADS  Google Scholar 

  20. V. N. Ryzhov and E. E. Tareyeva, “Two-stage melting in two dimensions: First-principles approach,” Phys. Rev. B, 51, 8789–8794 (1995).

    Article  ADS  Google Scholar 

  21. V. N. Ryzhov and E. E. Tareeva, “Microscopic description of two-stage melting in two dimensions,” JETP, 81, 1115–1123 (1995).

    ADS  Google Scholar 

  22. V. N. Ryzhov and E. E. Tareyeva, “Melting in two dimensions: First-order versus continuous transition,” Phys. A, 314, 396–404 (2002).

    Article  Google Scholar 

  23. L. M. Pomirchi, V. N. Ryzhov, and E. E. Tareeva, “Melting of two-dimensional systems: Dependence of the type of transition on the radius of the potential,” Theor. Math. Phys., 130, 101–110 (2002).

    Article  MATH  Google Scholar 

  24. E. S. Chumakov, Y. D. Fomin, E. L. Shangina, E. E. Tareyeva, E. N. Tsiok, and V. N. Ryzhov, “Phase diagram of the system with the repulsive shoulder potential in two dimensions: Density functional approach,” Phys. A, 432, 279–286 (2015).

    Article  MathSciNet  Google Scholar 

  25. V. N. Ryzhov, “Disclination-mediated melting of two-dimensional lattices,” Theor. Math. Phys., 88, 990–997 (1991).

    Article  Google Scholar 

  26. V. N. Ryzhov, “Dislocation-disclination melting of two-dimensional lattices,” Sov. Phys. JETP, 73, 899–905 (1991).

    Google Scholar 

  27. S. Prestipino, F. Saija, and P. V. Giaquinta, “Hexatic phase and water-like anomalies in a two-dimensional fluid of particles with a weakly softened core,” J. Chem. Phys., 137, 104503 (2012).

    Article  ADS  Google Scholar 

  28. P. Bladon and D. Frenkel, “Dislocation unbinding in dense two-dimensional crystals,” Phys. Rev. Lett., 74, 2519–2522 (1995).

    Article  ADS  Google Scholar 

  29. S. I. Lee and S. J. Lee, “Effect of the range of the potential on two-dimensional melting,” Phys. Rev. E, 78, 041504 (2008).

    Article  ADS  Google Scholar 

  30. S. Prestipino, F. Saija, and P. V. Giaquinta, “Hexatic phase in the two-dimensional gaussian-core model,” Phys. Rev. Lett., 106, 235701 (2011).

    Article  ADS  Google Scholar 

  31. R. Zangi and S. A. Rice, “Phase transitions in a quasi-two-dimensional system,” Phys. Rev. E, 58, 7529–7544 (1998).

    Article  ADS  Google Scholar 

  32. D. Frydel and S. A. Rice, “Phase diagram of a quasi-two-dimensional colloid assembly,” Phys. Rev. E, 68, 061405 (2003).

    Article  ADS  Google Scholar 

  33. D. E. Dudalov, Yu. D. Fomin, E. N. Tsiok, and V. N. Ryzhov, “Melting scenario of the two-dimensional coresoftened system: First-order or continuous transition?” J. Phys.: Conf. Ser., 510, 012016 (2014).

    Google Scholar 

  34. D. E. Dudalov, Yu. D. Fomin, E. N. Tsiok, and V. N. Ryzhov, “Effect of a potential softness on the solid–liquid transition in a two-dimensional core-softened potential system,” J. Chem. Phys., 141, 18C522 (2014).

    Article  Google Scholar 

  35. D. E. Dudalov, Yu. D. Fomin, E. N. Tsiok, and V. N. Ryzhov, “How dimensionality changes the anomalous behavior and melting scenario of a core-softened potential system?” Soft Matter, 10, 4966–4976 (2014).

    Article  ADS  Google Scholar 

  36. E. N. Tsiok, D. E. Dudalov, Yu. D. Fomin, and V. N. Ryzhov, “Random pinning changes the melting scenario of a two-dimensional core-softened potential system,” Phys. Rev. E, 92, 032110 (2015).

    Article  ADS  Google Scholar 

  37. J. Lee and K. J. Strandburg, “First-order melting transition of the hard-disk system,” Phys. Rev. B, 46, 11190–11193 (1992).

    Article  ADS  Google Scholar 

  38. H. Weber, D. Marx, and K. Binder, “Melting transition in two dimensions: A finite-size scaling analysis of bond-orientational order in hard disks,” Phys. Rev. B, 51, 14636–14651 (1995).

    Article  ADS  Google Scholar 

  39. C. H. Mak, “Large-scale simulations of the two-dimensional melting of hard disks,” Phys. Rev. E, 73, 065104 (2006).

    Article  ADS  Google Scholar 

  40. A. Jaster, “Orientational order of the two-dimensional hard-disk system,” Europhys. Lett., 42, 277–281 (1998).

    Article  ADS  Google Scholar 

  41. A. Jaster, “The hexatic phase of the two-dimensional hard disk system,” Phys. Lett. A, 330, 120–125 (2004).

    Article  ADS  Google Scholar 

  42. K. Bagchi, H. C. Andersen, and W. Swope, “Computer simulation study of the melting transition in two dimensions,” Phys. Rev. Lett., 76, 255–258 (1996).

    Article  ADS  Google Scholar 

  43. K. Bagchi, H. C. Andersen, and W. Swope, “Observation of a two-stage melting transition in two dimensions,” Phys. Rev. E, 53, 3794–3803 (1996).

    Article  ADS  Google Scholar 

  44. K. Binder, S. Sengupta, and P. Nielaba, “Liquid–solid transition of hard discs: First-order transition or Kosterlitz–Thouless–Halperin–Nelson–Young scenario?” J. Phys.: Condens. Matter, 14, 2323–2333 (2002).

    ADS  Google Scholar 

  45. R. K. Kalia and P. Vashishta, “Interfacial colloidal crystals and melting transition,” J. Phys. C, 14, L643–L648 (1981).

    Article  ADS  Google Scholar 

  46. J. Q. Broughton, G. H. Gilmer, and J. D. Weeks, “Molecular-dynamics study of melting in two dimensions: Inverse-twelfth-power interaction,” Phys. Rev. B, 25, 4651–4669 (1982).

    Article  ADS  Google Scholar 

  47. R. S. Singh, M. Santra, and B. Bagchi, “Anisotropy induced crossover from weakly to strongly first order melting of two dimensional solids,” J. Chem. Phys., 138, 184507 (2013).

    Article  ADS  Google Scholar 

  48. K. Wierschem and E. Manousakis, “Simulation of melting of two-dimensional Lennard-Jones solids,” Phys. Rev. B, 83, 214108 (2011).

    Article  ADS  Google Scholar 

  49. N. Gribova, A. Arnold, T. Schilling, and C. Holm, “How close to two dimensions does a Lennard-Jones system need to be to produce a hexatic phase?” J. Chem. Phys., 135, 054514 (2011).

    Article  ADS  Google Scholar 

  50. Yu. E. Lozovik and V. M. Farztdinov, “Oscillation spectra and phase diagram of two-dimensional electron crystal: ‘New’ (3+4)-self-consistent approximation,” Solid State Commun., 54, 725–728 (1985).

    Article  ADS  Google Scholar 

  51. Yu. E. Lozovik, V. M. Farztdinov, B. Abdullaev, and S. A. Kucherov, “Melting and spectra of two-dimensional classical crystals,” Phys. Lett. A, 112, 61–63 (1985).

    Article  ADS  Google Scholar 

  52. E. P. Bernard and W. Krauth, “Two-step melting in two dimensions: First-order liquid–hexatic transition,” Phys. Rev. Lett., 107, 155704 (2011).

    Article  ADS  Google Scholar 

  53. M. Engel, J. A. Anderson, S. C. Glotzer, M. Isobe, E. P. Bernard, and W. Krauth, “Hard-disk equation of state: First-order liquid–hexatic transition in two dimensions with three simulation methods,” Phys. Rev. E, 87, 042134 (2013).

    Article  ADS  Google Scholar 

  54. W. Qi, A. P. Gantapara, and M. Dijkstra, “Two-stage melting induced by dislocations and grain boundaries in monolayers of hard spheres,” Soft Matter, 10, 5449–5457 (2014).

    Article  ADS  Google Scholar 

  55. S. C. Kapfer and W. Krauth, “Two-dimensional melting: From liquid–hexatic coexistence to continuous transitions,” Phys. Rev. Lett., 114, 035702 (2015).

    Article  ADS  Google Scholar 

  56. W.-K. Qi, S.-M. Qin, X.-Y. Zhao, and Y. Chen, “Coexistence of hexatic and isotropic phases in two-dimensional Yukawa systems,” J. Phys.: Condens. Matter, 20, 245102 (2008).

    ADS  Google Scholar 

  57. W. Qi and M. Dijkstra, “Destabilisation of the hexatic phase in systems of hard disks by quenched disorder due to pinning on a lattice,” Soft Matter, 11, 2852–2856 (2015).

    Article  ADS  Google Scholar 

  58. M. Zu, J. Liu, H. Tong, and N. Xu, “Density affects the nature of the hexatic–liquid transition in two-dimensional melting of soft-core systems,” Phys. Rev. Lett., 085702 (2016); arXiv:1605.00747v2 [cond-mat.soft] (2016).

    Google Scholar 

  59. V. N. Ryzhov, “Statistical theory of crystallization in classical systems,” Theor. Math. Phys., 55, 399–405 (1983).

    Article  MathSciNet  Google Scholar 

  60. V. N. Ryzhov and E. E. Tareeva, “Towards a statistical theory of freezing,” Phys. Lett. A, 75, 88–90 (1979).

    Article  ADS  Google Scholar 

  61. V. N. Ryzhov and E. E. Tareeva, “Statistical theory of crystallization in a system of hard spheres,” Theor. Math. Phys., 48, 835–840 (1981).

    Article  Google Scholar 

  62. M. Baus, “The present status of the density-functional theory of the liquid–solid transition,” J. Phys.: Condens. Matter, 2, 2111–2126 (1990).

    ADS  Google Scholar 

  63. Y. Singh, “Density-functional theory of freezing and properties of the ordered phase,” Phys. Rep., 207, 351–444 (1991).

    Article  ADS  Google Scholar 

  64. V. N. Ryzhov and E. E. Tareeva, “Microscopic approach to calculation of the shear and bulk moduli and the frank constant in two-dimensional melting,” Theor. Math. Phys., 92, 922–930 (1992).

    Article  Google Scholar 

  65. V. N. Ryzhov and S. M. Stishov, “A liquid–liquid phase transition in the ‘collapsing’ hard sphere system,” JETP, 95, 710–713 (2002).

    Article  ADS  Google Scholar 

  66. V. N. Ryzhov and S. M. Stishov, “Repulsive step potential: A model for a liquid–liquid phase transition,” Phys. Rev. E, 67, 010201 (2003).

    Article  ADS  Google Scholar 

  67. S. M. Stishov, “On the phase diagram of a ‘collapsing’ hard-sphere system,” Phil. Mag. B, 82, 1287–1290 (2002).

    Article  ADS  Google Scholar 

  68. Y. D. Fomin, N. V. Gribova, V. N. Ryzhov, S. M. Stishov, and D. Frenkel, “Quasibinary amorphous phase in a three-dimensional system of particles with repulsive-shoulder interactions,” J. Chem. Phys., 129, 064512 (2008).

    Article  ADS  Google Scholar 

  69. S. V. Buldyrev, G. Malescio, C. A. Angell, N. Giovambattista, S. Prestipino, F. Saija, H. E. Stanley, and L. Xu, “Unusual phase behavior of one-component systems with two-scale isotropic interactions,” J. Phys.: Condens. Matter, 21, 504106 (2009).

    Google Scholar 

  70. P. Vilaseca and G. Franzese, “Isotropic soft-core potentials with two characteristic length scales and anomalous behaviour,” J. Non-Crystalline Solids, 357, 419–426 (2011).

    Article  ADS  Google Scholar 

  71. N. V. Gribova, Y. D. Fomin, D. Frenkel, and V. N. Ryzhov, “Waterlike thermodynamic anomalies in a repulsiveshoulder potential system,” Phys. Rev. E, 79, 051202 (2009).

    Article  ADS  Google Scholar 

  72. Yu. D. Fomin, E. N. Tsiok, and V. N. Ryzhov, “Inversion of sequence of diffusion and density anomalies in core-softened systems,” J. Chem. Phys., 135, 234502 (2011).

    Article  ADS  Google Scholar 

  73. Y. D. Fomin, E. N. Tsiok, and V. N. Ryzhov, “Core-softened system with attraction: Trajectory dependence of anomalous behavior,” J. Chem. Phys., 135, 124512 (2011).

    Article  ADS  Google Scholar 

  74. R. E. Ryltsev, N. M. Chtchelkatchev, and V. N. Ryzhov, “Superfragile glassy dynamics of a one-component system with isotropic potential: Competition of diffusion and frustration,” Phys. Rev. Lett., 110, 025701 (2013).

    Article  ADS  Google Scholar 

  75. Yu. D. Fomin, E. N. Tsiok, and V. N. Ryzhov, “Silicalike sequence of anomalies in core-softened systems,” Phys. Rev. E, 87, 042122 (2013).

    Article  ADS  Google Scholar 

  76. E. N. Tsiok, Yu. D. Fomin, and V. N. Ryzhov, “Influence of random pinning on melting scenario of twodimensional core-softened potential system,” arXiv:1608.05232v1 [cond-mat.soft] (2016).

    Google Scholar 

  77. V. N. Ryzhov and E. E. Tareyeva, “Bond orientational order in simple liquids,” J. Phys. C: Solid State Phys., 21, 819–824 (1988).

    Article  ADS  Google Scholar 

  78. V. N. Ryzhov, “Local structure and bond orientational order in a Lennard-Jones liquid,” J. Phys.: Condens. Matter, 2, 5855–5865 (1990).

    ADS  Google Scholar 

  79. J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids, Acad. Press, New York (1986).

    MATH  Google Scholar 

  80. R. Lovett, “On the stability of a fluid toward solid formation,” J. Chem. Phys., 66, 1225 (1977).

    Article  ADS  Google Scholar 

  81. V. N. Ryzhov, E. E. Tareeva, and Yu. D. Fomin, “Singularity of the ‘swallow-tail’ type and the glass–glass transition in a system of collapsing hard spheres,” Theor. Math. Phys., 167, 645–653 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  82. V. V. Brazhkin, Yu. D. Fomin, V. N. Ryzhov, E. E. Tareyeva, and E. N. Tsiok, “True Widom line for a square-well system,” Phys. Rev. E, 89, 042136 (2014).

    Article  ADS  Google Scholar 

  83. J. L. Colot and M. Baus, “The freezing of hard disks and hyperspheres,” Phys. Lett. A, 119, 135–139 (1986).

    Article  ADS  Google Scholar 

  84. M. Baus and J. L. Colot, “Thermodynamics and structure of a fluid of hard rods, disks, spheres, or hyperspheres from rescaled virial expansions,” Phys. Rev. A, 36, 3912–3925 (1987).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Vereshchagin Institute for High Pressure Physics, RAS, Moscow, Russia

    V. N. Ryzhov, E. E. Tareyeva, Yu. D. Fomin, E. N. Tsiok & E. S. Chumakov

Authors
  1. V. N. Ryzhov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. E. Tareyeva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yu. D. Fomin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. N. Tsiok
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. E. S. Chumakov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to V. N. Ryzhov or E. E. Tareyeva.

Additional information

This research is supported by a grant from the Russian Science Foundation (Project No. 14-12-00820).

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 191, No. 3, pp. 424–440, June, 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ryzhov, V.N., Tareyeva, E.E., Fomin, Y.D. et al. Renormalization group study of the melting of a two-dimensional system of collapsing hard disks. Theor Math Phys 191, 842–855 (2017). https://doi.org/10.1134/S0040577917060058

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577917060058

Keywords

Search

Navigation