1. 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

2. 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 Object of simulation Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation Conclusion qE(z) = q  z mgmg A monolayer of grains with periodical boundary conditions in the directions x and y.

3. 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 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)

4. 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 Basic equations Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation Conclusion С V =(  U/  T) V  V = n -1 (  P/  T) V  Т = T (  n/  P) T m – dimensionality of the system

5. 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 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,

6. 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 Approximations Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation Conclusion “Zero” approximation In case of T  0 U p  U 0, P p  P 0,  Т / T   Т 0 / T, where U 0, P 0 and  Т 0 / T can be easily computed for any known type of the crystal lattice

7. 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 Approximations Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation Conclusion [TLTT] H. Totsuji, M.S. Liman, C. Totsuji, and K. Tsuruta, Phys. Rev. E. 70, 016405 (2004) [HKDK] P. Hartmann, G.J. Kalman, Z. Donko and K. Kutasi, Physical Review E 72, 026409 (2005) B i = functions (Γ 2, κ 2 ) C i = polynomials (Γ 2, κ 2 )

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 Our approach Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation Conclusion “Jumps” theory: analogies between the solid and the liquid state of matter W a - the energy of “jump” activation - 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

9. 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 Our approach Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation Conclusion The energy density of analyzed systems The normalized value for the thermal component of the potential energy The pressure where

10. 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 Our approach Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation Conclusion The heat capacity where The thermal coefficient of pressure The normalized isothermal compressibility,

11. 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 Theories of 2D melting Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation Conclusion We considered two main approaches in the 2D melting theory that are based on unbinding of topological defects 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 (g s ) for pair correlation function g(r) is described by an exponential law [g s (r)  exp(-  r),   const], the bond orientational function g 6 (r) approaches a power law [g 6 (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.

12. 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 Numerical simulation: parameters Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation: parameters results comparison Conclusion The Langevin molecular dynamics method Various types of pair isotropic potentials  (r): qE(z) = q  z mgmg N p = 256..1024 l cut = 8r p.. 25r p β = 10 -2 V/cm 2..100V/cm 2

13. 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 Numerical simulation: results Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation: parameters results comparison Conclusion

14. 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 Numerical simulation: results Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation: parameters results comparison Conclusion Our approximation Yukawa system,

15. 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 Numerical simulation: results Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation: parameters results comparison Conclusion Our approximations

16. 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 Numerical simulation: results Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation: parameters results comparison Conclusion Our approximation Yukawa system,

17. 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 Numerical simulation: results Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation: parameters results comparison Conclusion Our approximation Yukawa system,

18. 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 Numerical simulation: results Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation: parameters results comparison Conclusion Yukawa system, Our approximation

19. 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 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

20. 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 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

21. 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 Numerical simulation: comparison Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation: parameters results comparison Conclusion Yukawa system, 1 – Our approximation 2 – HKDK 3 – TLTT

22. 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 Conclusion Introduction Basic equations Approximations Our approach Theories of 2D melting Numerical simulation 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.

23. 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 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).