TETRATIC PHASE IN THE PLANAR HARD SQUARE SYSTEM?
Wojciechowski Krzysztof W. 1*, Frenkel D. 2
1 Institute of Molecular Physics, Polish Academy of Sciences
Smoluchowskiego 17/19, 60-179 Poznań, Poland
2 FOM – Institute for Atomic and Molecular Physics
Postbus 41883, 1009 DB Amsterdam, The Netherlands
Received:
Rec. 4 December 2004
DOI: 10.12921/cmst.2004.10.02.235-255
OAI: oai:lib.psnc.pl:575
Abstract:
System of hard squares in two dimensions (2D) has been studied by Monte Carlo simulations. The simulations indicate that the isotropic fluid phase in this system does not freeze into a 2D ‘crystalline’ phase (of square lattice and quasi-long-range translational order) but transforms into an intermediate phase with the quadratic quasi-long-range orientational order (of coupled molecular axes and intermolecular bonds) and the translational order decaying faster than algebraically. The equation of state and the specific heat of the system are surprisingly well reproduced by smoothed version of the free volume theory in the whole density range.
Key words:
equation of state, hard convex body, liquid crystals, melting in two dimensions, quasi-long range order
References:
[1] B. J. Alder and W. G. Hoover, in Physics of Simple Liquids, Ed. H. N. V. Temperley, J. S. Rowlinson, and G. S. Rushbrooke, (North-Holland, 1968).
[2] J. Vieillard-Baron, J. Chem. Phys. 56, 4729 (1972).
[3] D. Frenkel and B. Mulder, Molec. Phvs. 55, 1171 (1985).
[4] A. Stroobants, H. N. W. Lekkerkerker, and D. Frenkel, Phvs. Rev. Lett. 57, 1452 (1987); Phvs.
Rev. A 36, 2929(1987).
[5] K. W. Wojciechowski, A. C. Brańka, and M. Parrinello, Molec. Phvs. 53, 1541 (1984).
[6] A. C. Brańka and K. W. Wojciechowski, Molec. Phys. 72, 941 (1991); ibid. 78, 1513 (1993).
[7] D. Frenkel, in: Liquids, Freezing and Glass Transition, Les Houches LI, Eds. J. P. Hansen,
D. Levesque and J. Zinn-Justin, Elsevier (1991).
[8] J. A. C. Veerman and D. Frenkel, Phys. Rev. A 45, 5632 (1992).
[9] K. W. Wojciechowski, D. Frenkel, and A. C. Brańka, Phvs. Rev. Lett. 66, 3168 (1991); Phvsica A 196, 519-545 (1993).
[10] K. W. Wojciechowski, Phys. Rev. B 46, 26 (1992).
[11] L. Onsager, Phys. Rev. 62, 558 (1942); L. Onsager, Ann. NY Acad. Sci. 51, 3441 (1949).
[12] K. W. Wojciechowski and J. Kłos, Nematic phase of cubic symmetiy: Solution of the Frenkel
3D-cross model, unpublished.
[13] R. Schneider, Convex bodies: The Brunn-Minkowski Theoiy Vol. 44 of Encyclopedia of Mathematics and its Applications, Ed. G.-C. Rota (Cambridge, 1993).
[14] T. Boublik, Mol. Phys. 29, 421 (1975).
[15] G. Tarjus, P. Viot, S. M. Ricci, and J. Talbot, Mol. Phys. 73, 773 (1991).
[16] K. W. Wojciechowski and D. Frenkel, On the phase diagram of two-dimensional hard rectangle system, unpublished.
[17] J. A. Zollweg and G. V. Chester, Phys. Rev B 46, 11186 (1992).
[18] J. Lee and K. J. Strandburg, Phvs. Rev. B 17, 11190 (1992).
[19] R. E. Peierls, Helv. Phys. Acta 7, 81 (1934).
[20] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part I (Pergamon, Oxford, 1980).
[21] N. D. Mermin, Phys. Rev. A 176, 250 (1968).
[22] The Mermin’s proof does not concern the case of the hard body systems. For hard discs the loga rithmic divergence has been demonstrated, in: D. A. Young, B. J. Alder, J. Chem Phys. 60, 1254 (1974).
[23] J. M. Kosterlitz and D. J. Thouless, J. Phvs. C 6, 1181 (1973); Prog. Low Temp. Phvs. B 7, 371 (1978).
[24] J. A. Barker and D. Henderson, Rev. Mod. Phvs. 48, 587 (1976).
[25] F. F. Abraham, Phvs. Rep. 80, 339 (1981).
[26] K. J. Strandburg, Rev. Mod. Phvs. 60, 161 (1988).
[27] M. A. Glaser and N. A. Clark, in Advances in Chemical Physics, Volume LXXXIII, Ed. I. Prigogine and S. A. Rice (Wilev, 1993).
[28] C. Udink and J. van der Elsken, Phvs. Rev. B 35, 279 (1987).
[29] C. Udink and D. Frenkel, Phys. Rev. B 35, 6933 (1987).
[30] D. R. Nelson and B. I. Halperin, Phvs. Rev. B 19, 2457 (1979).
[31] A. P. Young, Phys. Rev. B 19, 1855 (1979).
[32] R. Pindak et ai, Phys. Rev. Lett. 46, 1135 (1981); S. B. Dierker et al, Phys. Rev. Lett. 56, 1819
(1986); J. D. Brock et al., Phvs. Rev. Lett. 57, 98 (1986); A. Aharonv et al, Phvs. Rev. Lett. 46,
1012 (1986).
[33] H. Kleinert, Phys. Lett. A 130, 443 (1988).
[34] W. Janke and H. Kleinert, Phvs. Rev. Lett. 61, 2344 (1988).
[35] D. R. Nelson and B. I. Halperin, Phys. Rev. B 21, 5312 (1980).
[36] M. J. P. Gingras, P. C. W. Holdsworth, and B. Bergersen, Phys. Rev. A, 6786 (1990); see also the references therein.
[37] J. G. Kirkwood, J. Chem. Phys. 18, 380 (1950).
[38] K. W. Wojciechowski, On the free volume approximation for some anisotropic hard bodies, unpublished.
[39] A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988); ibid. 63, 1195 (1989).
[40] J. Lee and J. M. Kosterlitz, Phys. Rev. Lett. 65, 137 (1990).
[41] J. A. C. Veerman and D. Frenkel, Phys. Rev. A 41, 3237 (1990).
System of hard squares in two dimensions (2D) has been studied by Monte Carlo simulations. The simulations indicate that the isotropic fluid phase in this system does not freeze into a 2D ‘crystalline’ phase (of square lattice and quasi-long-range translational order) but transforms into an intermediate phase with the quadratic quasi-long-range orientational order (of coupled molecular axes and intermolecular bonds) and the translational order decaying faster than algebraically. The equation of state and the specific heat of the system are surprisingly well reproduced by smoothed version of the free volume theory in the whole density range.
Key words:
equation of state, hard convex body, liquid crystals, melting in two dimensions, quasi-long range order
References:
[1] B. J. Alder and W. G. Hoover, in Physics of Simple Liquids, Ed. H. N. V. Temperley, J. S. Rowlinson, and G. S. Rushbrooke, (North-Holland, 1968).
[2] J. Vieillard-Baron, J. Chem. Phys. 56, 4729 (1972).
[3] D. Frenkel and B. Mulder, Molec. Phvs. 55, 1171 (1985).
[4] A. Stroobants, H. N. W. Lekkerkerker, and D. Frenkel, Phvs. Rev. Lett. 57, 1452 (1987); Phvs.
Rev. A 36, 2929(1987).
[5] K. W. Wojciechowski, A. C. Brańka, and M. Parrinello, Molec. Phvs. 53, 1541 (1984).
[6] A. C. Brańka and K. W. Wojciechowski, Molec. Phys. 72, 941 (1991); ibid. 78, 1513 (1993).
[7] D. Frenkel, in: Liquids, Freezing and Glass Transition, Les Houches LI, Eds. J. P. Hansen,
D. Levesque and J. Zinn-Justin, Elsevier (1991).
[8] J. A. C. Veerman and D. Frenkel, Phys. Rev. A 45, 5632 (1992).
[9] K. W. Wojciechowski, D. Frenkel, and A. C. Brańka, Phvs. Rev. Lett. 66, 3168 (1991); Phvsica A 196, 519-545 (1993).
[10] K. W. Wojciechowski, Phys. Rev. B 46, 26 (1992).
[11] L. Onsager, Phys. Rev. 62, 558 (1942); L. Onsager, Ann. NY Acad. Sci. 51, 3441 (1949).
[12] K. W. Wojciechowski and J. Kłos, Nematic phase of cubic symmetiy: Solution of the Frenkel
3D-cross model, unpublished.
[13] R. Schneider, Convex bodies: The Brunn-Minkowski Theoiy Vol. 44 of Encyclopedia of Mathematics and its Applications, Ed. G.-C. Rota (Cambridge, 1993).
[14] T. Boublik, Mol. Phys. 29, 421 (1975).
[15] G. Tarjus, P. Viot, S. M. Ricci, and J. Talbot, Mol. Phys. 73, 773 (1991).
[16] K. W. Wojciechowski and D. Frenkel, On the phase diagram of two-dimensional hard rectangle system, unpublished.
[17] J. A. Zollweg and G. V. Chester, Phys. Rev B 46, 11186 (1992).
[18] J. Lee and K. J. Strandburg, Phvs. Rev. B 17, 11190 (1992).
[19] R. E. Peierls, Helv. Phys. Acta 7, 81 (1934).
[20] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part I (Pergamon, Oxford, 1980).
[21] N. D. Mermin, Phys. Rev. A 176, 250 (1968).
[22] The Mermin’s proof does not concern the case of the hard body systems. For hard discs the loga rithmic divergence has been demonstrated, in: D. A. Young, B. J. Alder, J. Chem Phys. 60, 1254 (1974).
[23] J. M. Kosterlitz and D. J. Thouless, J. Phvs. C 6, 1181 (1973); Prog. Low Temp. Phvs. B 7, 371 (1978).
[24] J. A. Barker and D. Henderson, Rev. Mod. Phvs. 48, 587 (1976).
[25] F. F. Abraham, Phvs. Rep. 80, 339 (1981).
[26] K. J. Strandburg, Rev. Mod. Phvs. 60, 161 (1988).
[27] M. A. Glaser and N. A. Clark, in Advances in Chemical Physics, Volume LXXXIII, Ed. I. Prigogine and S. A. Rice (Wilev, 1993).
[28] C. Udink and J. van der Elsken, Phvs. Rev. B 35, 279 (1987).
[29] C. Udink and D. Frenkel, Phys. Rev. B 35, 6933 (1987).
[30] D. R. Nelson and B. I. Halperin, Phvs. Rev. B 19, 2457 (1979).
[31] A. P. Young, Phys. Rev. B 19, 1855 (1979).
[32] R. Pindak et ai, Phys. Rev. Lett. 46, 1135 (1981); S. B. Dierker et al, Phys. Rev. Lett. 56, 1819
(1986); J. D. Brock et al., Phvs. Rev. Lett. 57, 98 (1986); A. Aharonv et al, Phvs. Rev. Lett. 46,
1012 (1986).
[33] H. Kleinert, Phys. Lett. A 130, 443 (1988).
[34] W. Janke and H. Kleinert, Phvs. Rev. Lett. 61, 2344 (1988).
[35] D. R. Nelson and B. I. Halperin, Phys. Rev. B 21, 5312 (1980).
[36] M. J. P. Gingras, P. C. W. Holdsworth, and B. Bergersen, Phys. Rev. A, 6786 (1990); see also the references therein.
[37] J. G. Kirkwood, J. Chem. Phys. 18, 380 (1950).
[38] K. W. Wojciechowski, On the free volume approximation for some anisotropic hard bodies, unpublished.
[39] A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988); ibid. 63, 1195 (1989).
[40] J. Lee and J. M. Kosterlitz, Phys. Rev. Lett. 65, 137 (1990).
[41] J. A. C. Veerman and D. Frenkel, Phys. Rev. A 41, 3237 (1990).
