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