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

V. N. Ryzhov   ; E. E. Tareeva   ; Yu. D. Fomin   ; E. N. Tsiok   ; E. S. Chumakov

Teoretičeskaâ i matematičeskaâ fizika, Tome 191 (2017) no. 3, pp. 424-440

Résumé

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.

Détail

Keywords: melting of two-dimensional system, Berezinskii–Kosterlitz–Thouless–Halperin–Nelson–Young theory, elastic modulus
Mots-clés : hexatic phase.

Bibliographie

[1] M. Alcoutlabi, G. B. McKenna, “Effects of confinement on material behaviour at the nanometre size scale”, J. Phys.: Condens. Matter, 17:15 (2005), R461–R524 | DOI

[2] S. A. Rice, “Structure in confined colloid suspensions”, Chem. Phys. Lett., 479:1–3 (2009), 1–13 | DOI

[3] L. B. Krott, M. C. Barbosa, “Anomalies in a waterlike model confined between plates”, J. Chem. Phys., 138 (2013), 084505, 12 pp. | DOI

[4] A. M. Almudallal, S. V. Buldyrev, I. Saika-Voivod, “Phase diagram of a two-dimensional system with anomalous liquid properties”, J. Chem. Phys., 137:3 (2012), 034507, 10 pp. | DOI

[5] L. B. Krott, J. R. Bordin, “Distinct dynamical and structural properties of a core-softened fluid when confined between fluctuating and fixed walls”, J. Chem. Phys., 139:15 (2013), 154502 | DOI

[6] L. B. Krott, M. C. Barbosa, “Model of waterlike fluid under confinement for hydrophobic and hydrophilic particle-plate interaction potentials”, Phys. Rev. E, 89:1 (2014), 012110, 11 pp. | DOI

[7] N. N. Bogolyubov, “Kvazisrednie v zadachakh statisticheskoi mekhaniki”, Sobranie nauchnykh trudov. Ctatisticheskaya mekhanika, v. 6, Ravnovesnaya statisticheskaya mekhanika: 1945–1986, Nauka, M., 2006, 236–360 | MR | Zbl

[8] N. D. Mermin, “Crystalline order in two dimensions”, Phys. Rev., 176:1 (1968), 250–254 ; Erratum, Phys. Rev. B, 20:11 (1979), 4762–4762 ; 74:14 (2006), 149902, 1 pp. | DOI | DOI | DOI

[9] M. Kosterlitz, D. J. Thouless, “Ordering, metastability and phase transitions in two-dimensional systems”, J. Phys. C, 6:7 (1973), 1181–1203 | DOI

[10] B. I. Halperin, D. R. Nelson, “Theory of two-dimensional melting”, Phys. Rev. Lett., 41:2 (1978), 121–124 | DOI | MR

[11] D. R. Nelson, B. I. Halperin, “Dislocation-mediated melting in two dimensions”, Phys. Rev. B, 19:5 (1979), 2457–2484 | DOI

[12] A. P. Young, “Melting and the vector Coulomb gas in two dimensions”, Phys. Rev. B, 19:4 (1979), 1855–1866 | DOI

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

[14] K. Zahn, G. Maret, “Dynamic criteria for melting in two dimensions”, Phys. Rev. Lett., 85:17 (2000), 3656–3659 | DOI

[15] P. Keim, G. Maret, H. H. von Grünberg, “Frank's constant in the hexatic phase”, Phys. Rev. E, 75:3 (2007), 031402, 6 pp. | DOI

[16] S. Deutschländer, T. Horn, H. Löwen, G. Maret, P. Keim, “Two-dimensional melting under quenched disorder”, Phys. Rev. Lett., 111:9 (2013), 098301, 5 pp. ; Erratum, 111:25, 259901, 1 \year 2013 pp. | DOI | DOI

[17] T. Horn, S. Deutschländer, H. Löwen, G. Maret, P. Keim, “Fluctuations of orientational order and clustering in a two-dimensional colloidal system under quenched disorder”, Phys. Rev. E, 88:6 (2013), 062305, 9 pp. | DOI

[18] S. T. Chui, “Grain-boundary theory of melting in two dimensions”, Phys. Rev. B, 28:1 (1983), 178–194 | DOI

[19] W. Janke, H. Kleinert, “Monte Carlo study of two-step defect melting”, Phys. Rev. B, 41:10 (1990), 6848–6863 | DOI

[20] V. N. Ryzhov, E. E. Tareyeva, “Two-stage melting in two dimensions: first-principles approach”, Phys. Rev. B, 51:14 (1995), 8789–8794 | DOI

[21] V. N. Ryzhov, E. E. Tareeva, “Mikroskopicheskoe opisanie dvukhstadiinogo plavleniya v dvukh izmereniyakh”, ZhETF, 108:6 (1995), 2044–2060

[22] V. N. Ryzhov, E. E. Tareyeva, “Melting in two dimensions: first-order versus continuous transition”, Physica A, 314:1–4 (2002), 396–404 | DOI

[23] L. M. Pomirchi, V. N. Ryzhov, E. E. Tareeva, “Plavlenie dvumernykh sistem: zavisimost roda perekhoda ot radiusa potentsiala”, TMF, 130:1 (2002), 119–130 | DOI | DOI | Zbl

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

[25] V. N. Ryzhov, “Disklinatsionnoe plavlenie dvumernykh reshetok”, TMF, 88:3 (1991), 449–458 | DOI | MR

[26] V. N. Ryzhov, “Dislokatsionno-disklinatsionnoe plavlenie dvumernykh reshetok”, ZhETF, 100:5 (1991), 1627–1639

[27] S. Prestipino, F. Saija, P. V. Giaquinta, “Hexatic phase and water-like anomalies in a two-dimensional fluid of particles with a weakly softened core”, J. Chem. Phys., 137:10 (2012), 104503 | DOI

[28] P. Bladon, D. Frenkel, “Dislocation unbinding in dense two-dimensional crystals”, Phys. Rev. Lett., 74:13 (1995), 2519–2522 | DOI

[29] S. I. Lee, S. J. Lee, “Effect of the range of the potential on two-dimensional melting”, Phys. Rev. E, 78:4 (2008), 041504, 9 pp. | DOI

[30] S. Prestipino, F. Saija, P. V. Giaquinta, “Hexatic phase in the two-dimensional gaussian-core model”, Phys. Rev. Lett., 106:23 (2011), 235701, 4 pp. | DOI

[31] R. Zangi, S. A. Rice, “Phase transitions in a quasi-two-dimensional system”, Phys. Rev. E, 58:6 (1998), 7529–7544 | DOI

[32] D. Frydel, S. A. Rice, “Phase diagram of a quasi-two-dimensional colloid assembly”, Phys. Rev. E, 68:6 (2003), 061405, 16 pp. | DOI

[33] D. E. Dudalov, Yu. D. Fomin, E. N. Tsiok, V. N. Ryzhov, “Melting scenario of the two-dimensional core-softened system: first-order or continuous transition?”, J. Phys.: Conf. Ser., 510:1 (2014), 012016 | DOI

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

[35] D. E. Dudalov, Yu. D. Fomin, E. N. Tsiok, V. N. Ryzhov, “How dimensionality changes the anomalous behavior and melting scenario of a core-softened potential system?”, Soft Matter, 10:27 (2014), 4966–4976 | DOI

[36] E. N. Tsiok, D. E. Dudalov, Yu. D. Fomin, V. N. Ryzhov, “Random pinning changes the melting scenario of a two-dimensional core-softened potential system”, Phys. Rev. E, 92:3 (2015), 032110, 5 pp. | DOI

[37] J. Lee, K. J. Strandburg, “First-order melting transition of the hard-disk system”, Phys. Rev. B, 46:17 (1992), 11190–11193 | DOI

[38] H. Weber, D. Marx, K. Binder, “Melting transition in two dimensions: a finite-size scaling analysis of bond-orientational order in hard disks”, Phys. Rev. B, 51:20 (1995), 14636–14651 | DOI

[39] C. H. Mak, “Large-scale simulations of the two-dimensional melting of hard disks”, Phys. Rev. E, 73:6 (2006), 065104, 4 pp. | DOI

[40] A. Jaster, “Orientational order of the two-dimensional hard-disk system”, Europhys. Lett., 42:3 (1998), 277–281 | DOI

[41] A. Jaster, “The hexatic phase of the two-dimensional hard disk system”, Phys. Lett. A, 330:1–2 (2004), 120–125 | DOI

[42] K. Bagchi, H. C. Andersen, W. Swope, “Computer simulation study of the melting transition in two dimensions”, Phys. Rev. Lett., 76:2 (1996), 255–258 | DOI

[43] K. Bagchi, H. C. Andersen, W. Swope, “Observation of a two-stage melting transition in two dimensions”, Phys. Rev. E, 53:4 (1996), 3794–3803 | DOI

[44] K. Binder, S. Sengupta, P. Nielaba, “liquid-solid transition of hard discs: first-order transition or Kosterlitz–Thouless–Halperin–Nelson–Young scenario?”, J. Phys.: Condens. Matter, 14:9 (2002), 2323–2333 | DOI

[45] R. K. Kalia, P. Vashishta, “Interfacial colloidal crystals and melting transition”, J. Phys. C, 14:22 (1981), L643–L648 | DOI

[46] J. Q. Broughton, G. H. Gilmer, J. D. Weeks, “Molecular-dynamics study of melting in two dimensions. Inverse-twelfth-power interaction”, Phys. Rev. B, 25:7 (1982), 4651–4669 | DOI

[47] R. S. Singh, M. Santra, B. Bagchi, “Anisotropy induced crossover from weakly to strongly first order melting of two dimensional solids”, J. Chem. Phys., 138:18 (2013), 184507 | DOI

[48] K. Wierschem, E. Manousakis, “Simulation of melting of two-dimensional Lennard-Jones solids”, Phys. Rev. B, 83:21 (2011), 214108 | DOI

[49] N. Gribova, A. Arnold, T. Schilling, C. Holm, “How close to two dimensions does a Lennard-Jones system need to be to produce a hexatic phase?”, J. Chem. Phys., 135:5 (2011), 054514 | DOI

[50] Yu. E. Lozovik, V. M. Farztdinov, “Oscillation spectra and phase diagram of two-dimensional electron crystal: ‘new’ $(3+4)$-self-consistent approximation”, Solid State Commun., 54:8 (1985), 725–728 | DOI

[51] Yu. E. Lozovik, V. M. Farztdinov, B. Abdullaev, S. A. Kucherov, “Melting and spectra of two-dimensional classical crystals”, Phys. Lett. A, 112:1–2 (1985), 61–63 | DOI

[52] E. P. Bernard, W. Krauth, “Two-step melting in two dimensions: first-order liquid-hexatic transition”, Phys. Rev. Lett., 107:15 (2011), 155704, 4 pp. | DOI

[53] M. Engel, J. A. Anderson, S. C. Glotzer, M. Isobe, E. P. Bernard, W. Krauth, “Hard-disk equation of state: first-order liquid-hexatic transition in two dimensions with three simulation methods”, Phys. Rev. E, 87:4 (2013), 042134, 8 pp. | DOI

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

[55] S. C. Kapfer, W. Krauth, “Two-dimensional melting: from liquid-hexatic coexistence to continuous transitions”, Phys. Rev. Lett., 114:3 (2015), 035702, 5 pp. | DOI | MR

[56] W.-K. Qi, S.-M. Qin, X.-Y. Zhao, Y. Chen, “Coexistence of hexatic and isotropic phases in two-dimensional Yukawa systems”, J. Phys.: Condens. Matter, 20:24 (2008), 245102, 9 pp. | DOI

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

[58] M. Zu, J. Liu, H. Tong, N. Xu, “Density affects the nature of the hexatic-liquid transition in two-dimensional melting of soft-core systems”, Phys. Rev. Lett., 117:8 (2016), 085702, 5 pp., arXiv: 1605.00747 | DOI

[59] V. N. Ryzhov, “Statisticheskaya teoriya kristallizatsii v klassicheskikh sistemakh”, TMF, 55:1 (1983), 128–136 | DOI | MR

[60] V. N. Ryzhov, E. E. Tareeva, “Towards a statistical theory of freezing”, Phys. Lett. A, 75:1–2 (1979), 88–90 | DOI

[61] V. N. Ryzhov, E. E. Tareeva, “K statisticheskoi teorii kristallizatsii v sisteme tverdykh sfer”, TMF, 48:3 (1981), 416–423 | DOI | MR

[62] M. Baus, “The present status of the density-functional theory of the liquid-solid transition”, J. Phys.: Condens. Matter, 2:9 (1990), 2111–2126 | DOI

[63] Y. Singh, “Density-functional theory of freezing and properties of the ordered phase”, Phys. Rep., 207:6 (1991), 351–444 | DOI

[64] V. N. Ryzhov, E. E. Tareeva, “Mikroskopicheskii podkhod k vychisleniyu modulei uprugosti i modulya Franka v teorii dvumernogo plavleniya”, TMF, 92:2 (1992), 331–343 | DOI | MR

[65] V. N. Ryzhov, S. M. Stishov, “A liquid-liquid phase transition in the ‘collapsing’ hard sphere system”, ZhETF, 122:4 (2002), 820–823 | DOI

[66] V. N. Ryzhov, S. M. Stishov, “Repulsive step potential: a model for a liquid-liquid phase transition”, Phys. Rev. E, 67:1 (2003), 010201, 4 pp. | DOI

[67] S. M. Stishov, “On the phase diagram of a 'collapsing' hard-sphere system”, Phil. Mag. B, 82:11 (2002), 1287–1290 | DOI

[68] Y. D. Fomin, N. V. Gribova, V. N. Ryzhov, S. M. Stishov, D. Frenkel, “Quasibinary amorphous phase in a three-dimensional system of particles with repulsive-shoulder interactions”, J. Chem. Phys., 129:6 (2008), 064512 | DOI

[69] S. V. Buldyrev, G. Malescio, C. A. Angell, N. Giovambattista, S. Prestipino, F. Saija, H. E. Stanley, L. Xu, “Unusual phase behavior of one-component systems with two-scale isotropic interactions”, J. Phys.: Condens. Matter, 21:50 (2009), 504106 | DOI

[70] P. Vilaseca, G. Franzese, “Isotropic soft-core potentials with two characteristic length scales and anomalous behaviour”, J. Non-Crystalline Solids, 357:2 (2011), 419–426 | DOI

[71] N. V. Gribova, Y. D. Fomin, D. Frenkel, V. N. Ryzhov, “Waterlike thermodynamic anomalies in a repulsive-shoulder potential system”, Phys. Rev. E, 79:5 (2009), 051202, 6 pp. | DOI

[72] Yu. D. Fomin, E. N. Tsiok, V. N. Ryzhov, “Inversion of sequence of diffusion and density anomalies in core-softened systems”, J. Chem. Phys., 135:23 (2011), 234502 | DOI

[73] Y. D. Fomin, E. N. Tsiok, V. N. Ryzhov, “Core-softened system with attraction: trajectory dependence of anomalous behavior”, J. Chem. Phys., 135:12 (2011), 124512 | DOI

[74] R. E. Ryltsev, N. M. Chtchelkatchev, V. N. Ryzhov, “Superfragile glassy dynamics of a one-component system with isotropic potential: competition of diffusion and frustration”, Phys. Rev. Lett., 110:2 (2013), 025701, 5 pp. | DOI

[75] Yu. D. Fomin, E. N. Tsiok, V. N. Ryzhov, “Silicalike sequence of anomalies in core-softened systems”, Phys. Rev. E, 87:4 (2013), 042122, 5 pp. | DOI

[76] E. N. Tsiok, Yu. D. Fomin, V. N. Ryzhov, “Influence of random pinning on melting scenario of two-dimensional core-softened potential system”, arXiv: 1608.05232

[77] V. N. Ryzhov, E. E. Tareyeva, “Bond orientational order in simple liquids”, J. Phys. C: Solid State Phys., 21:5 (1988), 819–824 | DOI

[78] V. N. Ryzhov, “Local structure and bond orientational order in a Lennard-Jones liquid”, J. Phys.: Condens. Matter, 2:26 (1990), 5855–5865 | DOI

[79] J.-P. Hansen, I. R. McDonald, Theory of Simple Liquids, Academic Press, New York, 1986 | Zbl

[80] R. Lovett, “On the stability of a fluid toward solid formation”, J. Chem. Phys., 66:3 (1977), 1225 | DOI

[81] V. N. Ryzhov, E. E. Tareeva, Yu. D. Fomin, “Osobennost tipa ‘lastochkin khvost’ i perekhod steklo–steklo v sisteme kollapsiruyuschikh tverdykh sfer”, TMF, 167:2 (2011), 284–294 | DOI | DOI | Zbl

[82] V. V. Brazhkin, Yu. D. Fomin, V. N. Ryzhov, E. E. Tareyeva, E. N. Tsiok, “True Widom line for a square-well system”, Phys. Rev. E, 89:4 (2014), 042136, 6 pp. | DOI

[83] J. L. Colot, M. Baus, “The freezing of hard disks and hyperspheres”, Phys. Lett. A, 119:3 (1986), 135–139 | DOI

[84] M. Baus, J. L. Colot, “Thermodynamics and structure of a fluid of hard rods, disks, spheres, or hyperspheres from rescaled virial expansions”, Phys. Rev. A, 36:8 (1987), 3912–3925 | DOI