Thermodynamic functions of nonideal two-dimensional systems
with isotropic pair interaction
potentials
Xeniya G. Koss1,2
Olga S. Vaulina1
1JIHT RAS, Moscow, Russia
2MIPT, Moscow, Russia
Object of simulation
qE(z) = qz
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Introduction
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Basic equations
Approximations
Our approach
Theories of 2D
melting
Numerical
simulation
Conclusion
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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
Dust layers in the linear
electrical field*
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Introduction
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Basic equations
Approximations
Our approach
Theories of 2D
melting
Numerical
simulation
Conclusion
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  const  N p
Np
q  2  ' (ri ) / ri
i 1
*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
Basic equations

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Introduction
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Basic
equations
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Approximations
Our approach
Theories of 2D
melting
Numerical
simulation
Conclusion
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m
U  T  (m  1)n  (r ) g (r )r m1dr
2
0

(m  1)n 2  (r )
m
P  nT 
g
(
r
)
r
dr

m
r
0
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
Some useful parameters
U  (U  U 0 
•
Introduction
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Basic
equations
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Approximations
Our approach
Theories of 2D
melting
Numerical
simulation
Conclusion
m
T) /T
2
CV  CV  m / 2
q2

Trp
O.S. Vaulina and S.V. Vladimirov, Plasma Phys. 9, 835 (2002):
  1.5r   / 2T
*
2
p

For the Yukawa systems,
1
 fr
 
M
 / c  exp( r / rp )
 *  (1     2 / 2) exp(  )
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
“Zero” approximation
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Introduction
Basic equations
Approximations
Our approach
Theories of 2D
melting
Numerical
simulation
Conclusion
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. Koss
Workshop on Crystallization and Melting in Two-Dimensions, MTA-SZFKI, May 18, 2010
Approximations
[TLTT] H. Totsuji, M.S. Liman, C. Totsuji, and
K. Tsuruta, Phys. Rev. E. 70, 016405 (2004)
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Introduction
Basic equations
Approximations
Our approach
Theories of 2D
melting
Numerical
simulation
Conclusion
0.05  2  100
0.5   2  2
U  U 2TLTT  ( B12  B2 ) exp{2.55(20.18  0.050.18 )}
Bi = functions (Γ2, κ2)
[HKDK] P. Hartmann, G.J. Kalman, Z. Donko and
K. Kutasi, Physical Review E 72, 026409 (2005)
0.05  2  120
0  2  3
U  U 2HKDK  2 (C1  C 2 C32 / 3 )  U 0 /( T )   / 
Ci = polynomials (Γ2, κ2)
2   
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
Our approach
“Jumps” theory: analogies between the solid and the liquid
state of matter
Wa - the energy of “jump” activation
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•
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Introduction
Basic equations
Approximations
•
Our approach
•
Theories of 2D
melting
Numerical
simulation
Conclusion
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N  N1  N 2
U1  U 0  mT / 2
U 2  U 1  a1 f
 f   2  1
 1, 2 - the energy of state per one degree of freedom
 f  Wa  Qa  a 2Tc  a3 (T  Tc ) / 2
Tc - crystallization temperature
a1, 2 ,3 - 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
Our approach
The energy density of analyzed systems
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Introduction
Basic equations
Approximations
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Our approach
•
Theories of 2D
melting
Numerical
simulation
Conclusion
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a1 f
m
U  Ua  U0  T 
2
1  exp(  f / T )
The normalized value for the thermal component
of the potential energy
a1 f / T
m
U  (U  U 0  T ) / T 
2
1  exp(  f / T )
The pressure
Pa  P0  nT  n  U / m
where
  (* / rp )rp *
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
The heat capacity
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Introduction
Basic equations
Approximations
•
Our approach
•
Theories of 2D
melting
Numerical
simulation
Conclusion
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CV
a
m 0.5a1  ( f / T  0.5)U exp(  f / T )
 
2
1  exp(  f / T )
The thermal coefficient of pressure
V a  1  m 1 (CV a  m / 2)
The normalized isothermal compressibility
    
a 1
T
0 1
T
1

   0 (m  1)
a1  U


  ( f / T  0.5)
 U   
  02 1 U
 1  exp(  / T )
 
m

f


2
0
where  0   / m , 1  * (d 2 * / drp2 ) /( d* / drp ) 2  1
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
We considered two main approaches in the 2D melting
theory that are based on unbinding of topological defects
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Introduction
Basic equations
Approximations
Our approach
•
Theories of
2D melting
•
Numerical
simulation
Conclusion
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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-boundaryinduced 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
Numerical simulation:
parameters
•The Langevin molecular dynamics method
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•
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Introduction
Basic equations
Approximations
Our approach
Theories of 2D
melting
•
Numerical
simulation:
•
parameters
results
comparison
Conclusion
•Various types of pair isotropic potentials (r):
  c b1 exp( 1r / rp )  b2 (rp / r ) n exp(  2 r / rp )
qE(z) = qz
β = 10-2V/cm2..100V/cm2
Np = 256..1024
Np
lcut = 8rp .. 25rp

1
 fr
 
 0.04..4
M
mg
q  2  ' (ri ) / ri
i 1
*  1.5rp2  / 2T  1..250
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
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•
•
Introduction
Basic equations
Approximations
Our approach
Theories of 2D
melting
•
Numerical
simulation:
1
parameters
results
comparison
Conclusion
0
•
(a)
g(r/r p )
3

 /  c  exp( 4r / rp )
 3
2
 / c  exp( 3r / rp )  0.05rp / r
  0.12

0
1
2
r/r p
3
 / c  0.05(rp / r ) 3
  0 .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
Numerical simulation:
results
1,2
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•
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Introduction
Basic equations
Approximations
Our approach
Theories of 2D
melting
U
(b)
 2
1,0
 3
0,8
 4
Numerical
simulation:
parameters
results
comparison
Conclusion
Our
approximation
  5.5
0,6
0
50
100
 150
200
Yukawa system,  / c  exp( r / rp )
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
Our approximations
1,2
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U
Introduction
Basic equations
Approximations
Our approach
Theories of 2D
melting
Numerical
simulation:
parameters
results
comparison
Conclusion
1,0
 / c  exp( 2r / rp )
0,8
 / c  exp( 5.5r / rp )
P
0,6
 / c  exp( 3r / rp )  0.05rp / r
 / c  exp( 4r / rp )  0.01(rp / r )2
0,4
0,2
0
50
100
150
200
 / c  0.05(rp / r ) 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
Numerical simulation:
results
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•
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Introduction
Basic equations
Approximations
Our approach
Theories of 2D
melting
CV
Our approximation
2,5
2,0
parameters
results
comparison
Conclusion
 2
 2
  0.2
  5.5   2
(b)
Numerical
simulation:
 2
  5.5   0.2
1,5
0
50
100
*
150
Yukawa system,  / c  exp( r / rp )
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
V
4
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•
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Introduction
Basic equations
Approximations
Our approach
Theories of 2D
melting
•
Numerical
simulation:
•
parameters
results
comparison
Conclusion
Our approximation
 2
3
 3
2
 4
1
0
40
80
120
* 160
Yukawa system,  / c  exp( r / rp )   1.86
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
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•
•
•
•
Introduction
Basic equations
Approximations
Our approach
Theories of 2D
melting
•
Numerical
simulation:
•
parameters
results
comparison
Conclusion
 T
Our approximation
0,58
  0.23
0,56
  1.86
*
0,54
0
40
80
120
Yukawa system,
160
200
 / c  exp( 2r / rp )
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
2,0
U
1,8
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•
•
•
•
•
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Introduction
Basic equations
Approximations
Our approach
Theories of 2D
melting
Numerical
simulation:
parameters
results
comparison
Conclusion
2,4
(a)
C V
2,2
(b)
2,0
1,6
1,8
1,4
1,6
1,2
1,4
1,0
1,2
0,8
1,0
0,6
0,8
0,4
0
40
80
120


0,6
160
0
40
80
120

160
Yukawa system,  / c  exp( r / rp )
Our approximations
HKDK
TLTT
 1
 2
 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
Numerical simulation:
comparison
1
-0,6
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•
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Introduction
Basic equations
Approximations
Our approach
Theories of 2D
melting
Numerical
simulation:
parameters
results
comparison
Conclusion
-0,8
-1,0
10
U c / {T }

100
(c)
 
 1
 2
 3
 
-1,2
-1,4
 
-1,6
Our approximations
HKDK
TLTT
Yukawa system,  / c  exp( r / rp )
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
1
10
-0,6
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•
•
•
•
Introduction
Basic equations
Approximations
Our approach
Theories of 2D
melting
•
Numerical
simulation:
•
parameters
results
comparison
Conclusion

  1.84
U c / {T }
  0.92
1
-0,8
2
  0.23
3
-1,0
100
 
1 – Our approximation
2 – HKDK
3 – TLTT
-1,2
Yukawa system,
 / c  exp( 2r / rp )
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
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•
•
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 nonideal 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
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