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Xeniya G. Koss 1,2 Olga S. Vaulina 1 1 JIHT RAS, Moscow, Russia 2 MIPT, Moscow, Russia

Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials. Xeniya G. Koss 1,2 Olga S. Vaulina 1 1 JIHT RAS, Moscow, Russia 2 MIPT, Moscow, Russia. Introduction Basic equations Approximations Our approach Theories of 2D melting

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Xeniya G. Koss 1,2 Olga S. Vaulina 1 1 JIHT RAS, Moscow, Russia 2 MIPT, Moscow, Russia

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  1. Thermodynamic functions of non-ideal two-dimensional systemswith isotropic pair interaction potentials Xeniya G. Koss1,2 Olga S. Vaulina1 1JIHT RAS, Moscow, Russia 2MIPT, Moscow, Russia

  2. Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation Conclusion Object of simulation qE(z) = qz mg A monolayer of grains with periodical boundary conditions in the directions x and y. Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  3. Dust layers in the linear electrical field* • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation • Conclusion *O.S. Vaulina, X.G. Adamovich and S.V. Vladimirov, Physica Scripta 79, 035501 (2009) Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  4. Basic equations • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation • Conclusion СV =(U/T)V V= n-1 (P/T)V Т = T (n/P)T m – dimensionality of the system Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  5. Some useful parameters • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation • Conclusion O.S. Vaulina and S.V. Vladimirov, Plasma Phys. 9, 835 (2002): For the Yukawa systems, Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  6. Approximations “Zero” approximation In case of T 0 Up U0, Pp  P0, Т/ T Т0 / T, where U0,P0and Т0 / T can be easily computed for any known type of the crystal lattice • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation • Conclusion Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  7. Approximations [TLTT] H. Totsuji, M.S. Liman, C. Totsuji, and K. Tsuruta, Phys. Rev. E. 70, 016405 (2004) • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation • Conclusion Bi = functions (Γ2, κ2) [HKDK] P. Hartmann, G.J. Kalman, Z. Donko and K. Kutasi, Physical Review E 72, 026409(2005) Ci = polynomials (Γ2, κ2) Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  8. Our approach “Jumps” theory: analogies between the solid and the liquid state of matter Wa - the energy of “jump” activation • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation • Conclusion - the energy of state per one degree of freedom - crystallization temperature • coefficients dependent on the type of crystalline lattice • and on the total number of degrees of freedom Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  9. Our approach The energy density of analyzed systems • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation • Conclusion The normalized value for the thermal component of the potential energy The pressure where Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  10. Our approach The heat capacity • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation • Conclusion The thermal coefficient of pressure The normalized isothermal compressibility where , Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  11. Theories of 2D melting We considered two main approaches in the 2D melting theory that are based on unbinding of topological defects • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation • Conclusion • KTHNY theory: • two phase transitions from the solid to fluid state via “hexatic” phase. • The hexatic phase is characterized by • the long-range translational order combined with the short-range orientational order • the spatial reducing of peaks (gs) for pair correlation function g(r) is described by an exponential law [gs(r)  exp(-r), const], • the bond orientational function g6(r) approaches a power law [g6(r) r-,  > 0.25]. The theory of grain-boundary-induced melting: a single first-order transition from the solid to the fluid state without an intermediate phase for a certain range of values of the point-defect core energies. Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  12. Numerical simulation: parameters • The Langevin molecular dynamics method • Various types of pair isotropic potentials (r): • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation: parameters results comparison • Conclusion qE(z) = qz Np = 256..1024 β = 10-2V/cm2..100V/cm2 lcut = 8rp .. 25rp mg Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  13. Numerical simulation: results • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation: parameters results comparison • Conclusion Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  14. Numerical simulation: results Our approximation • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation: parameters results comparison • Conclusion Yukawa system, Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  15. Numerical simulation: results Our approximations • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation: parameters results comparison • Conclusion Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  16. Numerical simulation: results Our approximation • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation: parameters results comparison • Conclusion Yukawa system, Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  17. Numerical simulation: results Our approximation • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation: parameters results comparison • Conclusion Yukawa system, Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  18. Numerical simulation: results • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation: parameters results comparison • Conclusion Our approximation Yukawa system, Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  19. Numerical simulation: comparison • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation: parameters results comparison • Conclusion Yukawa system, Our approximations HKDK TLTT Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  20. Numerical simulation: comparison • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation: parameters results comparison • Conclusion Our approximations HKDK TLTT Yukawa system, Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  21. Numerical simulation: comparison • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation: parameters results comparison • Conclusion 1 – Our approximation 2 – HKDK 3 – TLTT Yukawa system, Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  22. Conclusion • The analytical approximation of the energy density for 2D non-ideal systems with various isotropic interaction potentials is proposed. • The parameters of the approximation were obtained by the best fit of the analytical function by the simulation data. • Based on the proposed approximation, the relationships for the pressure, thermal coefficient of pressure, isothermal compressibility and the heat capacity are obtained. • The comparison to the results of the numerical simulation has shown that the proposed approximation can be used for the description of thermodynamic properties of the considered non-ideal systems. • Introduction • Basic equations • Approximations • Our approach • Theories of 2D melting • Numerical simulation • Conclusion Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

  23. Thank you for attention! This work was partially supported by the Russian Foundation for Fundamental Research (project no. 07-08-00290), by CRDF (RUP2-2891-MO-07), by NWO (project 047.017.039), by the Program of the Presidium of RAS, and by the Federal Agency for Science and Innovation (grant no. МК-4112.2009.8). Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. Koss Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010

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