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 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.


Dust layers in the linear electrical field*

pN

iii rrq

1

/)('2




pNconst

*O.S. Vaulina, X.G. Adamovich and S.V. Vladimirov, Physica Scripta 79, 035501 (2009)


Basic equations

• Introduction• Basic

equations• Approximations• Our approach• Theories of 2D



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


Some useful parameters

• Introduction• Basic

equations• Approximations• Our approach• Theories of 2D



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*


Approximations




“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


Approximations




[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)


Our approach




“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


Our approach




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


Our approach




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,


Theories of 2D melting

• Introduction• Basic equations• Approximations• Our approach• Theories of

2D melting• Numerical


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.


Numerical simulation: parameters



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


Numerical simulation: results




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






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


0,2

0,4

0,6

0,8

1,0

1,2

0 50 100 150 200

U

P


)/5.5exp(/ pc rr




)/2exp(/ pc rr

rrrr ppc /05.0)/3exp(/

3)/(05.0/ rrpc

2)/(01.0)/4exp(/ rrrr ppc

Our approximations






1,5

2,0

2,5

0 50 100 150 *

C V

(b)

Our approximation


2

5.5

2

5.5

2.0

2.0

2

2






1

2

3

4

0 40 80 120 160 *

V Our approximation

Yukawa system, 86.1

2

3

4

)/exp(/ pc rr






0,54

0,56

0,58

0 40 80 120 160 200

*

T

Yukawa system, )/2exp(/ pc rr

Our approximation

23.0

86.1


Numerical simulation: comparison




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)


Our approximations 123

HKDKTLTT






-1,6

-1,4

-1,2

-1,0

-0,8

-0,61 10 100

U c / {T }

(c)


Our approximations

123

HKDKTLTT






-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


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.


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).

xeniya g. koss 1,2 olga s. vaulina 1 1 jiht ras, moscow, russia 2 mipt, moscow, russia

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