xeniya g. koss 1,2 olga s. vaulina 1 1 jiht ras, moscow, russia 2 mipt, moscow, russia
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
Thermodynamic functions of non-ideal two-dimensional systemswith isotropic pair interaction
potentials Xeniya G. KossXeniya G. Koss1,21,2
Olga S. VaulinaOlga S. Vaulina11
11JIHT RAS, Moscow, RussiaJIHT RAS, Moscow, Russia22MIPT, Moscow, RussiaMIPT, Moscow, Russia
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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) = qz
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. KossWorkshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010
Dust layers in the linear electrical field*
pN
iii rrq
1
/)('2
• Introduction• Basic equations• Approximations• Our approach• Theories of 2D
melting• Numerical
simulation• Conclusion
pNconst
*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. KossWorkshop 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
СV =(U/T)V V = n-1 (P/T)V
Т = T (n/P)T
0
1)()()1(2
drrrgrnmTmU m
0
2
)()()1( drrrgrr
mnmnTP m
m – dimensionality of the system
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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
TTmUUU /)2
( 0
2/mCC VV
Mfr
1Trp 2/5.1 2*
pTrq2
O.S. Vaulina and S.V. Vladimirov, Plasma Phys. 9, 835 (2002):
For the Yukawa systems, )/exp(/ pc rr
)exp()2/1( 2*
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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 Up U0, Pp P0,
Т / T Т0 / T,
where U0, P0 and Т0 / T
can be easily computed
for any known type
of the crystal lattice
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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)
10005.0 2
30 2
/)/()( 03/2
32122 TUCCCUU HKDK
12005.0 2
25.0 2
)}05.0(55.2exp{)( 18.018.022212 BBUU TLTT
2 /2
Bi = functions (Γ2, κ2)
Ci = polynomials (Γ2, κ2)
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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 matterWa - the energy of “jump” activation
21 NNN 2/01 mTUU
faUU 112
12 f
2,1
2/)(32 ccaaf TTaTaQW
cT3,2,1a
- 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. KossWorkshop 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 componentof the potential energy
The pressure
where
)/exp(121
0 Ta
TmUUUf
fa
)/exp(1/
/)2
( 10 T
TaTTmUUU
f
f
mUnnTPPa /0
** )/( pp rr
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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
)/exp(1)/exp()5.0/(5.0
21
TTUTamC
f
ffaV
)2/(1 1 mCm aV
aV
U
mm
UT
UaT
ff
Ta
T
120
0120
101
)1()/exp(1
)5.0/(
1
m/0 1)//()/( 2*2*2*1 pp drddrd,
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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 (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. KossWorkshop 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) = qz
mg
Np = 256..1024
lcut = 8rp .. 25rp
β = 10-2V/cm2..100V/cm2
4..04.01
Mfr
pN
iii rrq
1
/)('2
250..12/5.1 2* Trp
)/exp()/()/exp( 2211 pn
ppc rrrrbrrb
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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
0
1
2
3
0 1 2 3
g(r/r p )
r/r p
(a) )/4exp(/ pc rr
3
rrrr ppc /05.0)/3exp(/
12.0
3)/(05.0/ rrpc
5.0
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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, )/exp(/ pc rr
0,6
0,8
1,0
1,2
0 50 100 150 200
U (b)
2
3
4
5.5
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010
0,2
0,4
0,6
0,8
1,0
1,2
0 50 100 150 200
U
P
Numerical simulation: results
)/5.5exp(/ pc rr
• Introduction• Basic equations• Approximations• Our approach• Theories of 2D
melting• Numerical
simulation: parameters results comparison• Conclusion
)/2exp(/ pc rr
rrrr ppc /05.0)/3exp(/
3)/(05.0/ rrpc
2)/(01.0)/4exp(/ rrrr ppc
Our approximations
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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
1,5
2,0
2,5
0 50 100 150 *
C V
(b)
Our approximation
Yukawa system, )/exp(/ pc rr
2
5.5
2
5.5
2.0
2.0
2
2
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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
1
2
3
4
0 40 80 120 160 *
V Our approximation
Yukawa system, 86.1
2
3
4
)/exp(/ pc rr
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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
0,54
0,56
0,58
0 40 80 120 160 200
*
T
Yukawa system, )/2exp(/ pc rr
Our approximation
23.0
86.1
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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
0,4
0,6
0,8
1,0
1,2
1,4
1,6
1,8
2,0
0 40 80 120 160
U (a)
0,6
0,8
1,0
1,2
1,4
1,6
1,8
2,0
2,2
2,4
0 40 80 120 160
C V (b)
Yukawa system, )/exp(/ pc rr
Our approximations 123
HKDKTLTT
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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
-1,6
-1,4
-1,2
-1,0
-0,8
-0,61 10 100
U c / {T }
(c)
Yukawa system, )/exp(/ pc rr
Our approximations
123
HKDKTLTT
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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
-1,2
-1,0
-0,8
-0,61 10 100
U c / {T }
12
3
Yukawa system, )/2exp(/ pc rr
1 – Our approximation
2 – HKDK
3 – TLTT
84.1
92.0
23.0
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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.
Thermodynamic functions of non-ideal two-dimensional systems with isotropic pair interaction potentials, X. KossWorkshop 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).