# TRANSPORT OF MACROPARTICLES IN TWODIMENSIONAL YUKAWA SYSTEMS

O. S. Vaulina, I.E. Dranzhevski

Institute for High Energy Densities, RAS, Moscow, Russia

Abstract. Dynamics of macro- particles forming 2d- structures is numerically studied for pair potentials of Yukawa

type. The parameters responsible for the phase state and transport processes are determined and investigated.

Comparison of obtained results with data of numerical investigations for 3d- Yukawa systems is presented.

Keywords: Dusty Plasma, Numerical Simulation

PACS: 52.25. Ub, 52.25. Zb, 82.70. Db

INTRODUCTION

Dusty plasma (consisting of electrons, ions, neutral gas and solid macro-particles of micron sizes) is a

good experimental model for studying of non-ideal systems. The most of experimental investigations of dusty

plasma properties are performed in weakly- ionized plasma of gas discharges. The dust particles immersed in

gas discharge acquire negative electric charges (~ 103-105e, where e is the electron charge) that are responsible

for the particle interactions. The combined effect of the particle interactions with themselves and the ambient

plasma can lead to formation of 3d- (three dimensional) or quasi 2d- (two dimensional) strongly coupled dust

structures. The quasi 2d- dust structures, which consist from 1 to 10 dust layers, are typical for plasma of radio

frequency (RF-) discharge. Here we present the numerical study of transport of macroparticles in quasi 2-d

systems consisted of a single dust layer trapped in external one-dimensional linear electric field.

It is generally assumed that dust particles in a plasma interact with one another through the intermediary of

screened Coulomb potential, U = (eZр)2 exp(-l/λ)/l, where l is the distance, λ is the screening radius, and Zр is

the dust charge. Two parameters responsible for the transport of particles in 2d- and 3d- systems were found [3-5]

for κ = lp/λ < 6 where lp = n-1/3 is the mean inter-particle distance, and n is the particle’s concentration. These

parameters are: the “screened” coupling parameter Γ* = a1 (eZр)

2 (1+κ+ κ2 /2) exp(-κ)/Tp lp, and the scaling factor ξ

= ω* / vfr , associated with the characteristic frequency vfr of the friction. Here Tр is the temperature of particles

with the mass mp, and the characteristic frequency ω* of dust-dust collisions is ω*= eZр [a2 (1+κ+ κ2 /2)exp(-

κ)/(πlр

3mp)]1/2. Here a1 = a2 ≡ 1 for 3d- problem; and a1 =1.5, a2 = 2 for the case of 2d- Yukawa systems.

NUMERICAL RESULTS

Numerical model and parameters of 2d- simulations. The particle’s dynamics is studied by Langevin

molecular dynamic (MD) method. The simulation technique is detailed in [6, 7]. The numerical study of transport

of macroparticles in single dust layer were performed for particles in field of gravity mpg balanced by onedimensional

linear electric field E(z) = βz (where β is the value of gradient of electric field) with periodic

boundary conditions in two different directions (x and y). The calculations are performed for Np = 256

independent particles with the cut-off of the pair potential: lcut = 8lp. To check the validity of our simulations we also

have performed the test calculations for Np = 625 independent particles with lcut = 14lp. The difference between the

results of simulations with different Np, and lcut is within numerical errors ± (1-3)%.

The transport characteristics were calculated for κ ≡ lp/λ = 2- 4. The coupling parameter Γ* is changed from ~

1 to ~250, the scaling factor ξ is varied from 0.04 to 4 typical for conditions of experiments. These calculations

show, that the characteristics of 2d- systems are independent on the value of β (from ~ 0.1 to ~ 50 V/cm2).

Pair correlation in 2d- and 3d- systems. The pair correlation functions of dust particles in plasma can be

obtained from simple calculation of particle’s positions registered by video camera and it is often used for

analysis of phase state in experimental dust structures. The pair correlation function g(l) is usually averaged

over time for correct calculation for a finite number of resisted particles. In our simulation the time-averaging

of g(l) was performed over 500-2000 timesteps. (This time increased with a decrease of Г*.)

The results of numerical simulation of 3d- systems demonstrate that the viscosity of buffer gas has almost

no effect on the correlation of particles, and that the effective coupling parameter, Г* = (Zр e)2 (1+κ+ κ2 /2) exp(-

κ) /(T lр) , may be used to analyze their phase state. In the case of κ < 6, the value of Г* fully defines the form of

the binary correlation function g(l) for liquid systems up to the crystallization point Γ*

c ≈ 102, where the body

centered cubic (bcc-) lattice is formed [2-4]. The calculations of pair correlations in 2d-systems also show that

the order in the system of macro-particles is practically independent on the friction (vfr) and is determined by Г*

for weakly correlated systems as well as for stronger coupled structures (Γ*~1-100).

The first maximum gmax of g(l) and the ratio of gmax to first minimum gmin of g(l≠0) versus Γ* are shown in

Figs. 1 a,b for 2-d structures and for 3-d systems. We can see that the difference between the pair correlations

functions for the 2d- , and 3d- systems is more pronounced for the ratio gmax/gmin. Notice that the results of our 2dcalculations

are in agreement with the numerical study of properties for strongly dissipative (ξ→0) 2d- colloidal

systems presented in Ref. [8]. The numerical simulations shown, that the pair correlation functions of 2-d

systems have two singular points. First of them (for Γ*~ 97-102, near the solid-liquid transition of 3d- Yukawa

systems) is the inflection point (in contrast with 3d- systems, where the gmax, and gmax/gmin values are abruptly

changed). It may be related to the special solid-liquid transition (the liquid-to-hexatic phase transition), and the

Γ* ≈ 100 may be a critical point where the formation of hexatic phase of solid occurs. We can easily see also

the second singular point (jumps of the gmax, and gmax/gmin values) for Γ*~ 160, and we can assume that Γ*

c (2d) ≈

160 is the crystallisation point of analysed 2d- systems in a solid with the hexagonal latex (the hexatic-to-solid

phase transition).

Eq. (17) is in accordance with above-mentioned “jump” theory, within the limits of which the known Andrade

semi-empirical formula is used for viscosity constans: η ∝ f(T) exp(E/T), where E is the activation energy of

self-diffusion, and f(T) is some function which exhibits a weaker temperature dependence than exponent.

Here we present a first data on the calculations of viscosity in dissipative (νfr ≠ 0) 2-d Yukawa systems.

The viscosity constants were calculated using the Green-Kubo relation. The numerical procedure was detailed in the

set of works [5, 9]. The normalized coefficient of viscosity ν* vs. Γ* are shown in Fig. 4 for the different

screening parameters κ, and value of ξ together with the date for disperse systems (νfr = 0, κ = 0.56) [9] as well as

with the approximations of the numerical results by Einstein –Stokes equations (Eq. (3) and Eq. (4)). One can easily

seen that the normalized viscosity ν* determines by the Γ* value, and it is practically not dependent on ξ for Γ*

from ~1 to ~100-120.

CONCLUSIONS

To conclude, here we studied the dynamics of macro-particles in 2d- dissipative non- ideal Yukawa

structures. We have introduced generalized dimensionless parameters responsible for the particle correlations,

Γ*, and for the scaling of dynamic processes, ξ, in dissipative systems. We demonstrated that these parameters,

together with the particle temperature, fully determine thermodynamic properties of the modeled systems since

they are responsible for the phase state as well as for the transport processes, such as the mass transfer and the

shear viscosity. Comparison of obtained results with data of numerical investigations for 3d- Yukawa systems

is presented. In contrast of 3d- systems we have found that there are two singular points for all transport

characteristics of 2-d Yukawa systems under study. First of them (for Γ*~ 100) may be related to the liquid-tohexatic

phase transitions, second (Γ*~ 160) is the crystallisation point of analysed systems in a solid with the

hexagonal latex (hexatic-to-solid phase transitions).We have shown that the mass-transfer processes in the liquid

dust systems at the small observation time t < tact (where tact ≈ 2/ ω

c) is similar to the mass-transfer processes in

the solid. Obtained results can be useful in an analysis of the kinetic processes at physically small times.We

have studied a validity of Einstein-Stokes formula for relation between the diffusion, and viscosity transport

constants in quasi 2d- dissipative, and dispersive structures. Finally, we note that results of the present study

can be used to develop new methods for passive diagnostics of physical properties of complex plasma without

disturbing the studied system.

Acknowledgments

This work was partially supported by the Russian Foundation for Fundamental Research (No. 04-02-

16362), CRDF (No. RU-P2-2593-MO-04), the Program of the Presidium of RAS.

REFERENCES

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